File Menu

The file menu contains the following items:

•	New

•	Open

•	Reopen

•	Save

•	Save As

•	Close

•	Append Results

•	Import Matrix

•	Export Matrix

•	Switch Matrix

•	Delete

•	Memory Requirements

•	Print

•	Print Setup

•	Font

•	DOS Shell

•	Exit


Open Project

File | Open | Project

A "Project" in PC-ORD is a set of files associated with each other, along with the options and settings at the time the "Project" is saved.  When you open a project, files are opened, the contents of the PC-ORD windows are restored, and your options and settings are restored from the last time you saved them.  Note that if you are working on only one project, this is unnecessary, because PC-ORD automatically "remembers" your settings and the contents of your windows.  If however, you are working on several projects, you can easily move between them, starting right from where you left off, by opening a saved project.

If you have a project open and you open a matrix file (spreadsheet), graph file, result file, or if you create a graph file or result file by analysis, then the project is considered to be modified.  When you exit PC-ORD or open a new project you will be asked whether to save the modified project.

You can reset all of the PC-ORD analysis defaults and graphing defaults to what they were when you installed the software by selecting File | Open Project, navigating to the "Working" folder, then selecting "Default.prj".  If you cannot find this file, minimize PC-ORD and use the Windows "Find" utility (Start | Find ) to locate the file.  Then open this file using Open Project.

In version 3 the default directory for open and save dialogs was the last directory accessed.  In version 4 the last directory comes from the "history" for each individual window (main matrix, second matrix, graphrow, etc.).  The reason for the change was that in version 3, the file name in the open/save dialogs was initialized blank while in version 4 we wanted to have the last-used file name available simply by pressing Enter.  To do that, the dialog directory also has to be initialized to the directory that has the last history filename instead of the last open/saved (default) directory.

When you open a new project, the "memory" of the default directory and all the other "histories" are cleared.  Then when you open a file in the new project the dialog opens in the working directory, since there are no histories and there now is no default directory.  The "working directory" is the directory specified in the "Start in" property associated with the PC-ORD file icon.

Every time you open or save a file in PC-ORD you write some "history" associated with that  window and when you write any history, that directory becomes the default directory for that window.  It also becomes the default directory for any other windows or file operations that do not have a history associated with it.

File Management Definitions for PC-ORD:

Default directory:  Within a given project, PC-ORD assumes it will open or save a file in the default directory.  The default directory is the folder last used in saving or opening the contents of a given window.  If a window has no previous use in the project the default directory is assigned to the default directory.  If no files have been opened in a project (i.e. you just started a new project), the default directory is the same as the working directory.

File history (= history list):  A series of "remembered" file names associated with a given window (main matrix, second matrix, graphrow, etc.) in a given project.  Histories are also saved for other file openings or savings, for example when you open other graphs: cluster analysis dendrograms, species-area curves, or NMS scree plots.

Project: a set of files associated with each other, along with the last-used options and settings at the time the "Project" is saved.

Working directory (= Start in directory, = Working folder):  The working directory is the folder specified under the "Start In" property associated with the PC-ORD icon.  You can change this folder to whatever you want.  Network users must, however, specify a directory to which they have "write" privileges.

File Menu


Save Project As

File | Save Project As

A "Project" in PC-ORD is a set of files associated with each other, along with the options and settings at the time the "Project" is saved.  When you save a project, saved files and the contents of the PC-ORD windows are recorded, and your options and settings are saved.  Note that if you are working on only one project, this is unnecessary, because PC-ORD automatically "remembers" your settings and the contents of your windows.  If however, you are working on several projects, you can easily move between them, starting right from where you left off, by first saving, then later opening a saved project.

File Menu


File New

File | New | Project starts a new project by clearing the file histories.  In other words, PC-ORD begins new lists of recently used files.  See Open Project for a description of how to use projects in PC-ORD.

File | New | Main Matrix and File | New | Second Matrix start a new matrix (spreadsheet) and opens it for editing.  This may be useful if you wish to type in a small matrix directly.  After saving the file, it can be reopened for use or editing with the File | Open  or File | Reopen command.  When entering a medium to large matrix, the flexibility and advanced features of spreadsheet software will normally be more convenient than data entry within PC-ORD.

File Menu


File Open

File | Open

The Open item on the file menu allows you to open existing projects or files.  You can open a:

•	Project

•	Main Matrix

•	Second Matrix

•	Graph Row file (ordination scores for rows in your main matrix)

•	Graph Column file (ordination scores for column variables in your main matrix)

•	Results file

•	Dendrogram

•	Species-area Curve

•	NMS Scree Plot

You can also open a file by clicking on Main:, Second:, Row, Column:, or Result: on the status line at the bottom of your screen.  Once in the dialog box you can select an option by either clicking on it, tabbing to it by pressing the Tab key, or pressing Alt + the highlighted letter of the item.

You can also open previously opened files or projects by selecting File | Reopen .

Hotkeys:

F7	Open Main
F8	Open Second
F9	Open Graph
F10	Open Result
More Hotkeys

Window

File Formats

File Menu


File Reopen

Using File | Reopen offers the same set of actions as File | Open except that Reopen provides a list of recently used files and projects from which you can select one to open.

File Menu


History List

To open the filename history list box select File | Reopen. Or, to access the only the most recent name in the history list, you can select the default file written into the file name box of a file dialog.

File Open

File Menu


File Close

The File | Close command closes a window.  It can be applied to the windows containing the main matrix, the second matrix, the result file, or the graph file.

File Menu


File Save

The File | Save command allows you to save an altered current project, main matrix, or second matrix, using the existing name.  For more information on projects, see Open Project.

File Menu


File Save As

The File | Save As command allows you to save the current project or the contents of a window to a disk file with a new name.  The following files are available:

•	Main Matrix

•	Second Matrix

•	Graph Row file (ordination scores for rows in your main matrix)

•	Graph Column file (ordination scores for column variables in your main matrix)

•	Results file

You can use any file name, as long as it is not one of PC-ORD’s reserved names.

Dendrograms, scree plots, and species-area curves are saved under the Graph menus.

Hotkeys:

Action	Default extensions:
F5  Save Main	*.WK1
F6  Save Second	*.WK1
F2  Save Graph	*.GPH
F3  Save Result	*.TXT
More Hotkeys

File formats are described elsewhere.

File Menu


File Append Results

You might wish to build an output file by appending a series of results files. Do this by saving a results file under a new name, then after each successive run, selecting File | Append Results (hotkey = F4).  This appends the latest “result.txt” to your named file.  Once you append the contents of the latest results file, Append Results is grayed out and not available.

Results File Format

File Menu


Import

File | Import Matrix

PC-ORD’s standard data format is the *.wk1 spreadsheet format.  You can import the following formats to the spreadsheet format:

•	Old PC-ORD Format

•	Compact Format

•	DECORANA/TWINSPAN Condensed Format  (= Cornell condensed format)

•	List (Database) Format

•	Comma Separated Values Format

Data Preparation

Open File

File Menu


Export

File | Export Matrix

You can export from the PC-ORD spreadsheet format to any one of the following formats:

•	Old PC-ORD Standard Format

•	Compact Format

•	DECORANA/TWINSPAN Condensed Format

•	List (Database) Format

•	Comma Separated Values Format

In each case the exported files are plain text files (ASCII files).  Note that if you wish to export to the compact format, there must be a Species File already in existence.

Data Preparation

File Formats

File Menu


Import Compact Format

Compact Format Setup

The Compact Format Setup dialog allows you to specify the options for reading your data and writing output.  Each of these options is described below.

Data Type

Three types of data are allowed: lists of individual trees and their sizes, quantitative lists of species, and presence lists of species.  See Compact Format Data Types

Aggregate Sample Units On Entry

This option can be used in two ways, depending on whether your data file contains subgroups defined by breakpoints.

a. If your file contains breakpoints, you can create an aggregated matrix where each row of the main matrix represents average species values within a subgroup.  This is useful if you have a hierarchical data structure.  For example, consider a data set consisting of species composition in 40 plots in each of 20 stands.  At the plot level you would have 40 x 20 = 800 sample units.  However, the data can be aggregated to the stand level by averaging the 40 sample units in each of the 30 stands, yielding a main matrix containing 20 stands.

b. You can also use subgroups if you want to summarize or compare different portions of a single data set (e.g. the upland vs. lowland parts of a woods).

Names for Rows

You are asked to provide a name that applies to your rows collectively.  For example, if your rows are named Plot1, Plot2, Plot3, etc., you will probably collectively call your rows "Plots".  Enter the name in the space provided.

Names for Columns

You are asked to provide a name that applies to your columns collectively.  For example, if your columns are acronyms for various species, you may wish to call your columns "Species".  Enter the name in the space provided.

File Menu

Compact Data Format


File Delete

You can quickly delete files that you no longer need by using File | Delete.  This helps clean up a directory listing and make more room on disk for new files.

File Menu


Memory Requirements

Selecting File | Memory requirements tells you whether or not a particular PC-ORD operation will run or not depending on the size of the matrices.  The current matrices are evaluated when you first open the dialog.  You can type in the number of rows and columns of other matrices and then click on Evaluate to see if they will run. Because the exact memory use will vary somewhat depending on the options selected, the results of this function should be used only as an approximate guide.

Memory requirements are not given for Compact-format Summary because the results are strongly dependent on whether there is a hierarchical structure defined by breakpoints and whether you choose to aggregate within that structure.

Tips for managing memory are described under Memory use.

File Menu


Switching Matrices

File | Switch Matrix

You can quickly move a main matrix to the second window, a second matrix to the main window, or swap the main and second if both are open.

Think of the main matrix as the active matrix.  Whichever matrix is designated as the main matrix is the subject of most of the operations in PC-ORD, whether it is data transformations, clustering, or import/export operations.  This option allows you to easily move data between the main and second matrices.

File Menu


Print

File | Print

If open, you can print the contents of Main: or 2nd: or Graph: or Results: file on the status line.

You can print an entire file or just a selection.  To print only a portion of a file, first select the text with the mouse or shift+cursor keys.  Then choose “Selection” in the Print dialog.  Your selected text is printed along with the appropriate column and row names.

Choose your printer, portrait or landscape orientation, and other options by selecting File | Print Setup

Ordination graphics are printed from the Graph menu under Print Graph.  Layout of graphics on the page are specified by Graph Page Setup.

Hotkeys:

Shift + F2	Print Graph
Shift + F3	Print Result
Shift + F5	Print Main
Shift + F6	Print Second

File Menu


Font

Font | Screen

Font | Printer

You can select the font used for matrices and other text.  These can be selected independently for display on your screen and display on your printer.  Fonts for graphics are selected under the Graph menu.

We strongly recommend using a non-proportional font, such as Courier or Letter Gothic, for displaying and printing your output files.  Because PC-ORD includes tabular output, built by spaces instead of tab characters, the formatting of the tables will be best with a non-proportional font.  If you wish to import these tables into a document that has proportional fonts, you will need to insert the tab characters or use the parsing function present in most spreadsheets and word processors.

File Menu


DOS Shell

File | DOS Shell

With the DOS Shell command, you can leave the program temporarily to perform a DOS command or run another program.  To return to PC-ORD, type EXIT at the DOS prompt.

File Menu


Exit

File | Exit

The Exit command terminates PC-ORD.  The status of your session is saved in the current working directory, so that when you reopen PC-ORD, your options and files will be set up as you left them.

File Menu


Reserved Filenames

PC-ORD reserves the following filenames for it's internal use:

advisor.tmp	choices.tmp	deco.fil
dendprep.den	graphcol.gph	graphrow.gph
dissim.tmp	groups.tmp	groups2.tmp
loglist.txt	montenms.tmp	pict.tmp
pcord.err	pcord4.ini	pcord4.tmp
prescrib.fil	result.txt	sfi.bin
spparea.spa	stress.str	temp.str
temp.wk1	temp2.wk1	temp.fil
untitled.wk1	work.wk1	work2.wk1
If you attempt to explicitly open one of these files from within PC-ORD, you will be prevented from doing so and given a warning that they are reserved for internal use.

File Menu


Hotkeys

A complete list of PC-ORD hotkeys follows.  Most hotkeys are used for file management..

F1	Help
F2	Save Graph
F3	Save Result
F4	Append Result
F5	Save Main
F6	Save Second
F7	Open Main	Shift + F2	Print Graph
F8	Open Second	Shift + F3	Print Result
F9	Open Graph	Shift + F5	Print Main
F10	Open Result	Shift + F6	Print Second
Ctrl + C	Copy
Ctrl + A	Select All
F2 or N	Next Overlay (Graph)
F3 or P	Previous Overlay (Graph)

File Menu


Getting Started

Using PC-ORD

Introduction

Help for Old Friends

Data Preparation

File Formats

Typical Flow of Analysis

Example of an Analysis

Managing Files and Projects

Managing Large Projects

Network Environments

System Requirements

Memory Use

Waiver

Orders

Problems

Frequently Asked Questions

PC-ORD Failure

Technical Support


Introduction

The PC-ORD System

PC-ORD performs multivariate analysis of ecological data.  Its Windows interface provides easy-to-use mouse control and windowed output.  In addition to utilities for transforming data and managing files, PC-ORD offers many ordination and classification techniques not available in major statistical packages.

PC-ORD is intended for use by those with little or no previous computing experience.  However, rank beginners will need to learn how to handle a microcomputer and printer.

Virtually any multivariate data set consisting of a set of entities, each with a number of measured attributes, is adaptable to PC-ORD.  The possible formats for entering these data can be found in Data Preparation.  In addition to this main matrix, a second matrix can be entered for analysis of its relationship to the first matrix.  The inclusion of this second matrix  greatly enriches the system's analytical capabilities.  Both matrices are displayed in PC-ORD in file viewing windows.

In community ecology, the first matrix will often contain species abundance data in a set of samples, while the second matrix contains environmental data for the samples.

In studies of resource partitioning, the first matrix contains relative use of a set of resources by various species, subspecies, or individuals.  The second matrix could contain characteristics of the resources.

In taxonomy, the main matrix is usually a variety of traits measured for many individuals.  Various classifications of the individuals (at different taxonomic levels or according to different authors) can be used as the secondary matrix.

When running PC-ORD, think of the main matrix as the "hotseat."  Whichever matrix is in the hotseat is the subject of most of the operations in PC-ORD, whether it is data transformations, clustering, or import/export operations.  You can use File|Switch Matrix to easily move matrices into and out of the hotseat.

The PC-ORD help system provides brief descriptions of the programs available in PC-ORD, focusing on the mechanics of running the programs. It does not say much about why you would run a particular analysis, it being your responsibility to know what is appropriate or useful for your data set and why.  Some useful references to help you decide what is appropriate for your data are Beals (1984), Gauch (1982), Tabachnik and Fidell (1989), and Greig-Smith (1983), and Ludwig and Reynolds (1988).  Gauch's book is the best reference for the techniques developed at Cornell, and is a good introduction to community analysis in general.  Beals (1984) contains a penetrating critique of distance measures and ordination techniques from an ecological viewpoint.  Clarke (1993) is an excellent summary of the use of nonparametric techniques in ecology, for example non-metric multidimensional scaling.  Tabachnik and Fidell (1996) is a good practical guide to standard multivariate analyses.

Typical Flow of Analysis

Getting Started


Managing Files and Projects

Normally you will want to keep your data and result files in folder that is separate from the folder that contains PC-ORD.  If you have more than one project in which you are using PC-ORD, you can keep the files from each project in a separate folder or put them in a single folder.  PC-ORD helps you resume work where you left off with the Open Project and Save Project menu items.  Details of how PC-ORD manages folders (directories) and file management definitions are given under Open Project.

Windows 95/NT

Installing PC-ORD puts the PC-ORD icon in the Programs section of the Start Menu. If you wish to create an icon for a given project, first create a “shortcut” on your desktop or in a folder (see your Windows help system if you don’t know how to do this).  Then:

•	Rename the shortcut icon to tell you what project it represents. (Right click, rename)

•	Now set the working directory for that project. (Right click on icon, click on properties, click on shortcut tab, then in the “Start in” space, type in the path to the folder containing the data files for that project, for example, c:\forests\edgedata)

From then on, when you click on this shortcut icon, PC-ORD will look in the folder you specified.  It will also remember the history of files, graph setup, etc., so that when you start PC-ORD from that icon, you will be right where you left off.

Windows 3.x

Version 4 of PC-ORD no longer supports Windows 3.x.  If you are using Windows 3.x, you should either upgrade your operating system or use PC-ORD version 3.

Network Environments

Getting Started


Network Environments

PC-ORD runs fine from a file server in a network environment, but you must set up the software correctly.  Because network environments vary, your network administrator may need to help you set up PC-ORD.   Generic instructions for a typical network setup are given below.  These presume that PC-ORD has already been installed on your network by your network administrator.

1.	Right click on some free space on the desktop.

2.	Select:   New | Shortcut

3.	For “command line” enter the path for pcord4.exe.  Your network administrator can provide this path.

4.	Click on “Finish”.  There should now be the PC-ORD Windows icon (two overlapping circles) on the desktop.

5.	Right click on the PC-ORD icon.

6.	Select:  Properties | Shortcut

7.	Under “Start in” enter your working directory (where you will keep your PC-ORD data files and temporary files).

IMPORTANT POINTS

1.  Each user needs a unique working directory.  Call this the "user's directory."  The user's directory can be either on a local drive or on a network drive.  However, the user's directory must be unique to that user or workstation and the user must have write access to that directory.

2. The "Start In" directory MUST be changed to that unique user's directory. Install PC-ORD in the normal way.  Then right-click the PC-ORD icon, select "properties", and type in the name of the user's directory into the "start in" box.  Conflicts will result if multiple users are writing to the same directory.

Recommended setup:

1.  Executable files are installed on network directory and shared by multiple users.

2.  Users keep data files and temporary files on their own hard disks or in their own directory on a network drive.

For class use:

1.  Same as above, but each user should copy the shared data files into their own directories.

2.  In computer classrooms, users may want to define their "Start in" directory as the floppy drive (usually the A drive).  This allows users to preserve their own settings from one session to the next, regardless of which machine they are using.  (Note, however, that the same thing is accomplished by having users use their own logical drive on the network, if such is available.)

Getting Started

Managing Files and Projects


System Requirements

Operating PC-ORD version 4 requires the hardware and software listed below.

Hardware Required

80486 or higher CPU

8 MB RAM

6 MB of available hard disk space

Operating System

Windows 95, 98, or NT

Spreadsheet Software Recommended

A spreadsheet program capable of writing files in WK1 format is recommended.  This includes Lotus 1-2-3, Excel, and Quattro.

Use With Memory-resident Programs

Getting Started


Memory Use

Most of the programs in PC-ORD for Windows can accept a matrix up to 32,000 rows by 32,000 columns, provided that you have adequate memory installed in your machine.

In PC-ORD 2.x and earlier versions, memory use was limited to 640 kb.  With the first windows version (PC-ORD version 3.x), all of your available memory is usable by PC-ORD if needed, subject to a limit of 16 mb for a single matrix.  This allowed much larger data sets.  Few users with 16 mb or more of RAM had a data set too large for PC-ORD.  With PC-ORD 4.x all of the available memory can be used and there is no longer a limitation to single matrices being less than 16 mb.  In this version, matrix size is limited by the maximum matrix size possible with the *.wk1 file format:  32,000 rows by 32,000 columns.

You can evaluate your memory needs with the Memory Requirements utility provided under the File menu.

If you do encounter an “out of memory” error, and you do not have an exceptionally large data set, then a memory allocation error may have occurred.  In this case, close out all programs, go through your normal shut-down procedure, restart, then try to rerun your analysis in PC-ORD.

Getting Started


Orders

Multivariate Analysis on the PC-ORD System Order Form (Print in fixed pitch font, such as Courier, to maintain format.)

 -----------------------------------------------------

|  [ ] Regular new user  $299  . . . . . . . ________ |

|                                                     |

|  [ ] Student new user $199 . . . . . . . . ________ |

|                                                     |

|  [ ] Upgrade $99 . . . . . . . . . . . . . ________ |

|                                                     |

|  [ ] Site or Network license $299 base . . ________ |

|      plus $100 per simultaneous user                |

|                      ______ Users x $100 = ________ |

|  [ ] Network Upgrade $99 base . . . . . . .________ |

|      plus $50 per simultaneous user                 |

|                      ______ Users x $50  = ________ |

|                                                     |

|      Shipping and handling . . . . . . . .   $10.00 |

|      Overseas add $10.00 . . . . . . . . .          |

|                                            ---------|

|                                Total . . . $        |

 -----------------------------------------------------

 Method of Payment:

 _         _               _

|_| Visa  |_| MasterCard  |_| Purchase Order #___________

 _

|_| Check enclosed, payable to MjM Software

Outside the USA: please use credit card or bank draft in US dollars

Card #______________________________ Exp.________________

Phone No.________________________________________________

 -------------------------------------------------------------

| Send order to:          MJM Software Design                 |

|                            PO Box 129                       |

|                      Gleneden Beach, OR 97388               |

|                                                             |

| For fastest service, fax this order form to: 1-541-764-3935 |

| Or for credit card orders, call toll free: 1-800-690-4499   |

 -------------------------------------------------------------

Please rush me PC-ORD 4.0 for Windows via first class Mail to:

Name: ___________________________________________________________

      ___________________________________________________________

Address:_________________________________________________________

City:__________________ State:_____ Zip (or postal code):________

Country (if outside USA): _______________________________________

Getting Started


Waiver

Although the software has been carefully tested, the user accepts and uses the program material as it is, at the user's own risk, relying solely upon his/her own inspection of the program material and without reliance upon any representation or description concerning the program material.  Neither MjM Software Design nor its individual staff members make any expressed or implied warranty of any kind with regard to the program material, and shall have no liability or responsibility to any recipient with respect to any liability, loss or damage directly or indirectly arising out of the use of the disk and the programs and/or subroutines contained on the disk, including, but not limited to, any loss of business or loss of data or other incidental or consequential damages.  By using this software you agree to this.

Getting Started


Use With Memory-resident Programs

Some PC users routinely load memory-resident programs (such as network drivers) when they boot up.  You may find that this prevents some of your larger data sets from loading into memory.  If you encounter this problem, reboot but do not load the memory-resident programs.  You can check the amount of free memory (RAM) by selecting File | Memory Requirements in PC-ORD.

Getting Started


PCORD Failure

If a PC-ORD program fails for any reason, an error message is printed in a dialog box.  Additional messages or hints to the problem may appear in the result file.  By far the most common problems encountered with PC-ORD result from errors in the data.

Occasionally conflicts over system resources will be encountered in Windows.  If you receive a message that is a “General protection fault” or that a program performed an “illegal operation,” you should exit PC-ORD, close any unnecessary programs that are open, then restart PC-ORD.  In some circumstances you may need to reboot your computer.  If the problem persists, note the exact steps that lead to the problem, then contact MjM Software.

The output from the ordination programs tends to be relatively bulky compared to your original data matrix.  This means that if you are doing a number of analyses and working from a floppy disk (not recommended but sometimes it’s helpful on multi-user stations) it's a good idea to check your available disk space periodically.  If it's getting full, print, then delete, your output file.

If the disk receiving your output fills up, the program running at the time will abort.  Print your output file, then delete it and any other unnecessary files, then rerun the program.

Technical Help

Frequently Asked Questions

Getting Started


Frequently Asked Questions

The best source for the latest "Frequently Asked Questions" is the MjM website.  Some FAQ are included here, grouped by subject.

General and misc.

Data Structure

Graphs

CCA

Technical Help

PCORD Failure

Getting Started

Example Analysis

Typical Flow of Analysis


FAQ General and misc.

Empty Variables Error

Q. I received an error message "Error: Empty variables; attempt to  divide by zero." What's wrong?

A. If your spreadsheet has rows or columns that are all zeros, many analyses will fail because in the underlying math, something is divided by sums or sums of squares that equal zero. In general you should check for empty rows and columns before running analyses. The  easiest way to spot these is to run the Row & Column Summary under  "Summary".  A related problem is when a column has a variance of zero  (all values are the same in that column). Any analysis requiring  division by the variance or standard deviation of that column will,  therefore, produce an error message.

Delete Rows/Columns from Both Matrices

Q. I deleted rows from my main matrix, but when I performed an analysis, I received an error message that the second matrix didn't match the main matrix.  How can I delete the corresponding rows from the second matrix?

A. Use "File | Switch Matrix " to swap the main and second matrices.  Delete the corresponding rows with "Modify | Delete Rows."  Then swap the matrices back to their original positions.

Analyze Second Matrix

Q. How do I get row and columns summaries, outlier analyses, etc. on the second matrix?

A. Use "File | Switch Matrix" to swap the main and second matrices.  Think of the main matrix as the active matrix.  This is the matrix on which the operations are performed, unless an operation specifically requires the second matrix.

Delete Rare Species

Q. Is there an easy way to delete rare species from a matrix?

A. Assume that species are columns.  Select "Modify | Delete columns | fewer than N non- zero values".  Then you enter a value for N.  If N=1, then empty columns are deleted.

Out of Memory

Q. I am running a large matrix and I get the "out of memory” error.  What can I do?

A. In the DOS version PC-ORD was limited to using 640K of memory.  But the Windows version can use ALL of your available memory. If you have plenty of memory and you get this error, your operating system has encountered some conflict in system resources and you may need to reboot.

There are several steps you can take to maximize the amount of available memory (check "Memory Requirements" under the File menu).

•	Avoid loading other memory-resident software, such as network drivers, if possible.

•	Reduce the size of your matrix by:

1) analyzing subsets of the data,

2) aggregating the observations to a higher level in your sampling design, or

3) eliminating rare species (say those in less than 5% of your sample units).

•	Add more memory (RAM) to your machine.

•	See additional suggestions for using large data sets.

Second Matrix and Graph File do not Match

Q. We get the following error message. "The number of items in your secondary matrix does not match the number of items in your graph file. Secondary matrix will be ignored.  This error came after doing the ordination and while trying to graph the ordination.

A. The most common cause of this problem is when one or more sample units (rows) are deleted from the main matrix. If you do this, you also need to delete the corresponding rows from the second matrix.  The easiest way to do that is to switch matrices (File | Switch), delete the offending rows, then switch them back.  Check the firs  lines for the data files (main, second) and the second line in the graph file.  They should all contain the same number.

Editing Spreadsheets

Q. Why doesn't PC-ORD allow cell-by-cell editing of spreadsheets?  I have to go back and forth between PC-ORD and a spreadsheet program when I am cleaning my datA.

A. PC-ORD doesn't attempt to duplicate the spreadsheet functions.  However the PC-ORD version 4 allows simple editing of the main and second matrices.  More complicated editing and sorting is best done in your spreadsheet software.  It is so easy to go back and forth between applications using Windows that there should not be a  great inconvenience.  You should note, however, that some spreadsheet programs, such as Excel, keep the file open, unless explicitly closed. This means that while a file is open in Excel it is not available to PC-ORD.

Editing and Searching Result Files

Q. How can I edit, search in, or insert comments in my result files?

A. Simple editing and commenting of your result file is possible directly within PC-ORD.  More functionality is provided on every single Windows computer -- just open the result file with the notepad or other text editor.  You can do this without closing down PC-ORD.

Convert Output to Tables

Q. How can I convert tables in PC-ORD’s output files into tables in my word processor?

A. Your word processor probably has functions for converting tabular text files into tables.  For example, using Microsoft Word, you can highlight a table in a PC-ORD output file, then select Table | Convert Text To Table, to convert the output into a standard tabular format.  Be sure to set the delimiter to a space character and adjust the desired number of columns.

Negative CV's

Q. In Row and Column Summary I obtained a negative coefficient of  variation (CV). How can that be?

A. One can get wild results like CV = -833% if the data include negative numbers, or were standardized in some way that produces negative numbers (such as relativization by standard deviates).  But the CV is always non-negative if the data are all non-negative.

Resampling Tests

Q. Are resampling tests (e.g., jackknife, bootstrap) provided for ordination procedures other than CCA?

A. Resampling procedures are used in PC-ORD for several other applications besides CCA (NMS, Indicator species analysis, Mantel test).  Jackknife estimates of species diversity are included.

Mac Version?

Q. Are versions of PC-ORD available for the Macintosh?

A. No.  At present there are only DOS and Windows versions of PC-ORD.

Frequently Asked Questions

Getting Started


FAQ on Data Structure

Analyses And Transformations of the Second Matrix

Q. How do I get row and columns summaries, outlier analyses, etc. on  the second matrix?

A. Use File|Switch Matrix to swap the main and second matrices.  Think of the main matrix as the active matrix -- the matrix on which  the operations are performed, unless an operation specifically  requires the second matrix.

Spreadsheet Columns

Q. How many columns of data can I use?  My spreadsheet program allows a maximum 255 columns.  I want to enter a data set with more species than that.

A. You can use up to 32,000 columns if your machine can handle it.  But most spreadsheet software is limited to 255 columns.  You can circumvent that by using compact format or database format.  PC-ORD imports these into a spreadsheet.  Or, depending on the shape of your data matrix, it may be advantageous to enter the transpose of your matrix in a spreadsheet program.  For example, if you had 100 plots and 300 species, you could enter the species as rows and the plots as columns, then transpose the matrix in PC-ORD.  If you have a very large data set we recommend either compact format or database format.

To use the database format, first export your data from your database to a delimited text file.  PC-ORD expects three columns, in the order:

      SampleUnitID     SpeciesID     Abundance

Then PC-ORD will build a matrix from this.  (This is new in version 4).  Both PC-ORD Versions 3 and 4 can write huge matrices by importing from compact format.

Both compact format and database (list) format are described in detail in the on-line help system and the manual.

Delete Rare Species

Q. Is there an easy way to delete rare species from a matrix?

A. Assume that species are columns. Select Modify | Delete columns | fewer than N non-zero values. Then you enter a value for N.  If N=1 then empty columns are deleted.

Delete Rows From Second Matrix

Q. I deleted rows from my main matrix, but when I performed an analysis, I received an error message that the second matrix didn't  match the main matrix. How can I delete the corresponding rows from  the second matrix?

A. Use File | Switch to swap the main and second matrices. Delete the corresponding rows with Modify | Delete Rows. Then switch the matrices back to their original positions.

Missing Data

Q. What can I do about missing data (cells in my matrices for which I have no data)?

A. You must take steps to replace missing cells with numerical values, or you must delete the offending rows or columns. If you ignore warnings about missing data, they are most likely to be interpreted as zeros, which may or may not be reasonable.  For more information see Missing Values.

Error Message About A Cell Outside Data Matrix

Q. When PC-ORD reads my data I get an error message about a cell that is outside of my data matrix. Why?

A. There is something in your spreadsheet outside the matrix. Even if a cell looks blank, there may be something there. For example, if you go to a cell in a spreadsheet, hit the space bar, then move to another cell, the cell will look blank, but there are actually data written in that cell: first a code saying there is a label field, next the "blank" character. When you put the cursor on the cell in question the spreadsheet will display one of the characters that signals a label field: usually ' " or ^. This is displayed not in the cell itself, but in the separate display that shows the contents and any imbedded codes in the current cell. You can get rid of these by erasing the cell in question or a whole block of cells.

Subplots In Each Plot

Q. I have 40 plots and 60 subplots in each plot. Each of the plots has been recorded at 3 dates. How can I enter and analyze this large data set in PC-ORD?

A. Your data set has a nested structure which is best handled in PC-ORD with the compact data format using "break groups." If each combination of date/plot/subplot is analyzed separately, then you will have way too many rows for analysis (40 x 60 x 3 = 7200 sample units). We suggest combining some of the sample units -- essentially collapsing the nested structure somewhat. How you do this depends on the question of interest. For example, if you are interested in the differences between dates, you might aggregate the 60 subplots within each combination of date/plot, leaving you with a manageable 80 sample units. This would be done by creating a compact format file. If you already have the data in spreadsheets, you can export the data to compact format, then use a text editor to append the break groups that you choose, then import the data into PC-ORD, selecting aggregation of subsamples. The groups of plots to be combined are separated by the keyword "break". However you set up your main matrix, you should set up a corresponding second matrix that codes for each level in your design.

Using Numbers as Row Labels

Q. I wish to label my sample units with numbers, but PC-ORD insists on character data in the first column. It seems that the only way to get number labels into PC-ORD is to either enter them in compact format and import, or reformat each cell by hand in Quattro or Lotus, or insert a character before the first number in Excel.

A. This is a problem with the Excel export. It is too smart for its own good, converting numbers defined as characters into numbers defined as numbers. We could change it to allow numbers in that field, but at some risk of clarity -- some people will surely omit the row labels and the analyses will be based on one too few columns.

Compact Format vs. Spreadsheet

Q. I have a large data set in a spreadsheet program. The manual states that the compact data format should be used to optimize storage of this type of data. Do I need to convert my spreadsheet to compact format?

A. Compact format is mainly a time saver with data entry. You CAN use it to store data in a more compact way, but unless you are low on disk space, it is usually more convenient to keep the data in spreadsheet format. There are, however, a number of situations in which using compact format can be very helpful. If you are managing large data sets with many subsets, you will find compact format convenient for combining subsets in various ways. You can easily combine different subsets of a large data set, just by appending the compact-format text files to each other. But if you (a) have plenty of disk space, and (b) do not need to fiddle a lot with subsets of data, and (c) do not need compact format for data entry, then you should use spreadsheet format.

Export Compact Format with Breakpoints

Q. I have a large spreadsheet data set that I wish to export to compact format. How can I insert breakpoints in the compact format file?

A. When you export to compact format, breakpoints are not inserted. If you wish to insert those, do so with a text editor after you have created the compact file.

Enhancements of TWINSPAN

Q. I would like to see dendrograms produced by TWINSPAN as well as other enhancements to the program.  Will TWINSPAN be improved in PC-ORD?

A.  We decided not to spend much time trying to upgrade Twinspan, other than expanding its memory capabilities, because of serious, fundamental flaws with the concepts behind the method and the resulting poor performance of Twinspan with heterogeneous data sets (see for example van Groenewoud 1992, Belbin, L. and C. McDonald.  1993). The problem is how well the underlying structure of the data is represented by the two-way table produced by Twinspan.  Have you tried cluster analysis? If you are not familiar with cluster analysis, we recommend using the following options: Ward's method of linkage and Euclidean distance. If you get groups of sample units that are acceptable, then save the classification as a new variable (see option under cluster analysis), then run Indicator Species Analysis to tell you which species are most strongly separated among the groups.

Append Matrix

Q. How can I append one community matrix to another in PC-ORD?

A. If the columns of the two matrices match, it is fairly easy to cut and past the matrices together, using your spreadsheet program.  If, however, the column variables are in a different order or differ in number, it may be easiest to append the matrices in compact format. First, export each matrix to PC-ORD compact format. To do this you will first need to create a species file. Then combine the files by cutting and pasting with your word processor or text editor. Save the new file as a text file, then import this combined compact-format file to a *.wk1 file with PC-ORD.

Convert Database Format to Spreadsheet

Q. How can I convert my data in a database to wk1 spreadsheet format?

A. PC-ORD version 4 allows you to import/export files in a list (or database) form.  You can choose your field delimiters.  Under file formats, see List (Database) Format.

Frequently Asked Questions

Getting Started


FAQ on Graphing

Edit and Save Graphics

Q. How can I edit and save graphic images of ordinations and overlays?

A. Graph an ordination.  Then select File | Save Graphic As.  The graph is saved as a Windows metafile (*.wmf).  This file can then be inserted into many word processors and other programs where you can easily resize and edit it.

3-D Plots Possible?

Q. Is it possible to generate 3-dimensional (XYZ) ordination diagrams in PC-ORD?

A. No. You can plot axes 1 vs 2, 1 vs 3, and 2 vs 3, but not 1 vs 2 vs 3. We have not seen a good way of doing 3D ordination plots. If you know a good way, let us know!

View Two Graphs at Once

Q. How can I view two graphs at once?

A. Yes.  You can start two instances of PC-ORD, putting each in a window occupying half of your screen. Then put up a graph in each one. There is some potential for confusion if the two "sessions" are working from the same directory. So it is safest to run each session from a different icon, the "Start In" property of each icon pointing to different directories.

Rotation of Axes

Q. Does the new version permit rotation of PCA axes?

A. Both the DOS and Windows versions allow rotation of axes from ANY of the ordination methods. Varimax rotation is, however, available only with NMS in both versions.

Frequently Asked Questions

Getting Started


FAQ on CCA

How CCA In PC-ORD Differs

Q. How does CCA in PC-ORD differ from that available in other statistical packages?

A. CCA is not available in major statistical software packages (do not confuse it with canonical correlation, an older eigenanalysis technique based on a linear response model). CCA in PC-ORD does not have some of the options available in another implementation (such as inclusion of covariates and detrending), but we think you will be pleased with the spreadsheet data format, modern interface, integrated graphing, modest price, and our fully integrated set of data transformations and other multivariate tools.

CCA in PC-ORD is much easier to use than CANOCO v. 3.1 but doesn't have some of the bells and whistles. For example, CANOCO allows for inclusion of covariates while CCA in PC-ORD does not. On the other hand, CCA in PC-ORD is integrated into a relatively comprehensive package of community analysis, providing many more classes of analyses than CANOCO.  The graphics in PC-ORD are more flexible and polished than from CANOCO.  PC-ORD has stringent criteria for convergence of the solution, in response to criticism of the question of "tolerance" in CCA (see Oksanen and Minchin, 1997).  PC-ORD offers two kinds of monte carlo hypotheses for CCA while CANOCO has only one.

CCA: Cannot Solve Weighted Least Squares

Q. When trying CCA I get the message "cannot solve weighted least squares, please check contents of matrices." What should I check for?

A. CCA cannot solve the regression equation under several circumstances. One common cause is when two or more variables are essentially identical or identical. Another possibility is that the same variables are included in both the main and second matrices.

CCA Option Not Given In Ordination Menu

Q. I want to run CCA but that option is not given in the drop- down Ordination menu. What should I do?

A. You must have both main and second matrices opened in order to run CCA.

Order of Variables in Correspondence Analysis

Q. Concerns have be raised about results from correspondence analyses depending on order in which variables are entered - is this a problem? If so, has it been fixed?

A. This was mainly a problem with DCA and TWINSPAN, less so with CCA. The Windows version has implemented Oksanen and Minchin's (1997) "super strict" criteria for stability of CCA, DCA, and TWINSPAN.

Frequently Asked Questions

Getting Started


Descriptive Title

Most of the analytical procedures request that you enter a descriptive title for your results.  Click on the input box and type in a phrase that will help you document your output.  Whatever you enter will be used as a title on your output.

Getting Started

How to Use Help


Help For Old Friends

Changes in version 4.x (Windows 32-bit version) from 3.x (Windows 16-bit version):

General

•	32-bit Windows 95/98/NT native

•	Allows data matrices larger than 16 MB

•	New manual

•	More distance measures

•	Edit spreadsheets

•	Organize files within multiple projects

New graphics

•	Publication-quality dendrograms for cluster analysis

•	Publication-quality species-area curves

•	Confidence bands in species-area curves

•	Scree plot in NMS

Enhanced ordination graphics

•	Reflect ordinations (in addition to rotation)

•	Faster graphing of large data sets

•	Fit envelope to side graphs in overlays

•	Optional one-dimensional ordination

New analyses

•	Autopilot mode for NMS

•	Species ordinations for NMS, BC, PCA

•	Blocked MRPP (for randomized block designs)

•	Predictive-mode NMS (NMS Scores)

•	Species List Broken Down by Groups

•	Weighted Averaging ordination

Enhanced import/export

•	Export ordination scores to spreadsheet

•	Add ordination scores to second matrix

•	Import/export database format

•	Import/export comma-separated values (csv) format

Details

•	Default graphics files changed from *.WMF (Windows Metafiles) to *.EMF (Windows Enhanced Metafiles); *.BMP output also available.

•	Increased table size in Twinspan (up to 9999 species)

•	Simpson's index of diversity

•	Improved random number generator

•	Up to 1000 blocks and 100 groups in blocked MRPP and MRPP

•	More flexible matrix multiplication

•	Option for rank-transformed distances in MRPP

•	Save a series of grouping variables from cluster analysis.

Changes in version 3.x (Windows version) from 2.x (DOS version):

•	Windows-native software (runs under Windows 3.1, Windows 95, and Windows NT)

•	All available memory can be used, allowing very large data sets (matrices up to 16,000 by 16,000 if you have enough memory).

•	New routines:

•	Indicator Species Analysis

•	Species Listing

•	Publication-quality graphics and completely reworked graphing routines including vector-based graphics, option to save graphics in Windows Metafiles (*.WMF), flexible labelling, much expanded printer support, and more.

•	Mantel Test

•	Additions to old routines

•	Monte Carlo testing in CCA

•	Chance-corrected within-group agreement added to MRPP

•	Species and sample unit codes up to 8 characters (formerly 6 characters)

•	Data and graph file formats are identical between version 2.x and 3.x

Some details:

•	The “Classify” menu has been renamed “Groups ” and now includes Indicator Species Analysis and Mantel Tests.

****************

Users of the old versions of PC-ORD (1993 version and older) will easily adjust to the versions 2.x or 3.x by keeping the following in mind:

•	The new standard data format is a *.wk1 spreadsheet.

•	Fortran format specifications are no longer necessary.

•	The main and second matrices are now placed in separate files.

•	The main and second matrices now have identical formats.

•	Old data files can be imported to spreadsheets by selecting File | Import. (Note that if you import an old PC-ORD file containing both main and second matrices, then two spreadsheet files are created, one for each matrix.)

Getting Started

Introduction


Modify Data

Data modifications are saved to a temporary file called  Work.wk1 replacing  the original matrix data file and Work2.wk1 replacing the second matrix,  if any.  This does not affect your original raw data file.  All of the data modifications EXCEPT matrix multiplication are performed only on the main matrix.  Data modifications include relativizations and standardizations, multiplication of two matrices, deletion of columns or rows with fewer than x nonzero entries, deletion of specific columns or rows, arithmetic operations on columns or rows, transposing the main matrix, and shuffling (randomizing) the data.

You can also modify your data with your spreadsheet software, but be sure to close the file in PC-ORD before modifying and saving it with other software.

Deciding the fate of temporary and modified files is discussed below under Saving Modifications.

Standardize

Relativization

Beals Smoothing

Multiply Matrices

Transpose

Delete With Fewer Than N Nonzero

Delete Rows or Columns That You Specify

Multiply or Add a Constant

Shuffle


Domains and Ranges

Some operations are unreasonable or even impossible for certain types of data.  The Table below lists the kinds of data that are potentially usable for each transformation.

Table.  Domain of input and range of output from transformations.

	Transformation	Reasonable and   acceptable    domain of x	Range of f(x)
MONOTONIC TRANSFORMATIONS
	x0  (power)	all	0 or 1 only
	x½  (power)	non-negative	non-negative
	log(x)	positive	all
	(2/π)⋅arcsin(x)	0 < x < 1	0 to 1 inclusive
	(2/π)⋅arcsin (x½)	0 < x < 1	0 to 1 inclusive
SMOOTHING

	Beals smoothing	0 or 1 only	0 to 1 inclusive
ROW/COLUMN RELATIVIZATIONS
	general	non-negative	0 to 1 inclusive
	by maximum	non-negative	0 to 1 inclusive
	by mean	all	all
	by standard deviates	all	generally between -10 and 10
	binary by mean	all	0 or 1 only
	rank	all	positive integers
	binary by mean	all	0 or 1 only
	ubiquity	non-negative	non-negative
	information function of ubiquity	non-negative	non-negative


Picklists

A number of dialogs under the Modify Data (and elsewhere in PC-ORD) use "picklists" to select particular rows or columns, or sets of rows or columns.  To select a single item in a picklist, click on that item, then click “OK.”

To select a block of items in the picklist, click on the first item in the block, then hold down the Shift key and click on the last item, then click “OK.”

To select a number of different items in the picklist, hold down the Ctrl key while clicking on the items you want to select.  Then click “OK.”

Getting Started


Standardize

Modify Data | Standardizations

Standardizations are applied to each element of your data matrix, independent of the other elements.

In the descriptions of all standardizations, Xij = the original value in row i and column j of the data matrix and b = the adjusted value that replaces Xij.

Power Transformation

Logarithmic Transformation

Arcsine Transformation

Arcsine Square Root Transformation

Modify Data


Relativization

Modify Data | Relativizations

Relativizations adjust matrix elements by a row or column standard (e.g., maximum, sum, mean, etc.).  You will be asked whether it should be applied to rows or columns.

Most of these relativizations result in numbers that vary between 0 and 1.  Some relativizations produce both positive and negative numbers.

In all cases, Xij = the original value in row i and column j of the data matrix and b = the adjusted value that replaces Xij.  Bear in mind that some operations are unreasonable or even impossible for certain types of data (see Domains and Ranges).

General Relativization

Relativization by Maximum

Adjustment to Mean

Adjustment to Standard Deviate

Binary with Respect to Mean

Rank Adjustment

Binary with Respect to Median

Variates by Ubiquity in Rows/Cols

Information Function of Ubiquity


Power Transformation

Modify Data | Standardizations | Power Transformation

Enter the parameter (exponent) for the transformation.  p = 0.0 gives presence/absence, p = 0.5 gives square root, etc.  The square root transformation is similar in effect but less drastic than the log transformation.  Special treatment of zeros is not necessary.

where 0 <= p <= 9,  and p is any real number.

Using picklists

Modify Data

Standardizations


Log Transformation

Modify Data | Standardizations | Logarithmic Transformation (base 10)

This transformation can be applied to individual columns or all columns  at once.  Note that to log-transform data containing zeros, a small number should be added to all data points.  With count data it is common to add one, so that b=log(0+1)=0.

A log transformation is often useful when there is a high degree of variation within attributes or when there is a high degree of variation among attributes within a sample.

Using picklists

Modify Data

Standardizations


Arcsine Transformation

Modify Data | Standardizations | Arcsine Transformation

Data must range between zero and one, inclusive.  If they are not, you should relativize (general relativization or relativization by maximum) before selecting this option.

The constant 2/π scales the result of arcsin(x) [in radians] to range from 0 to 1, assuming that 0 < x < 1.

Using picklists

Modify Data

Standardizations


Arcsine Squareroot Transformation

Modify Data | Standardizations | Arcsine Squareroot Transformation

This transformation is recommended by many statisticians for proportion data (Sokal and Rohlf 1981).  As for the arcsine transformation, data must range between zero and one, inclusive.  If they are not, you should relativize (General Relativization or Relativization by Maximum) before selecting this option. The arcsine squareroot is multiplied by 2/π for the same reason as for the arcsine transformation.

Using picklists

Modify Data

Standardizations


General Relativization

Modify Data | General Relativization

If p = 1, relativization is by row or column totals.  This is appropriate if you plan to use a city-block distance measure, such as the Sorensen coefficient.  If p = 2, you are "standardizing by the  norm" (Greig-Smith 1983, p.248).  Using p = 2 is the  Euclidean equivalent of relativization by row or column totals.

Modify Data

Relativizations


Relativization by Maximum

Modify Data | Relativization by Maximum

where rows (i) are samples and columns (j) are species, Xmaxj is the largest value in the matrix for species j.  This adjustment tends to equalize common and uncommon species.  Very rare species can cause considerable noise in subsequent analyses if not omitted.

Modify Data

Relativizations


Adjustment to Mean

Modify Data | Adjustment to Mean

This relativization produces positive and negative numbers.

If relativized by rows, the means are row means; if by columns, the means are column means.  This is a centering procedure which can have detrimental effects on analysis of community data.  It tends to give more emphasis to values of zero.  Also, more variable species are reduced in  importance relative to more constant species, a questionable procedure ecologically.

Modify Data

Relativizations


Adjustment to Standard Deviate

Modify Data | Adjustment to Standard Deviate

where Si is the standard deviation of row or column i.

The transformed values represent the number of standard deviations away from the mean.  As for all of the row/column relativizations, this transformation can be applied to either rows or columns.  Note that this transformation is NOT compatible with some distance measures, such as Sorenson's (1-2w/(a+b)).  See additional cautions under Adjustment to Mean.

Modify Data

Relativizations


Binary with Respect to Mean

Modify Data | Binary with Respect to Mean

b = 1 if x > x   b = 0 if x ≤ x

The transformed values are zeros or ones.  An element is assigned a zero if its value is less than or equal to the row or column mean.  The element is assigned a one if it is greater than the row or column mean.  This transformation can be used to emphasize the optimal parts of a species range, at the same time equalizing to some extent the weight given to rare and dominant species.

Modify Data

Relativizations


Rank Adjustment

Modify Data | Rank Adjustment

Matrix elements are assigned ranks within rows or columns such that the row or column totals are constant.  Ties are assigned the average rank of the tied elements.  For example, the values 1, 3, 3, 9, 10 would receive ranks 1, 2.5, 2.5, 4, 5.

This transformation should be applied with caution.  For example, most community data have many zeros.  These zeros are counted as ties.  Because the number of zeros in each row or column will vary, zeros will be transformed to different values, depending on the number of zeros in each row or column.  For example, the values 0, 0, 0, 0, 6, 9 would receive the ranks 2.5, 2.5, 2.5, 2.5, 5, 6 while the values  0, 0, 6, 9 would receive the ranks 1.5, 1.5, 3, 4.

Modify Data

Relativizations


Binary with Respect to Median

Modify Data | Binary with Respect to Median

b = 1 if x > median, b = 0 if x ≤ median

The transformed values are zeros or ones.  An element is assigned a zero if its value is less than or equal to the row or column mean.  The element is assigned a one if it is greater than the row or column median.  This transformation can be used to emphasize the optimal parts of a species range, at the same time equalizing to some extent the weight given to rare and dominant species.

The Rank Adjustment caution also applies to this relativization, because it is based on ranks.

Modify Data

Relativizations


Weighting by Ubiquity

Modify Data | Variates Weighted by Ubiquity in Rows or Columns

For example, if rows are samples, columns are species, and relativization is by columns, more ubiquitous species are given more weight.

where

Assuming the conditions stated above, Nj is the number of samples in which species j occurs and N is the total number of samples.

Modify Data

Relativizations


Information Function of Ubiquity

Modify Data | Information Function of Ubiquity

*

where

and

Nj and N are as defined under Weighting by Ubiquity

To illustrate the effect of this relativization, assume that rows are samples, columns are species, and relativization is by columns.  Maximum weight is applied to species occurring in half of the samples because those species have the maximum information content, according to information theory.  Very common and rare species receive little weight.  Note that if there are empty columns, the transformation will fail because the log of zero is undefined.

Modify Data

Relativizations


Multiply Matrices

Modify Data | Multiply Matrices

This procedure is included mainly for weighted averaging ordination and experimental purposes.  Matrix multiplication greatly increases the analytical possibilities when you have two related matrices (main matrix and secondary matrix).  For example, suppose you are doing a standard community analysis on  species composition and environmental data.  Your main matrix contains the abundance of each species in each plot and your secondary matrix contains the environmental parameters for each plot.  The most common analytical approach is to ordinate plots in species space.  That is, plots are arranged by their similarities in species composition.  Environmental variables from the secondary matrix are then overlaid graphically on the ordination or correlated with axis scores.  The purpose of this procedure is to see how species distributions are related to environmental factors.

But why not ordinate species directly in environmental space?  First you would need a matrix of species scores for each environmental variable.  One way to do this is to multiply your main matrix (A) by your secondary matrix (E).  Actually, to make the matrix algebra come out right, you must post-multiply the transpose of the main matrix (rows and columns switched) by the environmental matrix:

matrix of species x environmental matrix = A'E

(The transpose of A is indicated by A'.)  The resulting matrix can now be used as the main matrix, allowing you to ordinate directly species in environmental space (or environmental variables in species space).  There are serious problems with this approach.  For example, all of the environmental variables are included, whether or not they are related to species distribution and abundance.

There must be both main and secondary matrices (otherwise, you don't have two matrices to multiply).  It is important to give some thought to prior data transformations.  Because matrix multiplication involves adding the products of many numbers, the resulting numbers can becoming very small or very large.

A worse problem is that the resulting matrix can be nearly meaningless if careful attention is not paid to scaling your variables.  For example, if one of the environmental variables ranges from 1000 to 3000 and the other environmental variables range from 0 to 1, the large numbers will  completely obscure the small numbers, resulting in a matrix dominated by a single environmental variable.  This problem could be avoided by standardizing or relativizing the variables in the two matrices, so that each variable is given equal weight.

It is important that you examine the resulting matrices critically to be sure they contain what you think they contain.  See GreigSmith (1983 pp. 229 and 278) for further cautions about ordinations based on environmental data.

Matrix Multiplication Setup

Modify Data


Matrix Multiplication Setup

Before multiplying your main and second matrices you must select several options as detailed below.

Multiplication Type

You can calculate the matrix product AE or the product A`E.  With a given pair of main and second matrices, only one of these is possible, according to the rules of matrix algebra.  If the number of rows is the same in your main and second matrices, then only A'E is possible.  If the number of columns in the main matrix equals the number of rows in the second matrix, only AE is possible.

Where the result is stored is optional. The first option calculates A`E and uses it to replace E.  However, if you are interested in an ordination of A`E, use A`E to replace the main matrix (A).  You can do this with the second or third options.  These replace A with A`E (Option 2) or its transpose (A`E)` (Option 3).  With the third option environmental parameters end up as rows of the main matrix.

If you calculate AE you can write the product into the spreadsheet used as the second matrix (the fourth option), or you can write the product into a graph file (the fifth option). This allows you to graphically display the result for the first three columns of the resulting matrix, using Graph | Graph Ordination.  Any additional columns beyond the third column of data are ignored by the graphing procedure.

Normalize matrices before multiplication

If you choose this option, the rows of your main and secondary matrices are standardized by their standard deviates.  In other words, each value in a row is expressed as the number of standard deviations away from the row mean.  Do not choose this option if you also choose to divide by the column totals after multiplication (see below).  If you wish to apply some other kind of normalization, use Modify data | Relativizations before requesting matrix multiplication.

Division after multiplication

Three options are available:

•	None –  No division after multiplication results in a simple sums of the products in the resulting matrix.

•	By N = x samples" –  "x samples" is the number of rows (usually sample units) in your main matrix.  Elements of the new matrix are filled with average cross-products of two vectors rather than the sum of the cross-products.  This option can be used to either (a) in cases where the average is conceptually more desirable than the sum or (b) help to keep small the elements in the resultant matrix without affecting the weighting of individual rows or columns.

•	By column totals – Column totals are from the main matrix.  This results in a weighted averaging step and can be used to produce a weighted averaging ordination .  Be sure not to normalize your matrix before selecting this option because you will be attempting to divide by zero.

It is important that you critically examine the resulting matrices to be sure they contain what you think they contain.  Check the dimensions of the resulting matrix and examine some elements of the matrix.  See GreigSmith (1983: pp. 229 and 278) for further cautions about ordinations based on environmental data.


Transpose

Modify Data | Transpose Main Matrix

By transposing your main matrix, you can treat your former entities as attributes and vice-versa.  For example, starting with the standard community matrix having rows as samples and columns as species, the normal analysis is to ordinate samples in species space.  If you wish to  ordinate species in sample space (the transpose analysis), transpose your main matrix by selecting this option.  Then run the ordination program.

Modify Data


Delete Rows

Modify Data | Delete Rows

Rows are deleted from the main matrix.  You can delete rows that you specify or delete rows with fewer than N nonzero values   Use the latter option to delete rows that are empty or nearly empty, using the criterion N, which you can set.

Note that these operations apply only to the main matrix.  If you wish to delete corresponding rows or columns in the second matrix, you should switch the matrices, perform the operation, then switch them back.

Modify Data

Delete Rows with Fewer than N Nonzero Values

Delete Rows that You Specify


Delete Columns

Modify Data | Delete Columns

Columns are deleted from the main matrix.  You can delete columns that you specify or delete columns with fewer than N nonzero values.  Use the latter option to delete columns that are empty or nearly empty.  You set the cutoff number, N.

If you wish to delete empty columns, select “Delete with fewer than N nonzero” then enter a 1.  If your columns are species and you wish to delete rare species, say those occurring in less than 3 sample units, set N=3.

Note that these operations apply only to the main matrix.  If you wish to delete corresponding rows or columns in the second matrix, you should switch the matrices, perform the operation, then switch them back.

Modify Data


Delete With Fewer Than N Nonzero

Modify Data | Delete Rows  | with Fewer than N Nonzero Values

Modify Data | Delete Columns | with Fewer than N Nonzero Values

This is a useful way of reducing the bulk of your data set without losing much information.  It is a good way of deleting empty rows or columns.  It is an effective way of reducing the noise introduced by rare species.

Modify Data


Delete Rows or Columns That You Specify

Modify Data | Delete Rows | that You Specify

Modify Data | Delete Columns | that You Specify

This is useful for eliminating anomalous samples or species.  It can also be used to partition your matrix, for example, to delete all the shrubs and herbs, leaving just the trees.

Using picklists

Modify Data


Multiply or Add a Constant

Modify Data | Multiply or Add a Constant

This can be useful in several ways.  One common use would be to convert proportion data (range = 0 to 1) to percents (range = 0 to 100) by multiplying by 100.  Another use is to replace the values for a variable by some function of that variable; this may require more than one pass through the option.  For example, equations of the form y = ax+b will require two passes: one pass for the slope a and one for the intercept b.  Note that to subtract, you can add a negative number; to divide, multiply by the reciprocal of the constant that you wish to divide by.

Using picklists

Modify Data


Shuffle

Modify Data | Shuffle

Select Shuffle to randomize your data within columns:  values in columns are randomly reassigned to new positions in the same column.  The result is a data set that has the same column totals, the same matrix total, and the same number of elements containing zeros.

Why would you want to shuffle your data?  It is useful as an educational tool for showing how multivariate methods can appear to detect pattern from nonsense.  It is more interesting than using a table of random numbers as input data, because a random numbers table is unlike typical  community data.  For example, most community data sets have a large proportion of zeros in the matrix and a very skewed distribution that is neither flat nor normal.  Those characteristics (proportion of zeros and frequency distribution within columns) are preserved by shuffling.  Two characteristics of the data are destroyed: the resemblance's among columns (usually species) and the row totals.

Your data should be arranged with sample units as rows and species as columns (but feel free to experiment with other arrangements and types of data).  Only the main matrix is shuffled.

Modify Data


Beals Smoothing

Modify Data | Beals Smoothing

where S is the number of species in sample i, Mjk is the number of samples with both species j and k, and Nk is the number of samples with species k.  (See example below.)

The purpose of this transformation is to relieve the "zero-truncation problem" (Beals 1984; McCune 1994).  This problem is most severe in heterogeneous community data sets with a large number of zeros in the data matrix (i.e., most samples contain a fairly small proportion of the  species).  Beals smoothing (also known as the "sociological favorability index") replaces presence/absence or other binary data with quantitative values that represent the "favorability" of each sample for each species, regardless of whether the species was present in the sample.  The index evaluates the favorability of a given sample for species i, based on the whole data set, using the proportions of joint occurrences between the species that do occur in the sample and species i.

This transformation is essentially a smoothing operation designed for community data (McCune 1994).  As with any numerical smoothing, it tends to reduce the noise in the data by enhancing the strongest patterns in the data.  In this case the signal that is smoothed is the pattern of joint occurrences in the data.  This is an extremely powerful transformation that is particularly effective on heterogeneous or noisy data.  Caution is warranted, however, because, like any smoothing function, this transformation can produce the appearance of reliable, consistent trends even from a series of random numbers.

This transformation should not be used on data sets with few zeros.  It also should not be used if the data are quantitative and you do not want to lose this information.

Beals smoothing can be slow to compute.  If you have a large data set and a slow machine, be sure to allocate plenty of time.  The more non- zero values in your data, the longer it takes.  This transformation is intended for zero-laden matrices.

Because Beals smoothing is applied by first calculating joint occurrences between species, you will be asked whether your rows or columns represent species.

Modify Data


Example of Beals Smoothing

Data matrix A before transformation (3 sample units x 5 species):

	sp1	sp2	sp3	sp4	sp5	Si
SU1	1	0	1	1	1	4
SU2	0	0	0	1	0	1
SU3	1	1	0	0	0	2
Nj	2	1	1	2	1
si = number of species in sample unit i.

Nj = number of samples with species j.

Construct matrix M, where Mjk = number of samples with both species j and k.  Note that where j = k, then Mjk = Nj).

	Species k
		1	2	3	4	5
	1	2
Species j	2	1	1
	3	1	0	1
	4	1	0	1	2
	5	1	0	1	1	1

Construct new matrix B containing values transformed with Beals smoothing function:

Data after transformation (B):

	sp1	sp2	sp3	sp4	sp5
SU1	0.88	0.13	0.75	0.88	0.75
SU2	0.50	0.00	0.50	1.00	0.50
SU3	1.00	0.75	0.25	0.25	0.25

Example for sample 1 and species 2:

b1,2 = 1/4 (1/2 + 0/1 + 0/2 + 0/1)

b1,2 = 0.25 (0.5)

b1,2 = 0.125 (rounded to 0.13 in matrix above)

Example for sample 3 and species 2:

b3,2 = 1/2 (1/2 + 1/1)

b3,2 = 0.5 (1.5)

b3,2 = 0.75

Beals Smoothing

Modify Data


Saving Modifications

Several dialogs are used to handle saving or discarding temporary files:

•	Transformed matrix

•	Result File Not Saved

•	Main Matrix File Not Saved

•	Second Matrix File Not Saved

•	Graph File Not Saved

•

These are described below.

Modify Data


Transformed Matrix

After a matrix is altered you are asked whether to replace your existing matrix with the newly altered matrix: "Do you wish to use temp.wk1 as the new main matrix?"  Temp.wk1 is a temporary file containing the transformed data.  Choose “OK” to accept the new matrix, “discard” to revert to your old matrix.

Modify Data


Result File Not Saved

Most PC-ORD operations create a result file.  If a result file already exists, PC-ORD displays a dialog entitled "Results file not saved."  You are asked if you want to save the old one before the new result file is created.

You can:

•	Save the old result file – you will be asked for a file name

•	Append the old result file to a pre-existing named file– you will be asked for a file name

•	Discard the old result file.

Getting Started


Main Matrix File Not Saved

If your main matrix is a temporary file (work.wk1), rather than a named file, before it is replaced by another matrix, the "Transformed Matrix" dialog gives you a chance to:

•	save it as a named file, then move your new matrix into its place as the main matrix,

•	cancel the operation, or

•	discard the information in work.wk1 and replace it with the new matrix.

Getting Started


Second Matrix File Not Saved

If your second matrix is a temporary file (work2.wk1), rather than a named file, before it is replaced by another matrix, the "Transformed Matrix" dialog gives you a chance to:

•	save it as a named file, then move your new matrix into its place as the second matrix,

•	cancel the operation, or

•	discard the information in the temporary second matrix (work2.wk1) and replace it with the new matrix.

Getting Started


Graph File Not Saved

Ordinations create graph files.  If a graph file already exists, you are asked what to do with the old one before the new graph file is created.  This results in dialogs entitled "Graph Row File Not Saved" and/or "Graph Column File Not Saved"

You can:

•	save the old graph file – you will be asked for a file name

•	cancel the operation, or

•	discard the old graph file.

Getting Started


Summary

PC-ORD provides several tools for descriptive statistics and examining the basic characteristics of your data.

You may choose Compact-Format Data Summary if your data are in PC-ORD compact format.  (All of the other items below require the standard PC-ORD spreadsheet format.)  Your data can be summarized or rewritten in a number of different formats.  This is a useful tool for producing readable listings for error checking.  If you have defined a hierarchical structure in your data with breakpoints, you can produce summaries at both levels.

Row and Column Summary produces basic descriptive statistics for rows, columns, or both rows and columns.

Outlier Analysis is an important step for any multivariate analysis, because outliers can have a very strong influence on your results.

Species-Area Curves are useful for evaluating adequacy of subsampling.  Because we almost always underestimate diversity in our sampling, jackknife estimators are also included in this program, to provide an estimate of the number of species that were missed.

Species Lists can be produced from your main matrix, either for individual sample units or for your whole data set.  This requires that you have a species file, essentially a dictionary for decoding your species acronyms.


Row and Column Summary

Summary | Row and Columns Summary

Rows and columns can be summarized with basic descriptive statistics, and diversity measures (see table below).

Statistics calculated for each row and column include the mean, standard deviation, sum, minimum, and maximum, skewness, and kurtosis.  The skewness statistic in particular may be useful in evaluating univariate normality and examining the consequences of data transformations.

Four measures of diversity are also calculated:

S  =  richness  =  number of nonzero elements in the row.  If you have a normal community matrix then S is species richness (the number of species in a sample unit).

H' =  Shannon diversity

where	       pi  = importance probability in column i

= matrix elements relativized by row totals (see Greig-Smith 1983, p.163; based on Shannon and Weaver 1949).

E = evenness (equitability) = H' / ln(richness) (see Pielou 1969, p. 233).

D = Simpson's index of diversity for an infinite population. This is the complement of Simpson's original index. It is the likelihood that two randomly chosen individuals will be different species.

The following statistics are also reported:

•	Number of cells in main matrix

•	Percent of cells empty

•	Matrix total

•	Matrix mean

•	Variance of totals of rows or columns

•	CV (coefficient of variation) of totals of rows or columns  (useful for evaluating the variation among rows or columns as an indication of possible need for relativization).  Note: if you obtain a negative CV it is likely that your data contain negative numbers or that they were standardized in some way that produces negative numbers.  Conceptually the CV is applicable only to non-negative numbers.

This summary is produced only for the main matrix.  To summarize  your second matrix, select File | Switch to switch positions of  the matrices, then perform the row and column summary,  then switch them back.

Table.  Parameters in the statistical summaries.  S, E, H, and D are diversity statistics if rows are sample units.

Parameter	Sample unit summary	Species summary
Mean	Average species value in sample unit.	Average value across sample units.
Standard deviation	Variation in species within sample unit (N-1 as divisor).	Variation in species values across sample units (N-1 as divisor).
Sum	Total of species values in sample unit.	Total for a species across sample units.
Minimum	Lowest species value in sample unit (usually zero).	Lowest value for a species across sample units.
Maximum	Highest species value in sample unit.	Highest value for a species across sample units.
Skewness	Skewness of distribution of species values in a sample unit.	Skewness of distribution of values for a given species, across sample units.
Kurtosis	Kurtosis (peakiness) of distribution of species values in a sample unit.	Kurtosis of distribution of values for a given species, across sample units.
CV of totals	Coefficient of variation (100*standard deviation/mean) of the sample unit TOTALS as a percentage.	Coefficient of variation of the species TOTALS as a percentage.
S	Species richness in a sample unit.	Number of occurrences of a species.
E	Evenness of non-zero species values in a sample unit.  E  = H / ln(S)	Evenness of non-zero species values across sample units.   E = H / ln(S)
H	Diversity of species in a sample unit (Shannon index of information content).	Information content of a species across sample units.  (Only non-zero values considered)
E	Evenness of non-zero species values in a sample unit.  E  = H / ln(S)	Evenness of non-zero species values across sample units.   E = H / ln(S)
D	Simpson's index of species diversity in a sample unit	Ubiquity of a species across sample units (maximal for a species equally abundant in all sample units).

Row and Column Summary Setup

Summary


Compact Format Data Summary

Summary | Compact-Data Format Summary

The compact-format summary accomplishes a number of summarization tasks starting with a data file in compact format.  The purpose of the compact format is to make data entry and data checking faster.  It will save you time if you have a sparse data matrix, as is typical of community ecology.  Most sample units (plots) have only a small number of species compared to the total number of species in the sample.  In other words, the data matrix has many zeros.  The compact format eliminates the need to enter those zeros.

In addition, the compact format allows you to enter individual tree data, then aggregate the data to produce summaries for stands of trees (see example)

Compact Data Format

Tree Data Summaries

Compact Format Setup

Data Preparation

Summary


Compact Format Setup

The Compact Format Setup dialog allows you to specify the options for reading your data and writing output.  Each of these options is described below.  Output options include row summaries of the main matrix (Write Statistical Summaries of Main Matrix), a readable listing of the data (Write Sample Lists for Verification), and rewriting the data in list format (Write Data in List Format).

Data Type

Three types of data are allowed: lists of individual trees and their sizes, quantitative lists of species, and presence lists of species.  See Compact Format Data Types.

Aggregate Sample Units On Entry

If your compact file contains groups of data defined by breakpoints, you have options on whether or not these groups are (a) used to aggregate your data and (b) used to produce separate summaries for each group.

a. If your file contains breakpoints, you can create an aggregated matrix where each row of the main matrix represents average species values within a subgroup.  This is useful if you have a hierarchical data structure.  For example, consider a data set consisting of species composition in 40 plots in each of 20 stands.  At the plot level you would have 40 x 20 = 800 sample units.  However, the data can be aggregated to the stand level by averaging the 40 sample units in each of the 30 stands, yielding a main matrix containing 20 stands.

b. You can produce separate data summaries for each subgroup of data defined by breakpoints.  This is useful if you want to summarize or compare different portions of a single data set (for example, the upland vs. lowland parts of a woods).

Names for Rows

You are asked to provide a name that applies to your rows collectively  For example, if your rows are named Plot1, Plot2, Plot3, etc., you might collectively call your rows "Plots".  Enter the name in the space provided.

Names for Columns

You are asked to provide a name that applies to your columns collectively.  For example, if your columns are acronyms for various species, you might call your columns "Species".  Enter the name in the space provided.

Write Statistical Summaries of Main Matrix

Statistical summaries of the sample units and species of your main matrix are written to an output file.  Parameters included in these summaries are the mean, standard deviation, sum, minimum, maximum, richness, evenness, and diversity.  The parameters are calculated for each row separately.  Average values for each parameter are also given.  If you have breakpoints defined in your compact data file, you will be asked whether you want your data to be treated as a whole (i.e., ignore the breakpoints) or whether each subgroup of data should be summarized independently.

Write Sample Unit Lists for Field and Office Verification

This option produces a readable listing of your data.  This listing is a convenient way of checking your data for errors.  It eliminates the need for mentally converting species codes into species names.  The listing is also useful if your sample units are permanent, as one can revisit the sample units with data in hand in a readable form.

Separate Summary for Each Group

Groups of data may be defined by breakpoints, each of which may be analyzed separately.  You can use breakpoints to summarize or compare different portions of a single data set (for example, the upland versus lowland parts of a woods).

This is also useful if you have a hierarchical data structure.  For example, consider a data set consisting of species composition in 40 plots in each of 20 stands.  At the plot level you would have 40 x 20 = 800 samples.  You could select this option to produce a separate summary for each stand.

Write Data In List Format

In some cases you may wish to have a listing of your data where each observation occupies its own row, as opposed to a matrix format.  For example, if you have data on individual trees, you may want a listing of tree species and sizes that you could import into another program for the analysis of size distributions by species.

The exact format of the list will depend on the data type and whether or not you have set breakpoints.  The list is written into your results file.

Summary

Compact Format Data Types

Breakpoints


Summary of Individual Tree Data

Individual tree data in compact format are handled as a special case.  This applies only if you have collected diameter or basal area data in fixed-area plots. You must specify the units and the plot size in hectares.

The summarization of individual tree data produces a table containing results by species: frequency, basal area, density, relative frequency, relative density, relative dominance, importance value, variance in density, and the variance to mean ratio.  See the Example of Tree Summary and the table below.

Importance values have been calculated in various ways.  PC-ORD calculates an importance value (IV) as the average of relative frequency, relative density, and relative dominance.  If you wish to analyze a matrix of importance values, you can either build a new spreadsheet by entering IV's or you can use your spreadsheet software to average the component matrices.   To do the latter, first create separate matrices that you wish to combine into importance values.  Then add the matrices using your spreadsheet.  For example, in PC-ORD you could create a matrix based on basal areas, then relativize it by plot totals.  Then create another matrix based on densities (stem counts), and relativize that.  Then using your spreadsheet software, bring both matrices into a single spreadsheet, copy a formula to add the corresponding elements of the two matrices and divide by two.

You have the option of excluding trees below a minimum size.  If you wish to do so, just enter the cutoff point using the same units as selected above.

Table. Parameters included in PC-ORD summaries of individual-tree data.  N = number of sample units, A = sample area = N * sample unit area; Cij = number  of trees of species j in sample unit i.

_______________________________________________________________________

_______________________________________________________________________

Compact Format

Compact Data Types


Example Tree Summary

The following tables are produced by selection a summary from compact format, specifying the tree data type.

Quadrat Size =    .0300 hectares

_____________________________________________________________

                          No.   BA    %     Density Basal Area

   Species         Freq. Trees (dm2) Freq.  Trees/ha  dm2/ha

_____________________________________________________________

ACER SACCHARUM         3     4  12.94  7.89      3.51  11.35

CARYA SPP.             2     3  18.23  5.26      2.63  15.99

CORNUS FLORIDA        36    84 129.03 94.74     73.68 113.18

FRAXINUS PENNSYLVAMN.  5     9  78.36 13.16      7.89  68.74

LIRIODENDRON TULIPIF.  1     1   3.37  2.63       .88   2.95

NYSSA SYLVATICA       10    22  82.04 26.32     19.30  71.97

OXYDENDRON ARBOREUM    1     1   1.23  2.63       .88   1.08

QUERCUS ALBA           7     9  62.15 18.42      7.89  54.52

QUERCUS RUBRA          4     4  41.95 10.53      3.51  36.80

QUERCUS VELUTINA      17    27 201.55 44.74     23.68 176.80

SASSAFRAS ALBIDUM      3     5  11.59  7.89      4.39  10.17

QUERCUS FALCATA        8     9  81.90 21.05      7.89  71.84

JUNIPERUS VIRGINIANA   6     8  12.46 15.79      7.02  10.93

ULMUS RUBRA            1     3   4.00  2.63      2.63   3.51

FRAXINUS AMERICANA     1     1   5.90  2.63       .88   5.17

PINUS ECHINATA         5     9  71.13 13.16      7.89  62.39

____________________________________________________________

 Totals              110   199  817.82         174.56 717.38

___________________________________________________________________

_________________________________________________________________

                   |    Relative %   |          Variance

       Species      Freq.  Dens.  Dom.  I.V.(%) no. trees   V/M

_________________________________________________________________

ACER SACCHARUM        2.73   2.01   1.58   2.11    .1508   1.43

CARYA SPP.            1.82   1.51   2.23   1.85    .1287   1.63

CORNUS FLORIDA       32.73  42.21  15.78  30.24   2.5491   1.15

FRAXINUS PENNSYLVAN   4.55   4.52   9.58   6.22    .5100   2.15

LIRIODENDRON TULIPI    .91    .50    .41    .61    .0263   1.00

NYSSA SYLVATICA       9.09  11.06  10.03  10.06   1.6017   2.77

OXYDENDRON ARBOREUM    .91    .50    .15    .52    .0263   1.00

QUERCUS ALBA          6.36   4.52   7.60   6.16    .3478   1.47

QUERCUS RUBRA         3.64   2.01   5.13   3.59    .0967    .92

QUERCUS VELUTINA     15.45  13.57  24.65  17.89   1.1302   1.59

SASSAFRAS ALBIDUM     2.73   2.51   1.42   2.22    .2795   2.12

QUERCUS FALCATA       7.27   4.52  10.01   7.27    .2397   1.01

JUNIPERUS VIRGINIANA  5.45   4.02   1.52   3.67    .2788   1.32

ULMUS RUBRA            .91   1.51    .49    .97    .2368   3.00

FRAXINUS AMERICANA     .91    .50    .72    .71    .0263   1.00

PINUS ECHINATA        4.55   4.52   8.70   5.92    .5640   2.38

_______________________________________________________________

                                   All species    6.6721   1.27

   Number of quadrats =     38

   Empty quadrats     =      0

   Quadrat size       =       .030 hectares

   Area sampled       =      1.140 hectares

   BA/tree            =      4.110 sq. dm

   BA/hectare         =      7.174 sq. m/hectare

   Trees/hectare      =    174.561

   Trees/quadrat      =      5.237

Individual Tree Data

Summary of Individual Tree Data

Summary


Row and Column Summary Setup

All three options, Only Rows, Only Columns, Both Rows and Columns give statistical summaries and diversity statistics for your choice of rows or columns.

Row and Column Summary

Outlier Analysis

Summary


Outlier Analysis

Summary | Outlier Analysis

Outliers can profoundly influence multivariate analysis, yet they are seldom discussed in the open.  But the effects of outliers won't go away unless we look for them, then deal with them when we find them.  There are a number of ways of doing this, and the topic is covered well by Tabachnik and Fidell (1989).  Some of our multivariate methods are relatively resistant to distortions induced by non-normal distributions, particularly when we are not seeking exact probabilities for hypothesis tests.  Outliers and extreme non-normality, however, can obscure the information carried by the bulk of the data.

Outlier analysis in PC-ORD provides graphical and tabular results.  See Example of Outlier Analysis.

Outlier Analysis Setup

Summary


Outlier Analysis Setup

Outliers can be evaluated for either rows or columns of the main matrix.  You select a distance measure, then the program calculates the average distance of each entity from every other entity.  A frequency distribution of these distances is constructed and outliers are flagged, according to criteria that you set.  If you find outliers in your data, you may want to take steps to reduce their impact on the analysis (see Tabachnik and Fidell 1989).

Outlier Distance Measure

Outlier Results

Select Cutoff for Outliers

Example of Outlier Analysis

Outlier Analysis

Summary


Outliers Distance Measure

All of the standard distance measures in PC-ORD are available.  See Distance Measures.

Outlier Analysis Setup

Outlier Analysis

Summary


Outlier Results

You may choose one or all of the following ways of listing the results of the outlier analysis.  See examples of outlier analysis.

•	Graph Frequency Distribution:  A frequency distribution of average distance of each entity to every other entity is written to the results file.

•	Write List of Outliers:  A list of any items identified as outliers is written to the results file.

•	Write Ranked List of All Sample Units:  All items are listed, using the same format and statistics as in the preceding option.

Outlier Analysis Setup

Outlier Analysis

Summary


Select Cutoff for Outliers

You can set the cutoff for defining what is or is not an outlier.  Specify the number of standard deviations away from the mean that should qualify as an "outlier" (default = 2 standard deviations).

Outlier Analysis Setup

Outlier Analysis

Summary

Example of Outlier Analysis


Example of Outlier Analysis

Examples of two of the three portions of results from outlier analysis are listed below.  The third portion is simply a list of all items ranked in order of average distance to other items.

Graph Frequency Distribution Example

Frequency distribution of average distances    N =  92  PLOTS

Distance*      Frequency   (each "X" represents one entity)

-------------------------------------------------------------

.97878 |XXXX  50600 200600 (Two plots were identified as outliers

.96840 |XXX            and their 6-character codes are

.95801 |XXXXXX         printed. In this case, the codes are

.94762 |XXX            numerical coordinates indicating the

.93723 |XXXXXX         plot locations.)

.92685 |XXXXXX

.91646 |XXXXXX

.90607 |XXXXXXXXXX

.89569 |XXXXXXXXXX

.88530 |XXXXXX

.87491 |XXXXXXXXXX

.86453 |XXXXXXXXX

.85414 |XXXXXXX

.84375 |X

.83337 |XXXX

----------------------------------------------------

* Distances at left are lower end of that bin`s range.

Statistics for average distances for each of N =  92  PLOTS

  1-2W/A+B = Distance measure

    .90692 = Grand mean

    .03933 = Standard deviation

   2.00000 = Cutoff number of standard deviations used to flag outliers

List of Outliers Example

---------------------------------------

      ENTITY      AVERAGE     STANDARD

RANK   NAME       DISTANCE   DEVIATIONS

---------------------------------------

   1   50600      .98917     2.09118

   2  200600      .98832     2.06956

---------------------------------------

Statistics for average distances for each of N = 92 PLOTS

Distance measure:  Sorensen (Bray & Curtis)

.90692 = Grand mean

.03933 = Standard deviation

2.00000 = Cutoff number of standard deviations used to flag outliers

Outlier Analysis Setup

Outlier Analysis

Summary


Species-area Curves

Summary | Species-area Curves

Species-area curves are used (among other things) to evaluate the adequacy of sample size in a community data set.  The version in PC-ORD is not intended for the island-biogeographic use of species area curves, where the sample units vary in size.  Normally, you will want to apply this program to preliminary sampling to help you determine sample size.  A relatively homogeneous area (e.g., a "stand" of vegetation) is sampled with a large number of sample units.  This sample is subsampled to determine the average number of species as a function of size of the subsample.

Subsampling is repeated 500 times for each subsample size.  If fewer than 2000 subsamples are possible, all possible subsamples are used.  One side effect of all this subsampling is that the program will be very slow if you have a large data set and a slow computer.

This procedure reports a table of number of species and average distance for each subsample size.  The standard deviations for these values are also reported (see example below).  Note that for the last item in the list, no standard deviations are reported because there is only one possible subsample of size N from a sample of N items.

Example numerical output from Species-Area Curve.  In this case there are 100 sample units.

---------------------------------------------

Number   Number of Species     Distance

   of     ----------------- -----------------

  Plots    Average  St.Dev.  Average  St.Dev.

--------- -------- -------- -------- --------

        1    10.21   1.9348  28.8256 12.91994

        2    16.48   3.5745  20.4528  7.76168

        3    20.46   3.8492  16.4983  5.97781

        4    23.05   3.4739  14.4119  4.52582

        5    25.24   2.9953  13.5049  4.78323

        6    27.84   3.7598  11.7690  3.41383

etc.

       98    40.00   0.0000   0.4090  0.14633

       99    40.00   0.0000   0.3165  0.14494

      100    40.00            0.0000

---------------------------------------------

In addition to the classical species-area curve, the program also calculates average distance between subsamples and the overall species composition, in relation to subsample size.  The utility of this curve is that unlike the species-area curve, the curve of average distance incorporates information on species abundance, not just presence-absence.  This will be particularly useful if your goal is to determine a subsample size that gives you a consistent species composition, rather than a consistent species list.

The basic procedure for the calculations is:

1. Calculate the centroid for the full data set.  This centroid is defined by the abundance of each species, averaged across all sample units.

2. Select a random subsample of a given size (say 5 sample units).

3. Calculate the centroid for the subsample.

4. Calculate the distance between the centroid of the subsample and the centroid of the full sample.

5. Repeat steps 2-4, 500 times, accumulating the average distance.

6. Repeat steps 2-5 for every possible subsample size (1 through N-1 where N is the number of sample units).

Interpretation of the distance curve is similar to the species-area curve.  Look for a plateau where increases in sample size give only marginal decrease in the distance between the subsample and the whole sample.  There are, however, with a two important differences: (1) The species area curve rises to a plateau, while the distance curve falls to zero, hopefully with flattening before zero is reached, and (2) The distance curve must fall to zero when the full sample size is reached, while the species area curve has no set maximum value that is applicable across all data sets.

If you choose a proportional distance measure (such as Sorensen or Jaccard), a crude plot of the species area curve is provided with the initial output written into the result file.  But for all distance measures you can easily make a publication-quality plot, including confidence bands (plus or minus two standard deviations) by running Graph | Species-Area Curve after you run Summary | Species Area Curve.

Under the Species-Area choice PC-ORD also provides two jackknife estimators of species richness.

Your data should be arranged with sample units as rows and species as columns.  Only the main matrix is used.

Graph Species Area Curve

Summary


Jackknife Estimators of Species Richness

Summary | Species-Area Curve

Under the Species-Area choice PC-ORD also provides two jackknife estimators of species richness.  Palmer (1990, 1991) compared several ways of estimating species richness of an area when it is subsampled with smaller sample units.  Included in these comparisons were two jackknife estimators, nonparametric resampling procedures. The number of observed species in a subsample will typically be smaller than the true number of species.  These jackknife estimators produce more accurate and less biased estimates, at least when subsampling a restricted area.

Palmer warned that these estimators may not be appropriate when sampling large heterogeneous regions, because the estimated number of species can never be twice the number of observed species.  Palmer (1995) also warned that these estimators are highly sensitive to the number of rare species observed.

The first-order jackknife estimator (Heltshe and Forrester 1983; Palmer 1990) is

Jack1 = S + r1(n-1)/n

where S = the observed number of species, r1 = the number of species occurring in one sample unit, and n = the number of sample units.

The second-order jackknife estimator (Burnham and Overton 1979; Palmer 1991) is

Jack2 = S + r1(2n-3)/n - r2(n-2)^2/(n(n-1))

where r2 = the number of species occurring in exactly two sample units.  This estimator can have disadvantageous properties with small samples sizes.  For example, the estimated species richness will be lower than the observed specie richness when r1 = 0 and r2 > 0.

The following is an example of the output from the jackknife estimates:

Estimates of total number of species:

85.0 = Number of species observed

103.6 = First-order jackknife estimate

114.4 = Second-order jackknife estimate

Additional information used for jackknife estimates:

0 = Number of columns in matrix with no positive values

19 = Number of species with only 1 occurrence

8 = Number of species with only 2 occurrences

Graph Species Area Curve

Summary


Species Area Curve Distance Measure

In addition to the classical species-area curve, PC-ORD  calculates average distance between subsamples and the overall species composition. Like the curve for species number, average distance is calculated for each possible size of subsample.  The standard distance measures in PC-ORD are available (see distance measures).

If the average distance between a subsample and the whole sample is small, then one can conclude that the subsample is nearly as effective for characterizing the community as is the whole sample.  Like the species area curve, this is most informative if one has oversampled the community, compared to the expected subsample size.

Graph Species Area Curve

Summary


Species Lists

Perhaps the most basic summary of community data is a species list.  You can readily produce species lists from your spreadsheet data, provided that you have set up a species file.  The species file contains your species acronyms and full species names, as they will appear in your lists.  The order of species in your species file determines the order of species in your lists.

You can request a species list for each sample unit, or a species list for your combined sample units.  If you choose the latter, you can include key summary statistics, such as the number and proportion of sample units containing each species.

Species List Setup

Species List Summary Statistics

Summary


Species List Setup

To obtain species lists, you must specify the following:

•	Species File  (If your data were originally in compact format, then you already have this file).  The order of species in this file determines the order of species in your lists.

•	Descriptive Title for your lists

•	Aggregate or not

•	Broken down by grouping variable from second matrix.  If you select this checkbox you will be prompted to select a categorical variable from your second matrix.  PC-ORD will then build a table of aggregate characteristics broken down by group.  For example, assume you have data from three regions in a single data set.  You have a categorical variable called "region" in your second matrix.  Selecting this option produces a table of species x region.  The cells of the table specify any combination of the following: average abundance, number of sample units in which each species is present (raw frequency), and percent frequency.  See Species List Summary Statistics.

•	Whether or not to include the following in your list

•	Species code number

•	Species code name (the 8-character acronym)

•	Full species name

•	If you do not aggregate your data into one list, you may choose to list the abundance value for each species in each site.  If you do aggregate your data into one list, you can specify which summary statistics to include in the list.

Species Lists

Summary


Species List Summary Statistics

If you have chosen to aggregate your data into a single list, you have the option of including any and all of the following summary statistics in your species list.  Just click on the check box to include or exclude the following statistics.  In all cases we assume that your rows are sample units and your columns are species.

•	Average abundance

•	Counts (number of nonzero values for each row)

•	Percent frequency (percentage of sample units containing a given species)

If you are interested in other summary statistics, see Row and Column Summaries.

Species List Setup

Species Lists

Summary


Species List Example

The goal was to contrast vegetation in 63 burned quadrats with 36 unburned quadrats.  In this case we checked "Aggregate sample units" and "Broken down by grouping variable from second matrix."  Two aggregate statistics were selected: "averages" (the average abundance) and "counts" (raw frequency)  within each group.  Species are listed in the same order as in the species file, not the data matrix.  Species are identified by their code number, acronym, and full name.

Matrix contents:    99 Quadrats

             by    104 Species

Species file:

D:\SWANBURN\Swanburn.spp

Groups were defined by values of: BURN

        -----------------------

        GROUP     Code     Size

        -----------------------

            1        1       63

            2        0       36

        -----------------------

--------------------------------------------------------------------

                            BURN              1              0

                                      Average Count  Average Count

--------------------------------------------------------------------

   2 Abilas   Abies lasiocarpa          0.000     0    1.172    18

   3 Larocc   Larix occidentalis        0.010     1    0.192     4

   4 Piceng   Picea engelmannii         0.131    11    0.010     1

   5 Pincon   Pinus contorta            0.111     5    0.000     0

   6 Pinmon   Pinus monticola           0.000     0    0.131     5

   7 Pinpon   Pinus ponderosa           0.091     4    0.000     0

   8 Psemen   Pseudotsuga menziesii     0.000     0    0.121     2

  11 Acergl   Acer glabrum              0.091     6    0.121     7

  12 Adebic   Adenocaulon bicolor       0.081     8    0.273    16

  13 Amealn   Amelanchier alnifolia     0.323    23    0.263    13

  90 Anamar   Anaphalis margaritacea    0.020     1    0.000     0

  14 Antluz   Antennaria luzuloides     0.192    17    0.000     0

  82 Antmic   Antennaria microphylla    0.030     3    0.000     0

  15 Apoand   Apocynum androsaemifolium 0.111    10    0.000     0

…

etc.

…

 111 Oryasp   Oryzopsis asperifolia     0.687    35    0.273    15

 112 Phlpra   Phleum pratense           0.343    14    0.000     0

 113 Poapra   Poa pratensis             0.303    14    0.000     0

--------------------------------------------------------------------

Species Lists

Summary


Graph Menu

Several kinds of graphs are available from the Graph menu item on the main PC-ORD screen.  In each case, you need to have run a procedure that produces the results for the graph.

If you have run an ordination (BC, CCA, DCA, NMS, RA) you can graph the ordination by selecting Graph | Graph Ordination.

If you have run cluster analysis, you can produce a high-quality graphics rendition of the dendrogram by selecting Graph | Cluster Dendrogram.

Species-area curves along with confidence bands can be graphed with Graph | Species-area Curves.

If you have run non-metric multidimensional scaling (NMS), you can plot "stress" versus number of dimensions, along with a comparison of results from runs with randomized and real data.  This is known as a "scree plot."  Select Graph | NMS Scree Plot.


Graph Cluster Dendrogram

High-quality dendrograms are produced with the Graph | Cluster Dendrogram option.  But you must first run Cluster Analysis.  This procedure writes a result file as well as another text file, dendprep.den, that contains the numerical specifications for the dendrogram.  Then select Graph | Cluster Dendrogram.  If you save the numerical specification for the dendrogram, the *.den file, using Graph | File | Save File As, then you can reopen your dendrogram at any time by simply invoking Graph | Cluster Dendrogram, then File | Open,  and selecting the *.den file you wish to graph.

The dendrogram from Cluster Analysis shows a hierarchy of relationships among the rows in the main matrix.  Each end branch of the dendrogram is labeled with the name of that row.  The dendrogram is scaled in two ways: by Wishart's "objective function" describing the amount of information remaining, and by the objective function converted into a scale of percentage of information remaining.  These are described further under Cluster Distance Measures.

Menu items:

•	File (including printing controls)

•	Edit (copy to clipboard)

•	Groups

•	Options (Preferences) for Dendrogram

Graph File Menu

Graph Menu


Dendrogram Preferences

Graph | Cluster Dendrogram | Options, Preferences

Format tab

Graph title

Distance Scale

Information Scale

Row Labels

Print black and white

Save black and white

Background color

Group Preferences

Styles tab

Line Weights

Line Spacing

Fonts tab

Fonts

Dendrogram Pages

Graph Cluster Dendrogram

Graph Menu


Dendrogram Distance Scale

Graph | Cluster Dendrogram | Options, Preferences | Format tab

By default, two parallel axes are provided for a dendrogram, but you can also choose just one of those axes or neither.  One horizontal axis of the dendrogram is the distance between groups expressed as Wishart's Objective Function (see Cluster Analysis).  The other axis is also based on Wishart's Objective Function, but it is converted to a Percentage of Information Remaining.  Choose whether or not to include the axis for Wishart's Objective Function by clicking the checkbox “Distance Scale” under Options | Preferences in the Graph | Cluster Dendrogram menu.  You can also select the font size and type.

Dendrogram Preferences

Graph Cluster Dendrogram

Graph Menu


Dendrogram Information Scale

Graph | Cluster Dendrogram | Options, Preferences | Format tab

By default, two parallel axes are provided for a dendrogram, but you can also choose just one of those axes or neither.  One horizontal axis of the dendrogram is the distance between groups expressed as Wishart's Objective Function (see Cluster Analysis).  The other axis is also based on Wishart's Objective Function, but it is converted to a Percentage of Information Remaining.  Choose whether or not to include the axis for percentage of information remaining by clicking the checkbox “Information Scale” under Options | Preferences in the Graph | Cluster Dendrogram menu.  You can also select the font size and type.

Dendrogram Preferences

Graph Cluster Dendrogram

Graph Menu


Dendrogram Row Labels

Graph | Cluster Dendrogram | Options, Preferences | Format tab

By default each terminal branch of the dendrogram from cluster analysis is labeled with the name of a row from the main matrix.  To turn these labels on or off use the checkbox on the Format tab of Options | Preferences under the Graph | Cluster Dendrogram menu.  You can also change the font of these labels.

You can color-code or symbol-code your dendrogram labels by selecting a grouping variable.  This requires one or more categorical variables in your second matrix.

The corresponding legend is draggable in the graph.  Characteristics of the legend and selection of the symbol and/or colors are also set on the Format tab.

The grouping variable can be an a priori variable or you can generate it for a particular level in the dendrogram.  To do this, select "Add group membership variable to second matrix" in the Cluster Setup dialog when you run cluster analysis.  Then select this variable under the Groups item on the Graph | Cluster Dendrogram menu

Dendrogram Preferences

Graph Cluster Dendrogram

Graph Menu


Dendrogram Line Spacing

Graph | Cluster Dendrogram | Options, Preferences | Styles tab

You can save paper and make very compact dendrograms by adjusting the spacing between the horizontal lines that form the "stems" of the dendrogram. This is be controlled with the Line Spacing edit box on the Format tab of Options | Preferences under the Graph | Cluster Dendrogram menu.  Acceptable values range from 1 - 99 pixels.

This option affects both the appearance on screen and on paper.  You can, however, force the dendrogram onto one or any other number of printed pages via the Dendrogram Pages dialog.

Dendrogram Preferences

Graph Cluster Dendrogram

Graph Menu


Dendrogram Pages

Graph | Cluster Dendrogram | File | Print

When you print a dendrogram PC-ORD uses the "Dendrogram Pages" dialog to suggest a number of printed pages, taking into consideration all of the sizing options that you have selected.  You can, however, select a smaller or larger number than this, and PC-ORD will attempt to accommodate your request.

For efficient, attractive use of space on your paper, you should specify both the number of pages and the Page Setup.

This dialog affects only the printed dendrogram.  Layout on the screen is controlled by the Dendrogram Line Spacing option.

Graph Cluster Dendrogram


Graph Species Area Curve

Publication-quality species -area curves are produced by first selecting Summary | Species-area Curve from the main menu in PC-ORD.  This procedure writes a result file as well as another text file, spparea.spa, that contains the numerical specifications for the graph. Then select Graph | Species Area Curve.  If you save the numerical specification for the graph, the *.spa file, using Graph | File | Save File As, then you can reopen your graph at any time by simply invoking Graph | Species Area Curve, then File | Open,  and selecting the *.spa file you wish to graph.

By default these graphs contain two vertical scales, plotting average number of species and average distance as a function of the number of sample units included (see Summary | Species Area Curves).  To help you appreciate the sampling error, curves representing plus or minus two standard deviations (based on the multiple subsamples of a given size) are also plotted.

Menu items:

•	File (including printing controls)

•	Graph Edit (copy to clipboard)

•	Options (Preferences) for Species Area Curves

Graph Menu


Species Area Curve Preferences

Graph | Species Area Curves | Options, Preferences

Format tab

Graph title

Species axis title

Distance axis title

Number of subplots axis title

Tick marks

Tick labels

Label every other tick

Show legend

Show legend frame

Background color

Print black and white

Save black and white

Curves to plot

Scale maxima

Styles tab

Line Weights

Colors

Line types for curves

Fonts tab

Fonts

Graph | Species Area Curves

Graph Menu


Species axis title

Graph | Species Area Curves | Options, Preferences | Format tab | Species axis title

The left-hand vertical axis of the species-area curve is the average number of species. You can turn this axis title on and off by clicking the checkbox “Species Axis Title” under Options | Preferences in the Graph | Species Area Curve menu.  This title can be repositioned by dragging it to a new location.  You can also select the font size and type.  You can also click on the axis title and edit it to fit your needs.

Species Area Curve Preferences

Graph | Species Area Curves


Distance axis title

Graph | Species Area Curves | Options, Preferences | Format tab | Species axis title

The right-hand vertical axis of the species-area curve is the average distance between the whole sample and subsamples of the indicated size.  You can turn this axis title on and off by clicking the checkbox “Distance Axis Title” under Options | Preferences in the Graph | Species Area Curve menu.  This title can be repositioned by dragging it to a new location.  You can also select the font size and type.  You can also click on the axis title and edit it to fit your needs.

Species Area Curve Preferences

Graph | Species Area Curves


Number of subplots axis title

Graph | Species Area Curves | Options, Preferences | Format tab | Number of subplots title

The horizontal axis of the species-area curve is the size of the subsample ("number of subplots").  You can turn this axis title on and off by clicking the checkbox “Number of Subplots Axis Title” This title can be repositioned by dragging it to a new location.  You can also select the font size and type.  You can also click on the axis title and edit it to fit your needs.

Species Area Curve Preferences

Graph | Species Area Curves


Species Area Curves to Plot

Graph | Species Area Curves | Options, Preferences | Format Tab | Curves to Plot

Select with checkboxes which curves to include in the graph:

•	Average number or species

•	Average distance

•	Species confidence bands (plus or minus two standard deviations from the curve of average distance).

•	Distance confidence bands (plus or minus two standard deviations from the curve of average distance).

Species Area Curve Preferences

Graph | Species Area Curves


Species Area Scale Maxima

Graph | Species Area Curves | Options, Preferences | Format Tab | Scale maxima

The axis scales on the species area curves are set automatically by PC-ORD.  While the minimum is always zero, you can set the maximum on the two vertical scales: average number of species and average distance.  You will find this option useful if you are trying to produce comparably scaled graphs from different data sets.

Species Area Curve Preferences

Graph | Species Area Curves


Graph NMS Scree Plot

Publication-quality scree plots for NMS (Non-metric multidimensional scaling) are produced by first selecting Ordination | NMS from the main menu in PC-ORD.  This procedure writes a result file as well as another text file, stress.str, that contains the numerical specifications for the graph. Then select Graph | NMS Scree Plot.  If you save the numerical specification for the graph, the *.str file, using Graph | File | Save File As, then you can reopen your graph at any time by simply invoking Graph | NMS Scree Plot, then File | Open,  and selecting the *.str file you wish to graph.

A scree plot describes the improvement in fit as dimensions are added to a description of a multivariate data set.  It is called a "scree plot" because it often resembles a scree slope, the inclined pile of broken rock that accumulates at the base of large cliffs.  Each additional dimension tends to contribute less to the description of the data than the previous dimension, forming a curve similar to a negative exponential function.  Of course with real data, many variations are possible.

The scree plot is an important tool in selecting the appropriate dimensionality in NMS (or any other ordination technique; see Jackson (1993) for applications to PCA).  In NMS we plot final stress against dimensionality, while in other ordination methods one could plot variance explained or eigenvalues as a function of dimensionality.

The decision of appropriate dimensionality is critical in NMS, more so than other ordination techniques, because the axes with NMS are not invariant with respect to adding axes to the solution.  In contrast, with PCA and other ordination techniques, whether you use the third axis or not does not affect the definition of the second axis.

The decision on dimensionality in NMS is greatly helped by including results of Monte Carlo tests.  In PC-ORD, this is shown by graphing the range of final stress values obtained with randomized data at each dimensionality.

Menu items:

•	File (including printing controls)

•	Graph Edit (copy to clipboard)

•	Options (Preferences) for Scree Plot

Graph Menu


NMS Scree Plot Preferences

Graph | NMS Scree Plot | Options, Preferences

Format tab

Graph title

Stress Axis Title

Dimensions Axis Title

Tick marks

Tick labels

Label every other tick

Show legend

Show legend frame

Background color

Print black and white

Save black and white

Curves to Plot

Scale Maximum

Styles tab

Line Weights

Colors

Line types for curves

Fonts tab

Fonts

Graph NMS Scree Plot

Graph Menu


Stress Axis Title

NMS Scree Plot | Options, Preferences | Format tab | Stress axis title

The vertical axis of the NMS Scree Plot is the final stress. You can turn this axis title on and off by clicking the checkbox “Stress Axis Title” under Options | Preferences in the NMS Scree Plot menu.  This title can be repositioned by dragging it to a new location.  You can also select the font size and type.  You can also click on the axis title and edit it to fit your needs.

NMS Scree Plot Preferences

Graph NMS Scree Plot

Graph Menu


Dimensions Axis Title

NMS Scree Plot | Options, Preferences | Format tab | Dimensions axis title

The horizontal axis of the NMS Scree Plot is the number of dimensions in the solution. You can turn this axis title on and off by clicking the checkbox “Dimensions Axis Title” under Options | Preferences in the NMS Scree Plot menu.  This title can be repositioned by dragging it to a new location.  You can also select the font size and type.  You can also click on the axis title and edit it to fit your needs.

NMS Scree Plot Preferences

Graph NMS Scree Plot

Graph Menu


NMS Scree Plot Curves to Plot

NMS Scree Plot | Options, Preferences | Format tab | Curves to plot

Under the Format tab of the Options | Preferences dialog for Graph NMS Scree Plot you can select with checkboxes which curves and symbols to include in the graph:

•	Randomized data lines

•	Real data lines

•	Randomized data symbols

•	Real data symbols.

NMS Scree Plot Preferences

Graph NMS Scree Plot

Graph Menu


NMS Scree Plot Scale Maximum

NMS Scree Plot | Options, Preferences | Format tab | Scale maximum

The axis scales on the scree plots are set automatically by PC-ORD.  While the minimum is always zero, you can set the maximum on the vertical scale (stress).: You will find this option useful if you are trying to produce comparably scaled graphs from different data sets.

NMS Scree Plot Preferences

Graph NMS Scree Plot

Graph Menu


Graph Ordination

Graph | Ordination produces ordination diagrams and relates variables in both your main and second matrices to the ordination. You should select Graph | Ordination after running an ordination.  The menu items in for ordination graphs are:

•	File (including printing controls)

•	Graph Edit (copy to clipboard)

•	Graph (Type)

•	Simple Scatterplot

•	Overlay from Main Matrix

•	Overlay from 2nd Matrix

•	Joint Plot

•	Vector Overlay

•	Grid Overlay

•	Axes

•	Scaling

•	Rotate  (See also Reflecting Ordinations)

•	Groups

•	Statistics

•	Options

All of the ordination programs in PC-ORD write ASCII files that contain coordinates for graphing.  One of these files, GRAPHROW.GPH, contains the X, Y, and Z coordinates for the ordination; i.e., the position of each entity on each axis.  GRAPHROW.GPH is read for graphing and is not deleted on completion of the ordinations.

The second file, GRAPHCOL.GPH, contains coordinates for points representing the column variables in the main matrix.   This file is always produced by CCA, DCA, and RA.  It is optionally produced by Bray-Curtis Ordination, NMS, and PCA if you select the checkbox to calculate scores by a single weighted averaging step.

If you are new to ordination, read on...

Relating variables to your ordinations is the principal means for interpreting them.  You can relate variables from your main matrix and your secondary matrix (if you have one) to the ordination.  There are two main ways to relate a variable to an ordination: correlations and overlays.

The correlation coefficient provides a way of comparing positions of the sample units (row scores) on the ordination axes with the abundance of species or the size of secondary (often environmental).  If a variable has any linear relationship with an ordination axis, it is expressed in the correlation coefficient.  The squared values of the correlation coefficients express the proportion of variation in position on an ordination axis that is "explained" by the variable in question.

Overlays are a more flexible way of seeing if a variable is patterned on an ordination, because you are not limited to detecting only linear relationships.  Only one variable is overlaid at a time.  Each point on the ordination is substituted by a symbol or value representing the size of the variable (for example, abundance of a species or the size of an environmental variable).  Graphics overlays use variously sized symbols (for continuous variables) or various symbols or colors (for categorical variables; see variable type).

Graph Menu


Graph | File

The File menu under Graph contains the following items:

•	Open

•	Save Graphic

•	Save File As

•	Print Graph

•	Print Setup

•	Page Setup

•	Close

In addition, the Graph | File menu for ordinations has:

•	Save Scores As

•	Open Preferences

•	Save Preferences As

Graph Menu


Graph, File Open

Graph | File | Open reads the data from a dendrogram, scree plot, or species-area curve that you previously saved.  See Graph | File | Save File As.  For ordinations, this is accomplished by running an ordination or opening a graph file before selecting Graph | Ordination from the main PC-ORD menu.

Graph Menu


Graph Edit (Copy to Clipboard)

Graph | Edit | Copy

Select this option to copy a graph to the clipboard.  It is copied as a Windows metafile.  As such it can be pasted into documents or other software.

You can alter the labeling by dragging and editing graph labels.  You can also edit the contents of a graph by setting your preferences:

Graph Preferences

Dendrogram Preferences

NMS Scree Plot Preferences

Species Area Curve Preferences

Graph Menu


Save Graphic

File | Save Graphic

File | Save Graphic As

Select this to save the currently displayed graph as a Windows Metafile.  You will be prompted for a file name.  Version 3 of PC-ORD wrote Windows Metafiles with the *.WMF extension on the file name, while version 4 defaults to enhanced windows metafiles with the *.EMF extension.  A graph saved in this way can be brought into various graphics packages or editors, inserted into text, rescaled, edited, etc.  You cannot, however, reload a metafile into PC-ORD.

You can also save a graph as a bitmap image (extension *.BMP) as opposed to the vector-based images in metafiles.  Bitmap images are not resizable by other software without altering the clarity of the rendition.

Graph | File


Save Scores

File | Save Scores As | Rows

File | Save Scores As | Columns

If you select this option, the coordinates for points in ordination space are written into a file. If axes were rotated or rescaled in any way, the coordinates will reflect those rescalings.

Each ordination that you run rewrites the temporary graph files (graphrow.gph and graphcol.gph).  If you wish to use them for other purposes (say to transfer the ordination to another graphing program or statistical package), you can save the graph files in various ways:

Rows (points on ordination representing rows from the main matrix)

•	Text file (default extension = *.txt)

•	Spreadsheet – a new spreadsheet is opened, the axis scores are written to a new matrix, then saved with the name that you provide (default extension = *.wk1).

•	Add to Second Matrix – variables are appended to the end of the existing second matrix.  This option is available only if you have opened a second matrix before graphing your ordination.

Columns (points on ordination representing columns from the main matrix)

•	Text file (default extension = *.txt)

•	Spreadsheet – a new spreadsheet is opened, the axis scores are written to a new matrix, then saved with the name that you provide (default extension = *.wk1).

You will be prompted for a new file name, unless you are adding the variables to an existing spreadsheet.

Graph | File


Graph, Save File As

Graph | File | Save File As

A graph is built from a set of numerical results along with options (preferences) that you can set.  To save this combination as an actual graphic (like taking a snapshot of the screen) you use Graph | File | Save Graphic .  But to allow yourself to modify the appearance of the graph, while keeping the numerical results, you should use Graph | File | Save File As.  This applies to dendrograms, scree plots, and species-area curves.  Ordination coordinates, on the other hand are saved with Graph | File | Save Scores As .

In all cases under Graph | File | Save File As the files are text files, saved with the following default extensions:

*.gph = graph files

*.den = dendrogram files

*.spa = species - area curve files

*.str = NMS scree plot files

Graph | File


Open Preferences

Graph | File | Open Preferences

You can reuse your setup for ordination graphs by opening a “preference file” that you created previously.  A preference file contains none of your data, only settings for options that control the formatting of your ordination graphs.

Graph | File


Save Preferences As

Graph | File | Save Preferences

You can save your setup for ordination graphs in a “preference file.”  Think of it as a template that can be reused so that your preferences for displaying ordinations can be saved.  A preference file contains none of your data, only settings for options that control the formatting of your ordination graphs.  When you Save Preferences you also save your settings for axis scaling and the settings for joint plots.

The default extension for the preference file name is *.OPT.

Graph | File


Print Graph or Graph Statistics

Graph | File | Print

You can print your ordinations using whatever printers are installed in your Windows Print Manager.  You can also print statistics generated by the Graph routine by using File | Print in the Graph menu or clicking on the printer icon.

If you request printing and nothing happens or garbage emerges from your printer, be sure you have set the printer options correctly.

Graph | File


Print Setup

File | Print Setup

Graph | File | Print Setup

Specify your printer, paper orientation, and paper size and source by using Print Setup.  This is available under both the File menu (for printing your spreadsheets and result files) or the Graph | File menu (for printing graphics).

You can choose either the default printer or a specific printer.  You can select a specific printer if you have more than one printer set up with Windows.

Available paper sizes and sources depend on the specific printer.  Paper orientation can be set to either landscape or portrait. The font is set under File | Font | Printer.

PC-ORD uses the maximum printable area of the page when printing from the four viewing windows.  To set the margins when printing the matrices, result file, or graph file, copy the text or spreadsheet into your word processor and print from there.

The layout of a graphic on a printed page is set by Graph Page Setup.

Print Graph

File | Print


Graph Page Setup

Graph | File | Page Setup

•	Print % of page: Set this to the percent of the page you want to print. This defaults to 60%.  Set it to 50%, for example, if you want to print a graph that uses only half of the page.  Graphs are sized to the best fit of a page.  That means if a graph is proportionately wider than the page than the height of the page, the % of page will be taken from the width.  For the 50% of page example that means the graph will fill 4.25” of an 8.5” wide page (less the printer's unprintable margin) and the height will be proportionate to the width.

(Note: some printer drivers do not handle scaling very well.  Some HP inkjets are especially noted for this.  If printing a graph less than 100% of the page results in improper scaling, you can print at 100% and then reduce the graph via photocopy or other photo reduction. You also can check your printer manufacturer's web site for an updated driver for the printer.)

•	Top Margin %: The total top and bottom margin is the portion of the page that is not taken up by the graph.  Set top margin to what portion of the total margin you want at the top of the page.  The default is 50% which means the graph will be centered vertically on the page.

•	Left Margin %: This works the same as top margin except it applies to the left margin.  The default is 50% centering the graph horizontally.

Use File | Print Setup to orient the graph either in portrait or landscape mode.

Graph | File

Graph Menu


Exit Graph

Graph | File | Close

The graphing program ends and you return to the main PC-ORD menu.  The file(s) containing the basis for the graph  (GRAPHROW.GPH, GRAPHCOL.GPH, DENDPREP.DEN, SPPAREA.SPA, or STRESS.STR) are left intact.  However, the next procedure of the same type that you run will overwrite these files unless you save them (use Graph | File | Save As or Save Scores As).

If you exit the graphing program, then decide you want more graphs from the same ordination, you can re-use the graph files, as long they correspond to the main matrix.

Graph Menu


Select Axes

Graph Ordination | Axes

The ordinations are two-dimensional or one-dimensional.  When the input contains scores for three axes, three plots are generated: axes 1 vs. 2, 1 vs. 3, and 2 vs. 3.  You can select any pair of the first three axes.  You can freely move among these different pairs of axes by selecting from this menu or clicking the appropriate buttons on the toolbar.

One-dimensional ordinations are plotted with the ordination scores on the horizontal axis and rank on the second axis.  This spreads the points and their labels better than if the points are all plotted on a horizontal line.  Many options on the graph menu are disabled with one-dimensional ordinations (such as joint plots and rotation) because they do not apply to a one-dimensional array.

Graph Menu


Scaling Axes

Graph Ordination | Scaling

You may choose to scale your ordinations by one of three methods.  Your choices are:

•	Raw Scores.  Ordinations are plotted using the raw scores in the graph file(s).  The chief disadvantage with this approach is that with many ordination methods the scaling of the raw scores is fairly arbitrary and dependent on the units of the original data.  On the other hand, with methods in which the axes have already been scaled in some meaningful way, you may wish to use the raw scores to preserve the original scale.

•	Proportionate to longest axis (% of Max, default).  This method accurately portrays the similarity relationships among points, and is recommended for general use.  Each axis is scaled in proportion to the longest axes (usually the first). In other words, if the scores on the first axis ranged from 0 to 100 but ranged from 50 to 75 on the second, the scatter of points would occupy only 1/4 of the space on the second axis, reflecting the narrower spread on the second axis.

•	Minimum to maximum (Min to Max).  Use this method to maximize the spread of points on the graph.  Each axis is scaled from the lowest score on the axis to the highest score.  With this method, the points occupy the whole space on every axis, regardless of the range in scores.  This will usually distort the relationship between ordination distance and sample similarity, resulting in lower r2 values.

You can also reverse the scaling of the axes by selecting Reflect Horizontally or Reflect Vertically.  This creates a mirror image of your ordination diagram.

Graph Menu


Reflect Ordination Axes

Graph | Ordination | Scaling | Reflect Vertically

Graph | Ordination | Scaling | Reflect Horizontally

You can reverse the scaling of the axes by selecting Reflect Horizontally or Reflect Vertically.  This creates a mirror image of your ordination diagram.  The most common uses for this are:

•	Align two or more ordination diagrams to maximize their comparability. By using both rotation and reflection ordinations can easily be re-oriented, while rigidly maintaining the Euclidean distance among points in the ordination diagram.

•	Reverse the scale to improve the ease of communication.    For example, if score on an axis decreases with site moisture, you may wish to reflect the ordination so that score increases with moisture, facilitating referring to it as an axis representing "moisture".

Scaling Axes

Graph Menu


Graph Type

Graph Ordination | Graph

Select from the following graph types:

•	Simple scatterplot

•	Overlay from main matrix

•	Overlay from second matrix

•	Joint plot

•	Vector overlay

•	Grid overlay


Simple Scatterplot

Graph Ordination | Graph | Simple Scatterplot

The basic ordination diagram depicts points in a space defined by two ordination axes.  The points usually represent the rows in your main matrix, unless you ran CCA, DCA, or RA, in which case either columns or rows or both may be plotted.

You can control the symbol used to mark the points and whether or not those symbols are labeled.

Graph Type

Graph Menu


Overlay from Main Matrix

Graph Ordination | Graph | Overlay from Main Matrix

Graphic overlays of quantitative variables are convenient for relating individual variables to the ordination space.  There are three components to the output when you request an overlay:

1.  An ordination diagram where the ordination points have been replaced by various sized symbols, the size being proportional to the size of the variable.  These overlays are produced by dividing the total range of a given variable into equal-sized segments, each of which is represented by a different size symbol.  Thus the symbol size indicates the relative size of the given variable in a particular entity (remember, "entities" are the points being ordinated, be they samples or species).  The number of different symbol sizes depends on the resolution of your monitor.  The minimum value (often zero) is shown on an overlay as the smallest size for that symbol.  You can choose Symbols in the preference dialog.

2.  Side scatterplots of the overlay variable against the individual axes shown.  Two additional scatterplots are shown on the sides of the ordination overlay.  The scatterplot below the ordination is relates the overlay variable (vertical axis) to the horizontal axis of the ordination.  The scatterplot to the left of the ordination relates the overlay variable (horizontal axis) to the vertical axis of the ordination.  These scatterplots make the correlation statistics (below) more meaningful.  Furthermore these scatterplots display the relationship between the overlay variable and an ordination axis without making any assumptions about the form of the relationship.  You can include, if you wish, a simple linear regression line and/or an envelope that includes about 95% of the points.

The side scatterplots can be turned off in the preferences dialog.

3.  Correlation coefficients for the relationships depicted in the scatterplot.  You are given two types of correlation coefficients: Pearson's (the standard correlation coefficient) and Kendall's (a type of rank correlation).

Graph Type

Graph Menu


Overlay from 2nd Matrix

Graph Ordination | Graph | Overlay from Second Matrix

Variables in the secondary matrix can be used as ordination overlays.  Two kinds of overlays are produced, depending on whether a variable was declared as quantitative or categorical (see variable type).

Quantitative variables are handled in the same way as with overlays of variables from the main matrix.   When you select a variable for an overlay only the quantitative variables (those labeled with a “Q”) are shown in the picklist.

Categorical variables are used as overlays by coding with colors or symbols according to the category code numbers (see Graph Groups). These symbol and color codes are displayed on the basic scatterplot.  The use of symbols or colors or both is selected in the Groups box of the Graph Preferences Unlike overlays of quantitative variables, accessory scatterplots and correlations are not produced.

If you select a categorical variable to overlay and the categories are not shown, go to the Options | Preferences dialog and be sure that you have checked either Symbol Codes or Color Codes or both.  If you had previously selected a grouping variable, then the categorical overlay is temporarily substituted for your grouping variable.

Graph Type

Graph Menu


Joint Plot

Graph Ordination | Graph | Joint Plot

An increasingly popular method of showing the relationship between a set of variables (usually environmental variables) and ordination scores is a diagram of radiating lines (example below), often called a "joint plot."  The angle and length of the line tell the direction and strength of the relationship.  These lines are often referred to as “vectors”, but do not confuse them with successional vectors.  In PC-ORD these lines in a joint plot are constructed in a way that is applicable to all ordination methods.

A joint plot is similar to a biplot but does not follow the strict set of rules required for a true biplot (Jongman et al. 1987).  In both cases, it is an ordination diagram which shows two kinds of entities (often sample units and environmental variables).

With community data you can show sample units and environmental variables on a joint plot by having your environmental variables in the second matrix and the species data in the main matrix.  You can show sample units and species simultaneously by having your species data in BOTH your main and second matrices.  In other words, you would load your sample units x species matrix into both the main and second matrices before selecting "Graph."

The lines radiate from the centroid of the ordination scores.  For a given variable, the line forms the hypotenuse (h) of a right triangle with the two other sides being r values between the variable and the two axes.  The angle of the line is arccos(rx*|rx|/h), where rx is the correlation of the variable with the horizontal axis and h is proportional to sqrt((rx^2)^2 + (ry^2)^2).  Although the r values determine the relative scaling of the vectors, the absolute scaling is arbitrary and the Vector Scaling option is provided to set this scaling factor with methods other than CCA.

In the special case of CCA , the coordinates for the tips of the radiating lines representing the second matrix are read directly from the file graphcol.fil, rather than calculating the coordinates in the graphing routine. If you chose biplot scaling (centering and normalizing) rather than Hill’s scaling in CCA, then the resulting plot is a biplot.  Otherwise, it is a joint plot.

You can eliminate weak variables from your joint plot or biplot diagram.  Select Joint Plot Cutoff or Biplot Cutoff from the Graph submenu or click on the toolbar button that displays the current cutoff value.  Only variables from your secondary matrix with an r^2 (with scores on either axis) larger than the cutoff value are plotted.  The default cutoff is 0.2, but you may set this to any value from 0-1.

This dialog also contains an edit box for Vector Scaling where you can magnify or shrink the size of the joint plot vectors relative to the scatter of points.  Note, however, that this option is NOT applied when you are using CCA.  In that case,

the coordinates of the tips of the vectors have a fixed position as calculated by the CCA routine.  In all other cases, the length of the radiating lines in the joint has an arbitrary scaling relative to the scatter of points.


Grid Overlay

Graph Ordination | Graph | Grid Overlay

This option is specifically designed for artificial data sets based on a rectangular grid (see, for example, Minchin 1987b, Austin & Noy-Meir 1971, McCune 1994).  Use a grid for artificial two-dimensional data sets to see if ordination methods can reproduce the known underlying grid structure.  If you select a grid representation, then you are asked how many groups of rows are in your data.  One group of rows forms one line of grid points.

Graph Type

Graph Menu


Vector Overlay

Graph Ordination | Graph | Vector Overlay

Vectors can be used in any case where a number of sample units have been followed through time.  For example, say you are recording species abundance in a number of plots, and revisiting those plots every year.  By connecting the data points in temporal sequence, you can see the "trajectory" of a sample unit in species space.

Your data can be arranged in one of two ways: entities can be grouped by time or by sample unit (plot) as described below.

	Entities grouped by
Sequence in Data	Sample Unit	Time
1	Unit 1 t=1	Unit1 t=1
2	Unit 1 t=2	Unit 2 t=1
3	Unit 1 t=3	Unit 3 t=1
4	Unit 2 t=1	Unit 1 t=2
5	Unit 2 t=2	Unit 2 t=2
etc.	etc.	etc.

Graph Type

Graph


Graph Preferences

Graph ordination | Options, Preferences

Customize your ordination graphs by specifying the following options.  Once you have set up a graph the way you want it, you can save that set of options by selecting File | Save Preferences.  You can then apply those formatting options to future graphs by retrieving the set of options (File | Open Preferences).

The following options can be set:

Format tab

•	Graph Title

•	Axis Titles

•	Tick Marks

•	Tick Labels

•	Label Every Other Tick

•	Overlay Tick Marks

•	Hide Toolbar

•	Background Color

•	Print in black and white

•	Save in black and white

•	Label Points

•	Plot Points

•	Groups

•	Color code categories

•	Symbol code categories

•	Show legend

•	Show legend frame

•	Legend Symbol/Color

•	Quantitative overlays

•	Show side scatterplots

•	Show regression line

•	Show envelope line

Styles tab

•	Frame type

•	Symbols

•	Line weights

Font tab

•	Fonts

Graph Menu


Frame Type

Graph ordination | Options, Preferences | Styles tab | Frame Type

Three kinds of frames are available for your ordinations:

•	Box.  A rectangular frame that encloses all of the points in the ordination.  This is a traditional way of displaying ordination diagrams.

•	Standard XY.  One vertical axis on the left and one horizontal axis on the bottom.

•	Centered XY.  One vertical axis and one horizontal axis, their intersection at the centroid of the cloud of points.

Graph Preferences


Graph Symbols

Graph | Options, Preferences | Styles  tab | Symbols

You may choose the symbol type and size used to display the ordination points. When you select this option, a selection of symbol types and sizes is displayed.  Point and click on the symbol you wish to select.

These symbol are also used for overlays.

If your graph contains points for both rows and columns, you will normally select a different symbol for each set of points.  You may choose to display just the ordination points or to have those points labeled.

Graph Preferences


Group Preferences

Graph ordination | Options, Preferences | Format tab | Groups

If you have a categorical variable in your second matrix, you can display groups in your graphs by coding them with different colors or symbols.  The variable is selected under Groups in the Graph menu (See Graph Groups).  The association between particular values in your data and colors or symbols in the display is selected with Legend Symbol/Color.

In the Preferences dialog you select whether groups are displayed by color codes, symbol codes, or both.  Note that if NEITHER color codes nor symbol codes is selected, then groups will NOT be displayed.

You can also turn on or off a legend and a frame around the legend.

Graph Preferences

Dendrogram Preferences


Color code categories

Graph ordination | Options, Preferences | Format tab | Color code categories

This checkbox turns on color coding of points in ordination diagrams. Different values of categorical variables are drawn with different colors.  You can also display these as different symbols , or different colors and symbols simultaneously.  Select the variable that is the basis for the color coding by using Groups | Select Grouping Variable from the graph menu.  The association between the value of the variable and the color is controlled with Legend Symbol/Color.

Group Preferences

Graph Preferences


Symbol code categories

Graph ordination | Options, Preferences | Format tab | Symbol code categories

This checkbox turns on symbol coding of points in ordination diagrams. Different values of categorical variables are drawn with different symbols.   You can also display these as different symbols , or different colors and symbols simultaneously.  Select the variable that is the basis for the color coding by using Groups | Select Grouping Variable from the graph menu.  The association between the value of the variable and the symbols is controlled with Legend Symbol/Color.

Group Preferences

Graph Preferences


Line Weights

Graph | Options, Preferences | Styles tab | Line weights

Use this option to customize the line weights for various parts of a graph:

•	frame (axes),

•	tick marks on the axes,

•	the radiating lines in a joint plot,

•	regression lines in the supplementary overlay scatterplots,

•	vector overlays , and

•	grid overlays.

•	species-area curves

•	scree plot curves

•	dendrogram lines

•	legend frame

Graph Preferences

Dendrogram Preferences


Label Points

Graph ordination | Options, Preferences | Format tab | Label points

You can label each point in an ordination diagram with the corresponding label from your main matrix.  The font is selected under Options | Preferences | Font.

Labels can be used for the points representing rows in your main matrix (usually sample units).  If your ordination includes points for both the rows and columns of your main matrix, you can label either or both the rows and columns.  Or you can turn off labeling altogether.

These labels will typically overlap.  You can arrange the labels by clicking on them and dragging them to where you want them.

When ordinating many points, adding the labels to the diagram often creates a mess that is impossible to clean up.  In these situations, it is best to turn off labeling.  Furthermore, you can improve the speed of the graphing large data sets by turning off the labeling.

Graph Preferences

Graph Menu


Graph Title

Graph | Options, Preferences | Format tab | Graph Title

When you run an analysis you are asked for a descriptive title.  This title is use in the numerical output written into RESULT.TXT.  It is also used as the title of your graph.

You can turn the graph title on and off by checking the box labeled “Graph Title” in the Graph | Preferences.

This label can be reposition by clicking on the graph title and dragging it to a new location.  You can also select the font size and type.  The graph title can be edited by clicking on it, positioning the cursor, and making the necessary adjustments.

Graph Preferences


Tick Labels

Graph | Options, Preferences | Format tab | Tick Labels

You can turn tick labeling on and off by clicking the checkbox “Tick Labels” under Options | Preferences.  These labels can be repositioned by clicking on them and dragging them to a new location.  You can also select the font size and type.  You can also click on the tick labels and edit them to fit your needs.

By default ordination axes are not given tick marks or tick mark labels.  This is because most ordinations are interpreted in a relative way, and the units of the axes are often unimportant or arbitrary.  But in some ordination methods such as CCA and DCA, the axis labels and tick marks may carry some useful information.

If you request tick labels, PC-ORD dynamically selects a format for the labels.  If the numbers are very small decimals or very large, PC-ORD will display the labels in a compact exponent format, an “E field,” for example 1.23E+03 = 1230.

Be aware that depending on your selection for axis scaling the scale may or may not be the same as the original raw ordination scores.

Graph Preferences


Tick Marks

Graph | Options, Preferences | Format tab | Tick Marks

Tick marks are turned on and off with a check box in the Options | Preferences dialog box under the Graph Menu.

By default axes in ordination diagrams are not given tick marks or tick mark labels.  This is because most ordinations are interpreted in a relative way, and the units of the axes are often unimportant or arbitrary.  But in some ordination methods such as CCA and DCA, the axis labels and tick marks may carry some useful information.

Be aware that depending on your selection for axis scaling, the scale may or may not be the same as the original raw ordination scores.

Graph Preferences


Axis Titles

Graph | Options, Preferences | Format tab | Axis Titles

Ordination axes are titled by default with “Axis 1,” “Axis 2,” or “Axis 3” (or "Rank" on the vertical axis if you have a 1-D ordination)  You can turn axis titles on and off by clicking the checkbox “Axis Titles” under Options | Preferences in the Graph Menu.  These labels can be repositioned by clicking on them and dragging them to a new location.  You can also select the font size and type.  You can also click on the axis titles and edit them to fit your needs.  For example, you might wish to change “Axis 1” to “NMS Axis 1.”

Graph Preferences


Overlay Tick Marks

Graph Ordination | Options, Preferences | Format tab | Overlay Tick Marks

Tick marks can be used on the small scatterplots accompanying overlays of quantitative variables.  These can be turned on and off with a check box in the Options | Preferences dialog box under the Graph Menu.

Graph Preferences


Print in black and white

Graph | Options, Preferences | Format tab | Print black and white

If you check “Print in black and white” in the Options | Preferences dialog box under the Graph Menu, then colors on the screen will be converted to black and white.

Graph Preferences


Save in black and white

Graph | Options, Preferences | Format tab | Save black and white

If you check “Save in black and white” in the Options | Preferences dialog box under the Graph Menu, then:

•	colors on the screen will be converted to black and white in the resulting Windows metafile.

•	colors on the screen will be converted to black and white when graphics are copied into the clipboard.

Graph Preferences


Hide Toolbar

Graph | Options, Preferences | Format tab | Hide toolbar

You can hide the toolbar by checking “hide toolbar” in the Options | Preferences dialog under the Graph Menu.

Graph Preferences


Plot Points

Graph | Options, Preferences | Format tab | Plot Points

Some ordination methods (CCA, DCA, and RA) result in coordinates for both your rows and columns.  Other ordination can produce coordinates by a single after-the-fact weighted averaging step.  Using the Options | Preferences dialog box under the Graph Menu you can select whether to plot points representing your rows or columns or both.

If you ran one of the other ordination that produced scores for only the rows in your main matrix, then options requiring a second set of points will be “grayed out.”

Graph Preferences


Background Color

Graph | Options, Preferences | Format tab | Background Color

Under Background Color on the Format tab in the Options | Preferences dialog you can choose:

•	White

•	Black

•	Navy

The foreground color is black if you choose a white background.  Otherwise the foreground color is white.  Use white (with black foreground) to produce black-and-white graphics for publication.  Use the black or navy backgrounds to produce color slides for presentations.  If you are coding groups by colors you may wish to choose different symbol colors to contrast with your background.

Graph Preferences


Graph Fonts

Graph | Options, Preferences | Fonts tab

You can select font type and size under Font tab of the Options | Preferences dialog box under the graphing menus.  The standard Windows fonts are available, as are other third-party fonts that you have installed on your system.

Fonts are selected individually for  the following:

•	Graph title

•	Row labels (points on the ordination representing rows in your main matrix)

•	Column labels (points on the ordination diagram representing columns in your main matrix; note that these are not always present)

•	Axis titles

•	Joint plot labels

•	Tick labels

•	Correlation labels (the correlation statistics given with overlays of quantitative variables)

•	Legend Labels

If you desire very small fonts in your graphs, for example when labeling points in your ordination diagrams, most systems will have available “Small Fonts”  as small as 2 pt. in the list of available fonts.

Graph Preferences

Graphing


Rotating Ordinations

Graph ordination | Rotate

You can rotate the point cluster in an ordination.  The point cluster is rotated around the centroid in several ways (listed below).  The centroid is the “center of balance” of the cloud of points, the two-dimensional average position of the points.  You can also create a mirror image of an ordination by reflecting it horizontally or vertically.

•	By Angle  Rotates the point cloud by an angle that you specify.

•	By Angle Continuous  Rotates the point cloud incrementally.  You specify the increment, then repeatedly rotate by clicking on the “Next” button or pressing Enter.

•	By Variable Main Matrix  Rotates the point cloud to maximize the correlation of a single variable that you select from the main matrix.

•	By Variable Second Matrix.  Rotates the point cloud to maximize the correlation of a single variable with the horizontal axis.  You select this variable (only quantitative variables permitted) from the second matrix.  Note that this also aligns the corresponding radiating line in a joint plot with the horizontal axis.

Note that the classic varimax rotation is available in PC-ORD only as part of NMS ordination.

If you rotate by angle, you may notice that the radiating lines do not necessarily rotate at the same angle as the point cloud.  Remember that you specify the angle of rotation of the point cloud, not the radiating lines.  Note also that rotation changes the variance explained by the axes, so you will often want to rerun overlays, correlations, and percent of variance explained.

Graph Menu


Graph Statistics

Graph ordination | Statistics

Three kinds of statistics can be calculated in association with your graphs:

•	Correlations with main matrix

•	Correlations with second matrix

•	Variance Explained by Graph

All of these statistics change if you rotate the ordination.

Graph Menu


Correlations with main matrix

Graph ordination | Statistics | Correlations with main matrix

For each column in your main matrix, this option calculates correlation coefficients with each ordination axis, then produces a table of these correlation coefficients (example below).  These coefficients express the linear (Pearson's r) and rank (Kendall's tau) relationships between the ordination scores and the individual variables used to construct the ordination.  Note that these correlations are also given when you request overlays, but in that case the correlations are calculated only for the overlay variable.

The table of correlation coefficients can be quite helpful in providing a quick interpretation of the ordination.  In some cases, however, they can be misleading.  Two common problems with interpreting an ordination just on basis of the correlation coefficients are:

1.	Even a single outlier can have undue effect on the coefficients, often resulting in a very strong correlation that has little to do with the bulk of the data, and

2.	The coefficients will misrepresent non-linear relationships (for example, if a species has a humped, unimodal distribution along an ordination axis).

For a "normal" analysis (samples in species space), Pearson correlations (parametric) and Kendall correlations (nonparametric) are given for each species' abundance with each of the first three axes.  Because the Kendall correlations are based on ranks (for example, the ranking of stands by a species' abundance is compared with rankings on ordination axes), it is inappropriate to use these with binary data.  The values are listed whether or not you have binary data, so you must decide when they should be ignored.

If you are doing a transpose analysis of community data (species in sample space), these correlations are less useful than with the normal analysis.  Kendall and Pearson correlations are calculated for sample units against ordination axes.  The positions of species on the ordination axes are compared to their abundances in a given sample unit.  This is done for each sample unit.

Example of Output

     PEARSON AND KENDALL CORRELATIONS WITH ORDINATION AXES   N= 54

_______________________________________________________________________

     AXIS:       1                      2                      3

      ________________________________________________________________

        r    r-sq    tau       r    r-sq    tau      r    r-sq    tau

 _______________________________________________________________________

 Piceng  .510   .260   .435    -.106   .011  -.076    .481   .231   .348

 Pincon  .836   .700   .754    -.199   .040  -.251   -.410   .168  -.286

             .

             .

             etc.

             .

 Oryasp -.464   .216  -.235    -.286   .082  -.085    .184   .034   .152

 Phleum  .904   .818   .806    -.303   .092  -.235   -.250   .062  -.141

 Poapra  .811   .658   .686    -.323   .104  -.337   -.555   .308  -.365

 _______________________________________________________________________

Graph Statistics

Graph Menu


Correlations with second matrix

Graph ordination | Statistics | Correlations with second matrix

This procedure is identical to correlations with the main matrix, except that it uses the variables in the secondary matrix.  See example output under that topic.  Only variables in the secondary matrix that you designated as quantitative are used.

Graph Statistics

Graph Menu


Variance Explained by Graph

Graph ordination | Statistics | Percent of variance in distance matrix

This option (see example output below) evaluates how well the distances in the ordination space represent the distances in the original, unreduced space.  Some ordination methods provide similar evaluations in their numerical output, for example Bray-Curtis ordination and Principal Components Analysis.  Other methods, including Detrended Correspondence Analysis and Nonmetric Multidimensional Scaling, provide no such statistics.   For these methods, this option allows you to calculate an after-the-fact evaluation of the quality of the data reduction, along with an assessment of how the variance explained is distributed among the primary axes.

The standard method for calculating total variance (inertia) in RA (CA) and CCA were criticized by Økland (1999) as not accurately representing the effectiveness of these ordination methods at recovering ecological gradients.  As an alternative we suggest using an after-the-fact evaluation of how well distances in the ordination space match the relative Euclidean distances in the main matrix.  Although these methods have an intrinsic basis in the chi-squared distance measure, empirically we have found that the correlation between points in a CCA or RA ordination space and the points in the original space is much higher when the latter is measured as relative Euclidean distance than as chi-squared distance.

Table: Recommended methods for calculating variance in the main matrix explained by various ordination methods.  The after-the-fact method is that provided by the coefficient of determination between Euclidean distance in the ordination space and a distance matrix based on the main matrix.  The distance matrix can be calculated with various distance measures, but the choice of distance measure is important.  Our suggestion for distance measure is indicated in the table.

Ordination Method	Recommended Method for Variance Explained
Bray-Curtis	Built-in: residual distance over original distance; see Bray-Curtis ordination
Canonical Correspondence Analysis	After-the-fact with Relative Euclidean Distance
Detrended Correspondence Analysis	After-the-fact with Relative Euclidean Distance
Non-metric Multidimensional Scaling	After-the-fact with same distance measure as used in analysis
Principal Components Analysis	Built-in: ratio of eigenvalues to total variance; see PCA Output .
Reciprocal Averaging (Correspondence Analysis)	After-the-fact with Relative Euclidean Distance
Weighted Averaging	After-the-fact with Relative Euclidean Distance

The after-the-fact evaluation is performed simply by calculating the coefficient of determination (r2) between distances in the ordination space and distances in the original space.  Distances in the ordination space are ALWAYS measured with Euclidean distance, while you have a choice of distance measure for the original space.  You should select the distance measure most closely allied with the ordination method you are using (see Table above).  For Bray-Curtis or Nonmetric Multidimensional Scaling, choose the distance measure that you used to construct the ordination.  For PCA, the results from this option will not match the percentage of variance calculated from the eigenvalues.  PCA is typically based on correlations among the columns rather than distances among the rows.

One half of the distance matrix (not including the diagonal) is used in this calculation.  If there are N entities, then there are N*(N-1)/2 pairs of values used to calculate r2.

These r2 values are by no means the whole story in evaluating the effectiveness of an ordination.  For example, if a couple of outliers dominate the relationships in the data set, then a very high r2 may result.  This will give the appearance of an effective ordination, even though the ordination has been ineffective in describing the similarity relationships within the main cloud of points.  Other cases of trivial structure in the data can cause similarly high r2 values.

In some cases the increment to r2 as axes are added can be negative.  This can happen for several reasons.  The ordination method may be optimizing some criterion other than maximizing the percentage of variance extracted on each axis, for example minimizing stress in NMS (non-metric multidimensional scaling).  Comparisons based on that criterion should improve with the addition of further axes, but it may not improve the r2 between ordination distance and original distance.  The results also depend on how the axes are scaled.  For example, rescaling axes by "Min to Max" usually weakens the relationship between ordination distance and original distance and may result in negative increments to r2.

Axes calculated by some ordination techniques are not necessarily uncorrelated. In particular, NMS often generates axes that are weakly correlated.  Although this is rarely a serious problem, moderately correlated axes can cause problems in interpretation and rotation of ordinations. To help you appreciate this problem we added some statistics to the output in version 4.15.  Correlations are calculated between all pairs of axes.  This correlation, along with an orthogonality statistic are listed in a table, for example,

Axis pair     r     Orthogonality,% = 100(1-r^2)

  1 vs 2     0.068     99.5

  1 vs 3    -0.026     99.9

  2 vs 3     0.066     99.6

In this case, all the axes are essentially orthogonal -- statistically independent.  The orthogonality is simply the 1 minus r-squared, expressed as a percentage. As in previous versions of PC-ORD, the table of r-squared increments between the ordination space and the original space automatically adjusts for any lack of independence of the axes.

Example of Output

Coefficients of determination for the correlations between ordination distances and distances in the original n-dimensional space:

______________________________

             R SQUARED

       _______________________

AXIS   INCREMENT   CUMULATIVE

______________________________

  1       .427       .427

  2       .196       .623

  3       .061       .684

______________________________

Number of entities =  59

Number of entity pairs used in correlation =    1711

Distance measure for ORIGINAL distance:     1-2W/A+B

Distance measure for ORDINATION distance:  Euclidean

Graph Statistics

Graph Menu


Graph Groups

Graph ordination | Groups | Select Grouping Variable

You can display groups in your ordinations and dendrograms by coding them with up to 15 different colors or symbols. Groups in the Graph menu allows you to select a categorical variable in your second matrix that can be used to define the groups.  Do this by choosing Select Grouping Variable.  A list of the categorical variables in your second matrix is displayed.  Select the variable that you wish to use to define groups.

Note that you CANNOT overlay groups on your ordination or dendrogram if you don’t have a second matrix or if you have no categorical variables defined in your second matrix.

Choose whether to display the groups by color codes, symbol codes, or both on the Format tab of the Options | Preferences dialog (See Group Preferences).  Note that if neither color codes nor symbol codes are turned on, your groups will not be displayed.  You can associate particular symbols or colors with particular values of your categorical variable by selecting the  Legend Symbol/Color button in the Graph Preferences dialog.

Graph Menu


Legend Symbol/Color

Graph | Options, Preferences | Format tab | Legend Symbol/Color

This option applies if you have selected a grouping variable and are coding symbols in your graph by colors or symbol shapes.  Use the Legend Symbol/Color dialog to associate particular colors or symbols with particular categories.  At most 15 categories can be displayed at one time.

PC-ORD scans the selected categorical variable for all of its unique values, then ranks them by increasing size.  This ranking is the order PC-ORD uses to assign default symbols and/or color codes.  You can change the defaults in the Legend Symbol/Color dialog by clicking on the symbol or color associated with a particular rank.  A palette of choices will pop out.  Click on the color or symbol of your choice.

If you attempt to print colored symbols on a black-and-white printer, some printers will not print certain colors using a grayscale.  If some symbols that appear on your screen do not appear in the printed version, you can try the following:

•	Check Print in black and white in the Graph Preferences dialog.

•	Select filled symbols instead of unfilled symbols.

•	Use a different printer.

Graph Preferences


Show Side Scatterplots

Graph | Options, Preferences | Format tab | Show Side Scatterplots

When using quantitative overlays, the default is to show two additional scatterplots to the sides of the overlay.  These are called the “side scatterplots.”  Their purpose and interpretation are described under Overlay from Main Matrix.

The side scatterplots can be turned on and off by using the checkbox in the Graph | Preferences dialog.  If the side scatterplots are turned off, then the ordination plot will be larger.

Graph Preferences


Show Regression Line

Graph | Options, Preferences | Format tab | Show Regression Line

If you choose to show the side scatterplots on a quantitative overlay, you have the option of displaying a regression line on top of those scatterplots.  The least-squares linear regression line is fitted to the relationship of the quantitative overlay variable to the axis score.  This regression line corresponds to the r-value given with the side scatterplots.

Caution should be used in interpreting these linear fits.  In many cases the relationship between a variable and an ordination is clearly non-linear.  Frequently outliers have a strong influence on the regression line.

Graph Preferences


Show Envelope Line

Graph | Options, Preferences | Format tab | Show Envelope Line

If you choose to show the side scatterplots on a quantitative overlay, you can display envelopes fitted to the scatterplots.  The purpose of the envelope is to display the approximate upper bound of the overlay variable, with respect to an ordination axis.

Conceptually an upper bound makes more sense when overlaying species abundances than does traditional curve fitting.  We usually think of a species response to an environmental gradient as being hump-shaped.  The upper bound (or envelope) of that hump approximates the maximum performance of a species along an environmental gradient.  Because a species can low in abundance or even absent in optimal habitat, species response curves are typically "solid".    "Solid" in this context means that with sufficient sampling, we expect species responses to fill the space between zero abundance and the maximum possible abundance all along an environmental gradient.

Our inclusion of a fitted envelope allows you to summarize a species response along a gradient represented by an ordination axis.  This representation is extremely flexible, in that it makes no assumptions about the shape of a species response.  Linear, unimodal, bimodal, sigmoid, and other curve shapes are approximated based on the data, rather than any preconception.

The fit of the envelope can be tuned with:

Flexibility (1-100, default=5) controls the flexibility of the envelope.  This parameter is the inverse of the width of the moving window used to fit the curve (see below).

Standard deviation (0-3, default=2) controls the inclusiveness of the envelope.  Higher values are more inclusive.  A value of 2 will, on average, result in an envelope that includes about 95% of the points in the scatterplot.   If the standard deviation is set at zero, then the "envelope" simply becomes a moving average.

PC-ORD fits the envelope by using a moving window ("window" meaning a band across the ordination axis, rather than a visible box on your screen).  The width of the window is the axis length divided by the flexibility parameter.  The window is shifted along the axis using a fixed increment, which is 1/100 of the axis length.

Assume we are calculating the envelope for the horizontal ordination axis.  At each position of the window, we calculate the mean and standard deviation of the overlay variable for all points within the window.  The mean is weighted inversely proportional to the horizontal distance of each point from the vertical midline of the window, so that the weight is 1 on the vertical midline and 0 on the edges.  The intersection of the envelope with the midline of the window is the weighted average plus the user-specified number of standard deviations (usually 2).  The calculation of the standard deviations is also weighted, the weights applied as the variance is accumulated.

Cautions:

•	If the overlay variable takes just a few discrete values (such as presence-absence data or a small number of cover classes), the fitted envelopes are a poor representation of the relationship.  In this case, the envelopes should be turned off.

•	The upper bound will sometimes extend beyond the range of the points in the scatterplot, resulting in the curve being "off scale."  Off-scale portions of the curve are not displayed on your screen, but when they typically appear on printed output.

Some details of calculations:

•	At the ends of the scale the window starts half way off on the left side and ends up half way off on the right side.  That is, it starts with the midline at the left border and ends with the midline at the right border.

•	If there are no points within a broad region of the graph, no line is fitted to that portion.

Graph Preferences


Label Every Other Tick

Graph | Options, Preferences | Format tab | Label every other tick

Depending on the crowding of tick labels, you may wish to use this option to label every other tick mark.  This option does not apply, however, if your tick labels are very large or very small numbers.  In that case, scientific notation is used and only the first and last tick marks are labeled.

Graph Preferences

Graphing


Show legend

Graph | Options, Preferences | Format tab | Show Legend

Legends can be included or removed from a graph by using the checkbox on the Format tab under Options | Preferences.  The legend can also be repositioned by dragging it. The frame around the legend can be turned on or off with Show Legend Frame.

Graph Preferences

Graphing


Show legend frame

Graph | Options, Preferences | Format tab | Show legend frame

The frame around a legend on graphs can be turned on or off by using the checkbox Show Legend Frame on the Format tab under Options | Preferences.

Graph Preferences

Graphing


Line types for curves

Graph | Options, Preferences | Styles tab | Line types

Go to the Options, Preferences | Style tab under the Graphing menu to select line types for the curves in NMS scree plots (real and randomized data) and species area curves (average number of species, average distance, and confidence bands).

The following line types can be selected:  solid, dashed, dot, dash-dot, and dash-dot-dot.

Graph Preferences

Graphing


Symbol and curve colors

Graph | Options, Preferences | Styles tab | Symbol and curve colors

Use  the Options | Preferences | Styles tab under the Graphing menu to select colors for curves and symbols in species area curves and NMS scree plots.  Click on the colored square in the dialog to open a palette of colors to choose from.

Graph Preferences

Graphing


Groups

Several programs are provided in PC-ORD for the explicit purpose of creating or examining grouped data structures.

•	Cluster Analysis is used to produce a hierarchical classification of objects.

•	Multi-Response Permutation Procedures (MRPP) are used to test whether a priori groups differ in their position in a multidimensional space.

•	Blocked MRPP is used to analyze randomized block experiments or paired-sample data.  It is a variant of MRPP

•	Two-Way Indicator Species Analysis (TWINSPAN) is a method of simultaneously classifying both species and sample units.  In addition it produces two-way ordered tables.

•	Indicator Species Analysis assigns indicator value for each species (assumed to be columns in your main matrix), based the degree to which they discriminate among groups.  You can define groups with a variable from either the main or second matrix.

•	Mantel Tests are used to compare two distance (or dissimilarity) matrices.


Cluster Analysis

Groups | Cluster Analysis

The purpose of cluster analysis is to define groups of items based on their similarities.  Program CLUSTR performs eight variants on the general class of cluster analyses that are hierarchical, agglomerative, and polythetic.  In this context, "hierarchical" means that large clusters are composed of smaller clusters.  "Agglomerative" means that the analysis proceeds by joining clusters rather than by dividing clusters ("divisive" cluster analysis, e.g. TWINSPAN). "Polythetic" means that many attributes of the items are used to decide the optimum way to combine or divide clusters.  See Sneath and Sokal (1973) for a comprehensive discussion of clustering methods; Gauch (1982) and Greig-Smith (1983) have good discussions of clustering for ecologists.

The basic algorithm was written by W. M. Post and J. D. Sheperd, then adapted to PC's by B. McCune.

A publication-quality dendrogram is available in PC-ORD by running Graph | Cluster Dendrogram after running cluster analysis.

Cluster Setup

Cluster Warning

Cluster Chaining

Groups


Cluster Setup

Before running cluster analysis you must specify a distance measure, group linkage method, dendrogram options, and other output options.

Cluster Analysis


Cluster Distance Measure

The properties of the distance measures are briefly described in the help system.  For more information see Faith et al. (1987) ,  Greig-Smith (1983) and Beals (1984) and Ludwig and Reynolds (1988).  All distance measures are squared before cluster analysis.

Note that the "distance" on the dendrogram scale is Wishart's (1969) objective function, not a simple distance measure.  The objective function is a measure of information loss as agglomeration proceeds.

The objective function (E) is the sum of error sum of squares from each centroid to the items in that group:

where t indexes the T clusters and Et is the error sum of squares for cluster t:. Et  is calculated as

where xijt is the value of the jth variable for the ith point of cluster t containing kt points, and jt is the mean of the jth variable for cluster t.  Note that the deviations are zero at the beginning and the sum of squared deviations is maximal when all groups are fused.  At this point the objective function is equal to the sum of the squared deviations from the centroid of the whole point cloud in p-dimensional space.

Conceptually this is similar to an r2 value (coefficient of determination), but in this application maximizing r2 is not an objective.  Rather, you should think of the objective as a compromise between minimizing the number of groups and maximizing the information retained.

You can apply the objective function to any clustering method, but most statistical packages plot the dendrogram on a straight distance scale.  Depending on the linkage method, this often results in “reversals” where fusion of two groups occurs at a lower distance than a previous fusion.  This never happens when the dendrogram is scaled by the objective function.

The dendrogram is also scaled by percentage of information remaining.  When you begin clustering all information is present, but by fusing groups, information is gradually lost.  When all items are fused into a single group, no information remains.  If E is the objective function and SST is the total sum of squared deviations from the centroid when all items are fused into one group, then the percentage of information remaining is 100(SST-E)/SST.

Cluster Setup

Cluster Analysis


Group Linkage Method

The following linkage methods are available:

•	NEAREST NEIGHBOR (single linkage)

•	FARTHEST NEIGHBOR (complete linkage)

•	MEDIAN (WPGMC)

•	GROUP AVERAGE (UPGMA)

•	CENTROID (unweighted centroid)

•	WARD'S METHOD (hierarchical grouping)

•	FLEXIBLE BETA (flexible clustering)

•	McQUITTY'S METHOD (McQuitty's hierarchic classification)

Description of each of these methods for linking groups is beyond this help system.  The names for the methods follow Lance and Williams (1967) and Wishart (1978, 1969).  Ward's method is also known as the "error sum of squares" and was independently developed (and often attributed to) L. Orloci.  The names used by Sneath and Sokal (1973) for the same techniques are given parenthetically above.  Please consult Greig-Smith (1983), Sneath and Sokal (1973), Wishart (1978, 1969), or Lance and Williams (1967) for details of the methods.  Ward’s method is recommended as a general-purpose linkage method that minimizes distortions in the underlying space.

Some of the linkage methods are incompatible with some of the distance measures.  Incompatibility arises when Sorensen (Bray-Curtis) distance is selected but the subsequent clustering strategy recalculates intergroup distances as Euclidean (Median, Ward's Method, and McQuitty's Method). You are warned if you choose an incompatible method, but the analysis is allowed to proceed at your own risk.  The flexible beta method requires you to enter a beta value.

Cluster Setup

Cluster Analysis


Dendrogram Options

These options apply to the crude dendrogram provided in the result file.  Be sure to view the graphics-mode dendrogram (Graph | Cluster dendrogram).  You can control the appearance of the dendrogram in the result file by selecting its width, spacing, and scaling.  These options are described below.  The dendrogram scaling is described under the Cluster Distance Measure.

Dendrogram Width

If you have chosen the double-wide output, you can cut and paste the two pieces into a single dendrogram.  If you are clustering less than 25 items, single-width output is always used.

Dendrogram Spacing

You can set the dendrogram to single-spacing or double spacing.  Single spacing produces more compact output and should be used on large data sets.

Log Transform Dendrogram Scale

If most of the junctions in the dendrogram are compressed into the extreme left side of the dendrogram, you can get more resolution on the left side by using the log of the objective function to scale the dendrogram.

Cluster Setup

Cluster Analysis


Cluster Analysis Output Options

There are two main portions to the output: (1) an account of the level of similarity and participants in each cycle of clustering and (2) a dendrogram that shows the resulting hierarchy of clusters.  Several options pertain to the volume and form of the output, as described below.

Include Cluster Information With The Dendrogram

If this option is selected, then parts 1 and 2  as described above are listed in the output.

Write Distance Measure

Selecting this option causes the original distance matrix (at the start of clustering) to be included in the results file.  Only the lower triangle of the distance matrix is listed.

Add Group Membership Variable to Second Matrix

After examining your cluster analysis results, you may decide to save a new variable representing group membership at one or more levels in the dendrogram.  You can do this by rerunning cluster analysis and selecting this option.  You must have previously decided on a level at which to "prune" your dendrogram.  This level is specified by the number of groups remaining.  For example, if you decide that a five-group level provides a useful classification, you may request that membership in these groups be written into the second matrix.  You are asked to provide a name for the new variable.  If you do not have a second matrix opened, a second matrix will be created.

The number assigned to the group tells the row number of the first item assigned to that group.  This is of limited use to most users, but it is slightly more informative than just numbering the groups 1,2,3, etc.

Cluster Setup

Cluster Analysis


Group Membership Variable

Groups | Cluster Analysis

If you checked "Add Group Membership Variable to Second Matrix" in the Cluster Analysis output options , you will be asked for a "Group Level" and a "Group Membership Variable".

The group level specifies where the dendrogram should be pruned.  For example, if you select 5, the procedure will save integer codes for the 5-group level.  The number of groups can be any integer between 2 and one less than the number of items in the main matrix, inclusive.

The group membership variable is the name given to this new variable, when it is written as a new column into the second matrix.  The name of the variable must be 8 characters or less.

If you check the box for "Save all higher level groupings" then a series of variables is written into the second matrix.  Each variable represents the groupings at a given step in the clustering beginning with the group level that you selected.  For example, assume you are clustering 30 items and you are interested in some number of groups near 5, but you wish to study the group structure at a series of levels.  You should select a group level somewhat higher than your target level (say 8) and check the box to "save all higher level groupings."  This results in 7 new variables added to your second matrix, ranging from the 8-group level to the 2-group level.  The variables will be named using the "Group Variable Name" that you provided, suffixed by the number of groups at a given level.  For example, if you entered "test" as your group variable name, the new variables in your second matrix will be named test8, test7, test6, … , test2.

Cluster Analysis Output Options

Cluster Analysis


Cluster Flexible Beta Value

The flexible beta method requires a beta value.  This must be less than 1 and greater than or equal to -1. As beta approaches one, chaining will increase, giving a result similar to the nearest neighbor method.  As beta approaches -1, the results become similar to farthest neighbor.  Sneath and Sokal (1973) suggest that beta = -0.25 will behave similarly to Ward's method.

Cluster Setup

Group Linkage Method

Cluster Analysis


Dendrogram Chaining

The degree of chaining is calculated and written to the output file.  Chaining is the sequential addition of small groups to one or a few large groups. Dendrograms that are highly chained are usually undesirable as they are generally not helpful in defining subgroups.  You can use the percent chaining to help screen poor dendrograms.  Highly chained analysis are probably not worth printing or saving to the output file.  On the other hand, small differences in percent chaining at the low end of the scale are generally a poor criterion for selecting one linkage algorithm over another.

The method for calculating the degree of chaining compares the average path length of the dendrogram with the minimum possible (no chaining) and maximum possible (complete chaining) average path length. The path length for each item in the dendrogram is the number of nodes that connect it to the highest level in the dendrogram.  Complete chaining occurs when each fusion involves the addition of a single item to a single, increasingly large group.

Chaining can be controlled to some extent by selection of the group linkage method, although the tendency to chain is also in part a property of the data.

Cluster Setup

Group Linkage Method


Cluster Incompatibility Warning

If you select a distance measure that is incompatible with your choice of linkage method , you are warned of the problem.  Incompatibility of methods in this case means that the rules for the combinatorial equation (Wishart 1969) that is used to update the distance matrix will be violated.  You may go ahead and run the analysis but do so with the understanding the clustering may fail or the meaning of the results may be debatable!  Some users report good results despite the warning.

Note that the Euclidean and relative Euclidean distance measures are compatible with all of the linkage methods.  Sorensen (Bray-Curtis) distance is incompatible with Median, Ward's, and McQuitty's methods of linkage.

Cluster Setup

Cluster Analysis


MRPP

Groups | MRPP (Multi-Response Permutation Procedures)

MRPP is a nonparametric method for testing multivariate differences among pre-defined groups.  Help is available on the following topics:

•	Background

•	Grouping Matrix

•	Group Variable

•	Incompatibility Warning

•	MRPP Setup

•	Distance Measure

•	Group Weighting

•	MRPP Output

•	Exclude One or More Groups

Groups


MRPP Background

MRPP is a non-parametric procedure for testing the hypothesis of no difference between two or more groups of entities.  The groups must be a priori.  For example, one could compare species composition between burned and unburned plots to test the hypothesis of no treatment effect.  Discriminant analysis is a parametric procedure that can be used on the same general class of questions.  However, MRPP has the advantage of not requiring assumptions (such as multivariate normality and homogeneity of variances) that are seldom met with ecological community data.

A good introduction to the method is to read the appendix in Biondini et al. (1985).  More details can be found in Mielke (1984) and Berry et al. (1983).  See Zimmerman et al. (1985) for an ecological application similar to the example in the preceding paragraph.

ANOSIM (Analysis of Similarity; Clarke & Green 1988 ; Clarke 1993) is similar to MRPP in concept but uses a different test statistic.  The R statistic of ANOSIM is qualitatively similar in interpretation to the A statistic of MRPP.

Program MRPP was adapted by Bruce McCune and Joe Poston from subroutine MRPPF written by P. W. Mielke and K. J. Berry.  All calculations are performed with double precision.  Subroutine MRPPF uses an algorithm based on the first three moments of the sampling distribution of delta, the average within-group distance.

If you have a randomized block design or paired-sample design, you should use Blocked MRPP (MRBP).

Summary of the Method

1. Calculate distance matrix = D

2. Calculate average distance in each group i = xi

3. Calculate delta (the weighted mean within-group distance)

for g groups, where C is the weight which is dependent on number of items in the groups (normally Ci = ni/N, where ni is the number of items in group i and N is the total number of items).  Note that all ni > 2.  See the description of alternative weightings below.

For a given mean overall distance, smaller (more negative) values of δ indicate tighter clustering within groups.

4. Determine probability of a d this small or smaller.  The brute force method (applicable only to tiny data sets) is to calculate d for all possible partitions of the same sizes.  The number of possible partitions (M) for two groups is:

M = N!/(n1! * n2!)

For example, for n1 = n2 = 15, M = 1.55 x 108

Assume all partitions could have occurred with equal chance.  Then calculate the proportion of these that is smaller than the observed d.

p = (1 + no. smaller deltas) / total no. possible deltas

A more reasonable method that is applicable to medium or large data sets is to approximate the distribution of d from a continuous distribution (Pearson type III).  This distribution accommodates the fact that the underlying permutation distribution is often substantially skewed.  The Pearson type III distribution incorporates three parameters, the mean m, standard deviation s, and gamma g (skewness of d under the null hypothesis).

The test statistic, T is

T = (d - md)/sd = (observed - expected)/s.dev. of expected

where md and sd are the mean and standard deviation of d under the null hypothesis.  The p-value associated with T is determined by numerical integration of the Pearson type III distribution.

5. The p value is useful for evaluating how likely an observed difference is due to chance, but we also need a description of the effect size that is independent of the sample size.  This is provided by the chance-corrected within-group agreement (A):

For more information on the agreement statistic, see below under MRPP output.


Grouping Matrix

Select which matrix contains the variable that defines the groups.  The grouping variable can be in either the main or the second matrix.  In both cases, however, the test is applied to a distance matrix that is calculated from the main matrix.  If your grouping variable is in the main matrix, the grouping variable is excluded when calculating the distance matrix.

Indicator Species Analysis Setup

Indicator Species Analysis

MRPP Setup

MRPP


Group Variable

Select one variable that defines your groups.  For example, you may wanted to define two groups of sample units based on the presence/absence of a species of particular interest.  Or may want to define two groups based on substrate of limestone or acid rocks, coded as 1 and 2 in your secondary matrix.  Or you may have entered a treatment code (e.g. burned v. unburned) in your secondary matrix.

Say you have a main matrix of stands x species and you want to define your groups with the FIRE variable. It might contain ones, twos, or threes, depending on whether a sample unit was burned once, twice, or three times.  If you select FIRE and you are running MRPP will test the hypothesis of no difference in species composition among your three groups of sample units.  If you are running Indicator Species Analysis, each species will be evaluated for its ability to discriminate among the three fire treatments.

As many as 100 groups may be defined, each group defined by a unique integer.  Code numbers for groups may be any non-negative whole number and can occur in any sequence in the input data.  If the variable is not declared as a categorical variable, you receive the Grouping Variable Incompatibility Warning.  If your grouping variable includes real numbers with decimal points, the number is truncated at the decimal point. Note also that groups with a single member are not allowed, nor is it recommended to have groups with only a few members.

Grouping Matrix

Exclude One Or More Groups from Comparison

Indicator Species Analysis Setup

Indicator Species Analysis

MRPP Group Weighting

Blocked MRPP

MRPP


Grouping Variable Incompatibility Warning

MRPP and Indicator Species Analysis expect you to choose a categorical variable (See variable types) that defines your groups.  A warning is given if you select a grouping variable that is not categorical.  You can ignore the warning if you know that the variable contains whole number values that could be used to define groups.

If the column you selected contains decimal points, the decimal part is truncated when used to define groups.  In some circumstances this may be acceptable, but usually it indicates that you are trying to select a variable that does not define a discrete number of groups.

If the column you selected contains only decimals from 0.0 to 1.0,all of the values would be interpreted as "group 0" except for any values that were 1.0, which would be interpreted as "group 1".

Grouping Matrix

Group Variable

Indicator Species Analysis Setup

Indicator Species Analysis

MRPP Setup

Blocked MRPP

MRPP


MRPP Setup

In the MRPP Setup dialog you select the following:

•	Distance measure

•	Group weighting

•	Exclude One or More Groups

•	Rank Transform Distance Matrix (or not)

MRPP


MRPP Distance Measure

Distance measures available for MRPP are described under Distance Measures.  Zimmerman et al. (1985) recommended Euclidean distance over squared Euclidean distance.  Sorensen distance has seldom been used in published studies with MRPP and ecological data.  However, preliminary exploration has shown results very similar to Euclidean distance, but perhaps with some desirable de-emphasis of outliers.

MRPP Setup

MRPP


MRPP Group Weighting

A weighting factor is applied to the items in each group.  Most of the weightings are based on group size and are listed below.  In each case, n is the number of items in group and g is the number of groups.

•	n/sum(n).  A natural weighting recommended by Mielke (1984) and used in most recent applications of MRPP.

•	n-1/sum(n-1).  With squared Euclidean distance this weighting results in an MRPP statistic that is equivalent to a 2-sample t-test or one-way ANOVA F-test (Mielke et al. 1982; Zimmerman et al. 1985).  While this option accounts for degrees of freedom, this is a foreign concept to permutation procedures.

•	1/g.  Not recommended but available for experimentation.

•	n(n-1)/sum(n(n-1)).  Not recommended but available for experimentation.  Used in some early applications of MRPP.

MRPP Setup

Exclude One Or More Groups from Comparison

MRPP


Exclude One Or More Groups from Comparison

By default all groups are included in the comparison.  You may wish, however, to make comparisons between groups while excluding one or more of the groups from comparison.

If you select this option, you will be asked to select from a list of group numbers those groups that you wish to exclude.  Use your mouse to select the boxes to the left of the group numbers.  Or you can use your keyboard:

Space bar:  toggles on/off

+ or Insert key:  selects item

- or Delete:  deselects item

Group Variable

Indicator Species Analysis Setup

Indicator Species Analysis

MRPP Setup

MRPP

Blocked MRPP


Group Numbers

If you request that one or more groups be excluded, you are asked to provide the numbers for the groups to be excluded.  Use your mouse to select the boxes to the left of the group numbers.  The procedure expects integers representing categories.  If the grouping variable contains non-integers, they are truncated at the decimal point.

Exclude One Or More Groups from Comparison

Indicator Species Analysis Setup

Indicator Species Analysis

MRPP Setup

Blocked MRPP

MRPP


MRPP Rank Transform Distance Matrix

In the MRPP Setup dialog you have the option to rank transform the distance matrix.  This can be applied to any of the distance measures.  The rank transformation can help to correct the loss of sensitivity of distance measures as community heterogeneity increases.  It also subtly changes the null hypothesis being answered from " average within-group distance no smaller than expected by chance" to "no difference in average within-group rank of distances."  It also makes the MRPP results more analogous in theory to non-metric multidimensional scaling.

MRPP on ranked distances is more similar to ANOSIM (Clarke, K. R., and R. H. Green.  1988; Clarke, K. R.  1993) than is MRPP with raw distances.

In practice with community data, the test statistic, skewness of the test statistic under the null hypothesis, and the resulting p-value are often similar, whether the data are ranked or not.  The chance-corrected within-group agreement, however, is often considerably higher after the distance measure is converted to ranks.

The ranking procedure operates on non-redundant distances, i.e. one triangle of the square, symmetrical distance matrix.  Ties are assigned the average rank of the tied elements.  For example, the values 1, 3, 3, 9, 10 would receive ranks 1, 2.5, 2.5, 4, 5.  After elements are assigned initial ranks, they are adjusted by subtracting the rank of zero distance.  This results in all raw distances of zero being assigned a rank distance of zero.  For example, if there were five zero distances in the matrix, they would each initially be assigned a rank of 3, taking into account the five-way tie.  Then 3 is subtracted from each element in the matrix.

MRPP Setup

MRPP


MRPP Output

The test statistic is the difference between the observed and expected deltas divided by the square root of the variance in delta.  The test statistic describes the separation between the groups.  The observed delta (the average within-group distance) is compared to an expected delta, the latter calculated to represent the mean delta for all possible partitions of the data.  The variance and skewness of delta are descriptors of the distribution of all possible deltas corresponding to the possible partitions of the items.  See example output.

The question then becomes, is the observed delta at all unusual considering the distribution of possible deltas?  The probability value expresses the likelihood of getting a delta as extreme or more extreme than the observed delta, given the distribution of possible deltas.  This probability is given in your output, for example:

Probability of a smaller or equal delta, p =     .00124251

The statistic A is given as a descriptor of within-group homogeneity, compared to the random expectation.  This is known as chance-corrected within-group agreement.  In Mielke's publications it is written as a script R.  When all items are identical within groups, then the observed delta = 0 and A = 1, the highest possible value for A.  If heterogeneity within groups equals expectation by chance, then A = 0.  On the other hand, if there is LESS agreement within groups than expected by chance, then A < 0.

In community ecology values for A are commonly below 0.1, even when the observed delta differs significantly from the expected.  An A > 0.3 is fairly high.  Remember that for A = 1 all items must be identical within groups.

Statistical significance (small p) may result even when the "effect size" (A) is small, if the sample size is large.  For example, you may find that A = 0.01 may be statistically significant with N = 200.   In such cases you need to carefully consider the ecological significance of the result, not just the statistical significance.  On the other hand, with a small sample size, a large effect size is needed to achieve statistical significance.

MRPP Example Results

MRPP Setup

Exclude One Or More Groups from Comparison

MRPP


MRPP Example Results

Differences in communities among levels of Brojap

        Groups were defined by values of: Brojap

        Input data has:   19 STANDS   by  49 SPECIES

        Weighting option: C(I) = n(I)/sum(n(I))

        Distance measure: Sorensen (Bray-Curtis)

GROUP:  1

Code:   0

Size:  10      0.66267458    = Average distance

Members:

 STAND1   STAND2   STAND3   STAND4   STAND6   STAND8   STAND9   STAN12

 STAN13   STAN15

GROUP:  2

Code:   2

Size:   4      0.61875399    = Average distance

Members:

 STAND5   STAND7   STAN10   STAN19

GROUP:  3

Code:   1

Size:   5      0.54680432    = Average distance

Members:

 STAN11   STAN14   STAN16   STAN17   STAN18

        Test statistic: T =      -2.5874500

           Observed delta =      0.62293597

           Expected delta =      0.66249308

        Variance of delta =      0.23372499E-03

        Skewness of delta =     -0.70854412

        Chance-corrected within-group agreement, A =    0.05970947

          A = 1 - (observed delta/expected delta)

          Amax = 1 when all items are identical within groups (delta=0)

          A = 0 when heterogeneity within groups equals expectation by chance

          A < 0 with more heterogeneity within groups than expected by chance

        Probability of a smaller or equal delta, p =    0.01498651

MRPP Setup

MRPP


Blocked MRPP (MRBP)

Randomized block experiments or paired-sample data can be analyzed with a variant of MRPP called MRBP or blocked MRPP.  PC-ORD allows up to 1000 blocks and 100 groups.  Note that this analysis requires a balanced design:  there must be one sample unit for each combination of block and treatment.  The number of treatments must be the same among blocks and each treatment must be present in each block.  The method is explained by Mielke (1984, 1991), Mielke and Berry (1982), Mielke and Iyer (1982), and Biondini et al. (1988).  It has been applied ecological data by Biondini et al. (1988) and Zimmerman et al. (1985).

Given b blocks and g groups (treatments), the MRPP statistic is modified to:

where Δ(x,y) is the distance between points x and y in the p dimensional space.  Each of the p dimensions should be measured on the same scale or standardized to a common scale.  In community ecology each of the p dimensions would normally correspond to a different species or a different environmental variable.  Note that for paired-sample data, b is the number of pairs of observations and g = 2.  The combinatoric term is simply the number of items represented in the double summation.  Thus delta is the average distance between blocks within treatments, including only non-redundant off-diagonal elements of the distance matrix.  Normally delta is calculated after alignment of the blocks.

The null hypothesis assigns equal probabilities to each of the M = (g!)b possible allocations of the g p-dimensional measurements to the g treatments within each of the b blocks.  In other words, the observed values are randomly reassigned to different treatments in each block.

Like MRPP, small values of δ imply a concentration of treatments in the p dimensional space.  The added twist of MRBP is that these distances are summed with respect to the blocks, and that the user has the option of aligning blocks so that all treatments in a given block have a median of zero.

Equations for calculating the mean, standard deviation, and skewness of delta are relatively complex (see Mielke 1984) and are not reproduced here.  The test statistic, T, is based on these values as in MRPP and the p-value is again approximated from a Pearson type III distribution.  See example output.

Several options are available:

•	Group Variable

•	Blocking Variable

•	Exclude One or More Groups

•	Distance Measure

•	Average Distance Function Commensuration

•	Median Alignment Within Blocks

Groups Menu


Blocking Variable

Groups | Blocked MRPP

Select one variable that defines your blocks.  Each block must have one instance of each level of the grouping variable.  The design must be balanced.

As many as 1000 blocks may be defined, each block defined by a unique integer in the blocking variable.  Code numbers for blocks may be any non-negative whole number and can occur in any sequence in the input data.  If your blocking variable includes real numbers with decimal points, the number is truncated at the decimal point.

Grouping Matrix

Blocked MRPP


Blocked MRPP Distance Measure

Only Euclidean distance and squared Euclidean distance are available for blocked MRPP (See distance measures ).  Euclidean distance has been used most often.  If squared Euclidean distance is chosen, then the resulting statistics are comparable to permutation tests of Pearson correlation.  Mielke (1991) presents some interesting arguments against this.  Although methods dependent on the normal distribution require sample statistics based on squared Euclidean distances.  Because MRPP, MRBP, and related statistics are based on a permutation distribution, there is no necessity for choosing squared Euclidean distance.  Mielke (1991, pp. 56-57) argues that "absolute deviations" (meaning Euclidean distance) is more logical and intuitive than squared deviations (squared Euclidean distance).  Although Mielke's papers do not consider proportional city-block distance measures, there is an incompatibility with the Sørensen distance and median alignment, since alignment results in both positive and negative values.

Example output

Blocked MRPP


Average Distance Function Commensuration

This option equalizes the contribution of each variable to the distance function.  For each variable m sum of deviations (Devm) is calculated:

The exponent V is set to 2 for squared Euclidean distance or 1 for Euclidean distance.  Then each element x of the data matrix is divided by the sum of the deviations for the corresponding variable to produce the transformed value y:

Example output

Blocked MRPP


Median Alignment Within Blocks

If the median for each variable in each block is subtracted from the raw data for each block, then the medians are said to be aligned to zero for all blocks (Table below).  Usually alignment in a randomized block design is desirable.  But if the problem is conceptualized as paired agreement, say between model predictions and  observed data, then alignment is not performed.  In this case each event or observation represents a group and one block contains the observed data and one block contains the predicted data. Because the goal is an exact match of predicted and observed, rather than the two just being correlated, then the medians should not be aligned.  If, however, the goal is whether or not the two sets of numbers are correlated, apart from any exact agreement in value, then the blocks should be aligned.

Table: Comparison of raw data and data aligned within blocks to zero as input to Blocked MRPP.

	Raw Data			Aligned Data
	Block 1	Block 2		Block 1	Block 2
Group 1	4	9		1.5	1.5
Group 2	2	7		-0.5	-0.5
Group 3	3	8		0.5	0.5
Group 4	1	2		-1.5	-5.5
Median	2.5	7.5		0	0
Observed δ	5 = (5+5+5+1)/4			1 = (0+0+0+4)/4
Expected δ	4.375			2.225
Agreement (A)	0.086			0.556
p	0.184			0.016

Example output

Blocked MRPP


Blocked MRPP Example Results

Comparison of response to thinning within blocks

Analysis of randomized block data with MRBP:

    39 variables

     5 blocks

     4 groups

       Groups were defined by values of: Treat

       Blocks were defined by values of: StanBloc

       Input data has:    20 plots

                   by     39 attrib

OPTIONS

       Distance measure: Euclidean

       Median alignment performed

       No average distance function commensuration.

GROUP:   1

Code:    1

Size:    5

Members:

 BEV1-1   GNO2-1   DLI3-1   SAN4-1   NWA5-1

GROUP:   2

Code:    2

Size:    5

Members:

 BEV1-2   GNO2-2   DLI3-2   SAN4-2   NWA5-2

GROUP:   3

Code:    3

Size:    5

Members:

 BEV1-3   GNO2-3   DLI3-3   SAN4-3   NWA5-3

GROUP:   4

Code:    4

Size:    5

Members:

 BEV1-4   GNO2-4   DLI3-4   SAN4-4   NWA5-4

        Test statistic: T =      -4.1880177

           Observed delta =       6.9528671

           Expected delta =       7.4888375

        Variance of delta =      0.16378141E-01

        Skewness of delta =     -0.48910187

        Chance-corrected within-group agreement, A =    0.07156924

          A = 1 - (observed delta/expected delta)

          Amax = 1 when all items are identical within groups (delta=0)

          A = 0 when heterogeneity within groups equals expectation by chance

          A < 0 with more heterogeneity within groups than expected by chance

        Probability of a smaller or equal delta, p =    0.00041511

Blocked MRPP


TWINSPAN

Groups | TWINSPAN (Two-Way INdicator SPecies ANalysis)

TWINSPAN (Hill 1979b, Gauch and Whittaker 1981) simultaneously classifies species and samples.  The program is geared towards ecological data.  At its core, TWINSPAN is based on dividing a reciprocal averaging ordination space.  The procedure was described in accessible terms by Gauch (1982, pp. 201-203).

One of the most useful features of TWINSPAN is the final ordered two-way table.  Species names are arrayed along the left side of the table, while sample numbers are along the top.  The pattern of zeros and ones on the right and bottom sides define the dendrogram of the classifications of species and samples, respectively.  The interior of the table contains the abundance class of each species in each sample.  Abundance classes are defined by pseudospecies cut levels.

The analysis is performed with a modified version of program TWINSPAN from the Cornell Ecology Program Series (Hill 1979b).  Several of the non-essential options and memory limitations were removed.  More details and the full documentation can be obtained from the Cornell Ecology Program Series.

TWINSPAN inherited a number of faults from its parent method, reciprocal averaging (RA, CA, or correspondence analysis).  Like RA it performs poorly when there is more than one important underlying gradient.  This limitation is particularly awkward when trying to interpret the ordered table.  Unlike cluster analysis, which has no inherent reduction in dimensionality, TWINSPAN cannot effectively represent complex data sets in its one-dimensional framework (van Groenewoud 1992, Belbin and McDonald 1993).

Oksanen and Minchin (1997) pointed out the extremely lax criteria for stability of the RA solutions at the root of TWINSPAN.  This caused instability in the results depending on sample order.  Since TWINSPAN in PC-ORD package was based on the original Cornell Ecology Programs, it too had this problems.  The problems were fixed in the DOS version of PC-ORD, with version 2.11 and higher.  The Windows version of PC-ORD contained these remedies from the start.

We apply Oksanen and Minchin's (1997) "super strict" criteria of tolerance=0.0000001 and maximum number of iterations = 999, replacing the old criteria of tolerance=0.003 and maximum iterations = 5.  These changes will mean somewhat slower computations, but in most cases you will hardly notice a difference.

TWINSPAN used to be the only routine in PC-ORD that did not use dynamically allocated memory.  In other words, it required a fixed amount of memory.  In version 4 of PC-ORD this memory limitation was removed.

Help is available on:

•	TWINSPAN Setup

•	Pseudospecies cut levels

•	TWINSPAN Setup Parameters

•	TWINSPAN Example Results

Groups


TWINSPAN Setup

In the TWINSPAN setup dialog you specify the pseudospecies cut levels and other setup parameters, including the maximum number of indicators per division, maximum level of division, minimum group size for division, and the maximum number of species in the final tabulation.

TWINSPAN Example Results

TWINSPAN

Groups


Pseudospecies Cut Levels

The number of non-zero items in the data includes occurrences of "pseudospecies".  If your data set is too large, you may be able to overcome the limitation by reducing the number of pseudospecies cut levels (see Memory Requirements).

You should understand the idea of pseudospecies.  TWINSPAN does not analyze abundance data directly, but is based on presence/absence.  However, it approximates quantitative abundance data by creating a variable number of "pseudospecies" representing abundance classes.  The "pseudospecies cut levels" are used to define the ranges of the abundance classes.

Suppose you have data ranging from 0 to 100% cover.  You need to define cover classes from the data (max. number of classes = 9).  The default cut levels are 0, 2, 5, 10, and 20, and are reasonable for your data in this case.  These are cutoff points for pseudospecies 1, 2, 3, 4, and 5.  A species (say its code name is Testsp) that had an abundance of 8% in a given sample would be interpreted as "present" for three pseudospecies: Testsp 1, Testsp 2, and Testsp 3 (the digit after the species name indicates the pseudospecies level).  An input value of 1.5% would result in only one pseudospecies being "present", Testsp 1.

Selection of appropriate cut levels of pseudospecies is important for retaining the quantitative nature of your data.  For example, if you had relativized your data such that all of the values fell between 0 and 1, then use of the default cut levels would result in your data being treated as presence/absence. This would occur because only the first pseudospecies level would be used.  If you want to retain the quantitative information in your data, cut levels such as .0, .02, .05, .10, and .20 would be appropriate.

User Defined Cutoff Levels

You may choose to define your own pseudospecies cut levels.  Enter a maximum of nine integers or real numbers, in order from smallest to largest.  In most cases a zero will be the lowest (and first) level.  A maximum of nine levels can be defined.

TWINSPAN Setup

Setup Parameters

TWINSPAN


TWINSPAN Setup Parameters

The following parameters are set in the TWINSPAN setup dialog.

Maximum Number of Indicators per Division (default=5)

Indicator species are considered those at the ends of the reciprocal averaging axes.  They are used to further polarize the ordination.  Too many indicator species weakens the polarization while too few indicator species increases the risk of noise introduced by overemphasis on too little data.

Maximum Level of Divisions (default=6)

This option, along with minimum group size for division, are used to control how deeply the classification is nested.  Note that the default level is quite generous: an average final group size of three and 6 levels of nesting would accommodate 26 x 3 = 192 items.

Minimum Group Size for Division (default=5)

TWINSPAN will not divide groups when they become smaller than a set minimum size.  Reducing this minimum size is dangerous because the number of items becomes so small that the concept of seeking indicator species for division of the sample units weakens.

Maximum Number of Species In The Final Tabulation (default=200; maximum=9999)

If the number of species in the data exceeds the maximum allowed in the table, the first species to be eliminated are the rarest.  The final table is listed in sections that are 100 sample units wide.

TWINSPAN Setup

TWINSPAN


TWINSPAN Example Results

 Number of samples:    41

 Number of species:    58

 Length of raw data array:     1345 non-zero items

[Names of sample units and species acronyms are listed]

 Cut levels:

       0.0000      1.0000      2.0000      3.0000      4.0000

       5.0000      6.0000

 Options:

    Minimum group size for division =   5

    Maximum number of indicators per division = 5

    Maximum number of species in final table =  200

    Maximum level of divisions = 6

 Length of data array after defining pseudospecies:     1885

 Total number of species and pseudospecies:         216

 Number of species:    58

      (excluding pseudospecies and ones with no occurrences)

                  CLASSIFICATION OF SAMPLES

*************************************************************

DIVISION    1  (N=    41)         i.e. group *

Eigenvalue: 0.2771  at iteration   7

[TWINSPAN first ordinates by reciprocal averaging.  The first RA axis is divided near the middle.  In this case first division split the 41 sample units into two groups.  The eigenvalue from this first reciprocal averaging axis is reported.  The number of iterations required to reach a specified tolerance (stability in the ordination) is also reported.]

INDICATORS and their signs:

Acergl   1(+)

Maximum indicator score for negative group   0

Minimum indicator score for positive group   1

[In this case a single indicator species was found (Acergl).  Indicator species are defined by selecting a minimal set of species that will reproduce the ordination as closely as possible.  The purpose of defining the indicator species is to aid interpretation of the dichotomies.  Next, the division is described by listing the items in each group.  In this case, the "negative group" is the group usually lacking Acergl.]

ITEMS IN NEGATIVE GROUP   2  (N =   35)         i.e. group *0

83-1-B    83-3-B    83-5-B    83-6-B    83-7-B    83-8-B    84-1-B    84-3-B

84-4-B    84-5-B    84-7-B    84-8-B    84-9-B    85-1-B    85-4-B    85-5-B

85-6-B    85-8-B    85-9-B    87-3-B    87-4-B    87-5-B    87-7-B    87-8-B

89-1-B    89-3-B    89-4-B    89-6-B    89-7-B    89-9-B    91-1-B    91-4-B

91-6-B    91-7-B    91-9-B

BORDERLINE NEGATIVES    (N =    2)

85-8-B    91-1-B

ITEMS IN POSITIVE GROUP   3  (N =    6)         i.e. group *1

83-2-B    84-2-B    85-2-B    87-2-B    89-2-B    91-2-B

[This contrast based on the preliminary ordination is amplified by constructing a refined ordination of sites.  This is essentially a weighted averaging ordination based on the differential species.  Weights are assigned to species which are preferential to one side or the other of the division.  The position of the division is then refined by summing across sites the weights of the most preferential species.  The preferential species used in this ordination are reported below, along with their raw frequencies in the two groups.  For example:

Xerten  2( 16,  0)

means that pseudospecies 2 of "Xerten" occurred sixteen times in the negative group and zero times in the positive group.

NEGATIVE PREFERENTIALS

Amealn  1( 14,  1)  Aranud  1( 20,  1)  Cliuni  1( 15,  0)  Corcan  1( 12,  0)

Dishoo  1(  7,  0)  Linbor  1( 16,  1)  Rosa    1( 30,  1)  Smiste  1( 16,  0)

Spibet  1( 34,  2)  Vaccsp  1(  7,  0)  Viola   1( 11,  0)  Xerten  1( 16,  0)

Calrub  1( 16,  1)  Carcon  1(  7,  0)  Oryasp  1( 19,  0)  Amealn  2( 14,  1)

Aranud  2( 20,  1)  Cliuni  2( 15,  0)  Corcan  2( 12,  0)  Dishoo  2(  7,  0)

Linbor  2( 16,  1)  Rosa    2( 30,  1)  Smiste  2( 16,  0)  Spibet  2( 34,  2)

Vaccsp  2(  7,  0)  Viola   2( 11,  0)  Xerten  2( 16,  0)  Calrub  2( 16,  1)

Carcon  2(  7,  0)  Oryasp  2( 19,  0)  Amealn  3(  7,  0)  Aranud  3( 15,  0)

Berrep  3( 34,  2)  Linbor  3(  7,  0)  Rosa    3( 19,  0)  Spibet  3( 25,  1)

Xerten  3( 14,  0)  Cargey  3(  8,  0)  Oryasp  3( 12,  0)  Berrep  4( 18,  1)

Spibet  4(  8,  0)  Symalb  4( 12,  1)  Xerten  4(  9,  0)  Berrep  5( 10,  0)

Xerten  5(  7,  0)  Berrep  6(  7,  0)  Xerten  6(  7,  0)

POSITIVE PREFERENTIALS

Acergl  1(  0,  6)  Antluz  1(  6,  3)  Cirarv  1( 10,  4)  Cyn     1(  1,  2)

Epiwat  1(  7,  4)  Fraves  1(  5,  2)  Loncil  1( 11,  5)  Pacmyr  1(  5,  5)

Pruvul  1(  1,  2)  Rhamnu  1(  1,  6)  Taroff  1(  4,  2)  UnkCom  1(  2,  2)

Agr.sp  1(  3,  2)  Phleum  1(  4,  2)  Poapra  1(  4,  3)  Acergl  2(  0,  6)

Antluz  2(  6,  3)  Cirarv  2( 10,  4)  Cyn     2(  1,  2)  Epiwat  2(  7,  4)

Fraves  2(  5,  2)  Loncil  2( 11,  5)  Pacmyr  2(  5,  5)  Pruvul  2(  1,  2)

Rhamnu  2(  1,  6)  Taroff  2(  4,  2)  UnkCom  2(  2,  2)  Agr.sp  2(  3,  2)

Phleum  2(  4,  2)  Poapra  2(  4,  3)  Acergl  3(  0,  3)  Cirarv  3(  7,  3)

Cyn     3(  0,  2)  Loncil  3(  6,  4)  Pruvul  3(  1,  2)  Vertha  3(  7,  3)

Poapra  3(  3,  2)  Cirarv  4(  0,  2)  Fravir  4(  0,  2)  Poapra  4(  1,  2)

Cirvul  5(  5,  3)  Poapra  5(  0,  2)  Cirvul  6(  3,  2)  Cirvul  7(  0,  2)

NON-PREFERENTIALS

Berrep  1( 35,  5)  Cirvul  1( 31,  6)  Epipan  1( 14,  3)  Fravir  1( 21,  5)

Rubpar  1( 12,  3)  Symalb  1( 25,  3)  Vertha  1( 23,  5)  Cargey  1( 13,  2)

Berrep  2( 35,  5)  Cirvul  2( 31,  6)  Epipan  2( 14,  3)  Fravir  2( 21,  5)

Rubpar  2( 12,  3)  Symalb  2( 25,  3)  Vertha  2( 23,  5)  Cargey  2( 13,  2)

Cirvul  3( 19,  5)  Epipan  3(  7,  1)  Fravir  3( 10,  2)  Rubpar  3( 11,  1)

Symalb  3( 22,  3)  Calrub  3( 10,  1)  Cirvul  4( 10,  3)

    --------  E N D   O F   L E V E L   1 --------

[Similar output repeated for each division at each level.  Then the output is repeated for the species classification.]

*********************************************************

ORDER OF SPECIES INCLUDING RARER ONES

21  Epiang  |  9  Aranud  | 28  Linbor  |  5  Adebic  | 13  Ceasan

38  Shecan  | 47  Vaccsp  | 10  Arcuva  | 16  Cliuni  | 17  Corcan

30  Lonuta  | 37  Senpse  |  8  Apoand  | 19  Dishoo  | 39  Smiste

11  Astlae  | 34  Rosa    | 49  Viola   | 56  Oryasp  | 40  Spibet

 1  Piceng  |  3  Pinpon  | 27  Genama  | 53  Carcon  | 24  Filarv

20  Dissp   | 36  Rumace  | 43  Thalic  | 50  Xerten  |  6  Amealn

52  Calrub  | 12  Berrep  | 41  Symalb  | 54  Cargey  | 35  Rubpar

45  Triagr  | 22  Epipan  | 26  Fravir  | 48  Vertha  | 55  Fes/De

31  Pacmyr  | 44  Tradub  | 46  UnkCom  | 57  Phleum  | 15  Cirvul

14  Cirarv  | 23  Epiwat  | 29  Loncil  | 42  Taroff  |  7  Antluz

25  Fraves  |  2  Pincon  | 18  Cyn     | 32  Pruvul  |  4  Acergl

33  Rhamnu  | 51  Agr.sp  | 58  Poapra  |

ORDER OF SAMPLES

27 87-7-B   | 34 89-7-B   | 40 91-7-B   | 15 84-9-B   | 22 85-9-B

35 89-9-B   | 41 91-9-B   |  3 83-3-B   |  4 83-5-B   |  7 83-8-B

12 84-5-B   |  1 83-1-B   |  6 83-7-B   |  8 84-1-B   | 13 84-7-B

14 84-8-B   |  5 83-6-B   | 10 84-3-B   | 24 87-3-B   | 31 89-3-B

32 89-4-B   | 38 91-4-B   | 11 84-4-B   | 18 85-4-B   | 25 87-4-B

29 89-1-B   | 36 91-1-B   | 16 85-1-B   | 19 85-5-B   | 26 87-5-B

33 89-6-B   | 39 91-6-B   | 20 85-6-B   | 21 85-8-B   | 28 87-8-B

30 89-2-B   | 37 91-2-B   |  2 83-2-B   |  9 84-2-B   | 17 85-2-B

23 87-2-B   |

[In the final two-way ordered table (below), species names are on the left side of the table, while sample numbers are along the top (the digits stacked vertically).  The pattern of zeros and ones on the right and bottom sides define the dendrogram of the classifications of species and samples, respectively.  For example, the first division below separated the six sample units on the right (30, 37, 2, 9, 17, and 23) from the remainder.  The interior of the table contains the abundance class of each species in each sample. Abundance classes are defined by pseudospecies cut levels.  In this example, "Epiang" met the third cut level in sample unit 27.

A dendrogram can be drawn by hand using the following rules.  The first column of zeros and ones shows the first division -- rows receiving a zero were separated from rows receiving ones. Rows receiving the same digit are more similar to each other than rows in different groups. Moving to the right, the next column shows the next two divisions. Each of the preceding groups is again divided, as shown by the pattern of zeros and ones. You can continue these sequential divisions until you reach the rightmost columns -- these groups were too small to divide further. You can sketch in a traditional dendrogram to the right of the TWINSPAN table. But it differs a little from the dendrogram you would get by cluster analysis, because in TWINSPAN the finest branches of the tree usually have more than one item on them.]

TWO-WAY ORDERED TABLE

                2341234   1   11 12333112231123322233  12

                74052513472168345041281859669639018072973

  21  Epiang    333-------------------------------------2  000

   9  Aranud    35423233323342323333-----------------2---  00100

  28  Linbor    33333-223-2-22----2--2--2---23------2----  00100

   5  Adebic    ---2222-------2-2------------------------  001010

  13  Ceasan    -------22223----------------2------------  001010

  38  Shecan    ---4477----------------------------------  001010

  47  Vaccsp    --------2-222--2---2---------2-----------  001010

  10  Arcuva    ----264-------------22------------2------  001011

  16  Cliuni    23432-3---3--232-222-22------------------  001011

  17  Corcan    45423---23----23-----2-----------22------  001011

  30  Lonuta    ---------2-----2-2-----------------------  001011

  37  Senpse    334--34---------------------------2------  001011

   8  Apoand    --------2-2---------2-2-2----2-----------  001100

  19  Dishoo    ---2-23---------2-33--2------------------  001100

  39  Smiste    --3---22--2--222-22322--222--2-----------  001100

  11  Astlae    2------------------2----------2----------  001101

  34  Rosa      ---233332323-2-45334232232322333423-----2  001101

  49  Viola     -----22-222--------2--------2-332-3------  001101

  56  Oryasp    656-3-3-2--22---2-33-----33223332--------  001101

  40  Spibet    33444333633222263332-3222233333454332----  00111

   1  Piceng    ---------------------------2222----------  010000

   3  Pinpon    ------------------------------33---------  010000

  27  Genama    --------------------------2-------2------  010000

  53  Carcon    -------------------3-2---22--234---------  010000

  24  Filarv    -------------------2---------2-----------  010001

  20  Dissp     -22-----------------22--------23---------  01001

  36  Rumace    --------------------33--2----------------  01001

  43  Thalic    --------------------23-------------------  01001

  50  Xerten    ----23342-------3366777774---3-----------  01001

   6  Amealn    -----------2-2--2---332232-2-3333----2---  0101

  52  Calrub    ----233---2----2--4-33-23--234--324-----3  0101

  12  Berrep    4333434353434333357633333664665642434222-  0110

  41  Symalb    444-334---2-----2-4466344333336532333---4  0110

  54  Cargey    ---2--3------222--------3453--33-2322----  0110

  35  Rubpar    ----------------63-333633-----334-223---2  0111

  45  Triagr    -2----------------33-2----2-------32-----  0111

  22  Epipan    3---222-----------42-2--322--434--722---6  100

  26  Fravir    233-----2223-2------333233322222--344-222  100

  48  Vertha    222--33------2---22322222332-23222332-234  100

  55  Fes/De    2-----2------------------32---2---2-----3  100

  31  Pacmyr    ----------------------------22222--22222-  101

  44  Tradub    --------------------------2---22----2----  101

  46  UnkCom    -------------------2------2--------22----  101

  57  Phleum    -------------------------44----2-2-24----  101

  15  Cirvul    33-423-2222426632542--2223263333445323775  11000

  14  Cirarv    333--33------------2-----32---23---64--23  11001

  23  Epiwat    222------------2----------------222-2-222  11001

  29  Loncil    -------33-24-2--2--------33223-----33323-  11001

  42  Taroff    2-2------------------------2---2---22----  11001

   7  Antluz    ------2---2-------22------2-----2--22--2-  1101

  25  Fraves    ---2223---------------------------223----  1101

   2  Pincon    --------------------------------2---3----  11100

  18  Cyn       ----------------------------------233----  11100

  32  Pruvul    -------------------------------3---33----  11100

   4  Acergl    -----------------------------------223323  11101

  33  Rhamnu    ---------------------2-------------322222  11101

  51  Agr.sp    -32----------------2---------------2---2-  1111

  58  Poapra    334----------------------------2---55---2  1111

                00000000000000000000000000000000000111111

                00000000000000001111111111111111111001111

                00011111111111110000000001111111111

                   00001111111110000111110000000111

                       000000111    001110000011

                       000011            00111

                 ********** TWINSPAN completed **********

If you wish to use the resulting classification of samples in another program, it is very awkward to extract the classification from this output.  We therefore provide a separate file, SAMCLASS.TXT, containing the classification in machine-readable format.  When you run Twinspan this file appears in the default directory. The file is overwritten every time you run Twinspan, so if you wish to save it you should rename it.  A portion of an example is listed below:

    8  STAND8    000

   12  STAN12    000

    9  STAND9    0010

   14  STAN14    0010

   15  STAN15    0010

   11  STAN11    0011

   13  STAN13    0011

   16  STAN16    010

etc.

The file contains the sample unit number in the first column, then its name, then the 0/1 pattern containing the classification of that sample unit.  The interpretation is the same as in the two-way ordered table.

TWINSPAN Setup

TWINSPAN


Indicator Species Analysis

Indicator Species Analysis in PC-ORD applies Dufrene and Legendre’s (1997) method.  Help with Indicator Species Analysis is given in several parts:

•	Background

•	Dufrene and Legendre’s Method

•	Setup

•	Interpretation of Results

•	Example Output

Groups


Indicator Species Analysis -- Background

A very common goal in community analysis is to detect and describe the value of different species for indicating environmental conditions.  If environmental differences are conceptualized as groups of sample units, then Dufrene and Legendre's (1997) method of calculating species indicator values provides a simple, intuitive solution.  The method combines information on the concentration of species abundance in a particular group and the faithfulness of occurrence of a species in a particular group.   It produces indicator values for each species in each group.  These are tested for statistical significance using a Monte Carlo technique.

Groups are commonly defined by categorical environmental variables, levels of disturbance, experimental treatments, presence/absence of a target species, or habitat types.

Dufrene and Legendre (1997) also use this method to choose a stopping point in cluster analysis.  If groups are too finely divided then indicator values will be low.  On the other hand, if groups are too large, then the internal heterogeneity will reduce the indicator values.  Dufrene and Legendre found that indicator values peak at some intermediate level of clustering, and that this peak will vary by species.  Taken collectively, the method can be used to decide on an appropriate level of clustering for species data.

Dufrene and Legendre's Method

Indicator Species Analysis Setup

Indicator Species Analysis

Groups


Dufrene and Legendre's Method

The method assumes that two or more a priori groups of sample units exist, and that species abundances have been recorded in each of the sample units.

For each species, the following steps are made in the implementation in PC-ORD:

1. Calculate the proportional abundance of a particular species in a particular group relative to the abundance of that species in all groups.  Also express as a percentage and display the intermediate result.

Let

A = sample unit x species matrix

aijk = abundance of species j in sample unit (SU) i of group k

nk = number of sample units in group k

g = total number of groups

First calculate the mean abundance xkj of species j in group k:

Then calculate the relative abundance RAkj of species j in group k

2. Calculate the proportional frequency of the species in each group, i.e. the percentage of sample units in each group that contain that species.  Also express as a percentage and display the intermediate result.

First transform A to a matrix of presence-absence, B:

Then calculate relative frequency RFkj of species j in group k:

3. Combine the two proportions calculated in steps 1 and 2 by multiplying them.  Express the result as a percentage, yielding an indicator value IVkj for each species j in each group k.

4. The highest indicator value (IVmax) for a given species across groups is saved as a summary of the overall indicator value of that species.

5. Evaluate statistical significance of IVmax by a Monte Carlo method.  Randomly reassign SU's to groups a large number of times (default=1000).  Each time, calculate IVmax.  The probability of type I error is based on the proportion of times that the IVmax from the randomized data set equals or exceeds the IVmax from the actual data set.  The null hypothesis is that IVmax is no larger than would be expected by chance (i.e. that the species has no indicator value).

Indicator Species Analysis Setup

Indicator Species Analysis

Groups


Indicator Species Analysis -- Interpretation

The indicator values range from zero (no indication) to 100 (perfect indication).  Perfect indication means that presence of a species points to a particular group without error, at least with the data set in hand.

When more than two groups are defined, the IV for a particular species in a particular group is dependent on the set of sample units belonging to other groups.  In the example below, the indicator values in group 1 would be different if group 3 were omitted from the analysis.

Use the Monte Carlo test to evaluate the statistical significance of the maximum indicator value recorded for given species.  The probability of type I error is the proportion of times that the IVmax from the randomized data set equals or exceeds the IVmax from the actual data set.  The null hypothesis is that IVmax is no larger than would be expected by chance (i.e. that the species has no indicator value).

Indicator Species Analysis Example Output

Indicator Species Analysis

Groups


Indicator Species Analysis Setup

To run indicator species analysis you need to specify:

•	the grouping matrix (the matrix containing the variable which defines the groups),

•	the grouping variable, and

•	any groups that you wish to exclude from the analysis.  If you choose to exclude groups, you will be asked to enter group numbers to be excluded.

•	whether or not you want Monte Carlo tests of significance

If your grouping variable is not defined as a categorical variable, you are given a warning that digits to the right of the decimal point will be ignored in defining your groups.

Indicator Species Analysis Example Output

Indicator Species Analysis -- Interpretation

Indicator Species Analysis

Groups


Indicator Species Monte Carlo Test

You can test for statistical significance of IVmax with a randomization (Monte Carlo) test.  See elsewhere in this help system for a description of the method and interpretation.

You will need to specify the number of runs.  Because the calculations are straightforward, the default of 1000 runs will usually execute fairly quickly.


Indicator Species Analysis Example Output

Colorado 1992-1996 data

        Code numbers for excluded groups:       0       7      10      84

[Note that excluded groups are selected by the user.]

        Groups were defined by values of: Crew

        Input data has:    33 Stands   by    35 Species

Empty Species   [These are species that were all zeros in the data matrix]

   Imspla

   Phahir

   Phycae

   Pclchl

[In the following tables, "Sequence" is the order of occurrence of the groups in the data.  "Identifier" is the unique value coding that you assigned to a particular group.  "Avg" and "Max" refer to the indicator values for the given column (usually species) in your data.  "MaxGrp" is the group identifier for the group with the highest indicator value.]

RELATIVE ABUNDANCE in group, % of perfect indication

(average abundance of a given Species  in a given group of Stands

over the average abundance of that Species  in all Stands

expressed as a %)

                                Group

                    Sequence:   1   2   3

                  Identifier:   2  13   3

             Number of items:  15   5  13

    Column     Avg Max MaxGrp

   1 Bry        33  90     13  10  90   0

   2 Bryfus     33  41     13  25  41  35

   3 Cndcon     33  54      3   0  46  54

   4 Cetfen     33 100      2 100   0   0

   5 Cetpin     33 100      3   0   0 100

.

      etc.

      .

      .

  32 Usnhir     33  69      2  69   0  31

  33 Usnlap     33  43     13  30  43  27

  34 Xanfal     33  40      2  40  40  20

  35 Xanmon     33  36      3  30  35  36

      Averages  30  56         32  26  31

RELATIVE FREQUENCY in group, % of perfect indication

(% of Stands   in given group where given Species  is present)

                                Group

                    Sequence:   1   2   3

                  Identifier:   2  13   3

             Number of items:  15   5  13

    Column     Avg Max MaxGrp

   1 Bry         9  20     13   7  20   0

   2 Bryfus     32  40     13  27  40  31

   3 Cndcon     12  20     13   0  20  15

   4 Cetfen      2   7      2   7   0   0

   5 Cetpin      5  15      3   0   0  15

.

      etc.

      .

      .

  32 Usnhir     23  47      2  47   0  23

  33 Usnlap     51  60     13  47  60  46

  34 Xanfal     34  40      2  40  40  23

  35 Xanmon     75  80     13  67  80  77

      Averages  23  32         22  25  21

INDICATOR VALUES (% of perfect indication,

based on combining the above values for relative abundance

and relative frequency)

                                Group

                    Sequence:   1   2   3

                  Identifier:   2  13   3

             Number of items:  15   5  13

    Column     Avg Max MaxGrp

   1 Bry         6  18     13   1  18   0

   2 Bryfus     11  16     13   7  16  11

   3 Cndcon      6   9     13   0   9   8

   4 Cetfen      2   7      2   7   0   0

   5 Cetpin      5  15      3   0   0  15

.

      etc.

      .

      .

  32 Usnhir     13  32      2  32   0   7

  33 Usnlap     17  26     13  14  26  12

  34 Xanfal     12  16      2  16  16   5

  35 Xanmon     25  28     13  20  28  27

      Averages   9  17          9  11   8

MONTE CARLO test of significance of observed maximum

indicator value for Species

  1000 permutations.

Random number seed:        1878

---------------------------------------------

                         IV from

              Observed  randomized

             Indicator    groups

   Column    Value (IV) Mean  S.Dev    p *

------------- -------- ----- ------ ---------

   1 Bry         18.0   11.2   6.23  0.086

   2 Bryfus      16.4   22.1   8.41  0.792

   3 Cndcon       9.3   13.3   6.59  0.848

   4 Cetfen       6.7    9.2   4.77  1.000

   5 Cetpin      15.4   11.4   6.62  0.316

.

      etc.

      .

      .

  32 Usnhir      32.4   21.7   7.89  0.162

  33 Usnlap      25.9   28.0   7.93  0.466

  34 Xanfal      15.9   23.2   8.73  0.727

  35 Xanmon      27.7   33.5   5.76  0.963

-------------------------------------------

* proportion of randomized trials with indicator value

  equal to or exceeding the observed indicator value.

  p = (1 + number of runs >= observed)/(1 + number of randomized runs)

Indicator Species Analysis

Groups


Mantel Test

The Mantel test evaluates the null hypothesis of no relationship between two dissimilarity (distance) or similarity matrices.

Introduction to the Mantel test

Mantel Test Setup

Mantel Test Example Output


Introduction to the Mantel test

The Mantel test evaluates the null hypothesis of no relationship between two dissimilarity (distance) or similarity matrices.  The Mantel test is an alternative to regressing distance matrices that circumvents the problem of partial dependence in these matrices.  Note that Lefkovitch (1984) proposed a method for this same class of problems that is based on ranked distances.  Smouse et al. (1986) have extended the method to deal with more than two matrices at once (analogous to multiple regression), but this is not implemented in PC-ORD.

In many cases it may be desirable to evaluate the congruence between two distance (or similarity) matrices of the same dimensions.  For example, we may wish to evaluate the correspondence between two groups of organisms from the same set of sample units.  Or it may be desirable to compare community structure before and after a disturbance.  This can be done without any reduction of the underlying species space by using Mantel tests on two distance matrices.

In contrast to MRPP, which compares multiple groups of various sizes, each group consisting of separate sample units, Mantel tests are appropriate when more than one distance matrix is to be compared from the same set of sample units.

Because the cells of distances matrices are not independent of each other, we cannot accept the p-values from standard techniques that assume independence of the observations (for example, Pearson correlation). Nevertheless, the Pearson correlation can be used as a measure of the strength of relationship between two distance matrices.  In this context it is called the standardized Mantel statistic (r) (Sokal & Rohlf 1995) and r ranges from -1 to 1.  The Mantel method tests the significance of the correlation by a permutation method:  the rows and columns of one of the two matrices are permuted.  The rows and columns are permuted simultaneously, such that for each i, the ith row and ith column will correspond to the same case.  The diagonal elements always remain in the diagonal (but in different positions), but nondiagonal elements appear with equal probability in each of the n(n-1) nondiagonal positions.  After each permutation the Z statistic is calculated and the resulting values provide an empirical distribution that is used for the significance test.  The Z statistic is simply the sum of the products of the corresponding off-diagonal elements of the two matrices, also known as the Hadamard product of the matrices.

The method was originally developed to evaluate spatial and temporal clusterings of diseases, such as leukemia (Mantel 1967).  Sokal (1979) introduced the technique to systematics and quantitative biogeography.  Douglas and Endler (1982) give the mathematical details in a palatable form.

Two methods are available.  You can choose Mantel’s asymptotic approximation method or perform a randomization (Monte Carlo) test by conducting the actual permutations of the distance matrix as described above.

Most uses in ecology have been with non-experimental data, but there is no reason why it could not be used for experimental research.  See Sokal and Rohlf (1995) for example applications to other areas in biology.  For example, one matrix might be the “design matrix” containing zeros for elements that contain within-group distances and ones for between-group distances.  The other matrix might contain observed distances in an ecological or genetic space.  If the observed distances appear independent of the design matrix, then there is no evidence that the treatment resulted in a difference.

Because the method is relatively new to ecologists, a few examples are given below.  McCune and Allen (1985) used Mantel tests to compare site difference matrices with compositional dissimilarity matrices.  Although the site variables could be combined into a single composite site difference matrix, they did not do this, because site variables vary greatly in their importance to vegetation, and the inclusion of weak factors with strong on an equal basis can only weaken the site difference matrix.

Burgman (1987) compared four community matrices (total data, common plants only, rare plants only, and guilds) against each of three environmental matrices.

Tuomisto et al. (1995) compared similarity matrices based on different families of plants and trees in the Amazonian rainforest.

Mantel Test Setup

Mantel Test Example Output

Mantel Test


Mantel Test Setup

To run a Mantel Test you must have both a main and a second matrix.  Both matrices MUST have the same number of rows.  The second matrix can be either a matrix in the usual form or a distance matrix (explained below).

Distance Measure

You must specify a distance measure.  You can select the same distance measure to be applied to both matrices or a different measure for each matrix.

Type of matrices

Normally the two matrices will be typical data sets used in PC-ORD; for example, a community matrix of sample units x species and an environmental matrix of sample units x environmental variables.  In this case, the Mantel test evaluates whether the two distance matrices derived from the data matrices are independent or not.

If, however, one or both matrices is already a distance matrix, then select the appropriate checkboxes:

•	Main matrix is already a distance matrix.  (and/or)

•	Second matrix is already a distance matrix

Note that these must be square symmetrical matrices, with a value for each element of the matrices.  The main and second matrices must always have the same number of rows, regardless or whether or not they are distance matrices.

If you plan to enter a distance (or dissimilarity) matrix directly, you should set up a spreadsheet file just like a normal data matrix.  You need row and column names, but usually the row names will be the same as the column names for a distance matrix.

Output options

Either or both of the distance matrices can be listed with your output.

Write matrices in list format

If you wish to plot distances from one matrix against distances from the other matrix, select this option.  The list of distances will be written into the result file.  By importing this text file into a graphics or spreadsheet software, you can make a scatter plot.  This listing will also be useful if you wish to import the distance measures into other software if you need to calculate other statistics.

The distance matrices are listed along with the row and column indices, for example:

    Row Column           Matrix1           Matrix2

      2      1        0.47380951      494.62573242

      3      1        0.71428573      826.43261719

      3      2        1.00000000      338.17446899

      4      1        0.47619048      703.83947754

      4      2        0.55000001     1189.42480469

      4      3        0.85000002     1525.55688477

      5      1        0.52380955      737.47662354

      5      2        0.80000001      700.45306396

      5      3        0.55555558      914.60339355

etc.

Method for calculating p value

Two methods are available.  You can choose Mantel’s asymptotic approximation method (much faster with large data sets) or perform a randomization (Monte Carlo) test by conducting the actual permutations of the distance matrix as described above.

Mantel Test

Mantel Test Example Output


Mantel Test Example Output

DATA MATRICES

-----------------------------------------------------------

     Main matrix:

             19 STANDS   (rows)

             50 SPECIES  (columns)

                Distance matrix calculated from main matrix.

                Distance measure = SORENSEN

     Second matrix:

             19  STANDS  (rows)

              6  ENVIRON (columns)

                Distance matrix calculated from second matrix.

                Distance measure = SORENSEN

-----------------------------------------------------------

LISTING OF FIRST DISTANCE MATRIX

    1 STAND1

 0.000E+00 4.286E-01 2.308E-01 8.182E-01 6.296E-01 4.286E-01 5.333E-01

 8.667E-01 6.000E-01 6.774E-01 5.625E-01 7.931E-01 6.000E-01 6.129E-01

 6.774E-01 6.667E-01 7.500E-01 5.556E-01 7.222E-01

    2 STAND2

 4.286E-01 0.000E+00 5.652E-01 1.000E+00 9.167E-01 6.800E-01 7.037E-01

 8.519E-01 6.296E-01 8.571E-01 7.241E-01 8.462E-01 6.296E-01 7.143E-01

 7.857E-01 8.519E-01 8.095E-01 7.576E-01 7.576E-01

.

.

etc.

.

LISTING OF SECOND DISTANCE MATRIX

    1 STAND1

 0.000E+00 4.636E-03 3.652E-03 8.246E-02 8.383E-02 8.366E-02 7.859E-02

 2.197E-01 2.143E-01 2.235E-01 2.151E-01 2.777E-01 2.615E-01 2.631E-01

 2.653E-01 2.154E-01 2.095E-01 2.069E-01 2.027E-01

    2 STAND2

 4.636E-03 0.000E+00 9.846E-04 7.784E-02 8.775E-02 8.757E-02 8.254E-02

 2.153E-01 2.172E-01 2.191E-01 2.174E-01 2.734E-01 2.582E-01 2.638E-01

 2.632E-01 2.110E-01 2.124E-01 2.025E-01 2.056E-01

.

.

etc.

.

TEST STATISTIC: t-distribution with infinite degrees of freedom

           using asymptotic approximation of Mantel (1967).

           If t < 0, then negative association is indicated.

           If t > 0, then positive association is indicated.

STANDARDIZED MANTEL STATISTIC:     .481371 = r

          OBSERVED Z =   .2838E+02

          EXPECTED Z =   .2645E+02

       VARIANCE OF Z =   .1222E+00

 STANDARD ERROR OF Z =   .3496E+00

                   t =      5.4969

                   p =   .00000005

If you chose the randomization method, then the form of the results is somewhat different:

MANTEL TEST RESULTS: Randomization (Monte Carlo test) method

------------------------------------------------------------

       -.054504 = r = Standardized Mantel statistic

    .246553E+02 = Observed Z (sum of cross products)

    .256640E+02 = Average Z from randomized runs

    .264561E+01 = Variance of Z from randomized runs

    .198977E+02 = Minimum Z from randomized runs

    .299859E+02 = Maximum Z from randomized runs

           1000 = Number of randomized runs

            723 = Number of runs with Z > observed Z

              0 = Number of runs with Z = observed Z

            277 = Number of runs with Z < observed Z

        .278000 = p (type I error)

------------------------------------------------------------

p = proportion of randomized runs with Z more extreme

    or equal to the observed Z; for positive association,

    p = (1 + number of runs >= observed)/(1 + number of randomized runs)

Positive association between matrices is indicated by observed

     Z greater than average Z from randomized runs.

Note that in this case, the association is negative:  the observed Z is less than the average Z from randomized runs and the sign of the standardized Mantel statistic is negative.  Because it is negative, the p value is calculated using the number of randomized runs with Z less than or equal to the observed value.

Mantel Test Setup

Mantel Test


Window

PC-ORD typically uses four file viewing windows: main matrix, second matrix, graph file, and result file.  The main matrix is the active matrix for most operations, including data modifications, ordinations, cluster analysis, etc.  The second matrix contains a set of additional variables used by many procedures, including ordination overlays, CCA, MRPP, and the Mantel Test.  The graph file contains coordinates for graphing ordinations.  The result file receives text and numerical results from all PC-ORD procedures.

The following commands control the arrangement of windows:

•	Tile

•	Cascade

•	Arrange

•	Close All

The ordination procedures can produce two graph files, depending on which procedure and options you selected.  One of these contains coordinates for the rows of the main matrix (usually sample units), the other contains coordinates for the columns (usually variables).  You can use the graph file window  to display either one or both of these files using the following selections under the Window menu:

•	Graph Row File Only

•	Graph Column File Only

•	Both Graph Files

•	Split Horizontally or Split Vertically

The appearance of the contents of the windows can be altered by:

•	changing the font

•	changing the width of columns in the spreadsheets

Getting Started


Tile

Window | Tile requests that the file viewing windows be arranged on the desktop so that they do not overlap each other.  In contrast, if you choose to cascade your windows, they are arranged to form an overlapping series.

Window


Cascade

Window | Cascade arranges the file viewing windows in a stacked, overlapping series.  In contrast, if you choose to tile your windows, they are arranged so that they do not overlap each other.

Window


Arrange Windows

Window | Arrange puts the four windows back into their respective quadrants for easier viewing.  This is useful if you have moved the windows around and end up with scroll bars on the main window.  Or, you may have tiled less than four windows and want to put them back into the four quadrant positions.

Window


Close All Windows

Window | Close closes the file viewing windows.  If the contents of the window have not been saved, you will be prompted to decide their fate.

Window


Options

Options Menu

The Options menu contains settings for reminder messages:

Getting Started Message

•	Ordination Graph Message

•	NMS Scores Message

•	Dendrogram Message

•	Species-area Curves Message

•	NMS Scree Plot Message

The numerous other options available under PC-ORD are linked with other components of the menu system.

Getting Started


Getting Started Message

Options | Started Message

This message comes up when you first start PC-ORD.  Click the "Don't show this any more" check box or press the space bar so it has an "X" in the box to disable this message.  Click or press the space bar again to remove the "X".  Once you turn the message off you can turn the message back on by selecting Options | Getting Started.

Options


Ordination Graph Message

Options | Ordination Message

The Ordination Graph Message comes up after you run an ordination to remind you to run Graph | Ordination do display your ordination.  Click the "Don't show this any more" check box or press the space bar so it has an "X" to disable this message.  Once you turn the message off you can turn the message back on by selecting Options | Ordination Message.

Ordination

Options


NMS Scores Message

This message comes up when you select NMS Scores.  The purpose of the message is to let you know that you need to have ready a very specific set of files to run this procedure.  Click the "Don't show this any more" check box or press the space bar so it has an "X" in the box to disable this message.  Click or press the space bar again to remove the "X".  Once you turn the message off you can turn the message back on by selecting Options | NMS Scores Message

Options


Dendrogram Message

The Dendrogram Message comes up after you run a cluster analysis to remind you to run Graph | Cluster Dendrogram to display your dendrogram.  Click the "Don't show this any more" check box or press the space bar so it has an "X" to disable this message.  Once you turn the message off you can turn the message back on by selecting Options | Dendrogram Message.

Options


Species-area Curves Message

The Species-area Curves Message comes up after you run Summary | Species-area Curves to remind you to run Graph | Species-area Curves to display your species-area curves in graphics mode.  Click the "Don't show this any more" check box or press the space bar so it has an "X" to disable this message.  Once you turn the message off you can turn the message back on by selecting Options | Species-area Curves Message.

Options


NMS Scree Plot Message

The NMS Scree Plot Message comes up after you run NMS ordination to remind you to run Graph | NMS Scree Plot.  Click the "Don't show this any more" check box or press the space bar so it has an "X" to disable this message.  Once you turn the message off you can turn the message back on by selecting Options | NMS Scree Plot Message.

Options


Examples

Typical Flow of Analysis

Example of an Analysis


Typical Flow of Analysis

1.  Data entry and error checking

2.  Create .WK1 spreadsheet file

3.  Modify data (transformations, etc.)

4.  Run selected analysis

5.  Inspect and save or discard results

6.  Ordination (or other) graphing

7.  Inspect and save or discard results

8.  Print results

Getting Started

Example of an Analysis

Data Preparation


Example of an Analysis

To run a principal components analysis (PCA) of the demonstration data set, perform the following steps:

1. Open the file "food.wk1" as the main matrix.  This sample data set is provided to you with PC-ORD.  The data are food consumption ratings by 59 people for 16 different foods.  Select

Open | Main matrix

Then select "food.wk1".

2. Before you run PCA, the data should be standardized so that no person receives any more weight in the analysis than any other.  This is important because different people will apply the scale differently -- some people may use larger numbers for the same frequency of food use.  Do this with the following steps:

•	Modify | Relativization | General relativization

•	Enter "2" for the relativization parameter.  This standardizes by "the norm" (an appropriate relativization because we'll be using Euclidean distance implicitly in the PCA).

•	Select relativization by rows (people).

•	Accept the new matrix.  Note that the integers are now shown as decimals.  The sums of squares of the elements in each row now equal 1.0.

3. Now request a principal components analysis (Ordination | PCA), then select options.  For this data set the following options are suggested:

•	Use correlations in the cross-products matrix.

•	Request that the cross-products matrix be listed (this will contain a matrix of correlation coefficients among foods).

•	Descriptive title for your PCA could be "Standardized PCA, people in food space"

4. Inspect the output and save it in a file (File | Save as | Result).  Note that the ordination is done in two stages; the first (which you've already done) is the numerical calculations.  The second (which you're about to do) is producing graphs of the ordination.  You may have noticed the file graphrow.gph was also produced.  This is a text file with the coordinates of the items for graphing (in this case the coordinates of each person in a reduced food space).

5. Select Graph | Graph Ordination to graph your results.  The program reads data from your matrices and another file (graphrow.gph) which contains the coordinates of the food items on the first three principal components (axes).

6. You now are given the Graph menu.  Select Graph | Simple Scatterplot to see the basic ordination displayed.  You can experiment with other options in the graphing module, for example:

•	Request correlations between your columns of the main matrix (foods) and the ordination axes (Statistics | Correlations with Main Matrix).  The correlations will be written to the results file and to the screen.

•	Request overlays of your columns of the main matrix (foods) onto the ordination (Graph | Overlay from Main Matrix).  Larger symbols indicate larger values.  You can later find out which point represents which person by requesting labels (Options | Preferences) on your graph.  Which foods are strongly patterned on the graph (i.e. a gradient in symbol size)?  Foods that control a given axis will be strongly patterned along that dimension.

•	You cannot request correlations or overlays with the second matrix, because this data set does not have a second matrix.

7. Interpret the output.  Use overlays, correlations of foods with axes, and information derived from eigenvalues. For example...

•	What proportion of the variance was "explained" in the first several dimensions?

•	How many dimensions explain significantly more variance than expected by chance?  Evaluate this by comparing the "broken-stick" eigenvalues to the actual eigenvalues.  Eigenvalues larger than the broken-stick values suggest that the corresponding dimension (= "axis" or "component") explains more variance than expected by chance alone.

•	Describe the dominant "gradients" in eating habits.  Typically users describe a something like a "junk food to health food" gradient in eating habits.  You may want to experiment more with overlays and correlations.  You can rerun the analysis starting with step 3 above, or you can just redo the graphing by selecting Graph.

Getting Started

Examples

Typical Flow of Analysis


Acknowledgments

PC-ORD evolved through the efforts of many programmers and scientists.  Contributing programmers are listed under the description of each program.  In addition to the authors these contributors include H. G. Gauch, M. O. Hill, P. W. Mielke, W. M. Post, and J. Poston.  Additional contributions were made by T. F. H. Allen, S. M. Bartell, K. J. Berry, R. Howe, and J. Sheperd.  Because their programming efforts have been modified to fit PC-ORD, they should share any acclaim but are exempt from any gripes. E. W. Beals contributed ideas and rationale for many of the options.  His students in Community Analysis at the University of Wisconsin-Madison and McCune's students at Oregon State University, along with many PC-ORD users, suggested improvements for the programs.  In particular, Mike Kearsley, JeriLynn Peck, Eric B. Peterson, and Hong Qian contributed numerous suggestions and helpful testing for version 4.  Additional testing of earlier versions was done by J. Poston at Holcomb Research Institute, and students and faculty at Oregon State University.  C. J. F. Ter Braak patiently answered questions on details of CCA.  Marc Dufrêne helped cross check the results of the Indicator Species Analysis and he and Pierre Legendre kindly gave permission to begin working with the procedure before it was formally published.  JeriLynn Peck, Jim Rogers, and Susan Will-Wolf contributed editorial suggestions on earlier versions of the manual.


References

Armitage, P. D., D. Moss, J. F. Wright, and M. T. Furse.  1983

Austin, M. P. and I. Noy-Meir.  1971

Beals, E. W.  1960

Beals, E. W.  1965

Beals, E. W.  1973

Beals, E. W.  1984

Belbin, L. and C. McDonald.  1993

Benzécri, J.-P.  1973

Burgman, M. A.  1987

Berry, K. J., K. L. Kvamme, and P. W. Mielke, Jr.  1983

Biondini, M. E., C. D. Bonham, and E. F. Redente.  1985

Bray, J. R. and J. T. Curtis.  1957

Burnham, K. P. and W. S. Overton.  1979

Chardy, P, M. Glemarec, and A. Laurec.  1976

Clarke, K. R.  1993

Clarke, K. R., and R. H. Green.  1988

Curtis, J. T. and R. P. McIntosh.  1951

Douglas, M. E. and J. A. Endler.  1982

Dufrene, M. and P. Legendre.  1997

Faith, D. P., P. R. Minchin, and L. Belbin.  1987

Federal Interagency Committee for Wetland Delineation.  1989

Gabriel, K. R.  1971

Gauch, H. G., Jr. 1977

Gauch, H. G., Jr. 1982

Gauch, H. G., Jr. and R. H. Whittaker.  1981

Goodall, D. W. 1954

Greig-Smith, P.  1983

Gower, J. C. and D. J. Hand. 1996

Heltshe, J. F. and N. E. Forrester.  1983

Hill, M. O.  1973

Hill, M. O.  1979

Hill, M. O.  1979b

Hill, M. O., and H. G. Gauch.  1980

Jackson, D. A.  1993

Johnson, R. K.  1998

Jongman, R. H. G., C. J. F. ter Braak, and O. F. R. van Tongeren.  1987

Jongman, R. H. G., C. J. F. ter Braak, and O. F. R. van Tongeren.  1995

Karr, J. R.  1991

Kerans, B. L. and J. R. Karr.  1994

Kruskal, J. B.  1964a

Kruskal, J. B.  1964b

Kruskal, J. B. and M. Wish.  1978

Lance, G. N. and W. T. Williams.  1967

LeBlanc, F. and J. de Sloover.  1970

Lefkovitch, L. P.  1984

Ludwig, J. A., and J. F. Reynolds.  1988

Mather, P. M.  1976

Mather, P. M.  1976

McCune, B.  1994

McCune, B.  1997

McCune, B. and T. F. H. Allen.  1985

McCune, B. and E. W. Beals.  1993

McCune, B., J. Dey, J. Peck, D. Cassell, K. Heiman, S. Will-Wolf, and P. Neitlich.  1997a

McCune, B, J. Dey, J. Peck, K. Heiman, S. Will-Wolf. 1997b

Mielke, P. W., Jr.  1984

Mielke, P. W., Jr.  1991

Mielke, P. W., Jr., and K. J. Berry.  1982

Mielke, P. W., K. J. Berry, and J. G. Medina.  1982

Mielke, P. W. and H. K. Iyer.  1982

Minchin, P. R.  1987a

Minchin, P. R.  1987b.

Okland, R. H.  1996

Okland, R. H.  1999

Oksanen, J. and P. R. Minchin.  1997

Palmer, M. W.  1990

Palmer, M. W.  1991

Palmer, M. W.  1993

Palmer, M. W.  1995

Peters, R. H.  1991

Pielou, E.C.  1969

Prentice, I. C.  1977

Prentice, I. C.  1980

Roberts, D. W.  1986

Shannon, C. E. and W. Weaver.  1949

Sneath, P. H. A. and R. R. Sokal.  1973

Smith, B. T. et al. 1976

Smouse, P. E., J. C. Long, and R. R. Sokal.  1986

Sokal, R. R.  1979

Sokal, R. R. and F. J. Rohlf.  1981

Sokal, R. R. and F. J. Rohlf.  1995

Tabachnik, B.G. and L.S. Fidell.  1989

Tabachnik, B.G. and L.S. Fidell.  1996

Ter Braak, C. J. F.  1986

Ter Braak, C. J. F.  1988

Ter Braak, C. J. F.  1990

Ter Braak, C. J. F.  1994

Tuomisto, H., K. Ruokolainen, R. Kalliola, A. Linna, W. Danjoy, and Z. Rodriguez.  1995

van Groenewoud, H.  1992

Whittaker, R. H.  1967

Wiederholm, T.  1980

Wishart, D.  1969

Wishart, D.  1978

Zimmerman, G. M., H. Goetz, and P. W. Mielke, Jr. 1985


Armitage, P. D., D. Moss, J. F. Wright, and M. T. Furse.  1983

The performance of a new biological water quality score system based on macroinvertebrates over a wide range of unpolluted running-water sites.  Water Research 17:333-347.


Austin, M. P. and I. Noy-Meir.  1971

The problem of non-linearity in ordination: experiments with two-gradient models.  Journal of Ecology 59:763-773.


Beals, E. W.  1960

Forest bird communities in the Apostle Islands of Wisconsin.  Wilson Bulletin 72: 156-181.


Beals, E. W.  1965

Species patterns in a Lebanese Poterietum.  Vegetatio 13: 68-87.


Beals, E. W.  1973

Ordination: mathematical elegance and ecological naivete. Journal of Ecology 61:23-35.


Beals, E. W.  1984

Bray-Curtis ordination: an effective strategy for analysis of multivariate ecological data.  Advances in Ecological Research 14: 1-55


Belbin, L. and C. McDonald.  1993

Comparing three classification strategies for use in ecology.  Journal of Vegetation Science 4:341-348.


Benzécri, J.-P.  1973

 L'Analyse des Données.  Vol. 1: La Taxinomie.  Vol. 2: L'Analyse des Correspondance.  Dunod, Paris.


Burgman, M. A.  1987

An analysis of the distribution of plants on granite outcrops in southern western Australia using Mantel tests.  Vegetatio 71:79-86.


Berry, K. J., K. L. Kvamme, and P. W. Mielke, Jr.  1983

Improvements in the permutation test for the spatial analysis of the distribution of artifacts into classes.  American Antiquity 48: 547-553.


Biondini, M. E., C. D. Bonham, and E. F. Redente.  1985

Secondary successional patterns in a sagebrush (Artemisia tridentata) community as they relate to soil disturbance and soil biological activity.  Vegetatio  60: 25-36.


Bray, J. R. and J. T. Curtis.  1957

An ordination of upland forest communities of southern Wisconsin.  Ecological Monographs 27: 325-349.


Burnham, K. P. and W. S. Overton.  1979

Robust estimation of population size when capture probabilities vary among animals.  Ecology 60:927-936.


Chardy, P, M. Glemarec, and A. Laurec.  1976

Application of inertia methods to benthic marine ecology: practical implications of the basic options.  Estuarine Coastal Mar. Science 4:179-205.


Clarke, K. R.  1993

Non-parametric multivariate analyses of changes in community structure.  Australian Journal of Ecology 18:117-143.


Clarke, K. R., and R. H. Green.  1988

Statistical design and analysis for a 'biological effects' study.  Marine Ecology Progress Series 46:213-226.


Curtis, J. T. and R. P. McIntosh.  1951

An upland forest continuum in the prairie-forest border region of Wisconsin.  Ecology 32:476-496.


Douglas, M. E. and J. A. Endler.  1982

Quantitative matrix comparisons in ecological and evolutionary investigations.  Journal of Theoretical Biology 99:777-795.


Dufrene, M. and P. Legendre.  1997

Species assemblages and indicator species: the need for a flexible asymmetrical approach.  Ecological Monographs 67:345-366.


Faith, D. P., P. R. Minchin, and L. Belbin.  1987

Compositional dissimilarity as a robust measure of ecological distance.  Vegetatio 69:57-68.


Federal Interagency Committee for Wetland Delineation.  1989

Federal manual for identifying and delineating jurisdictional wetlands.  U. S. Army Corps of Engineers, U. S. Environmental Protection Agency, U. S. Fish and Wildlife Service, and U. S. D. A. Soil Conservation Service, Washington, D.C.  Cooperative technical publication.  76 pp. plus appendices.


Gabriel, K. R.  1971

The biplot graphical display of matrices with application to principal component analysis.  Biometrika 58:453-467.


Gauch, H. G., Jr. 1977

ORDIFLEX, Release B.  Ithaca: Ecology and Systematics, Cornell University.


Gauch, H. G., Jr. 1982

Multivariate Analysis in Community Ecology.  New York:  Cambridge University Press.


Gauch, H. G., Jr. and R. H. Whittaker.  1981

Hierarchical classification of community data.  Journal of Ecology 69: 135-152.


Goodall, D. W. 1954

Objective methods for the classification of vegetation. III.  An essay in the use of factor analysis.  Australian Journal of Botany 2: 304-324.


Gower, J. C. and D. J. Hand. 1996

Biplots.  Chapman and Hall, London. 277 pages.


Greig-Smith, P.  1983

Quantitative Plant Ecology. Third edition.  Oxford: Blackwell Scientific Publications.  359 p.


Heltshe, J. F. and N. E. Forrester.  1983

Estimating species richness using the jackknife procedure.  Biometrics 39:1-12.


Hill, M. O.  1973

Reciprocal averaging: an eigenvector method of ordination.  Journal of Ecology 61: 237-249.


Hill, M. O.  1979

DECORANA--a FORTRAN program for detrended correspondence analysis and reciprocal averaging.  Ithaca, NY.:  Ecology and Systematics, Cornell University.


Hill, M. O.  1979b

TWINSPAN--A FORTRAN program for arranging multivariate data in an ordered two-way table by classification of the individuals and attributes.  Ithaca, NY:  Ecology and Systematics, Cornell University.


Hill, M. O., and H. G. Gauch.  1980

Detrended correspondence analysis: an improved ordination technique.  Vegetatio 42: 47-58.


Jackson, D. A.  1993

Stopping rules in principal components analysis: a comparison of heuristical and statistical approaches.  Ecology 74:2204-2214.


Jongman, R. H. G., C. J. F. ter Braak, and O. F. R. van Tongeren.  1987

Data analysis in community and landscape ecology.  Pudoc, Wageningen.


Jongman, R. H. G., C. J. F. ter Braak, and O. F. R. van Tongeren.  1995

Data analysis in community and landscape ecology.  Cambridge University Press, Cambridge.


Johnson, R. K.  1998

Spatiotemporal variability of temperate lake macroinvertebrate communities: detection of impact.  Ecological Applications 8:61-70.


Karr, J. R.  1991

Biological integrity: a long-neglected aspect of water resource management.  Ecological Applications 1:66-84.


Kerans, B. L. and J. R. Karr.  1994

A benthic index of biotic integrity (B-IBI) for rivers of the Tennessee Valley.  Ecological Applications 4:768-785.


Kruskal, J. B.  1964a

Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.  Psychometrika  29:1-27.


Kruskal, J. B.  1964b

Nonmetric multidimensional scaling:  a numerical method.  Psychometrika  29:115-129.


Kruskal, J. B. and M. Wish.  1978

Multidimensional Scaling.  Sage Publications, Beverly Hills, California.


Mather, P. M.  1976

Computational methods of multivariate analysis in physical geography.  J. Wiley and Sons, London.  532 pp.


Lance, G. N. and W. T. Williams.  1967

A general theory of classification sorting strategies.  I. hierarchical systems.  Computer Journal 9: 373-380.


LeBlanc, F. and J. de Sloover.  1970

Relation between industrialization and the distribution and growth of epiphytic lichens and mosses in Montreal.  Canadian Journal of Botany 48:1485-1496.


Lefkovitch, L. P.  1984

A nonparametric method for comparing dissimilarity matrices, a general measure of biogeographical distance, and their application.  American Naturalist 123:484-499.


Ludwig, J. A., and J. F. Reynolds.  1988

Statistical Ecology.  New York: John Wiley and Sons.


Mantel, N.  1967

The detection of disease clustering and generalized regression approach.  Cancer Research 27:209-220.


McCune, B.  1994

Improving community analysis with the Beals smoothing function.  Ecoscience 1:82-86.


McCune, B.  1997

Influence of noisy environmental data on canonical correspondence analysis.  Ecology 78:2617-2623.


McCune, B. and T. F. H. Allen.  1985

Will similar forests develop on similar sites?  Canadian Journal of Botany 63:367-376.


McCune, B. and E. W. Beals.  1993

History of the development of Bray-Curtis ordination.  Pages 67-79.  In J. S. Fralish et al., editors, Fifty Years of Wisconsin Plant Ecology.  Wisconsin Academy of Sciences, Arts and Letters, Madison.


McCune, B., J. Dey, J. Peck, D. Cassell, K. Heiman, S. Will-Wolf, and P. Neitlich.  1997a

Repeatability of community data: species richness versus gradient scores in large-scale lichen studies.  Bryologist 100: 40-46.


McCune, B, J. Dey, J. Peck, K. Heiman, S. Will-Wolf. 1997b

Regional gradients in lichen communities of the southeast United States. Bryologist 100: 145-158.


Mielke, P. W., Jr.  1984

Meteorological applications of permutation techniques based on distance functions.  Pages 813-830.  In P. R. Krishnaiah and P. K. Sen, eds., Handbook of Statistics, Vol. 4.  Elsevier Science Publishers.


Mielke, P. W., Jr.  1991

The application of multivariate permutation methods based on distance functions in the earth sciences.  Earth-Science Reviews 31:55-71.


Mielke, P. W., Jr., and K. J. Berry.  1982

An extended class of permutation techniques for matched pairs.  Commun. Statist.-Theor. Meth. 11:1197-1207.


Mielke, P. W., K. J. Berry, and J. G. Medina.  1982

Climax I and II: distortion resistant residual analyses.  Journal of Applied Meteorology 21:788-792.


Mielke, P. W. and H. K. Iyer.  1982

Permutation techniques for analyzing multiresponse data from randomized block experiments.  Commun. Statist. A  11:1427-1437.


Minchin, P. R.  1987a

An evaluation of the relative robustness of techniques for ecological ordination.  Vegetatio 69:89-107.


Minchin, P. R.  1987b.

Simulation of multidimensional community patterns: towards a comprehensive model.  Vegetatio 71:145-156.


Okland, R. H.  1996
Økland, R. H.  1996

Are ordination and constrained ordination alternative or complementary strategies in general ecological studies?  Journal of Vegetation Science 7:289-292.


Okland, R. H.  1999
Økland, R. H.  1999

On the variation explained by ordination and constrained ordination axes.  Journal of Vegetation Science 10:131-136.


Oksanen, J. and P. R. Minchin.  1997

Instability of ordination results under changes in input data order: explanations and remedies.  Journal of Vegetation Science 8:447-454.


Palmer, M. W.  1990

The estimation of species richness by extrapolation.  Ecology 71:1195-1198.


Palmer, M. W.  1991

Estimating species richness: the second-order jackknife reconsidered.  Ecology 72:1512-1513.


Palmer, M. W.  1993

Putting things in even better order: the advantages of canonical correspondence analysis.  Ecology 74:2215-2230.


Palmer, M. W.  1995

How should one count species?  Natural Areas Journal 15:124-135.


Peters, R. H.  1991

A Critique for Ecology.  Cambridge University Press, Cambridge.  366 pp.


Pielou, E.C.  1969

An Introduction to Mathematical Ecology.  New York:  John Wiley and Sons.


Prentice, I. C.  1977

Non-metric ordination methods in ecology.  Journal of Ecology65:85-94.


Prentice, I. C.  1980

Vegetation analysis and order invariant gradient models. Vegetatio 42:27-34.


Roberts, D. W.  1986

Ordination on the basis of fuzzy set theory.  Vegetatio 66:123-31.


Shannon, C. E. and W. Weaver.  1949

The Mathematical Theory of Communication.  Urbana:  University of Illinois Press.


Smith, B. T. et al. 1976

Matrix eigensystem routines, EISPACK guide.  2nd ed.  Berlin:  Springer-Verlag.


Smouse, P. E., J. C. Long, and R. R. Sokal.  1986

Multiple regression and correlation extensions of the Mantel test of matrix correspondence.  Systematic Zoology 35:627-632.


Sneath, P. H. A. and R. R. Sokal.  1973

Numerical Taxonomy.  San Francisco:  W. H. Freeman.


Sokal, R. R.  1979

Testing statistical significance of geographic variation patterns.  Systematic Zoology 28:627-632.


Sokal, R. R. and F. J. Rohlf.  1981

Biometrics.  Second edition.  San Francisco:  Freeman.


Sokal, R. R. and F. J. Rohlf.  1995

Biometry.  Third Edition.  W. H. Freeman, San Francisco.


Tabachnik, B.G. and L.S. Fidell.  1989

Using Multivariate Statistics.  2nd ed. New York:  Harper and Row.


Tabachnik, B.G. and L.S. Fidell.  1996

Using Multivariate Statistics.  Third ed. New York:  Harper and Row.


Ter Braak, C. J. F.  1986

Canonical correspondence analysis: a new eigenvector technique for multivariate direct gradient analysis.  Ecology 67:1167-1179.


Ter Braak, C. J. F.  1988

CANOCO.  Agricultural Mathematics Group.  Technical Report LWA-88-02.  Wageningen, Netherlands.  95 pages.


Ter Braak, C. J. F.  1990

Update Notes: CANOCO Version 3.10.  Agricultural Mathematics Group.  Wageningen, Netherlands.  35 pages.


Ter Braak, C. J. F.  1994

Canonical community ordination.  Part I: Basic theory and linear methods.  Ecoscience 1:127-140.


Tuomisto, H., K. Ruokolainen, R. Kalliola, A. Linna, W. Danjoy, and Z. Rodriguez.  1995

Dissecting Amazonian biodiversity.  Science 269:63-66.


van Groenewoud, H.  1992

The robustness of correspondence, detrended correspondence and TWINSPAN analysis.  Journal of Vegetation Science 3:239-246


Wiederholm, T.  1980

Use of benthos in lake monitoring.  Journal of the Water Pollution Control Federation 52:537-547.


Whittaker, R. H.  1967

Gradient analysis of vegetation.  Biological Reviews 42:207-264.


Wishart, D.  1969

An algorithm for hierarchical classifications.  Biometrics 25:165-170.


Wishart, D.  1978

CLUSTAN Users Manual.  3rd edition.  Edinburgh:  Edinburgh University.


Zimmerman, G. M., H. Goetz, and P. W. Mielke, Jr. 1985

Use of an improved statistical method for group comparisons to study effects of prairie fire.  Ecology 66: 606-611.


Distance Measures

Most methods of multivariate analysis rely on some measure of distance between objects in an n-dimensional hyperspace.  For many data sets the choice of distance measure is quite important.  The distance measures available in PC-ORD are described below, along with some hints for  deciding when to use them.  More information can be found in Beals (1984), Greig-Smith (1983), and Ludwig and Reynolds (1988).

Sorensen Distance (= Bray & Curtis; 1-2w/a+b; Czekanowski; and City-block distance)

Relative Sorensen Distance

Jaccard Distance

Euclidean (Pythagorean) Distance

Relativized Euclidean Distance

Correlation

Chi-squared Distance

Squared Euclidean Distance

Domains and ranges of these distance measures are given in the table below.

Table:  Reasonable and acceptable domains of input data, x, and ranges of distance measures, f(x).

Name (synonyms)	Domain of x	Range of f(x)	Comments
Sorensen  (Bray & Curtis; Czekanowski)	x ≥ 0	0 ≤ x ≤ 1 (or 0 ≤ x ≤ 100%)	proportion coefficient in city-block space
Relative Sorensen (relativized Manhattan)	x ≥ 0	0 ≤ x ≤ 1 (or 0 ≤ x ≤ 100%)	proportion coefficient in city-block space; same as Sorensen but data points relativized by sample unit totals
Jaccard	x ≥ 0	0 ≤ x ≤ 1 (or 0 ≤ x ≤ 100%)	proportion coefficient in city-block space
Euclidean  (Pythagorean)	all	non-negative
Relative Euclidean  (Chord distance;  standardized Euclidean)	all	0 ≤ x ≤ √2 for quarter hypersphere; 0 ≤ x ≤ 2 for full hypersphere	Euclidean distance between points on unit hypersphere
Correlation distance	all	0 ≤ x ≤ 1	converted from correlation to distance; proportional to arc distance between points on unit hypersphere; cosine of angle from centroid to points.
Chi-squared	x ≥ 0	x ≥ 0	Euclidean but doubly weighted by variable and sample unit totals
Squared Euclidean	all	x ≥ 0


Euclidean Distance

Euclidean distance (ED) is measured as the p-dimensional version of the Pythagorean theorem.  The distance between objects i and h is

Euclidean distance is conceptually straightforward, but usually compares poorly with other distance measures when applied to ecological community data.  Euclidean distance tends to emphasize outliers as compared to Sorensen distance.  Euclidean distance tends to lose  sensitivity more rapidly than many other distance measures as the  heterogeneity of a data set increases.

Distance Measures


Sorensen Distance
Sorensen (Bray & Curtis) Distance

Sørensen distance, measured as percent dissimilarity (PD) is a  proportion coefficient measured in city-block space.  The distance (or dissimilarity) between items i and h is:

where there are p attributes of the objects.  This formula is sometimes written in shorthand as "1-2W/(A+B)", where W is the sum of shared abundances and A and B are  the sums of abundances in individual sample units.

Another way of writing this, where MIN is the smaller of two values is:

One can convert this dissimilarity (or any of the following proportion coefficients) to a percentage dissimilarity (PD):

The Sørensen coefficient (also known as the Czekanowski or Bray-Curtis coefficient) was originally applied to presence-absence data, but it works equally well with quantitative data.  It is most useful for ecological community data.  Justification for its use is primarily  empirical rather than theoretical (although it can be justified  theoretically as the intersection of two fuzzy sets (Roberts 1986)).  As compared to Euclidean distance it retains  sensitivity in more heterogeneous data sets and gives less weight to  outliers.

Distance Measures


Relative Sorensen Distance

Relative Sørensen distance ("relativized Manhattan" in Faith et al. 1987) is the same as Sørensen distance except that it builds in a standardization by sample unit totals, each sample unit contributing equally to the distance measure.  The Relative Sørensen distance deserves much more use than it has received, based on the positive results of testing by Faith et al. (1987).  It is calculated as:

Note that with standardization or relativization of the data before application of the distance measure, many of the city-block distance measures become mathematically equivalent to each other.  After relativization by sample unit totals, Sørensen distance = Relative Sørensen distance = Kulczynski coefficient = Manhattan distance.

Distance Measures


Jaccard Distance

Jaccard distance, (JD) is also a proportion coefficient measured in  city-block space.  The distance between objects i and h for p attributes  is measured as follows.

This formula is sometimes written in shorthand as "1-2W/(A+B-W)", where W  is the sum of shared abundances and A and B are the sums of abundances in  individual sample units.

In general the Jaccard distance has not performed as well as the Sorensen distance (see Beals (1984)), but you may wish to use it for experimental purposes.

Distance Measures


Correlation as a Distance Measure

The standard Pearson's product-moment correlation coefficient (r) can  be used as a distance measure by reversing its directionality and rescaling it from zero to one (maximum distance, no correlation):

Correlation distance = (1 - r)/2

where r is the correlation coefficient.  This distance is generally very poor with ecological community data, but with multivariate normal data and linear relationships it may be ideal.  With community data it has maximum sensitivity to the zero-truncation problem (Beals 1984) and performs adequately only with data sets with very low beta diversity.

Distance Measures


Relativized Euclidean Distance

Relativized Euclidean distance (RED) is similar conceptually to Euclidean distance, except that the data are normalized such that all of the data points fall on the surface of a unit quarter hypersphere.  Built into this distance measure is the adjustment so that the sums of squares for each row equals one.  For sample unit x species data, this effectively removes differences in overall abundances among sample units, instead focusing the analysis on the differences in relative abundances among species.  The formula in Ludwig and Reynolds (1988) is incorrect.

Relative Euclidean distance eliminates differences in total abundance (actually totals of squared abundances) among sample units.  The range of RED is 0 to the square root of 2 given all non-negative data.  Visualize the SU's as being placed on the surface of a quarter hypersphere with a radius of one.  RED is the chord distance between two points on this surface.

RED builds in a standardization.  It puts differently scaled variables on the same footing, eliminating any signal other than relative abundance.  Note that the correlation coefficient also accomplishes this standardization, but arccos(r) gives the arc distance on the quarter hypersphere, not the chord distance.  Also, with the correlation coefficient the surface is a full hypersphere, not a quarter hypersphere, and the center of the hypersphere is the centroid of the cloud of points, not the origin.

Distance Measures


Chi-squared Distance

Chi-squared distance is an implicit choice inherent in several ordination methods:  Reciprocal Averaging (RA) and its offspring,  Detrended Correspondence Analysis (DCA, DECORANA) and Canonical Correspondence Analysis (CCA), are all conceptually based on chi-squared distance (Chardy et al. 1976, Minchin 1987a), even though a distance matrix is not calculated.

Let	ah+ = total for sample unit h

ai+ = total for sample unit i

a+j = total for species j

then the chi-squared distance (Chardy et al. 1976) is

Note that this distance measure is similar to Euclidean distance, but it is weighted by the inverse of the species totals.  If the data are pre-relativized by sample unit totals, i.e. bij = aij/ai+, then the equation simplifies to:

The numerator is the squared difference in relative abundance.  It is expressed as a proportion of the species total (the denominator) and summed over all species.

Although Faith et al. (1987) found that this distance measure performed poorly, it is important to be aware of it, since it is the implicit distance metric in some of the more popular ordination techniques.

Minchin (1987a) offered the following critique of this distance measure:  "The appropriateness of Chi-squared distance as a measure of compositional dissimilarity in ecology may be questioned  (Faith et al. 1987).  The measure accords high weight to species whose total abundance in the data is low.  It thus tends to exaggerate the distinctiveness of samples containing several rare species.  Unlike the Bray-Curtis coefficient and related measures, Chi-squared distance does not reach a constant, maximal value for sample pairs with no species in common, but  fluctuates according to variations in the representation of species with high or low total abundances.  These properties of Chi-squared distance may account for some of the distortions observed in DCA ordinations."

Distance Measures


Squared Euclidean Distance

Squared Euclidean distance is simply the square of the Euclidean (Pythagorean) distance.  This distance measure is not widely used and is included primarily for experimental purposes.

Distance Measures


File Formats

DATA FILES

Data Preparation

Spreadsheet Data Format

Example

Example: Environmental Matrix

Compact Data Format

Species File

Example

Compact Data File

Example: Quantitative List of Species

Example: Presence List of Species

Example: List of Individual Trees

Old PCORD Standard Format

Example

DECORANA/TWINSPAN Condensed Format

List or Database Format

Comma Separated Values Format

MISSING VALUES

OUTPUT FILES

Results File Format

Graph File Format

Graphs as Enhanced Windows Metafiles

Coordinates File Format


Data Preparation

Use the text editor, word processor, or spreadsheet of your choice to enter and edit your data.  Details of the format for entering your data are provided below.

PC-ORD recognizes five data formats (listed below).  The preferred format is the spreadsheet format.  Other formats must be imported to the spreadsheet format (File | Import) before running your data analysis.

PC-ORD Data Formats

•	Spreadsheet Format

•	Compact Format

•	Old PCORD Standard Format

•	DECORANA/TWINSPAN Condensed Format

•	List (Database) Format

You can create your data set with almost any editor, word processor, or spreadsheet.  These three options are described below.

1. Use a spreadsheet:

•	Enter your data following the spreadsheet format rules.

•	Save your file in *.WK1 format.

2. For compact format, use an text editor, e.g., "Notepad" that comes with Windows.   Enter your data following the PC-ORD rules for one of the formats described above (PC-ORD compact format is recommended).

3. For compact format you can also use a word processor, but you must export your data to a text (ASCII) file:

a. Enter your data following the PC-ORD rules for one of the formats described above (PC-ORD compact format is recommended).

b. Save the file as a text file.  For example in Word, use  File | SaveAs,  then select *.txt as the file type, then enter a file name.

Getting Started


Graph File Format

All of the ordination programs write ASCII (text) files called GRAPHROW.GPH and GRAPHCOL.GPH.  These files contains the X, Y, and Z coordinates for the ordination.  The position of each entity on each of the first three axes is stored in GRAPHROW.GPH, while location of points representing the variables (columns) in the spreadsheet are in GRAPHCOL.GPH.  If only two axes were calculated the graph file will contain only two axes.  Some of the ordination programs will write more than three axes into a graph file, but only three axes are used by the graphing routine.

File Formats


Results File Format

The results from a given run are initially written into a file called "RESULT.TXT".  The result file contains the results from a data operation including sometime error messages.  You can save a result file with File|Save|Results.  The saved results can be used to start a new output file or can be appended to the end with the results of another data operation (File|Append Results).

Tables in the results file are structured with spaces rather than tabs.  Therefore, alignment of columns is achieved by selecting a fixed-pitch (non-proportional) font.  If your tables have a ragged right edge, choose a fixed-pitch font such as Courier or Letter Gothic.

You can import result files or parts of result files to spreadsheets.  Import the information as text.  You can then easily create tabular spreadsheets by "parsing" the text into columns.  For example, in Excel 7.0 this is done with the "Data | Text to Columns" function.

You can search within or edit results files.  Edit or add comments to results files simply by clicking the result window, then typing or editing as you desire.  Or you can open the result file with a text editor or word processor.  Because it is a simple text file, the result file can be opened with any editor, such as Notepad or Wordpad.  Or a full-fledged word processor.  All of those editors provide a search facility.

File Open

File Formats


Graphs as Windows Enhanced Metafiles

You can save graphics images from PC-ORD as Windows Metafiles or Enhanced Metafiles.  Select Graph | File | Save Graphic As.  The currently displayed graph is saved as a Windows Enhanced Metafile.  Windows Enhanced Metafiles are usually given the *.EMF extension on the file name.  Windows Metafiles have a *.WMF extension.

A graph saved in this way is a “vector-based” image that can be brought into various graphics packages or word processors, inserted into text, rescaled, or edited.

For example, in preparing graphs for publication you may wish to alter a graph that you produced from PC-ORD.  This is easily done since they are saved as Windows Metafiles.  Using a word processor or graphics package that accepts Windows Metafiles, you can resize the graph by grabbing the corner and dragging.  Note: if resizing changes the proportionality of the graph, you may introduce defects into the graph, such as missing lines.

Labels in the graphs are easily edited or repositioned at a later time using a word processor or graphics package.  However, some programs cannot edit correctly the vertical axis titles produced by PC-ORD.  You may find the axis label has been turned horizontal by your software.  In that case, if you wish to preserve the vertical axis labels produced by PC-ORD, you will should take care to produce graphs from PC-ORD that will need no further editing.

If the *.emf format is not read correctly by your other software, try saving the graphic in *.wmf format.  But in some cases the simplest solution is to copy the graphic to the clipboard, then paste it into your other document or software package.  Or, as a last resort, you can save the graphic in bitmap (*.bmp) format.  That surely will come in to most graphics packages, but you cannot resize or edit the graphic very well.

If you have a photo editing program such as Paint Shop Pro, you can try importing the *.emf file into it and saving in other formats.

File Formats


Coordinates File Format

A coordinates file is simply a graph file used by non-metric multidimensional scaling (NMS) for starting coordinates.

Graph File Format

File Formats


Spreadsheet Data Format

The standard data format for PC-ORD is a spreadsheet (see example).  Use this format for both your main and second matrices.  You can control the appearance of your spreadsheet in PC-ORD by selecting the font and resizing the columns.  Resizing is done by pointing with a mouse to the boundary between columns or rows, then dragging the cells larger or smaller.

There are several spreadsheet formats.  The one used by PC-ORD has the file extension "WK1".  This format was first used in Lotus 1-2-3 and is probably the most widely usable spreadsheet format.  Many different programs can import/export this format, including other spreadsheet programs (like Quattro and Excel), statistical and graphical software, such as SPSSPC and Sigmaplot, and word processing programs such as WordPerfect and Word.

When you use a spreadsheet program to enter the data, be sure that you write the data file in a WK1 format rather than WQ1, WKS, or other spreadsheet format.

The spreadsheet must be structured like the partial example below:

	A	B	C	D	E
1	120	plots
2	58	species
3		Q	Q	Q	etc.
4		Abigr	Abilas	Piceng	etc.
5	Plot1	23.4	33.2	4.7	etc.
6	Plot2	0	29.4	0	etc.
7	etc.	etc	etc.	etc.	etc.

Format Rules

1. Cells A1 and B1 indicate the number of entities (rows) and what the entities are (8-character maximum).  In this example the spreadsheet contains 120 plots.

2. Cells A2 and B2 indicate the number of attributes (columns) and what the attributes are (8-character maximum).  In this example the spreadsheet contains 58 species.

3. Row 3 contains single letters which indicate the variable type for each column variable.  There are three acceptable values for these fields:

Q = Quantitative

C = Categorical

M = Mixed quantitative and categorical

4. Row 4 contains the name for each column variable.  ONLY THE FIRST 8 characters are used.  Leading blanks and trailing blanks are trimmed off each name.

5. Rows 5 and below contain a row name in the first cell, followed by the numeric values for each column attribute for that row.  In the example above the values are the abundances of each species in that plot.  The row names, like the column names, are limited to 8 characters.  Note: Character data (label fields) are NOT allowed in the body of the matrix.

A Few More Notes

1.  Convert calculated fields in your spreadsheet to numeric fields before requesting a PC-ORD analysis.  For example, in Lotus 1-2-3 version 2 you convert calculated fields to numeric values by selecting "Range" then "Value" from the menu system.  For more information, see the documentation that came with your spreadsheet program.

2.  If using a spreadsheet in which WK1 is not the native format, be sure to export your data to a *.WK1 file (sometimes it will call this format "1-2-3 release 2").

3.  The row and column labels (in column A and row 4 respectively) must contain labels or text.  If you label your rows or columns with numbers you must convert these numeric cells to “labels” or “text” format (the term depends on your spreadsheet).  Note for Excel users:  some versions of Excel will NOT export correctly to *.WK1 a number declared as text.  In this case, preface the row or column labels with a letter, e.g. use P1, P2, P3...  instead of 1, 2, 3...

4.  There is no fixed maximum matrix size that is acceptable to PC-ORD.  This depends on the memory available on your computer, the number of rows relative to the number of columns, and the particular analysis that you request.  See Memory Use and Memory Requirements for more information.

5.  If you use your data in PC-ORD, then decide to alter your spreadsheet with your spreadsheet software, be sure to close the file in PC-ORD (or exit from PC-ORD) before modifying and saving it with other software.

Suggestions for Structuring Your Spreadsheets

Each spreadsheet contains a single matrix.  In many cases you will want to analyze the structure of one matrix relative to another.  For example, in community ecology, we normally have a main matrix consisting of the sample unit x species data and a second matrix of sample units x environmental variables.  To do this you should create two spreadsheets (separate files), but give both files the same row names.  For these matrices to be compatible for analysis, there should be the same number of rows in each.

Example of Spreadsheet

Example of Environmental Matrix

Compact Format

Old PCORD Standard Format

DECORANA/TWINSPAN Condensed Format

Data Preparation

File Open


Spreadsheet Column Width

You can resize the columns in your spreadsheets, as they show in the windows for the main and second matrix.  Move the mouse to the boundary line between columns.  The pointer will change to a different symbol.  Now drag the column wider or narrower.  The rest of the columns will change accordingly.

Spreadsheet Data Format


Example: PCORD Spreadsheet Format

The following matrix contains biomass data for 7 species of fish in each of 16 lakes. Frequently one will also have a second matrix containing environmental data for the same sample units (see Example Environmental Data).

     16 Lakes

      7 Fish

       Q      Q     Q      Q      Q      Q      Q

       N.Pike Cisco W.Suck S.Bass L.Bass Pumpki Bluegi

Allen    21.2      0     0      0      0      0      0

Clubho   13.2   0.58  0.42   0.44      0   0.06    0.3

Elizab   11.7   0.64     0      0      0      0      0

GunnS    15.7    2.6     0      0    0.3      0      0

GunnN    12.9   0.35  0.13      0   0.29   0.04   0.05

Highl   28.32   2.95     0      0      0   0.07    0.1

LittleB   9.5   1.38     1      0    0.2      0      0

Pine    24.33      0     0      0    0.1      0    0.3

Pughol  33.16      0   4.7      0   0.56   0.04   1.42

3Islan   23.9      0     3    3.6      0      0      0

East    19.63      0   0.2      0      0      0      0

Horshw      0      0     0      0      0      0      0

Labarg    8.3      0  4.15      0      0      0    0.1

Mary    23.08      0     0      0    0.8      0   0.18

Larson   0.42      0  0.02      0      0   0.14      0

LittlE  19.67      0     0      0   3.36      0   0.63

Spreadsheet Data Format

Example: Environmental Matrix


Example: Environmental Matrix

The following matrix contains values for 4 environmental variables for each of 16 lakes. Three of the environmental variables are quantitative, one is categorical (a water color category).

     16  Lakes

      4  Environ

         Q        Q        Q        C

         Area     MaxDep   Secchi   Color

Allen          56       25        4        1

Clubho        210       90       13        0

Elizab        193       42       11        0

GunnS          88       75       14        0

GunnN         342       39       11        2

Highla        102       38       12        1

LittlB        155       62       14        0

Pine           72       44       17        0

Pughol        152       25        6        2

3Islan        235       66       20        0

East           83       30       50        2

Horshw        141       11       70        2

Labarg         39       30       20        2

Mary          193       45       25        2

Larson        199      175       15        0

LittlE         61      100       15        0

Spreadsheet Data Format

Variable Types


Variable Types

PC-ORD recognizes three kinds of variables, designated by single letters:

Q = Quantitative

C = Categorical

M = Mixed quantitative and categorical

Every column in a PC-ORD spreadsheet must be assigned one of these types.  These values are entered on the third row of the spreadsheet.

Quantitative variables are measurement variables such that the individual values for a variable can be ranked or arranged in a linear sequence.  Most variables that you will use are quantitative.

Categorical variables are qualitative statements of group membership.  For example, they may represent experimental groups (treatment vs. control).  Another example of a categorical variable is rock type.  There is no single obvious way of ranking limestone, serpentine, granite, and basalt.  Although one could come up with various rankings on such characteristics as hardness or silica content, no single ranking accommodates all of these characteristics.

Categorical variables are used for overlays on your ordinations, while quantitative variables are used for correlations with ordination axes AND overlays. (See Greig-Smith 1983 p. 227-230, for examples of overlays on ordinations.)  Certain variables can only be considered categorical (see the rock-type example given above).  For these variables, it makes no sense to correlate them with ordination axes, while overlays are appropriate.  Overlays of categorical variables on ordinations will be cleanest if they are single-digit integers.

If you want to change the definition of a variable from quantitative to categorical, this can be done by editing your data file.

Mixed quantitative and categorical variables (M) should generally be avoided.  They are produced when you transpose a matrix containing some categorical and some quantitative variables (columns).  Because they are neither quantitative nor categorical, they are not be used for either overlays or correlations with ordination axes.

Spreadsheet Data Format


Compact Data Format

Compact format is a text (ASCII) file consisting of pairs of species codes and abundance's.  For large data sets the compact format is usually much easier to enter.  If you have nested sampling (e.g., subplots within plots or plots within stands), you can make the most of this structure by using the compact format.  Convert to standard PC-ORD format by selecting File | Import in the menu system.

When using the compact format, two files are required: a species file containing a list of species and codes, and the compact data file.  Three types of compact data files are allowed.  See Compact Format Data Types.

For most users, compact format is mainly a timesaver with entering large data sets.  You can use it to store data in a more compact way, but usually data files are very small compared to the large-capacity disk drives in common use.

Compact format can also be very useful if you wish to combine different subsets of a large data set, just by appending the text files to each other.  This is much easier than appending one matrix to another where the variables (usually species) do not match.  For example consider a data set with two subsets, one with 100 species and the other with 200 species.  Combining these subsets with a spreadsheet would involve many cut-paste operations.  With compact format, however, you let PC-ORD combine the two subsets into a single matrix.

While the compact format is truly compact, it is not very compatible with database software because the records are of variable length and the ends of the records are marked in a nonstandard way (a forward slash).  To link your PC-ORD spreadsheets with database programs, use import/export from list (database) format.

Spreadsheet Format

Data Preparation


Species File

Species codes used in a compact format data file are defined in a separate file (any name you like).  This file has a strict format for individual lines (species) but the species may be in any order.  You can use the same species file for many different data files.  See the example.

Three items (or "fields") must be specified in each line of the species file:

•	Species code number, 4-digit maximum (to be used when entering your data in compact format).

•	8-digit character code for species names (to be used as labels for the resulting matrix).  Short code names are recommended to reduce overprinting problems with ordination graphics.

•	The full species name (up to 30 characters long; to be used in the output from Summary).

Note the following rules for the species file:

•	Code numbers can be in any order.

•	One or more spaces must separate the three fields.

•	Code names must be eight characters or less (no imbedded spaces).

•	Code names and species names may be in any order.  (However, you will find it helpful if you keep the species in alphabetical order because most of the output from Summary is ordered by the sequence species in your species file.)

•	Upper case and lower case letters are allowed.

•	No fixed maximum number of species.  It is common to have lists of 500 or more species.

Compact Data Format


Example: Species File

  1 ACERUB ACER RUBRUM

  2 ACESAC ACER SACCHARUM

  3 CARGLA CARYA GLABRA

  4 CARLAC CARYA LACINIOSA

  5 CAROVA CARYA OVALIS

  6 CARSPP CARYA SPP.

  7 CARTOM CARYA TOMENTOSA

  8 CERCAN CERCIS CANADENSIS

  9 CORFLO CORNUS FLORIDA

 10 FRAPEN FRAXINUS PENNSYLVANICA

 11 LIRTUL LIRIODENDRON TULIPIFERA

 12 NYSSYL NYSSA SYLVATICA

 13 OXYARB OXYDENDRON ARBOREUM

 14 QUEALB QUERCUS ALBA

 15 QUERUB QUERCUS RUBRA

 16 QUEVEL QUERCUS VELUTINA

 17 SASALB SASSAFRAS ALBIDUM

 18 QUEFAL QUERCUS FALCATA

 19 PINVIR PINUS VIRGINIANA

 20 JUNVIR JUNIPERUS VIRGINIANA

 21 ULMRUB ULMUS RUBRA

 22 FRAAME FRAXINUS AMERICANA

 23 PINECH PINUS ECHINATA

Compact Format

Compact Format Data Types


Compact Data File

Each of the three compact data types (list of individual trees, quantitative list of species, and presence list) can be found in Compact Format Data Types.  You can define subgroups in compact data files with breakpoints.


Breakpoints

Subgroups of data in a compact data file are defined by inserting "breakpoints" into the data file.  Each breakpoint must contain two items, each on its own line:

1. the keyword "BREAK" or "break"

2. a name for the subgroup (maximum of 8 characters).

The subgroups need not be of the same size.

Insert a breakpoint at the front of each section of data.  Therefore, the first two lines in a file containing subgroups must contain the first breakpoint.

The following example is a short file in compact format containing two subgroups of data.

      BREAK

      STAND1

      plot1

      11 98  14 14  14 16  14 16.5  16 25  3 15.9  9 10 /

      plot2

       9 12   1 23 /

      plot3

      /

      plot4

      14  12.5 16  44 3  22.5 17  16.5 /

      plot5

      14  20 3  23.5 3  22 9  10 /

      BREAK

      STAND2

      plot1

       14  11 9  10.5 9  10  6  29.5 /

      plot2

       14  18.5 9  11 /

      plot3

      /

      plot4

        3  11 3 11.5 3  12 3  15 4  14.5 6  17 6  13 /

Compact Format

Data Preparation


Compact Format Data Types

The three compact format data types are described below.  The data can be a list of individual trees (species and sizes), quantitative lists of species present in each sample unit, or presence lists of species.

List of Individual Trees

The data for each sample unit occupy two or more lines in your data file.  The first line is an 8-character (any character) name for the sample unit, followed by one or more lines with data for that sample (see Import Basal Area or Count Data).  The data consist of a species code, then a diameter or basal area, another species code, another size, and so on, until the last tree in that sample unit has been entered.  Items are separated by one or more spaces or a comma.  Then a slash is added to signal the end of that sample unit.  See example.

Quantitative List of Species

The data for each sample unit occupy two or more lines in your data file.  The first line is an 8-character (any character) name for the sample unit, followed by one or more lines with data for that sample unit.  The data consist of a species code, then an abundance value, another species code, another abundance, and so on, until the last abundance is entered.  Omit species that did not occur in the sample unit.  Only one value is expected for a species in a given sample unit.  Items are separated by one or more spaces or a comma.  Then a slash is added to signal the end of that sample unit.  See example.

Presence List of Species

The data for each sample unit occupy two or more lines in your data file.  The first line is a 8-character (any character) name for the sample unit, followed by one or more lines with data for that sample unit.  The data consist of a list of species code numbers separated by one or more spaces or a comma.  Each species should have no more than one entry in a given sample unit.  Then a slash is added to signal the end of that sample unit.  See example.

Compact Tree Data Units

Spreadsheet Format

Old PC-ORD Standard Format

Data Preparation


Compact Tree Data Units
Compact Tree Data

The tree sizes may be in any one of the following units:

•	DBH (diameter at breast height), cm

•	DBH, inches

•	Basal area, dm2

•	Basal area, cm2

•	Basal area, in2

Calculations based on individual trees require that you specify your plot size.  Enter the area of a plot in hectares.

Compact Format

Import Basal Area or Count Data Example: List of Individual Trees

Data Preparation


Example: List of Individual Trees

The following two lines. . .

       Plot1

       11 98 14 14 14 16 14 16.5 16 25 3 15.9 9 10 /

say that in plot 1 there is a size=98 of species 11, three trees of species 14 (sizes = 14, 16, and 16.5), one size=25 of species 16, one size=15.9 of species 3, and one size 10 of species 9.

A clearer way for the user to write the same data is by couplets for individual trees.  Separate the species code from the size with a comma and separate the different trees with one or more spaces:

      Plot1

      11,98  14,14  14,16  14,16.5  16,25  3,15.9  9,10 /

Here is an example in compact format, data type 1:

      Plot1

      11 98  14 14  14 16  14 16.5  16 25  3 15.9  9 10 /

      Plot2

      9 12  1 23 /

      Plot3                  (Note that plot 3 is empty)

      /

      Plot4

      3 30.5  9 12 /

      .

      etc.

      .

      .

      Plot20

      14 38  6 14  6 24.5 /

Import Basal Area or Count Data

Compact Tree Data Units

Compact Format

Compact Format Data Types


Example: Quantitative List of Species

The following two lines . . .

     Plot1

     11 6   14 3   16 1   3 1   9 1 /

say that in plot 1 species 11 has abundance of 6, species 14 an abundance of 3, and species 16, 3, and 9 have abundances of 1.  The abundance measure can be of any type.

Another acceptable version of the same lines of data would be:

     Plot1

     11,6 14,3 16,1 3,1 9,1/

A series of plots put together would look like:

     Plot1

     11 6   14 3   16 1   3 1   9 1 /

     Plot2

     9  1   1 23/

     Plot3

     /

     Plot4

     3 5   9 1 /

     .

     etc.

     .

     .

     Plot20

     14 3   6 1/

Compact Format

Compact Format Data Types


Example: Presence List of Species

The following two lines . . .

      Samp1

      11 14 16 3 9 /

say that species 11, 14, 16, 3, and 9 were found in sample 1.

Combining a number of plots looks like this:

      Plot1

      11 14 16 3 9 /

      Plot2

      9 1 /

      Plot3

      /

      Plot4

      3 9 /

        .

      etc.

        .

        .

      Plot20

      14 6 /

Compact Format

Compact Format Data Types


Import Basal Area or Count Data

When importing compact data on individual trees, you can create a spreadsheet containing either basal area data or count data.  If you choose basal area, the basal area of all individuals in a sample unit are summed to produce a total basal area for a given species in a given sample unit.  Those values are then stored in the spreadsheet.

Alternatively, if you choose to use count data, each cell contains  the number of trees of a given species in a given sample unit.  Note that you can later convert these to densities if you wish by dividing by the plot area.  Do this by selecting Modify | Multiply by a constant.  Enter a constant equal to 1/plot area.

Compact Data Format

Compact Format Data Types

Compact Tree Data Units


Old PCORD Standard Format

The first PC-ORD format (version 1) was a text file containing one or two matrices with fortran formatting statements.  To run old data sets on the new version of PC-ORD you must import your data by selecting File|Import from the menu system.

The minimum standard data set (see example below) consists of a main matrix, four statements defining its size and contents, and names for the rows and columns.  In addition a second matrix that related to the main matrix could be included.  In the old format, the main and second matrices were included in a single file.  The rules for the old standard format are described in detail in the old PC-ORD manuals.

Spreadsheet

Compact Format

Data Preparation


Example: Old PCORD Standard Format

  19STANDS

  23SPECIES

 (1A6,24F3.0/26F3.0)

 (1A6,19F3.0)

 STAND1 2. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 2. 1. 0. 0. 0. 1. 0. 0. 0. 0. 0.

 STAND2 0. 0. 0. 0. 0. 0. 0. 2. 0. 0. 0. 0. 1. 0. 0. 0. 1. 0. 0. 0. 0. 0.

 STAND3 2. 1. 0. 0. 1. 0. 0. 0. 0. 0. 0. 1. 2. 0. 0. 0. 2. 0. 0. 0. 0. 0.

 STAND4 1. 3. 0. 0. 1. 0. 0. 0. 1. 0. 2. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

 STAND5 2. 0. 3. 2. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 1. 0. 0. 0.

 STAND6 2. 0. 2. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 1. 0. 1. 0. 0. 0.

 STAND7 1. 0. 3. 2. 2. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0.

 STAND8 0. 0. 0. 0. 0. 0. 1. 0. 2. 3. 1. 0. 1. 1. 0. 0. 0. 1. 0. 1. 0. 0.

 STAND9 3. 0. 0. 0. 0. 0. 0. 0. 1. 1. 0. 0. 1. 2. 1. 0. 0. 0. 1. 1. 0. 0.

 STAN10 1. 0. 2. 2. 1. 1. 0. 0. 1. 0. 0. 0. 1. 0. 0. 1. 0. 0. 0. 0. 0. 0.

 STAN11 2. 0. 0. 1. 0. 0. 0. 0. 3. 0. 0. 1. 1. 1. 1. 0. 0. 0. 0. 0. 1. 0.

 STAN12 1. 0. 0. 0. 0. 0. 1. 0. 1. 2. 1. 0. 1. 1. 0. 0. 0. 1. 0. 1. 1. 1.

 STAN13 1. 0. 0. 0. 0. 1. 0. 0. 2. 0. 0. 0. 1. 2. 1. 0. 1. 0. 1. 0. 0. 0.

 STAN14 0. 0. 0. 1. 0. 0. 0. 0. 3. 0. 0. 2. 1. 1. 1. 0. 0. 0. 1. 1. 0. 1.

 STAN15 2. 0. 0. 0. 0. 0. 0. 0. 2. 1. 0. 1. 1. 0. 1. 0. 0. 1. 0. 0. 0. 1.

 STAN16 1. 0. 0. 1. 1. 0. 0. 0. 1. 0. 3. 0. 1. 0. 0. 0. 0. 0. 1. 1. 0. 0.

 STAN17 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 2. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 2.

 STAN18 2. 0. 0. 1. 1. 0. 0. 0. 3. 0. 2. 0. 1. 0. 0. 0. 0. 0. 1. 0. 0. 1.

 STAN19 0. 0. 0. 2. 2. 0. 0. 0. 1. 1. 0. 0. 1. 0. 0. 0. 0. 0. 2. 0. 0. 2.

 Agspic

 Agrint

 Arilon

 Brojap

 Brotec

 Carfil

 ... etc...

 Vertha

 Carnut

 Melalb

 Koecri

  19 STANDS

   5 ENVIRON

 (F5.2,F5.1,F3.0,F5.0,F4.1)

 QQCQQ

 15.84  0.  0. 792. 7.

 23.44  0.  0. 792. 7.

 21.82  0.  0. 792. 7.

 34.77 11.  3. 900. 3.

 10.93 20.5 2. 911. 8.

 11.64 15.  1. 918. 8.

 11.43 16.  2. 906. 8.

 43.68  7.  4.1209. 6.

 14.2  11.  2.1218. 8.5

 46.48 12.  1.1215. 8.

 16.38  9.5 2.1224. 8.

 51.68 22.5 5.1350.10.

 22.31  1.4 0.1356.10.

 17.89  7.  2.1359. 8.5

 21.0  11.  2.1359. 8.

 29.2  21.  5.1195. 6.

 14.86 26.  4.1189. 7.

 23.99 18.5 4.1183. 7.

 13.03 18.5 4.1176. 7.

 smoist

 slope

 aspect

 elev

 topo

***** End of Example *****

Old PCORD Standard Format


DECORANA/TWINSPAN Condensed Format

DECORANA/TWINSPAN Condensed Format refers to the condensed format from the Cornell Ecology Program (CEP) Series.  This is a text (ascii) file.  The format is similar, but not exactly the same as the PC-ORD compact format.  To use these data in PC-ORD you must import your data to the spreadsheet format by selecting File|Import from the menu system.

A small example of this format is included below.  Unfortunately several variants on this format exist.  PC-ORD's file import function attempts to recognize these variants an import them accordingly.  Only one format is used on export.

Note that CANOCO relaxed a key CEP formatting requirement, so that some "CEP" files acceptable to CANOCO cannot be read by the original CEP programs.  CANOCO allows the number of data items per line to be written as a separate item on line 3, rather than insisting on this number right justified in spaces 69-70 on line 2.  PC-ORD will read files with either format but writes them in the original CEP condensed format.

This is the title line

(I3,5(1X,I3,1X,F8.3))                                                5

  1   1    2.000   5    1.000  12    2.000  13    1.000  17    1.000

  1  23    1.000  28    1.000  29    1.000  32    1.000  44    1.000

  2   8    2.000  13    1.000  17    1.000  23    1.000  28    1.000

  2  29    1.000  32    1.000  43    1.000

  3   1    2.000   2    1.000   5    1.000  12    1.000  13    2.000

  3  17    2.000  28    1.000  29    1.000  32    2.000  44    1.000

  4   1    1.000   2    3.000   5    1.000   9    1.000  11    2.000

  4  35    1.000  36    1.000

  5   1    2.000   3    3.000   4    2.000   5    1.000  16    1.000

  5  19    1.000  23    1.000  24    1.000  25    1.000  40    1.000

  5  44    1.000

  6   1    2.000   3    2.000   5    1.000  14    1.000  17    1.000

  6  19    1.000  23    1.000  24    1.000  26    1.000  28    1.000

  6  29    1.000  30    1.000  36    1.000  44    1.000

"

" etc.

"

 18  29    1.000  34    1.000  35    1.000  36    2.000  44    1.000

 18  47    1.000  48    2.000  49    1.000

 19   4    2.000   5    2.000   9    1.000  10    1.000  13    1.000

 19  19    2.000  22    2.000  28    1.000  29    1.000  34    3.000

 19  35    1.000  36    1.000  43    1.000  44    1.000  47    1.000

 19  48    2.000  49    1.000

 Agspic  Agrint  Arilon  Brojap  Brotec  Carfil  Danint  Elycin  Fesida  Fessca

 Poapra  Poasan  Achmil  Antsp   Arecon  Artdra  Artfri  Astsp   Balsag  Besrub

 Camrot  Cerarv  Chrvil  Chrvis  Cirund  Collin  Compal  Diaarm  Erisp   Gaucoc

 Geutri  Grisqu  Hiealb  Hypper  Litrud  Lupser  Micgra  Ortten  Pensp   Phahas

 Polsp   Stesp   Taroff  Tradub  Versp   Pergai  Vertha  Carnut  Melalb  Koecri

 STAND1  STAND2  STAND3  STAND4  STAND5  STAND6  STAND7  STAND8  STAND9  STAN10

 STAN11  STAN12  STAN13  STAN14  STAN15  STAN16  STAN17  STAN18  STAN19

Data Preparation

File Formats


List (Database) Format

Species data.  Many people using relational database software (such as Paradox or Access) store species data as a list of values rather than in matrix format.  For example, with a sample of species in multiple stands, one might include only the non-zero values along with identifiers for the sample unit and species, for example:

Stand1	Abilas	12
Stand1	Piceng	34
Stand2	Abilas	4
Stand2	Psemen	46
Stand3	Poptri	39
etc.	etc.	etc.

If you wish to import such data, first export your data from your database to a delimited text file.  PC-ORD expects three columns, in the order:

SampleUnitID     SpeciesID     Abundance

Rules:

•	The first row of the file may contain variable names.  (Indicate whether or not you did this by using the check box in the dialog for this procedure.)  These are read as labels ("strings" or alphanumerics).  The first field is a collective name for all the rows in the matrix you are building.  In the example above an appropriate name would be "Stands."  The second field is a collective name for all the columns in the matrix you are building, for example, "species".  The third field describes the numerical values, for example "abundance."  You may need to type in this row with a text editor if your database software does not write these (though most of them will).  Note that if you do not include variable names in the first row, but "Read field names from first line" is checked in the dialog box, the first line of your data will be misinterpreted as the row and column names.

•	In subsequent lines the first two fields are text labels, while the last value must be numeric.

•	Text labels may be delimited by double quotes, single quotes, or another delimiter from the list below.  Note however, that if there are spaces in your labels you should not select spaces as a delimiter.

•	Delimiters that may be used are tabs, commas, semi-colons, spaces, double quotes, or single quotes.

•	Labels longer than 8 characters are truncated after the eighth character.  Sample unit and species identifiers must, therefore, be unique within the first eight characters.

•	At most 128 characters are read per line.

•	The variable types are assumed to be "Q" (quantitative).  This is appropriate even if your data are presence/absence.  But if you wish to declare a variable as categorical, use Edit | Edit matrix to change the variable type from Q to C.

The data from the preceding might look like the following, after exporting to a text file:

"Stand1","Abilas",12

"Stand1","Piceng",34

"Stand2","Abilas",4

"Stand2","Psemen",46

"Stand3","Poptri",39

etc.

Then import this into a PC-ORD matrix with:

File | Import | List (database) format | Create species spreadsheet

The resulting spreadsheet in PC-ORD would be:

	Abilas	Piceng	Psemen	Poptri
Stand1	12	34
Stand2	4		46
Stand3				39
etc.

Some software calls this creating a "pivot table."

Environmental data. – Environmental data from databases are typically imported directly into PC-ORD without the need for creating a pivot table.  That is because a value is present for every environmental variable for each sample unit in the database.  This is in contrast to species data where the zero values are not stored.

The database might look like:

Stand	Slope	Moisture
Stand1	12	2.1
Stand2	0	9.4
Stand3	30	3.2
etc.	etc.	etc.

If you wish to import such data, first export your data to a delimited text file.  PC-ORD expects two or more columns, in the order:

SampleUnitID     Variable 1     Variable 2     etc.

Rules:

•	The first row of the file must contain variable names.  These are read as labels ("strings" or alphanumerics).  You may need to type in this row with a text editor if your database software does not write these (though most of them will).

•	In subsequent rows the first field is a label (usually a sample unit name), while the other fields must be numeric.  The number of fields read matches the number of labels in the first row.

•	Text labels may be delimited by double quotes, single quotes, or another delimiter from the list below.  Note however, that if there are spaces in your labels you should delimit them with quotes.

•	Delimiters that may be used are tabs, commas, semi-colons, spaces, double quotes, or single quotes.

•	Labels longer than 8 characters are truncated.  Sample unit identifiers must, therefore, be unique within the first eight characters.

•	At most 128 characters are read per line.  (If you need to import longer lines, export your database to spreadsheet format directly.)

•	The variable types are assumed to be "Q" (quantitative).  This is appropriate even if your data are presence/absence.  But if you wish to declare a variable as categorical, use Edit | Edit matrix to change the variable type from Q to C.

The data from the preceding might look like the following, after exporting to a text file:

"Stand","Slope","Moisture"

"Stand1", 12, 2.1

"Stand2", 0, 9.4

"Stand3", 30, 3.2

etc.

Then import this into a PC-ORD matrix with:

File | Import | List (database) format | Create list format spreadsheet

The resulting spreadsheet in PC-ORD would be essentially identical to the original appearance above:

	Slope	Moisture
Stand1	12	2.1
Stand2	0	9.4
Stand3	30	3.2
etc.	etc.	etc.

Example List Format

Data Preparation

File Formats


Example List Format

In this example values are integers, the fields are tab delimited, "strings" are quoted, and zeros omitted.  The data were exported by PC-ORD for a matrix of 5 plot by 60 species.  Note that it is NOT necessary to pad the label fields with blanks. For example, the first line could also be written "plots"	"species"	"Value".

"plots   "	"species "	"Value   "

"00800813"	"Bryfus  "	           3

"00800813"	"Cndcon  "	           3

"00800813"	"Cetpin  "	           3

"00800813"	"Cla     "	           3

"00800813"	"Clafim  "	           2

"00800813"	"Hypaus  "	           3

"00800813"	"Melexa  "	           3

"00800813"	"Melsol  "	           4

"00800813"	"Popamb  "	           4

"00800813"	"Physte  "	           2

"00800813"	"Usnlap  "	           4

"00900831"	"Melexl  "	           3

"00900831"	"Melsol  "	           3

"00900831"	"Phyads  "	           3

"00900831"	"Phyaip  "	           2

"00900831"	"Phybiz  "	           3

"00900831"	"Pclchl  "	           2

"00900831"	"Usn     "	           3

"00900831"	"Xanful  "	           2

"00900831"	"Xanmon  "	           3

"00900832"	"Bryfus  "	           1

"00900832"	"Clafim  "	           1

"00900832"	"Fpusor  "	           3

"00900832"	"Melsol  "	           3

"00900832"	"Phymex  "	           2

…

etc.

…

"00900811"	"Melsol  "	           3

"00900811"	"Phyads  "	           2

"00900811"	"Phybiz  "	           3

"00900811"	"Usn     "	           2

"00900811"	"Xanfal  "	           1

"00900811"	"Xanmon  "	           3

List (Database) Format

Data Preparation

File Formats


Comma Separated Values Format

An ASCII file of comma-separated values (*.csv format) can be used to import/export data from PC-ORD.  This file is essentially a text-file image of your spreadsheet, one line per row in the matrix, with commas separating adjacent values (see example ).  The rules for *.csv files are simple, yet rigid:

•	Commas delimit values of numbers and text.

•	Each line (record) in the file must have the same number of commas (specifying a rectangular array of numbers and labels).  Note that there are the same number of commas in each line as there are attributes (data columns) in the matrix.

•	Strings are not surrounded by quotes.

•	At most 60,000 characters are read from a single line of data.

Example CSV format

Data Preparation

File Formats


Example CSV format

The following *.csv file was produced by exporting a PC-ORD *.wk1 file from Excel:

19, STANDS ,,,,

5, ENVIRON,,,,

,Q,Q,q,Q,C

,smoist,slope,aspect,elev,topo

STAND1,15.84,0,0,792,7

STAND2,23.44,0,0,792,7

STAND3,21.81,0,0,792,7

STAND4,34.77,11,3,900,3

STAND5,10.93,20.5,2,911,8

STAND6,11.64,15,1,918,8

STAND7,11.43,16,2,906,8

STAND8,43.68,7,4,1209,6

STAND9,14.19,11,2,1218,8

STAN10,46.47,12,1,1215,8

STAN11,16.37,9.5,2,1224,8

STAN12,51.68,22.5,5,1350,10

STAN13,22.30,1.396,0,1356,10

STAN14,17.88,7,2,1359,8

STAN15,21,11,2,1359,8

STAN16,29.20,21,5,1195,6

STAN17,14.85,26,4,1189,7

STAN18,23.98,18.5,4,1183,7

STAN19,13.02,18.5,4,1176,7

Note the series of commas at the ends of the first two lines.  Remember that these are necessary to specify a rectangular matrix.

Comma Separated Values Format

Data Preparation

File Formats


Missing Values

PC-ORD has no explicit provision for handling missing values.  It is up to you to deal with these one way or another.  Consider the amount and pattern of missing data.

Amount

If you have large amounts of missing data the most basic question is, "Should the data be analyzed?"  If you have small amounts of missing data, you need to decide the best way to deal with it.

Pattern

If missing values are concentrated into particular rows or columns, the most reasonable solution is to delete the offending rows or columns.  In community ecology, it is very common to have missing values in certain environmental variables.  If missing values are few and well distributed and essentially random with respect to groups of data or underlying gradients, then it is often reasonable to take corrective steps.

Corrective steps

A simple correction is simply to delete rows or columns (cases or variables) that contain missing values.  If this is too painful, sacrificing hard-earned data, then you might consider estimation of missing values.  Consider using prior knowledge if you have a very good basis for making a well-educated guess.  Of course you need to be extremely cautious in doing so.

Another possible correction is to insert the mean(or median) for a given variable.  This is usually preferable to the estimation mentioned above, because no subjectivity is involved.  One might use the overall mean for a given variable or the mean for a given variable within a group.  Introduction of a few means usually has little effect on the outcome of an analysis.  It will, however, reduce the variance of the variable and will often weaken the correlation structure slightly.

A rather heavy-handed approach is to use regression, estimating the missing values based on another variable or set of variables.  If the independent variables are also part of the matrix being analyzed, then this introduces a circularity: you will be partially creating the correlation structure that you later detect!

If you do take corrective steps, it is best to repeat your analyses with and without the corrections.  Did the corrective steps alter your conclusions?  If so, you should try to understand why the results were altered and whether your corrective steps were justified.

For further discussion, see Tabachnik and Fidell (1989 or 1996).

File Formats


Edit

The Edit menu provides facilities for simple editing of spreadsheets in PC-ORD, as well as selecting and copying information into the clipboard.  The Edit menu includes:

•	Edit Matrix

•	End Editing Matrix

•	Matrix Parameters

•	Copy

•	Select All

Getting Started

Spreadsheet Data Format

Modify Data


Edit Matrix

Edit | Edit Matrix | Main Matrix

Edit | Edit Matrix | Second Matrix

PC-ORD provides a very simple matrix editor.  This is most useful for small corrections or changes in a matrix.  We recommend, however, that you use a spreadsheet for more extensive editing of your matrices.  Edit Matrix is useful for correcting typographical errors, entering small data sets, or other small changes, but it does not provide the full functionality of spreadsheet software.

If you wish to change only the matrix specification cells for your matrix, select Edit | Matrix Parameters.

Note that many manipulations of the matrices, such as deleting rows or columns, transformations, and matrix multiplication are available under the Modify menu.

After you invoke Edit Matrix you will need to select End Editing Matrix to save your work and return to the menu system.

Edit menu


End Editing Matrix

Edit | End Editing Matrix

End Editing Matrix completes the Edit Matrix process.  Your work is saved and you are returned to the menu system.

Edit menu


Edit Matrix Parameters

Edit | Matrix Parameters allows you to change only the matrix specification cells (the first two rows of your spreadsheet) for either your main matrix or your second matrix.  A dialog allows you to change the names or numbers of items.

Edit menu


Copy

Edit | Copy extracts pieces of your output or input files for inclusion in reports or manuscripts.  First select text in any one of the four windows of PC-ORD.  Then select Edit | Copy or press Ctrl-C to copy the selection to the clipboard.  After you switch tasks to the destination software, select Paste in that software.

Edit menu


Select All

Edit | Select All copies into the clipboard the whole file shown in the active window.  The contents of the clipboard can be pasted into another document.

Edit menu


Help

Selecting Help from the PC-ORD menu gives the following selection:

•	Contents – Gives the Table of Contents for the help system.

•	Topic Search – Allows the user to search for particular topics in the help system.

•	How to Use Help – Gives basic instructions for using the help system.

•	Technical Support – How to obtain technical support.

•	Internet Support

•	About – Lists the active version number and gives a suggested citation for PC-ORD.


Topic Search

Help | Topic Search allows the user to search for particular topics in the help system.  For help on how to search by topic or index entries, press F1 or click on Help.

Help


How to Use Help

Help | How to Use Help gives basic instructions for using the help system.  For Help on Help, Press F1 or click on Help above this message.

Help

Getting Started


Technical Support

Help | Technical Support describes how to obtain technical support.  Assistance with the software is available in the form of written response to registered user's questions, preferably by email.  Our software support pertains only to the programs obtained directly from us.

Very often, if you are experiencing unexpected behavior, someone else will have detected it and the problem will already be fixed.

Before you contact us…

•	Please download the latest fixes from our website, and verify that the problem still exists in the latest version.

•	Check the Frequently Asked Questions section in the manual and/or website.

•	Determine what version you are using by selecting Help | About in the PC-ORD menu or inspecting the header at the top of your output files. Please include the version number in your correspondence with us, specifying not just "version 4" but "version 4.03" or whatever exact version you are using.

MjM Website

http://www.pcord.com

Email

Internet: mjm@centurytel.net

Mail

MjM Software Design

PO Box 129

Gleneden Beach, OR 97388

U.S.A.

Fax

541-764-3935

Phone (orders only)

1-800-690-4499

Frequently Asked Questions

PCORD Failure

Internet Support

Getting Started

Help


Internet Support

Help | Internet Support

PC-ORD maintenance fixes made since you received the software can be obtained for free.  Download improved modules from the MjM website:

http://www.pcord.com

Fixes will be posted periodically as needed.

Frequently Asked Questions

Technical Support

Getting Started

Help


About

Help | About gives basic information about PC-ORD.  It lists the active version number and gives a suggested literature citation for PC-ORD.

Help


Ordination

A variety of ordination techniques can be selected under the ordination menu.  In general, ordination is the ordering of objects along axes according to there resemblances.  The major objective is to achieve an effective data reduction, expressing many-dimensional relationships in a small number of dimensions.  This amounts to extracting the strongest correlation structure in the data (using correlation in the broad sense).  The correlation structure is used to positions objects in the ordination space.  Objects close in the ordination space are generally more similar than objects distant in the ordination space.

The available ordination techniques are listed below.

Bray-Curtis (=Polar Ordination)

Canonical Correspondence Analysis (CCA)

Detrended Correspondence Analysis (DCA, DECORANA)

Non-metric Multidimensional Scaling (NMS)

NMS Scores

Principal Components Analysis (PCA)

Reciprocal Averaging (RA) (Correspondence Analysis)

Weighted Averaging Ordination

Graphing


Bray-Curtis Ordination

Ordination | Bray-Curtis

PC-ORD performs Bray-Curtis ordination in its infinite variety (Beals 1984, McCune and Beals 1993).  In contrast to principal components analysis, it is not available in the major statistical packages.  Earlier implementations, such as ORDIFLEX (Gauch 1977), used a more primitive form of this method.  One notable computational difference between the two is that PC-ORD perpendicularizes successive axes by basing them on residual distance matrices rather than on the original distance matrix (method of Beals 1965).  PC-ORD also offers flexibility in distance measures, methods of endpoint selection, and type of geometry used at various points in the computations.  The basic algorithm was originally written by W. M. Post, with input from E. Beals and T. F. H. Allen.  Later modifications were made by R. Howe and B. McCune.

Graphs and correlations with ordination axes are then available through the  Graph menu. Simultaneous plotting of column variables (usually species) in the ordination space is available if you select Calculate Scores by Weighted Averaging in the Bray-Curtis Setup dialog.  The scores are written into the temporary file GRAPHCOL.GPH.

The cumulative variance explained by the ordination is based on the ratio of the sum of squares of the residual distance matrix (SSR) to the sum of squares of the original distance matrix (SST):

Cumulative variance explained, % = 100(1 - SSR/SST)

Bcord Setup

Ordination


Bray-Curtis Setup

The following options are selected when running a Bray-Curtis ordination:

Distance Measure

Endpoint Selection Method

Axis Projection Geometry

Residual Distance

Number of Axes

List Residual Distance Matrix

List Distance Matrix

Calculate Scores by Weighted Averaging


Bray-Curtis Distance Measures

The eight standard distance measures in PC-ORD are all available for Bray-Curtis ordination.  These are described under Distance Measures.

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


Endpoint Selection Method

•	Bray and Curtis' original (1957) method (maximum range).  This method is highly susceptible to outliers--beware!  This option is very similar (but not identical) to Bray and Curtis' original method.  The first endpoint chosen has the highest sum of distances with other entities.  The second endpoint has the highest distance from the first endpoint.  (If more than one pair share this distinction, the chosen pair is the first pair encountered.)  In contrast to Bray and Curtis' original method, successive endpoints are chosen by finding the highest distance from the residual distance matrix (Bray and Curtis did not calculate residual distances).

•	Variance-regression (Beals 1984). The first endpoint has the greatest variance in distances (or residual distances).  Distances between each point and the first endpoint are then regressed against distances between each point and trial endpoints.  The trial endpoint with the lowest regression coefficient is chosen as the second endpoint.

•	Minimum deviation (summing minimum residual distance, Beals 1984).  All pairs are considered as endpoints. The pair that minimizes the sums of residual distances defines the axis.  You must choose a geometry for minimum deviation endpoints (either city-block or Euclidean geometry).  This is a time-consuming method; if you have a slow machine and are running a large data set, come back a day later!

•	Subjective.  With this method you supply

1. The number of subjective axes,

2. The names of the entities to be used as subjective endpoints (these are chosen from a picklist) and

3. A non-subjective endpoint method IF the number of axes you requested exceeds the number of subjective axes requested.

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


Axis Projection Geometry

In city-block-space, a point is placed on an axis at the midpoint of the overlap between the distances from the two endpoints.  The usual method, however, is Euclidean (geometry given in Bray and Curtis [1957 ], algebra given in Beals [1960] and many later sources).

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


Calculation of Residual Distances

In city-block space, a residual distance between two points is calculated as the value from the distance matrix for the two points, less the absolute value of the difference between the two points on the ordination axis. In Euclidean space, the residual distance is the square root of the difference between the value from the distance matrix for two points squared, less the squared difference between the positions on the ordination axis (i.e., the usual method).  See

Distance Measures

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


Number of Bray-Curtis Axes

At most nine axes are calculated.  If there are fewer than 10 entities, the maximum number of axes is the number of entities less 1.

As many axes as you request will be written into the graph file.  Note, however, that only the first three axes can be graphed automatically using PC-ORD’s graphing routine.  If you want ordinations that include the fourth or higher axes, you must plot them by hand, use another graphics package, (using the coordinates in the graph file), or edit the graph file to remove earlier axes.

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


List Bray-Curtis Residual Distance Matrix

You can have the residual distance matrix listed for an axis of your choice.  The residual distance matrix represents the information remaining in the distance matrix after extraction of that axis.

Distance matrices are labeled only by row.  The column sequence is in the same order as the rows.  The numbers are listed in scientific notation as E Fields.

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


List Bray-Curtis Distance Matrix

The distance matrix is written to the results file.  The output from this option can be voluminous if you have a large data set.  Distance matrices are labeled only by row.  The column sequence is in the same order as the rows.  The numbers are listed in scientific notation as E Fields.

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


E Fields

Numbers are sometimes listed with the computer version of scientific notation.  This is used for maximum flexibility in portions of the output.  The numbers following the E are exponents.  For example,

           .7071E+01 = 7.071

           .9169E+00 =  .9169

           .1234E-05 =  .000001234.


Geometry for Minimum Deviation Endpoints

The minimum deviation method of endpoint selection minimizes the sums of residual distances.  You must choose a geometry for calculating the residual distances that you wish to minimize.  See Distance Measures.

Calculation of Residual Distances

Endpoint Selection Method

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


Bray-Curtis Subjective Axes

Enter the number of subjective axes you would like to specify.  If you request fewer subjective axes than the total number of axes to be printed, you will be asked to choose an automatic method for the remaining axes.  The available methods are the Bray-Curtis original, variance-regression, and minimum deviation.

Endpoint Selection Method

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


Bray-Curtis Non-Subjective Endpoint Method

You requested fewer subjective axes than the total number of axes requested.  Therefore you must choose an automatic method for the remaining axes.  The available methods are the Bray-Curtis original, variance-regression, and minimum deviation (see endpoint selection methods).

Bray-Curtis Setup

Bray-Curtis Ordination


Bray-Curtis Subjective Endpoints

Select two endpoints to define this subjective axis.  When selecting two or more items from a picklist, hold the Ctrl key down and click on the items you wish to select.

Bray-Curtis Setup

Bray-Curtis Ordination

Ordination


Canonical Correspondence Analysis (CCA)

Ordination | Canonical Correspondence Analysis (CCA)

CCA seeks structure in the main matrix in such a way as to maximize the strength of the relationship with the second matrix.  Normally the main matrix contains abundances of species in a set of sample units while the second matrix contains environmental variables measured in the same sample units.  Click here for an introduction to the method.

CCA Introduction

When to Use CCA

CCA Setup

CCA Output

CCA Example Output

Ordination


CCA Introduction

Canonical correspondence analysis (CCA; ter Braak 1986, 1994) is unique among the ordination methods in PC-ORD in that the ordination of the main matrix (by reciprocal averaging) is constrained by a multiple regression on variables included in the second matrix.  In community ecology, this means that the ordination of samples and species is constrained by their relationships to environmental variables.  While this facilitates interpretation, the method presumes that you have measured meaningful environmental variables.  In contrast, performing an ordination on just the community data, then secondarily relating the ordination to the environmental variables, allows an expression of pure community gradients, followed by an independent assessment of the importance of the environmental variables.  See Okland (1996) for a discussion of the contrasting goals of CCA and most other ordination techniques.

Thus, two corresponding matrices are required for CCA.  In the main matrix, the rows are normally sites (plots, etc.) and the columns are species.  In the second matrix, the rows are the same as in the main matrix, but the columns are environmental variables.  Note that the number of environmental variables must be less than the number of sites.

CCA is most likely to be useful when: (1) species responses are unimodal (hump-shaped), and (2) the important underlying environmental variables have been measured.  Note that condition 1 causes problems for methods assuming linear response curves (PCA) but causes no problems for CCA, according to ter Braak (1986, 1994).  Condition 2 results from the environmental matrix being used to constrain the ordination results, unlike any other ordination technique apart from Canonical Correlation.  For this reason, CCA has been called a method for "direct gradient analysis” (ter Braak 1986).

CCA is currently one of the most popular ordination techniques in community ecology.  It is, however, one of the most dangerous in the hands of people who do not take the time to understand this relatively complex method.  The dangers lie principally in several areas: (1) Because it includes multiple regression of community gradients on environmental variables, it is subject to all of the hazards of multiple regression.  These are well documented in the statistical literature, but often not fully appreciated by newcomers to multiple regression.  (2) As the number of environmental variables increases relative to the number of observations, the results become increasingly dubious, even though an appearance of very strong relationships is inevitable.  (3) Statistics indicating the "percentage of variance explained" can be calculated in several ways, each for a different question, but users frequently confuse these statistics when reporting their results.  (4) Users frequently interpret CCA as a description of community structure per se (see When to Use CCA).

CCA is not available in the major statistical packages.  The implementation in PC-ORD includes the basic technique with several scaling options.  Detrending and inclusion of covariates are not available in the PC-ORD version of CCA.  PC-ORD uses an iterative method of CCA following the steps in ter Braak (1986).

Problems with stability of CCA were described by Oksanen and Minchin (1997).  In accordance with their recommendations, we now impose more strict criteria for convergence on a stable solution (see “Iteration Report” under CCA Output).

CCA does not explicitly calculate a distance matrix.  But CCA, like CA and DCA, is implicitly based on the chi-squared distance measure where samples are weighted according to their totals (Chardy et al. 1976; Minchin 1987a).  This gives high weight to species whose total abundance in the data matrix is low, thus exaggerating the distinctiveness of samples containing several rare species (Faith et al., 1987; Minchin 1987a).

CCA Setup

Canonical Correspondence Analysis (CCA)


When to Use CCA

CCA is widely misused.  Part of this results from users not understanding the associated statistics, but the most basic problem is not understanding that CCA diagrams generally do not depict well community similarities. To help users understand the situations in which CCA is or is not appropriate, please consult the following table.

Table 1.  Questions about the community (A) and environmental or experimental design (E) matrices that are appropriate for using CCA.

Subject matrix	OK to use CCA	Not OK to use CCA
A only	Not applicable	What are the strongest gradients in species composition?
Ho:  no linear relationship A ·  E	Are any aspects of community structure related to these environmental variables?	Are the strongest community gradients related to these environmental variables?
Describe A ·  E	How is the community structure relative to these environmental variables?	How are the strongest gradients in community structure related to these environmental variables?
Relationships among rows or columns of A	What is the relationship among sites or species in this particular environmental space?	What is the relationships among sites or species in species space?

CCA Setup

Canonical Correspondence Analysis (CCA)


CCA Setup

In the CCA Setup dialog you specify the following options:

•	CCA Scaling Options

Row and Column Scores Standardized by...

Scaling of Ordination Scores

•	Scores for Graphing

•	List Correlation Coefficients for Second Matrix

•	CCA Monte Carlo Test

Select null hypothesis


CCA Scaling Options

The options in the CCA Setup dialog control the scaling of the ordination axes.  Six combinations of scaling options are provided by the two parts to the dialog.

Row and column scores standardized by...

"Centering and normalizing" (default) gives biplot scaling where site scores are rescaled such that the mean is zero and the variance is one.  With "Hill's method," site scores are rescaled such that the standard deviation ("tolerance" of ter Braak) is on average about 1 unit, and a species' response curve can be expected to rise and decline over an interval of about 4 units (i.e. ± 2 standard deviations).  It has been argued that this scaling makes the resulting ordination more interpretable ecologically.  One drawback is that this rescaling prevents interpretation of the resulting ordination as a biplot in the strict sense (Jongman et al. 1995.

Scaling of ordination scores

This set of three choices controls the relative scaling of species versus sites (sample units), as set by the parameter alpha.

If you choose to "optimize rows" (usually sites) then site scores are weighted mean species scores (alpha = 1).  Distances between sites approximate their chi-square distances (Jongman et al. 1995).

If you choose to "optimize columns" (usually species), then species scores are weighted mean site scores (α = 0, the default).   Distances between species then approximates their chi-square distances.  If both species points and biplot points for environmental variables are present in the ordination, then this choice allows a direct spatial interpretation of the relationship between environmental and species points.  The biplot approximates weighted averages of species with respect to environmental variables.

If a "compromise" is chosen then α = 0.5.

Combining the two sets of choices above results in the following constants used to rescale particular axes.  Lambda (λ) is the eigenvalue for the given axis.  Alpha (α)  is selected as described above.

	Biplot scaling	Hill's scaling
Constant for rescaling species scores
Constant for rescaling site scores

Canonical Correspondence Analysis (CCA)

CCA Output

CCA Example Output

Ordination


CCA Scores for Graphing

CCA calculates two sets of scores for the rows in your main matrix (usually sites).  You may select either set for graphing.  The two sets are:

•	Scores for rows in your main matrix are linear combinations of the columns in your second matrix (usually environmental variables).  These are the LC scores of Palmer (1993).

•	Scores for rows in your main matrix are derived from the columns in your main matrix (usually species).  These are the WA scores of Palmer (1993).

Early programs for CCA used the second option.  More recently ter Braak (1994) recommended the first option.  This is the default in PC-ORD.  The first option represents the best fit of species abundances to the environmental data.  The second option best represents the observed species abundances.

This option influences the resulting ordination more than any other option in CCA.   CCA is highly sensitive to noisy or irrelevant environmental data if one selects the option that final site scores are linear combinations of environmental variables (McCune 1997).  Its conceptually equivalent in simple regression to showing a scatterplot of “predicted” values rather than the observed data.  Use of LC scores will thus distort the representation of community relationships unless the data are noiseless.  Because environmental data in community ecology usually contain measurement error, CCA with LC scores is inappropriate where the objective is to describe community structure (McCune 1997; see further discussion of this point in Økland (1996)).  CCA can be appropriate, however, if the objective is to describe community variation with respect to a particular set of measured environmental variables. CCA is insensitive to noise if the final scores are derived from the species scores (WA scores), but in this case it is not a “direct gradient analysis” in that the axes are not combinations of environmental variables.

CCA Setup

Canonical Correspondence Analysis (CCA)


List Correlation Coefficients for Second Matrix, CCA

Choose this option if you want a matrix of correlation coefficients printed.  It results in a listing of the simple Pearson correlation between all pairs of variables in your second matrix.

CCA Setup


CCA Monte Carlo Test

Two kinds of Monte Carlo (randomization) tests are available with CCA in PC-ORD.  The type of randomization depends on the hypothesis tested:

•	Ho:  No linear relationship between matrices.  For this hypothesis, the rows in the second matrix are randomly reassigned within the second matrix.  This destroys the relationship between the main and second matrices, but it leaves the correlation structure of the variables in the second matrix intact.

•	Ho:  No structure in main matrix and therefore no linear relationship between matrices.  For this hypothesis, elements in the main matrix are randomly reassigned WITHIN columns.  This preserves the column totals and the matrix totals, but alters the row totals and destroys the relationships among columns in the main matrix (usually species).

For both hypotheses, you must select the number of runs, i.e. the number of randomizations.  The larger the number of runs, the more precision you can obtain in the resulting p-values.

You must also specify how to obtain the random number “seeds” to initialize the random number generator.

Your output will contain a table at the end showing how the observed eigenvalues and species-environment correlations compare with those from randomized runs (see example output).

For example, to evaluate the significance of the first CCA axis, a p-value is computed that is simply the proportion of randomized runs with an eigenvalue greater than or equal to the observed eigenvalue:

If:

n = the number of randomizations (permutations) with an eigenvalue greater than or equal to the corresponding observed eigenvalue.

N = the total number of randomizations (permutations)

then

p = (1 + n)/(1 + N)

This p value is the probability of type I error for the null hypothesis that you selected.

CCA Setup

Canonical Correspondence Analysis (CCA)


CCA Output

1. Iteration report

Convergence on a stable solution is evaluated in relationship to a specified "tolerance" value.  The results are stated either as having reached this tolerance in a certain number of iterations (or "loops") or as having failed to reach this tolerance after 999 iterations.  CCA in PC-ORD ver. 2.10 and lower used the tolerance criterion (10^-10) in equation 8.13 on p. 79 of the manual for CANOCO ver. 2.1.  This is the weighted mean squared difference between site scores of the previous iteration and the site scores of the current iteration.  Each term in the summation is weighted by fraction of the total abundance that is present at that site.  Beginning with PC-ORD ver. 2.11 and ver. 3.00, the tolerance criterion was made more strict, namely 10^-13.

2. Total variance in the species data

The total variance (or "inertia" in the terminology of ter Braak, 1990) in the species data is a statement of the total amount of variability in the community matrix that could potentially be "explained."  The ratio of an eigenvalue to the total variance indicates the proportion of variance explained by a given ordination axis.  See criticism of this approach by Økland (1999).  We offer as an alternative an after-the-fact correlation between ordination distance and distances in the unreduced species space.

3. Axis summary statistics  (see example)

•	Eigenvalue.  The eigenvalue represents the variance in the community matrix that is attributed to a particular axis (but see Økland 1999).

•	Variance in species data.  The percent of variance in the community matrix that is explained by each axis is reported.  For a particular axis, this value is calculated as 100 times the ratio of the eigenvalue to the total variance.

•	Pearson Correlation, species-environment.  These are standard correlation coefficients between sample scores for an axis derived from the species data (the WA scores) and the sample scores that are linear combinations of the environmental variables (the LC scores).  There are dangers for interpreting this statistic.  If the number of environmental variables is large relative to the number of sample units, a large correlation coefficient will result, regardless of any underlying biological meaning (McCune 1997).  The "species-environment correlation" coefficient is a misnomer, is often misleadingly high, and if reported should be accompanied by a significance test.  It would be better named the "LC-WA correlation."  A better (but still imperfect) measure of the strength of the species environment relationship is the proportion of variance in the species matrix that is explained by the environmental matrix.

•	Kendall (Rank) Correlation, species-environment.  This statistic is a rank correlation coefficient for the same relationship as in the preceding paragraph.

4. Multiple regression results  (see example)

The regression results report the effectiveness of the environmental variables in structuring the ordination and describe the relationships of the environmental variables to the ordination axes.

Canonical coefficients are conceptually similar to the usual regression coefficients and, as described above, are found by a weighted least squares regression procedure. Standard deviations express the variability of each of the environmental variables The canonical coefficients differ from regression coefficients because the regression is made within the iterations of the weighted averaging, rather than an after- the-fact regression of site scores on a set of environmental variables.  Although the interpretation is similar to multiple regression, canonical coefficients have a larger variance than regression coefficients (ter Braak 1988).   Because of this, canonical coefficients cannot be tested for significance with a Student's t-test.

The canonical coefficients represent the unique contribution of individual variables, as opposed to the simple correlation coefficient between a variable and an ordination axis.  In just the same fashion as multiple regression, the canonical coefficients are unstable if the environmental variables are moderately to strongly intercorrelated (multicollinear).  Simple correlations avoid this problem.

The canonical coefficients are presented in two forms.  The standardized coefficients are based on the environmental variables centered and standardized to unit variance.  The coefficients are also given for the variables in their original units.

5. Final scores for sites and species

Ordination scores (i.e. coordinates on ordination axes; see example) are given for each site and each species.  Like the parent method, RA, these scores are superimposable on the same ordination diagram.  The scaling of these scores depends on the options selected for rescaling the ordination axes.

Two sets of site (row) scores are listed, one set in which site scores are linear combinations of environmental variables (the “LC scores” of Palmer 1993) and one set in which site scores are produced by weighted averaging with the species matrix (the WA scores of Palmer 1993).  Ter Braak (1994) and Palmer describe the derivation and meaning of these scores.

Note that: (1) Each site point falls approximately on the centroid of the species points that occur at that site, so that one may infer which species were most likely to have occurred at a given site.  (2) The species points represent their optima, the peaks of the hump-shaped response surfaces to the environmental gradients (assuming unimodal humps and that important environmental variables were measured).

Weights for sites and species are included in the tables of final scores.  Sites and species are weighted by their totals.  These weights can be useful in interpreting the ordination.  For example, species at the edges of the ordination often have low weights and therefore have little influence on the analysis (ter Braak 1988).

6. Biplot scores for environmental variables

The environmental variables are often represented as lines radiating from the centroid of the ordination.  The biplot scores (see example) give the coordinates of the tips of the lines.  Rules for constructing and interpreting biplots are given in Jongman et al. 1995).  Given the usual methods of scaling the axes of the community ordination (mean score = 0), then the origin of the environmental lines coincides with the origin of the community matrix.

The biplot scores are included in the output tables.  They are also written at the end of the file graphcol.fil, where they are used in constructing the actual biplot in the graphing routine.

The longer the environmental line, the stronger the relationship of that variable with the community.  The position of species points relative to the environmental lines can be used to interpret their relationships.  Imagine the environmental line extended through the whole ordination space.  If perpendiculars are run from each species point to the environmental line, then one can see (1) the approximate ranking of species response curves to that environmental variable and (2) whether a species has a higher-than-average or lower-than-average optimum on that environmental variable (higher than average if the species falls on the same side of the centroid as the tip of the line).     The coordinates for the environmental points are based on the "inter-set correlations," i.e. the simple correlations between the environmental variables and the ordination axes, weighted by the eigenvalues of those axes (ter Braak 1986, 1988)

7. Correlations of environmental variables with ordination axes.

Remember that CCA generates two sets of site scores: (1) the WA scores obtained by weighted averaging of the species scores and (2) the LC scores obtained by regressing the WA scores on the environmental variables then calculating fitted values from the regression equation.  Ter Braak (1986) calls correlations of environmental variables with the WA scores the "interset correlations" and correlations with the LC scores the "intraset correlations."  The intraset correlations will almost always be higher than the interset correlations, because of the dependence of the LC scores on the environmental variables that is built-in in the regression process.  Therefore, the intraset correlations may be helpful as indicators of which environmental variables were more important in structuring the ordination, but they cannot be viewed as an independent measure of the strength of relationship between communities and those environmental variables.

The interset correlations have the same problem, but in a weaker form:  because the ordination axes were constrained by the environmental variables, the inter-set correlations of the environmental with the ordination will be higher than for an unconstrained ordination (i.e. indirect gradient analysis).

For consistency in interpretation, it is reasonable to use the intraset correlations if the LC scores were used as the final scores, and to use the interset correlations if the WA scores were used as the final scores.

Canonical Correspondence Analysis (CCA)

CCA Example Output

CCA Setup

Ordination


Example of CCA Output

AXIS SUMMARY STATISTICS

Total variance ("inertia") in the species data:   2.901

---------------------------------------------------------

                                 Axis 1   Axis 2   Axis 3

---------------------------------------------------------

Eigenvalue                         .423     .336     .231

Variance in species data

     % of variance explained       14.6     11.6      8.0

     Cumulative % explained        14.6     26.1     34.1

Pearson Correlation, Spp-Envt*     .972     .951     .897

Kendall (Rank) Corr., Spp-Envt     .825     .848     .544

---------------------------------------------------------

* Correlation between sample scores for an axis derived from the species data and the sample scores that are linear combinations of the environmental variables.

MULTIPLE REGRESSION RESULTS:

Regression of STANDS in SPECIES space on ENVIRON

----------------------------------------------------------------

                       Canonical Coefficients

         ----------------------------------------------

              Standardized           Original Units

         ----------------------------------------------

Variable  Axis 1  Axis 2  Axis 3   Axis 1 Axis 2 Axis 3  S.Dev

----------------------------------------------------------------

1 smoist   .015   -.060   -.062     .001  -.005  -.005  .122E+02

2 slope    .233   -.450    .174     .032  -.061   .024  .735E+01

3 aspect  -.341    .240   -.146    -.211   .149  -.091  .161E+01

4 elev    -.291   -.057    .103    -.002   .000   .001  .189E+03

5 topo     .034    .212    .112     .025   .156   .083  .135E+01

----------------------------------------------------------------

Scores which are linear combinations of ENVIRON

FINAL SCORES and raw data totals (weights) for  19 STANDS

-------------------------------------------------------------

                                                    Raw Data

               Axis 1       Axis 2       Axis 3      Totals

--------------------------------------------------------------

 1 STAND1      .081841      .076631     -.067482     12.0000

 2 STAND2      .083046      .070388     -.076791      9.0000

 3 STAND3      .082789      .071719     -.074807     14.0000

    “

    " etc.

    "

18 STAN18     -.030652     -.039708     -.007681     24.0000

19 STAN19     -.030963     -.030347      .004816     24.0000

--------------------------------------------------------------

FINAL SCORES and raw data totals (weights) for  50 SPECIES

----------------------------------------------------------

                                                  Raw Data

              Axis 1       Axis 2       Axis 3     Totals

----------------------------------------------------------

1 Agspic      .182352      .030334     -.010032    23.0000

2 Agrint      .264496     -.313354      .540362     4.0000

3 Arilon      .649725     -.236825     -.123769    10.0000

4 Brojap      .106945     -.210955     -.069684    13.0000

5 Brotec      .287086     -.147118      .063540    12.0000

    "

    " etc.

    "

48 Carnut    -.315682     -.281733      .035865     5.0000

49 Melalb    -.190182     -.397242     -.057754     3.0000

50 Koecri     .232855     -.277317     -.069220     1.0000

----------------------------------------------------------

CORRELATIONS AND BIPLOT SCORES for   5  ENVIRON

----------------------------------------------------------

             Correlations*               Biplot Scores

Variable Axis 1  Axis 2  Axis 3     Axis 1  Axis 2  Axis 3

----------------------------------------------------------

1 smoist  -.447   -.086   -.335      -.383   -.061   -.184

2 slope   -.258   -.855    .398      -.221   -.608    .218

3 aspect  -.689   -.641   -.008      -.590   -.455   -.004

4 elev    -.865   -.015    .519      -.740   -.011    .284

5 topo    -.051    .422    .837      -.044    .300    .459

----------------------------------------------------------

 * Note: Correlations are "intraset correlations" of ter Braak (1986).

Monte Carlo test: null hypothesis is no relationship between matrices

MONTE CARLO TEST RESULTS -- EIGENVALUES

------------------------------------------------------------

                              Randomized data

            Real data    Monte Carlo test,  499 runs

           ------------  ---------------------------

  Axis     Eigenvalue    Mean    Minimum   Maximum       p

------------------------------------------------------------

     1          .423      .264      .170      .422     .0020

     2          .336      .191      .122      .322     .0020

     3          .231      .142      .089      .223     .0020

------------------------------------------------------------

p = proportion of randomized runs with eigenvalue greater

    than or equal to the observed eigenvalue; i.e.,

p = (1 + no. permutations >= observed)/(1 + no. permutations)

Canonical Correspondence Analysis (CCA)

CCA Output

CCA Setup

Ordination


Detrended Correspondence Analysis (DCA, DECORANA)

Ordination | Detrended Correspondence Analysis (DCA, DECORANA)

Detrended correspondence analysis (DCA; Hill and Gauch 1980) is an eigenanalysis ordination technique based on reciprocal averaging (RA; Hill 1973).  DCA is geared to ecological data sets and the terminology is based on samples and species.  If you have other kinds of data, DCA may not be for you.  If you want to use it for other types of data you must translate the terminology in the output to match your particular data set.

Like RA, DCA ordinates both species and samples simultaneously. The arch that is almost inevitable with more than one dimension in RA is squashed with DCA.  This is done by dividing the first axis into segments, then setting the average score on the second axis within each segment to zero. Another fault of RA, the tendency to compress the axis ends relative to the middle, is also corrected with DCA. This is done by rescaling the axis to equalize as much as possible the within-sample variance of species scores along the sample ordination axis.

Also like its parent technique, DCA implicitly uses a chi-squared distance measure.  Although DCA and CA do not require an explicit calculation of a matrix of chi-squared distances, it can be shown that RA is equivalent to an eigenanalysis of distances in a space defined by the chi-squared distance metric (Chardy et al. 1976).

DCA has been quite popular in ecology.  Recently its faults have been made more apparent.  Minchin (1987a) found lack of robustness and erratic performance of DCA as compared to NMS using a wide variety of simulated data sets based on a range of assumptions about the underlying distributions of species along environmental gradients.  He attributed this poor performance to two factors, properties of the implicit dissimilarity measure, and the behavior of the detrending and rescaling processes.  The underlying chi-square distance measure has been criticized by others (see Clarke 1993 and references therein).  Minchin, as well as others, (e.g. Beals 1984; Ter Braak 1986) have also expressed reservations about the desirability of detrending and rescaling all ordinations to fit a single mold.

DCA is performed with a modified version of DECORANA from the Cornell Ecology Program series (Hill 1979).  Earlier versions of DECORANA were unstable because of lax criteria for stability and a bug in the rescaling algorithm (Oksanen and Minchin 1997).  These problems have been corrected in PC-ORD. We now apply Oksanen and Minchin's "super strict" criteria of tolerance=0.0000001 and maximum number of iterations = 999.  The bug in the rescaling algorithm that is present in DECORANA beginning with the Cornell Ecology Program series and repeated by the versions in CANOCO and PC-ORD has been corrected in PC-ORD.  This bug caused sensitivity of ordination results to sample order, mainly on the third axis and higher.

DCA Setup

DCA Output

DCA Example

Ordination


DCA Setup

The following items are selected in the DCA Setup dialog.

Downweighting of Rare Species

In some cases you may wish to give rare species less weight.  This can be done here or with data standardizations or relativizations.  If you select downweighting of rare species in DCA, then the abundances of species rarer than Fmax/5 (where Fmax is the frequency of the commonest species) are downweighted in proportion to their frequency.  Species commoner than Fmax/5 are not downweighted.

Rescale Axes

The default is to rescale your axes. If you turn rescaling off, the first axis of DCA is identical to that of RA.  The default is recommended.

In DCA the axes are scaled by a non-linear rescaling based on the average standard deviation of species turnover (see Gauch 1982).  The PC-ORD version of DCA follows the original version of DECORANA in multiplying the standard deviations by 100 and shifting the scales such that all scores are positive.

Rescaling Threshold

Axes shorter than the threshold value will not be rescaled.  The default value is 0.

Number of Segments

The default value is 26, and it is recommended that you not vary it.

DCA Example

Detrended Correspondence Analysis (DCA)

DCA Output

Ordination


DCA Output

The first part of the output is a record of the options you selected.  Then, for each axis the following information is given (see example):

List of residuals.  The eigenvectors are found by iteration and the residuals indicate how close a given vector is to an eigenvector.  A vector is deemed to be an eigenvector if it has a residual of less than 10^-7.  If this value is not reached in 999 iterations, the current trial vector is taken to be the eigenvector and a warning is printed that the residual exceeds the desired tolerance.

Eigenvalue.  The eigenvalue for the first axis has the traditional interpretation of eigenvalues.  However, the processes of rescaling and detrending destroys the correspondence between the eigenvalue and the structure along that axis.  Thus eigenvalues in DCA cannot be interpreted as proportions of variance explained.  To evaluate the effectiveness of your ordination, we recommend an after-the-fact coefficient of determination between Relative Euclidean distance in the unreduced species space and Euclidean distance in the ordination space (Graph | Ordination | Statistics | Percent of Variance ).

Length of segments.  These lengths indicate the degree of rescaling that occurred.  For example, small values for the middle segments suggests that a gradient was compressed at the ends before rescaling.  The rescaling routine equalizes the length of the segments, and these new segment lengths are then printed.

Following the axis-by-axis output, species and sample scores for each axis are printed.  Graphs and correlations with ordination axes are then available through Graph in the menu.

DCA Example

Detrended Correspondence Analysis (DCA)

DCA Setup

Ordination


DCA Example

Alaskan Epiphytes

Number of non-zero data items:      563

No downweighting

Axes are rescaled

Number of segments: 30

Threshold:  0.00

---------- Axis 1 ----------

0.0470212474 = residual at iteration   0

0.0035524392 = residual at iteration   1

0.0000393731 = residual at iteration   2

0.0000004670 = residual at iteration   3

0.0000000229 = residual at iteration   4

0.1880371273 = eigenvalue

Length of gradient:     2.383

Length of segments:  0.18  0.22  0.25  0.25  0.23  0.22  0.21  0.19  0.18  0.17

Length of segments:  0.15  0.14

Length of gradient:     2.496

Length of gradient:     2.632

Length of segments:  0.16  0.18  0.20  0.20  0.20  0.20  0.20  0.20  0.19  0.19

Length of segments:  0.18  0.18  0.17  0.17

Length of gradient:     2.671

---------- Axis 2 ----------

0.0280125942 = residual at iteration   0

0.0034079745 = residual at iteration   1

0.0002317748 = residual at iteration   2

0.0000110374 = residual at iteration   3

0.0000008595 = residual at iteration   4

0.0000000466 = residual at iteration   5

0.0930673778 = eigenvalue

Length of gradient:     1.148

Length of segments:  0.07  0.07  0.08  0.09  0.11  0.12  0.14  0.15  0.16  0.16

Length of gradient:     1.302

Length of gradient:     1.304

Length of segments:  0.11  0.12  0.13  0.14  0.14  0.14  0.14  0.14  0.13  0.12

Length of gradient:     1.315

---------- Axis 3 ----------

0.0157066677 = residual at iteration   0

0.0036390589 = residual at iteration   1

0.0003493103 = residual at iteration   2

0.0000179558 = residual at iteration   3

0.0000016785 = residual at iteration   4

0.0000000898 = residual at iteration   5

0.0523688756 = eigenvalue

Length of gradient:     1.158

Length of segments:  0.12  0.13  0.14  0.13  0.13  0.12  0.11  0.10  0.09  0.09

Length of gradient:     1.199

Length of gradient:     1.208

Length of segments:  0.10  0.12  0.14  0.14  0.13  0.13  0.12  0.12  0.11  0.10

Length of gradient:     1.197

Alaskan Epiphytes

SPECIES SCORES

N    NAME     AX1  AX2  AX3 |    RANKED 1      |    RANKED 2

                            |   EIG=0.188      |   EIG=0.093      |

 1 ALSA       149   79    6 | 19 PASQ      476 | 26 RARO      923 |

 2 BRYOC      -26  245  -10 |  6 CECA      452 | 17 PAHY      702 |

 3 BRspp      120  146   68 | 13 HYIN      365 | 23 PLHE      302 |

 4 CAHU       301  218  149 | 24 PLLA      353 | 29 USSPP     254 |

 5 CALO       257  206  208 | 17 PAHY      309 |  2 BRYOC     245 |

 6 CECA       452  226  -23 | 11 HYEN      302 |  6 CECA      226 |

 7 CECH       206  201  212 |  4 CAHU      301 | 20 PASPP     223 |

 8 CLRA      -174   -5  183 | 26 RARO      297 |  4 CAHU      218 |

 9 CLSPP       -8   25  280 | 14 HYOC      280 | 12 HYIM      218 |

10 HYDU         0  -41  -55 | 25 PLNO      277 |  5 CALO      206 |

11 HYEN       302  -42   46 |  5 CALO      257 |  7 CECH      201 |

12 HYIM       248  218  408 | 22 PLGL      256 | 16 PASA      182 |

13 HYIN       365   90  -13 | 12 HYIM      248 |  3 BRspp     146 |

14 HYOC       280   24  -50 | 23 PLHE      232 | 18 PASU      143 |

15 HYVI       175   41  300 |  7 CECH      206 | 28 USLO      143 |

16 PASA       186  182  284 | 16 PASA      186 | 13 HYIN       90 |

17 PAHY       309  702 -207 | 21 PAHYP     180 | 19 PASQ       80 |

18 PASU       177  143 -191 | 18 PASU      177 | 25 PLNO       80 |

19 PASQ       476   80  -60 | 15 HYVI      175 |  1 ALSA       79 |

20 PASPP       32  223 -214 |  1 ALSA      149 | 15 HYVI       41 |

21 PAHYP      180 -162 -177 |  3 BRspp     120 |  9 CLSPP      25 |

22 PLGL       256  -30  211 | 20 PASPP      32 | 14 HYOC       24 |

23 PLHE       232  302   67 | 28 USLO       10 |  8 CLRA       -5 |

24 PLLA       353 -115  -29 | 27 SPGL        0 | 27 SPGL      -18 |

25 PLNO       277   80   84 | 10 HYDU        0 | 22 PLGL      -30 |

26 RARO       297  923  829 |  9 CLSPP      -8 | 10 HYDU      -41 |

27 SPGL         0  -18   99 | 29 USSPP     -19 | 11 HYEN      -42 |

28 USLO        10  143  270 |  2 BRYOC     -26 | 24 PLLA     -115 |

29 USSPP      -19  254  -19 |  8 CLRA     -174 | 21 PAHYP    -162 |

Alaskan Epiphytes

SAMPLE SCORES - WHICH ARE WEIGHTED MEAN SPECIES SCORES

N    NAME     AX1  AX2  AX3 |    RANKED 1      |    RANKED 2

                            |   EIG=0.188      |   EIG=0.093      |

 1 Plot01     181   31   76 | 28 Plot28    267 | 36 Plot36    131 |

 2 Plot02     167   58   89 | 24 Plot24    239 |  8 Plot08    130 |

 3 Plot03       0    0  111 | 50 Plot50    233 | 37 Plot37    129 |

 4 Plot04      95   58   74 | 29 Plot29    227 | 30 Plot30    118 |

 5 Plot05     131    9   61 | 31 Plot31    206 |  9 Plot09    107 |

 6 Plot06     116   88  116 |  8 Plot08    201 | 33 Plot33    102 |

 7 Plot07     138   93   64 | 27 Plot27    198 | 21 Plot21    101 |

 8 Plot08     201  130  119 | 14 Plot14    197 | 44 Plot44    100 |

 9 Plot09     131  107   80 | 25 Plot25    196 | 16 Plot16     99 |

10 Plot10     177   80   94 | 13 Plot13    195 | 28 Plot28     98 |

11 Plot11     181   93   60 | 15 Plot15    186 | 13 Plot13     96 |

12 Plot12     121   85   60 | 21 Plot21    181 | 48 Plot48     95 |

13 Plot13     195   96   46 | 11 Plot11    181 | 11 Plot11     93 |

14 Plot14     197   22   39 |  1 Plot01    181 | 46 Plot46     93 |

15 Plot15     186   61   64 | 34 Plot34    179 |  7 Plot07     93 |

16 Plot16     111   99   39 | 10 Plot10    177 | 43 Plot43     90 |

17 Plot17     147   83   54 | 36 Plot36    170 | 22 Plot22     89 |

18 Plot18     113   67   70 | 49 Plot49    170 | 29 Plot29     89 |

19 Plot19     162   84   42 | 30 Plot30    168 | 31 Plot31     89 |

20 Plot20     108   47   49 |  2 Plot02    167 |  6 Plot06     88 |

21 Plot21     181  101   63 | 42 Plot42    164 | 40 Plot40     87 |

22 Plot22      92   89   81 | 19 Plot19    162 | 45 Plot45     87 |

23 Plot23      68   71   84 | 40 Plot40    161 | 32 Plot32     86 |

24 Plot24     239   72  106 | 39 Plot39    160 | 12 Plot12     85 |

25 Plot25     196   64  115 | 47 Plot47    157 | 19 Plot19     84 |

26 Plot26     132   59   32 | 45 Plot45    151 | 17 Plot17     83 |

27 Plot27     198   65   53 | 41 Plot41    151 | 10 Plot10     80 |

28 Plot28     267   98   58 | 17 Plot17    147 | 35 Plot35     77 |

29 Plot29     227   89   67 | 43 Plot43    144 | 42 Plot42     74 |

30 Plot30     168  118   42 | 44 Plot44    140 | 24 Plot24     72 |

31 Plot31     206   89   47 | 37 Plot37    138 | 23 Plot23     71 |

32 Plot32     133   86   43 |  7 Plot07    138 | 47 Plot47     69 |

33 Plot33      83  102   50 | 35 Plot35    135 | 39 Plot39     68 |

34 Plot34     179   57   48 | 32 Plot32    133 | 18 Plot18     67 |

35 Plot35     135   77   43 | 26 Plot26    132 | 27 Plot27     65 |

36 Plot36     170  131   53 | 46 Plot46    132 | 25 Plot25     64 |

37 Plot37     138  129   65 |  9 Plot09    131 | 38 Plot38     61 |

38 Plot38      98   61   63 |  5 Plot05    131 | 15 Plot15     61 |

39 Plot39     160   68   66 | 48 Plot48    123 | 26 Plot26     59 |

40 Plot40     161   87   77 | 12 Plot12    121 |  4 Plot04     58 |

41 Plot41     151   57  101 |  6 Plot06    116 |  2 Plot02     58 |

42 Plot42     164   74   97 | 18 Plot18    113 | 41 Plot41     57 |

43 Plot43     144   90   16 | 16 Plot16    111 | 34 Plot34     57 |

44 Plot44     140  100  100 | 20 Plot20    108 | 50 Plot50     54 |

45 Plot45     151   87   18 | 38 Plot38     98 | 20 Plot20     47 |

46 Plot46     132   93   27 |  4 Plot04     95 |  1 Plot01     31 |

47 Plot47     157   69   68 | 22 Plot22     92 | 14 Plot14     22 |

48 Plot48     123   95   66 | 33 Plot33     83 |  5 Plot05      9 |

49 Plot49     170    3    0 | 23 Plot23     68 | 49 Plot49      3 |

50 Plot50     233   54   57 |  3 Plot03      0 |  3 Plot03      0 |

Detrended Correspondence Analysis (DCA)

DCA Setup

Ordination


Non-metric Multidimensional Scaling (NMS)

Ordination | NMS

Non-metric Multidimensional Scaling (NMS, MDS, NMDS, or NMMDS) is an ordination method that is well suited to data that are nonnormal or are on arbitrary, discontinuous, or otherwise questionable scales.  For this reason, NMS should probably be used in ecology more often than it is.  This method can be used both as an ordination technique and as a method for assessing the dimensionality of a data set (plot minimum stress against k, the number of dimensions in the ordination space).  NMS differs fundamentally in design and interpretation from other ordination techniques (Kruskal and Wish 1978; Clarke 1993).  See Clarke (1993) for an excellent summary of the advantages and uses of NMS.  If you are unfamiliar with NMS, be sure to read the warnings section at the end of this topic.

An advantage of NMS is that, being based on ranked distances, it tends to linearize the relation between environmental distance and sociological distance (Beals 1984), relieving the "zero-truncation problem", a problem which plagues all ordinations of heterogeneous data sets.  Possible disadvantages include difficulties in detecting discontinuities and failing to find the best solution (minimum stress) because of intervening local minima.  Nevertheless, NMS has performed well with both simulated gradients and field data.

NMS is an iterative search for a ranking and placement of n entities on k dimensions (axes) that minimizes the stress of the k-dimensional configuration.  The calculations are based on an n x n distance matrix calculated from the n x p-dimensional main matrix, where n is the number of rows and p is the number of columns in the main matrix.  "Stress" is a measure of departure from monotonicity in the relationship between the dissimilarity (distance) in the original p-dimensional space and distance in the reduced k-dimensional ordination space.

Graphical output is provided from two sources:  graphs generated as NMS is running and graphs generated by Graph in the PC-ORD menu.  The latter includes the usual options for overlays of attributes or variables from the secondary matrix. Simultaneous plotting of column variables (usually species) in the ordination space is available if you select Calculate Scores by Weighted Averaging in the NMS Setup dialog.

Program NMS in PC-ORD is largely based on Mather's program NMMDS (Mather 1976; includes listing of source code).  The central computational algorithm (steepest descent minimization to find minimum stress) in NMMDS is based on Kruskal (1964b).

Global NMS is the form of NMS implemented in PC-ORD.  Local NMS (Prentice 1977, 1980; Minchin 1987a) uses a somewhat different criterion than described above. For each sample unit, the rank order of distances to all other sample units in the ordination space is matched as well as possible with the n-dimensional dissimilarities between that sample unit and the other sample units.  This allows the possibility that the rate of increase in sociological distance with environmental distance may differ in different parts of the ordination space.  The consequences of the choice of local vs. global NMS have not been fully explored in the literature.

Individual rows of your main matrix become points in the ordination space.  For a transpose NMS analysis (columns instead of rows), first transpose your matrix.

The Basic Method

Definitions:

X = coordinates of n sample units or entities in a k-dimensional space

Δ = matrix of dissimilarity coefficients from the original data

δij = elements of Δ

D = matrix of interpoint distances in the k-space

dij = elements of D

Procedure:

1. Calculate dissimilarity matrix Δ

2. Assign sample units to starting configurations in the k-space (define an initial X).

3. Normalize X by subtracting the column means and dividing by the standard deviations:

4. Calculate D, containing Euclidean distances between sample units in k-space

5. Rank elements of Δ in ascending order

6. Put the elements of D in the same order as Δ

7. Calculate Dhat (containing elements ij, these being the result of replacing elements of D which do not satisfy the monotonicity constraint.  See Kruskal and Wish (1978) or Mather (1976) for the method of calculating the replacement values.  The amount of movement required to achieve monotonicity is measured by the difference between d and hat.  The sum of these squared differences is the basis for evaluating stress (the opposite of "fit").

8. Calculate raw stress, S*

Note that S* measures the departure from monotonicity.  If S* = 0 there is perfect monotonicity.

9. Because raw stress would be altered if the configuration of points were magnified or reduced, it is desirable to standardize ("normalize") it.  This can be done in several ways, but PC-ORD follows Mather (1976) in using:

In some versions of NMS the standardized stress is expressed on a scale from 0-1.  PC-ORD follows Mather (1976) in rescaling stress from 0-100.

10. Now we try to minimize S by changing the configuration of the sample units in the k-space.  This is done with the steepest-descent minimization algorithm.   Other algorithms are available that accomplish the same thing.  For the method of steepest descent, we first calculate the "negative gradient of stress" for each point i (i.e. the direction of movement of each point i in the lth dimension of the k-space in which stress will decrease fastest).  The gradient vector (i.e. a vector of length k containing the movement of each point h in each dimension l) is calculated by:

The letter i indexes one of the n points, g indexes another of the n points, l indexes a particular dimension, and h indexes a third point, the point of interest which is being moved on dimension l.  So ghl indicates a shift for point h along dimension l.  The Δhi and Δhj are Kroekner deltas which have the value 1 if i and h or h and j indices are equal, otherwise they have the value 0.  If the strict lower triangles of the distance matrices have been used then the Kroekner deltas can be omitted.

11. The amount of movement in the direction of the negative gradient is set by the step size, α, which is normally about 0.2 initially.  The step size is recalculated after each step such that the step size gets smaller as reductions in stress become smaller.  The manner in which the step size is adjusted is complex and need not concern most users.

12. The iterations proceed (go to step 3) until either (a) a set maximum number of iterations is reached or (b) a criterion of stability is met.  At the stopping point, the stress should be acceptably small.  One way to assess this is by calculating the relative magnitude of the gradient vector g:

Mather (1976) suggested that mag(g) reach a value of 2-5% of its initial value for an arbitrary configuration.  This is often not achievable.

Suggested Procedure

The following procedure determines appropriate dimensionality, statistical significance, and seeks to avoid local minima.  An alternative to this procedure is to use the autopilot mode with NMS in PC-ORD, which automates this process.

1. Preliminary runs:  request a 6-dimensional solution, stepping down to a 1-dimensional solution, an instability criterion of 0.0005, and 200 iterations, 10 runs with your real data and 20 runs with randomized data.  The latter provide the basis for a Monte Carlo test of significance at each dimensionality.

2. Scree plot.  Plot the final stress vs. the number of dimensions; see Graph | NMS Scree Plot .  Do this in PC-ORD by selecting Graph | NMS Scree Plot just after running NMS.

3. Select a number of dimensions beyond which additional dimensions provide only small reductions in stress.  Check for a better-than-random solution by using the results of the Monte Carlo test.  Note that the first axis with randomized community data is often nearly as strong or stronger than the real data, even when the pattern in the real data is strong.  This is because the randomization scheme creates rows with unequal abundances – some rows can result that have much higher or lower totals than any rows in the real data.  Thus a 1-D NMS solution from the shuffled data tends to describe variation in row totals.

4. Check either (a) the plot of stress vs. iteration for stability of the solution at the number of dimensions you decided on or (b) the final instability value for the chosen solution, as listed in the numerical output from NMS.

5. Rerun NMS requesting the following

•	the number of dimensions you decided on,

•	the starting configuration that worked best

•	no step-down in dimensionality

•	one real run

•	no Monte Carlo test (no randomized runs).

6. Calculate the correlation between distance in the ordination space and distance in the original p-dimensional space.  You can do this as an option in the graphing program in PC-ORD (Graph | Ordination | Statistics | Percent of Variance).  Select the same distance measure for that statistic as you did for the NMS ordination.

Warnings

Although NMS is a very powerful and useful program, it is important to understand how its use and interpretation differ from other ordination methods.

1. The first dimension in a 2-D solution is not the same as the first dimension in a 3-D solution or a 1-D solution.  For a given number of dimensions, the solution for a particular axis is unique.

2. Axis numbers are arbitrary, so that the percent of variance on a given axis does not necessarily form a descending series with increasing axis number.

3. An appropriate number of dimensions (axes) should be determined by plotting or tabulating final stress vs. the number of dimensions.  Choose a number of axes beyond which reductions in stress are small.  However, graphics in PC-ORD are limited to three axes.

NMS Setup

NMS Autopilot

Ordination


NMS Autopilot

In version 4 of PC-ORD an "autopilot" mode was added to NMS to assist the user in making multiple runs, choosing the best solution at each dimensionality, and testing for significance.  Everything that is done in autopilot mode can also be done by adjusting the parameter settings and option selections in PC-ORD.  When you select autopilot mode all options are set automatically except for the distance measure, which you still must select.  For community data we recommend Sorensen (Bray-Curtis) distance.

Table:  Thoroughness settings for autopilot mode in NMS in PC-ORD version 4.

	Thoroughness setting
Parameter	Quick and  dirty	Medium	Slow and thorough
Maximum number of iterations	75	200	400
Instability criterion	0.001	0.0001	0.00001
Starting number of axes	3	4	6
Number of real runs	5	15	40
Number of randomized runs	20	30	50

Autopilot Procedure

Autopilot in NMS uses the following procedure:

Multiple runs with real data.  Perform x real runs, each with a random starting configuration.  A “run” consists of a series of solutions, stepping down in dimensionality from the highest number of axes to one axis.

For each dimensionality, the best starting configuration from the x real runs is saved to disk.  These files are named config1.gph, config2.gph, etc., for the 1-dimensional, 2-dimensional, etc. configurations respectively

Multiple runs with randomized data.  Perform y runs with randomized data.  Before each run the data from the main matrix are shuffled within columns.  Each run uses a different random starting configuration.. Statistics on the final stress for each dimensionality are accumulated.

Select the best solutions.  The best solution is defined by a particular starting configuration and number of dimensions.  PC-ORD selects the best solution for each dimensionality as that with the lowest final stress from a real run.

Select the dimensionality.  PC-ORD selects the dimensionality by comparing the final stress values among the best solutions, one best solution for each dimensionality.  Additional dimensions are considered useful if they reduce the final stress by 5 or more (on a scale of 0-100).  PC-ORD selects the highest dimensionality that meets this criterion.

At that dimensionality, the final stress must be lower than that for 95% of the randomized runs (i.e. p <= 0.05 for the Monte Carlo test).  If this criterion is not met, PC-ORD does not accept that solution or it chooses a lower-dimensional solution, providing that it passes the Monte Carlo test.

NMS Setup

Non-metric Multidimensional Scaling (NMS)

Ordination


NMS Setup

The NMS setup dialog allows you to select the distance measure, the method for obtaining starting coordinates, parameters controlling the flow of the analysis, and output options.

NMS Autopilot

NMS Distance Measure

NMS Parameters

NMS Output Options

Calculate Scores for Column Variables by Weighted Averaging

NMS Starting Coordinates

NMS Random Numbers

NMS


NMS Distance Measure

All of the standard distance measures in PC-ORD are available for NMS:   See Distance Measures.

NMS Setup

NMS


NMS Starting Coordinates

The starting configuration can be generated by random numbers (the default) or you can supply the starting configuration (Read from Graph File).  If you choose the latter option, you enter into the edit box the name of  file containing the starting configuration (Coordinates File).

Follow these steps to use a starting configuration:

1. Run an ordination on your data set using another method.

2. Select File | Save | Graph after the ordination and save the ordination scores by renaming GRAPHROW.GPH to a name of your choice, e.g.: SCORES.GPH.

3. Select NMS from the main menu.

4. Select "Read from Graph file" to supply your own starting configuration.

5. When prompted, enter the file name where you saved the ordination scores (SCORES.GPH in this example).

NMS Setup

NMS


NMS Parameters

Number of Axes (max. = 6, default = 3)

This value (k) is critical because it sets the dimensionality of the solution.  A measure of goodness of fit (the stress) of the k-space configuration is provided.  This value is important because axis scores with NMS depend on the number of axes selected.  See warnings in NMS.

Number of Runs with Real Data

You can make multiple runs with real data when different random starting configurations are used (default = 10).  The purpose of multiple runs with real data is to help you avoid local minima by trying repeatedly for the lowest possible stress.  The only penalty for setting this number high is the computation time.  If you choose more than one run, the best run (lowest final stress) is rerun and the coordinates written to the graph file.

Stability Criterion

Instability is calculated as the standard deviation in stress over the preceding x iterations, where x is set by the user (see next section).  If the instability is less than the cutoff value, iterations are stopped and the solution is considered final.  Instabilities of 0.005 are usually achieved rapidly, while instabilities of 0.0001 are usually reached slowly or not at all.  Note that an instability of zero can never be reached, unless the finals stress is zero.  Zero instability is not achieved because NMS always nudges the configuration in the direction of the gradient vector.  So even when a local minimum is found, NMS is exploring for better solutions by making small adjustments in the configuration.   The degree of stability that can be attained is therefore set by the algorithm that determines the magnitude of the gradient vector.

Iterations to Evaluate Stability

Instability of the solution is determined be evaluating fluctuations in stress over some number of iterations (default = 10).  Setting this number too low will result in a premature conclusion of a stable solution.  Setting the number too high may result in unnecessary iterations.

Maximum Number of Iterations (max. = 500, default = 100)

The ideal number varies depending on the speed of your computer, the number of entities being ordinated, your patience, and whether or not you provide a starting configuration.  You have the option of plotting of stress as a function of iteration, which helps you to evaluate whether or not you had reached a minimum stress.

In versions 3 and lower of PC-ORD the number of iterations was fixed by the user.  In version 4, the maximum number of iterations is used only if a predefined criterion of stability (see above) is not met

This change has several beneficial effects:

•	The computer doesn’t waste time calculating many additional iterations that are not improving the solution.  This time saved can be used more profitably to explore starting configurations and perform more precise Monte Carlo tests.

•	Tracking instability allows the program and the analyst to readily detect and avoid unstable solutions, where stress oscillates between low and high stress over a small number of iterations.

•	The analyst can tune the thoroughness of the search for good solutions to the goals of the analysis, the available time, and the speed of the computer.  The method allows relatively rapid assessments with large data sets or slow, exhaustive searches if time allows.

Step Down in Dimensionality

NMS will produce a series of ordinations, stepping down in dimensionality, or holding the number of dimensions constant.  If you select this checkbox, NMS first constructs and ordination with the maximum number of dimensions requested above, then redoes the ordination with a smaller number of dimensions, reusing the same starting coordinates.  Normally you will want either to hold the dimensions constant or step down by one dimension per cycle; but this can be increased (see below).

Initial Step Length (rate of movement toward minimum stress)

The default value of 0.2 should be changed judiciously, if at all.  This step length is used as a starting point, and is modified by the program, depending on the rate at which adjustments are being made toward minimum stress.  See Mather (1976) for more details.

Starting Coordinates

NMS in PC-ORD allows you to specify a file containing starting coordinates or to generate them by random.  For more information see NMS Starting Coordinates.

NMS Setup

NMS


NMS Output Options

The following output is always produced:

•	List of options selected.

•	Final stress after the requested number of iterations.

•	Graph files

Because users normally make multiple NMS runs within a single request for analysis, a procedure must be used for selecting which run to write to files for graphing.  The following table describes the rules PC-ORD uses for making this decision.  The symbol "-" indicates that the option is irrelevant given the choices listed for other options.

If you select…	Then…
Auto- pilot	Number of Real Runs	Number of Randomized Runs	Step down in Dimensions	The solution written to the graph file is based on…
On	-	-	-	Final stress, monte carlo results, and amount of reduction in stress with addition of axes
Off	1	-	yes	Maximum number of dimensions requested
Off	1	-	no	Single solution that you requested
Off	> 1	≥ 19	yes	Final stress, monte carlo results, and amount of reduction in stress with addition of axes
Off	> 1	≥ 19	no	Final stress, monte carlo results, and amount of reduction in stress with addition of axes
Off	> 1	< 19	yes	Final stress and amount of reduction in stress with addition of axes
Off	> 1	< 19	no	Final stress

Write Distance Matrix

The lower triangle of the distance matrix is written to the output file.  Beware of voluminous output if you are using a large data set.  The matrix is printed row-wise, with individual rows wrapping across the page.

Write Starting Coordinates

The initial configuration is listed for the number of dimensions selected for the number of axes.  If you chose a random configuration, you see the results of the random number generator.  Otherwise, you see the initial values read from a graph file.

List Stress, etc. for Each Iteration

The following values are available at each step toward minimizing the stress:  stress, step length (starting value set by step length) and the relative magnitude of the gradient vector g ("MAG(G)"; see Mather 1976, p.336).  The stress value, instability, and MAG(G) can be used to indicate when a minimum has been found; however, they do not ensure that the minimum is global and not local.  This listing is not printed by default in the PC-ORD version to reduce paper waste.

Note that instability is not listed for the first nine steps, because its calculation is based on the fluctuations in stress in the latest 10 steps (see NMS Parameters for a description of the instability statistic).

Example:

**************** 2-dimensional solution ****************

List of stress, step length, and magnitude of the gradient

    vector at each iteration.

  Step    Stress Instability  StepLength    Mag(G)

     1  43.23488                 0.20000   0.000468189

     2  41.28541                 0.20000   0.000040363

     3  41.37618                 0.17045   0.000084628

     4  41.01664                 0.09110   0.000012026

     5  40.94319                 0.08100   0.000040712

     6  40.68351                 0.07230   0.000039850

     7  40.33906                 0.06715   0.000044444

     8  39.95139                 0.08115   0.000034286

     9  39.55381                 0.08778   0.000046967

    10  39.16057  0.01083044     0.08976   0.000066676

    11  38.72273  0.00875749     0.08268   0.000068978

    12  38.28082  0.00994596     0.07732   0.000063339

    13  37.79143  0.01077547     0.08166   0.000062975

    14  37.20531  0.01192640     0.10161   0.000091705

"

etc.

Write Final Configuration

With this option, the final configuration is written for a given number of dimensions.  This "final configuration" consists of the scores of each object on each of the k dimensions.  Because you can reduce the number of dimensions by one with each cycle, there are typically several "final" configurations, one for each number of dimensions.  For example, if the starting number of dimensions is three, and the dimensions are reduced by one each cycle, then three "final" configurations are written.

Example:

Final configuration (ordination scores) for this run

    Plots             Axis

Number Name              1          2

     1 ASH921       0.2432     1.0168

     2 ASH922       0.3328     0.6600

     3 ASH923       0.2822     1.1272

     4 DEY248      -0.1676    -0.4281

"

etc.

Plot Distance vs. Dissimilarity

Stress can be visualized as the departure from monotonicity in the following plot:  distance in the original n-dimensional space (dissimilarity) vs. distance in the ordination space.  This is known as a Shepard diagram.  This plot can be obtained if you have less than 45 objects (<1000 non-redundant distances) being ordinated (larger numbers result in so much overprinting that the sense of the plot is obscured).  Each asterisk on the plot represents a given pair of objects.  The ranked distance of that pair is indicated by the number next to the asterisk.

If you have more than 45 objects, or if you wish to make a higher quality graph, you can use the values written in the file “shepard.txt” in the default directory.  For example, you can copy the text from this file into a spreadsheet, then parse the fixed-width fields into separate columns, then make a scatterplot using the graphing functions in your spreadsheet software.  Note that this file is overwritten each time you use NMS in PC-ORD, so be sure to rename the file if you wish to keep it.

The file contains three columns: “pair,” a sequential numbering of pairs of values from the distance matrix, ranked by increasing distance; “original” the distance in the original p-dimensional space; and “ordination,” the Euclidean distance in the ordination space.

The plot of distance vs. dissimilarity and the associated file, shepard.txt, are produced only if you are doing a single NMS run.  In other words, you must have set the “number of real runs” with autopilot off.  Usually you would do this after you had settled on a particular starting configuration.

Monte Carlo Test

A checkbox on the NMS Output tab in the NMS Setup dialog provides for a Monte Carlo test.  If you select this option, you will be asked in a subsequent dialog for the number of randomized runs.  The Monte Carlo test randomizes your data by randomly reallocating elements within columns of your main matrix.  The randomization is the same as is used in the data shuffling procedure (see Modify data | Shuffle data).

Compare the final stress in your real data with that from multiple runs with randomized data.  This helps you to evaluate the strength of the patterns in your data.  This comparison is aided by graphing a scree plot (see Graph | NMS Scree Plot).

The Summary of Multiple Runs (below) includes a p-value based on the Monte Carlo test.

Varimax Rotation

NMS, like the other ordination programs in PC-ORD, writes a file of coordinates for the ordination graphing.  For NMS, you can write this file with either the unrotated, raw scores, or the scores after varimax rotation.  The purpose of varimax rotation is to maximize the loadings of individual variables on the dimensions of the reduced ordination space.  It is an orthogonal rotation, in that the perpendicularity of the coordinate system is preserved.  The version of varimax rotation used in NMS is a simultaneous rotation, in that all axes that define the ordination space are rotated simultaneously.  Varimax rotation has had wide use in the social sciences but has seldom been used in ecology.  Consult textbooks on multivariate analysis for more information on varimax rotations.

Before version 4.07, varimax rotation was applied to the first NMS run at the highest dimensionality requested.  Beginning with version 4.07 varimax rotation is applied to final run, if more than one “real run” is requested.  Those rotated scores are then written into the graph file.  If a single run is requested, varimax rotation is applied at the highest dimensionality requested, and those scores are written into the graph file.  If NMS autopilot is used, varimax rotation is not used.  You can, however, rerun the best solution from the autopilot and request varimax rotation at that time.

Run Log

Each entry in the log represents a separate ordination with the indicated number of dimensions.  In the example the user requested 20 real runs and 50 randomized runs, each run stepping down from 6 dimensions to 1 dimension.  Thus a total of 6 x (20 + 50) = 420 ordinations were performed.  The best ordinations with non-randomized data are indicated at right, along with the names of the file containing the scores.  Ordinations not meeting the instability criterion are flagged with an asterisk (*).

See Example Run Log

Plot Stress vs. Iteration

A plot of stress vs. iteration number (example below), for every ordination space that is constructed.  If you selected more than 100 iterations then this plot is broken into sections that are printed consecutively. To print it on regular-width paper, select a small font.

Summary of Multiple Runs

If you request more than one run, final stress values are summarized the end of the output.  The form of this summary depends on the options you selected.  A typical example is shown below.

You can evaluate whether NMS is extracting stronger axes than expected by chance by requesting a Monte Carlo test.  You select the number of runs used to conduct the test.  Compare the stress obtained with your data set with multiple runs of randomized versions of your data.  The data are randomly shuffled within columns after each run, using the same procedure as described under Shuffle.  A table is produced (see example below) allowing you compare the stress obtained with your real data against the stress obtained for randomized data.

In the example below we can conclude that each of the 1-D to 6-D solutions provide significantly more reduction in stress than expected by chance, accepting a probability of Type I error < 0.05. You can also compare the observed stress to the range and mean of stress obtained in the randomized runs.

Because NMS is computationally intensive, you will find that it is time-consuming to request even 20 Monte Carlo runs for a moderately large data set.

Note that the randomized main matrix is NOT stored, so that your spreadsheet files remain the same, whether you request the Monte Carlo runs or not.

STRESS IN RELATION TO DIMENSIONALITY (Number of Axes)

--------------------------------------------------------------------

          Stress in real data          Stress in randomized data

               20 run(s)               Monte Carlo test,   50 runs

      -------------------------  -----------------------------------

Axes  Minimum     Mean  Maximum  Minimum     Mean  Maximum      p

--------------------------------------------------------------------

   1   37.877   43.282   54.519   34.057   46.506   55.077    0.0392

   2   15.447   20.311   31.323   21.059   26.510   50.552    0.0196

   3    9.705   10.848   22.658   13.626   18.827   42.837    0.0196

   4    6.316    9.704   19.674    9.877   15.622   29.405    0.0196

   5    4.783    7.382   19.645    7.517   13.511   29.443    0.0196

   6    3.278   10.742   32.050    5.471   12.089   42.422    0.0196

--------------------------------------------------------------------

p = proportion of randomized runs with stress < or = observed stress

i.e., p  = (1 + no. permutations <= observed)/(1 + no. permutations)

Calculate Scores for Column Variables by Weighted Averaging

Simultaneous plotting of column variables (usually species) in the ordination space is available if you select Calculate Scores by Weighted Averaging in the NMS Setup dialog.  The scores are written into the temporary file GRAPHCOL.GPH.

NMS Setup

NMS Distance Measure

NMS Parameters

Non-metric Multidimensional Scaling (NMS)

Ordination


NMS Random Numbers

If you use random starting coordinates, Mather (1976) recommends that the program be run several times to ensure as much as possible that the minimum of stress is not a local minimum.

If you select a random starting configuration, the random number generator needs two integers to start from (the "seeds").  You are given a choice about how to generate the random numbers. (See Random Number Seeds.)  For experimental reasons you may wish to generate the same set of random configurations for different sessions.  Or you may wish to have each set of runs unique.

NMS Starting Coordinates

NMS Setup

Non-metric Multidimensional Scaling (NMS)


Random Number Seeds

Random number “seeds” are the initial numbers used by a random number generator to produce the next random number.  (You can probably already see why these are called “pseudo-random” number generators.)

There are two options for the source of the random number seeds:

•	Use time of day.  The computer will use its internal clock (using the minutes and seconds) to get the starting integers.  This results in a different set of random numbers every time you run the procedure.

•	User supplied.  For experimental reasons you may wish to generate the same set of random numbers for different set of runs.  To do this, you supply the same seeds.

CCA Monte Carlo Test

Indicator Species Monte Carlo Test

Mantel Test Setup

NMS Setup


Monte Carlo Test, Number of Runs

For any Monte Carlo (randomization) test, you must specify the number of “runs.”  Each run consists of a separate randomization or permutation of your data.  The exact nature of the randomization depends on which analysis you are running.

A large number of runs is generally desirable, but an acceptable value will depend on the speed of your computer, the size of your data set, and the desired precision of the resulting p-value.  Note that the p-value for a randomization test can be no smaller than 1/N where N is the total number of runs.

CCA Monte Carlo Test

Indicator Species Monte Carlo Test

Mantel Test Setup

NMS Setup


NMS Run Log Example

(Use wide window and small font to view.)

--------------------------------------------------------------------

    Random Start Dimen-    Final  Iter-              Best for   File

Run  data? file?  sions   stress ations Instability* x axes  saved**

--------------------------------------------------------------------

  1     0     0      6    8.176   300   0.02549171*

  1     0     0      5    8.976   300   0.03224550*

  1     0     0      4   11.207   300   0.01644603*

  1     0     0      3   10.659   300   0.02026778*

  1     0     0      2   15.448   300   0.01661373*

  1     0     0      1   40.934   144   0.00002360

  2     0     0      6   32.050   300   0.10159633*

  2     0     0      5   13.506   300   0.03019547*

etc.

  4     0     0      2   15.448   300   0.01661373*

  4     0     0      1   41.927    37   0.00008217

  5     0     0      6    3.790   300   0.01875078*

  5     0     0      5    4.783   158   0.00009957     5  CONFIG5.GPH

  5     0     0      4    6.324    51   0.00009260

  5     0     0      3    9.705   221   0.00001200     3  CONFIG3.GPH

  5     0     0      2   19.253   300   0.01661376*

  5     0     0      1   37.918   300   0.00818721*

  6     0     0      6    7.961   300   0.02378091*

  6     0     0      5    4.851    84   0.00009817

etc.

 19     0     0      2   19.253   300   0.01661375*

 19     0     0      1   39.483   300   0.01505782*

 20     0     0      6    8.008   300   0.02885656*

 20     0     0      5    4.825   300   0.02713503*

 20     0     0      4   12.785   300   0.02365956*

 20     0     0      3   10.055   115   0.00009711

 20     0     0      2   15.448   300   0.01660817*

 20     0     0      1   40.953   218   0.00001230

--------------------------------------------------------------------

 21     1     0      6    8.171   300   0.02415815*

 21     1     0      5   12.215   300   0.06933992*

 21     1     0      4   16.046   300   0.03037714*

 21     1     0      3   16.468   300   0.00400411*

 21     1     0      2   24.723   300   0.00064202*

 21     1     0      1   46.027   300   0.00013140*

 22     1     0      6   10.429   300   0.00720586*

 22     1     0      5   14.386   300   0.02538743*

etc.

 69     1     0      2   26.872   300   0.01056049*

 69     1     0      1   40.785   300   0.00038601*

 70     1     0      6   11.972   300   0.05243187*

 70     1     0      5   13.094   300   0.01491859*

 70     1     0      4   11.637   300   0.00019174*

 70     1     0      3   18.914   300   0.00481004*

 70     1     0      2   23.794   300   0.00748585*

 70     1     0      1   39.339   300   0.00158772*

--------------------------------------------------------------------

Random data: 0 = not randomized, 1 = randomized

Start file: 0 = random starting coordinates, 1 = read from file

Seeds: initial seeds for random number generator

* Stability criterion not met.

**To run a single NMS ordination that repeats the best result, specify this file as the starting configuration, rather than using a random start.  It is best to save this file under a new name, to avoid its being overwritten by the next NMS run.  To do this, open the file using File | Open | Graph Row file, then File | Save as | Graph Row file (then specify new name).

NMS Output Options

NMS Setup

Non-metric Multidimensional Scaling (NMS)


NMS Scores

NMS Scores provides a prediction algorithm for non-metric multidimensional scaling (NMS).  This is not prediction in the sense of forecasting, but rather statistical prediction in the same way as using multiple regression to estimate a dependent variable.  NMS Scores calculates scores for new items based on prior ordinations.

A criticism of ordination in general is that practitioners never take it seriously, as evidenced by the rarity of its use to predict anything or estimate scores for new data points (Peters 1991).  While predictions are easily made with equations resulting from eigenanalysis (e.g. the eigenvectors resulting from PCA), it is more difficult with some other methods, such as NMS.

NMS Scores fills this need by providing a method for calculating the positions of new points in an existing ordination, without altering the positions of the original points.  Conceptually, it is easiest to think of prediction as using a model to estimate certain outputs, given a set of inputs.  In regression the model is the linear equation, the input is a set of values for the independent variables, and the output is a estimate of the dependent variable.  In NMS Scores, the model is defined by a set of ordination scores for one to three axes, along with the original data used to derive the scores.  The input data are one or more new sample units in the main matrix.  These must share some or all of the variables with the original data set (for community data this means a set of overlapping but not necessarily the same species).  The output consists of ordination scores (positions on the ordination axes) for each of the new sample units.

Thus, to run NMS Scores you need to have three files ready:

1. The calibration data set (*.wk1)

2. The calibration scores (*.gph)

3. A PC-ORD spreadsheet with new data points that you need to score (*.wk1)

The model is applied by an iterative search for the position of best fit for each of the new data points, one at a time.  The data points in the original model are held fixed in the ordination space, rather than adjusting their positions as in regular NMS.

As an example of predictive-mode NMS, see McCune et al. (1997a , 1997b).  In this case the model was a two dimensional NMS ordination related to a regional climatic gradient and air quality.  The model was used to assign scores to new data points that were not part of the calibration data set.

Calibration Data Set

Calibration Scores

NMS Scores Setup

NMS

Ordination


Calibration Data Set

The calibration set is a spreadsheet (*.wk1) in standard PC-ORD format.  It should be the same data set that you ordinated to obtain the calibration scores.  The calibration data set is not displayed, but is read in the background by PC-ORD.

The calibration data set is one of the two components of the model, the other being the calibration scores.

Calibration Scores

NMS Scores Setup

NMS Scores


Calibration Scores

The calibration scores are a graph file (usually *.gph) produced by ordinating the calibration data set.  Therefore, the number of rows in the calibration scores must match exactly the number of items in the calibration data set. These scores are the coordinates of the row items in the calibration data set on one to three ordination axes.  The calibration scores are not displayed, but are read in the background by PC-ORD.

Together the calibration scores and the calibration data set define the model that is used to assign positions to new data points.  The new data points are contained in the main matrix.

Calibration Data Set

Calibration Scores

NMS Scores Setup

NMS Scores Example Output

NMS Scores


NMS Scores Setup

Ordination | NMS Scores | NMS Scores Setup

The NMS Scores dialog contains the following options:

•	NMS Scores Method

•	NMS Scores Distance Measure

•	Flag for Poor Fit

•	NMS Scores Extrapolation Limit

•	List Intermediate Results

Calibration Data Set

Calibration Scores

NMS Scores Example Output

NMS Scores


NMS Scores Method

NMS Scores Setup | NMS Scores Method

Two methods are available for finding scores for new points:

•	Simultaneous axes (default):  For a k-dimensional solution, the position (score) is sought that will minimize stress in the overall k-dimensional configuration.  This is done in a hierarchical way, increasing the focus in three steps.  The basic method is detailed below under "one axis at a time" except that trial scores are sought from all combinations of positions on the k axes.  This makes it slower to seek positions on three axes at once than on three axes one at a time.  The computation time increases with the square of the number of axes, while with "one axis at a time" the computation time increases linearly with the number of axes.  Choose "simultaneous axes" if you wish to optimize the overall position of the fitted points in the ordination space.  Choose "one axis at a time" if you seek the best fit with respect to a single axis.  For example, if one axis in particular is of interest for predictions, and the other axes are irrelevant to your specific purpose, select "one axis at a time."

•	One axis at a time:  For each axis, the position (score) is sought that will minimize stress in the overall 1-dimensional configuration.  This is done in a hierarchical way, increasing the focus in three steps.  In the first step, the axis plus a 20% margin on both ends of the axis is divided into 29 equally spaced trial points (20 intervals on the axis + 4 intervals at each end).  The overall stress is calculated for each of the 29 possible positions for the trial point.  The point with the lowest stress is selected, then used as the basis for refining the position of best fit.  The interval is divided by 5 and 5 trial points spaced by that interval are examined on each side of the previous best point.   Again, the best point is found, the interval is divided by 5 and the final best point is found.

Example:  Assume an axis is 80 units long, margin = 20%, focus factor = 5,  number of focus levels = 3.  In the table below the units for interval width and range are always in the original axis units.

Focus level	Range examined	Intervals	Points	Interval Width
1	80 + 2*(0.2*80) = 112	20+4+4=28	29	112/28 = 4
2	2*5*4/5 = 8	2*5=10	11	4/5 = 0.8
3	2*5*(0.8/5) = 1.6	2*5=10	11	0.8/5 = 0.16

NMS Scores Setup

NMS Scores


NMS Scores Distance Measure

NMS Scores Setup | NMS Scores Distance Measure

Choose from the standard selection of distance measures available in PC-ORD.  All of the standard distance measures are available except for Chi-square distance.  Normally you should choose the same distance measure as used for the ordination of the calibration data set.

NMS Scores Setup

NMS Scores


Flag for Poor Fit

NMS Scores Setup | Flag for Poor Fit

Two kinds of poor fit are screened by PC-ORD:  (1) points that fall beyond an extrapolation limit, and (2) final configurations exceeding a final stress criterion.  In both cases a nonzero "flag" is written into the output file (see table below).

Flag	Meaning
0	Criteria for acceptability were met
1	Final stress is too high
2	Score is too far off low end of gradient
3	Score is too far off high end of gradient
In the Flag for Poor Fit dialog, you select the method for setting the final stress criterion:

•	Autoselect:  This method is helpful only if you are fitting a moderate to large number of new points.  The criterion stress level is two standard deviations above the average final stress resulting from fitting each new point.  Assuming the final stresses are normally distributed, on average 2.5% of the points will be flagged with a 1.  If you choose "autoselect" and only one new point is being fitted, then a standard deviation cannot be calculated and no flags will be set.

•	User defined:  You will be prompted to enter the final stress criterion between 1 and 100 inclusive.  This is very difficult to do without prior knowledge of the range in final stress obtained with a given configuration.  You might, however, find it useful to first run NMS Score with the Autoselect option, then choose a criterion value based on the final stress values obtained.  The user-defined criterion is blank by default because there is no way PC-ORD can, in advance, assign a reasonable value for all data sets. The user must enter some value to proceed.

NMS Scores Example Output

NMS Scores Setup

NMS Scores


NMS Scores Extrapolation Limit

NMS Scores Setup | NMS Scores Extrapolation Limit

Two kinds of poor fit are screened by PC-ORD:  (1) points that fall beyond an extrapolation limit, and (2) final configurations exceeding a final stress criterion.  In the first case a flag=2 (score too low) or flag=3 (score too high) is written into the output file (see table under Flag for Poor Fit).

The default extrapolation limit is 5% of the axis length (minimum = 0%, maximum = 50%).  For example, if the calibration scores range from 0-100, and the extrapolation limit is 5% then points that score greater than 105 are flagged with a 3 in the output file.  Scores less than -5.0 are flagged with a 2 in the output file.

NMS Scores Setup

NMS Scores


NMS Scores List Intermediate Results

NMS Scores Setup | NMS Scores List Intermediate Results

Select this option to display the stress calculated for each trial position for each point.  The coordinates of each trial position are listed, along with the overall stress associated with that configuration.  The information is repeated for each level of focussing, as the procedure homes in on the position of least stress.

NMS Scores Setup

NMS Scores


NMS Scores Example Output

The following options were selected:

     Distance measure   = Sorensen

     Method for fitting = one axis at a time

     Flag for poor fit  = autoselect (2 stand.dev. > mean stress)

     Extrapolation limit, % of axis length:     5.000

Calibration data set: D:\FHM\DATA\94\SEModel.wk1

Contents of data matrices:

    85 Plots    and    87 Species in calibration data set

    34 plots    and   140 species in test data set

    85 scores from calibration data set

Title from file with calibration scores: SEModel.wk1, rotated 155 degrees clockwise

    63 species  in test data set are not present in calibration data

AXIS SUMMARY STATISTICS

--------------------------------------------------------------------

                                         Axis:         1         2

Mean stress                                        49.833    50.251

Standard deviation of stress                        0.351     0.391

Stress used to screen poor fit (flag = 1)          50.536    51.033

Cutoff score, low end-of-axis warning flag (2)     -6.473   -12.895

Cutoff score, high end-of-axis warning flag (3)   106.473   112.895

Minimum score from calibration data                 6.559     1.619

Maximum score from calibration data                93.441    98.381

Range in scores from calibration data              86.881    96.761

--------------------------------------------------------------------

BEST-FIT SCORES ON EACH AXIS

--------------------------------------------------

     Axis:           1                    2

Item       Score    Fit  Flag   Score    Fit  Flag

--------------------------------------------------

3008281   25.673  49.713    0  90.253  50.085    0

3008367   44.440  50.058    0  74.190  50.582    0

3008372   36.099  50.141    0  80.964  50.437    0

etc.

3108526    2.563  49.505    0   8.199  50.007    0

3108564   50.000  49.953    0  87.543  50.491    0

3108651    0.825  49.327    0 114.830  49.714    3

3108684    4.995  49.328    0 111.347  50.148    0

3108732  121.416  48.633    3 -17.539  48.957    2

3208432   14.379  49.840    0  16.714  50.243    0

3208436   56.429  49.988    0  66.062  50.480    0

---------------------------------------------------

NMS Scores Setup

NMS Scores

Non-metric Multidimensional Scaling (NMS)


Principal Components Analysis (PCA)

Ordination | PCA

Principal components analysis (PCA) is the basic eigenanalysis technique.  It was first used in ecology by Goodall (1954).  It maximizes the variance explained by each successive axis.  Although it has severe faults with many community data sets, it is probably the best technique to use when a data set approximates multivariate normality.

The eigenanalysis subroutine was adapted from subroutine EIGEN in the program ORDIFLEX (Gauch 1977) where it was adapted from subroutines in EISPACK (Smith et al. 1976).  Double precision is used in the eigenanalysis routine.

The PC-ORD version of PCA uses the R-route (see Greig-Smith, 1983:  top of p. 246). You have the option of centering your data by column means.  Eigenvectors and eigenvalues are found using the p x p cross-products matrix, where p is the number of columns.  Scores on each axis are written into a file (graphrow.gph) for graphing.

Computation time goes up exponentially with the size of p.

Information on PCA can be found in almost any textbook on multivariate statistics.  Sometimes it is considered a special case of factor analysis.  Tabachnik and Fidell 1989 and Ludwig and Reynolds (1988) contain good introductions to PCA.  Consult Jackson (1993) for advice on significance testing and interpreting PCA.  Beals (1984) discussed problems in applying PCA to ecological data.

PCA Setup

PCA Output

Ordination


PCA Setup

Select the type of cross-products matrix to use and whether or not to list it in the results file.  More information is given below.

Cross-products Matrix

Three options are available for the cross-products matrix:

•	Correlation (i.e., centered + standardized by standard deviation Greig-Smith 1983, p. 249).  The cross-products matrix will contain Pearson correlation coefficients among all column variables.  This choice is used to produce a "standardized" PCA.  Use of the correlation coefficient puts the column variables on equal footing with each other.

•	Variance/covariance (centered by columns--see Greig-Smith 1983, p. 253).  The cross products matrix will contain variances in the diagonals and covariances off the diagonal.  This centers the data but does not standardize the column variables.  Unless the data are standardized prior to running PCA, the column variables will contribute to the analysis in proportion to their variances (not normally a desirable characteristic).

•	Noncentered (Row 1 in Table 23 of Greig-Smith 1983, p. 248).  The cross-products matrix is neither centered nor standardized.  Interpretation is typically a mindbender, although some authors report gaining insights through noncentered PCA in special cases.

Note: many options other than those listed here are possible by prior modification by Standardizations and Relativizations)

List the Cross-products Matrix

You may choose to list the cross-products matrix in your result file.  Beware of voluminous output with large data sets.

Calculate Scores for Column Variables by Weighted Averaging

Simultaneous plotting of column variables (usually species) in the ordination space is available if you select Calculate Scores by Weighted Averaging in the PCA dialog.  The scores are written into the temporary file GRAPHCOL.GPH.  Note that these scores are not the same as the traditional biplot scores based on the linear relationships of the column variables to the ordination axes.  For more information, see section on calculating scores by weighted averaging.

PCA Output

PCA

Ordination


PCA Output

Output from Principal Components Analysis consists of the cross-products matrix (if requested), the first ten eigenvalues, the first six eigenvectors, and entity scores for the first six axes.

The cross-products matrix is labeled only by row.  The columns are in the same order as the rows.  Numbers are in double-precision scientific notation, the number after the "D" being the exponent.  Interpretation is similar to single-precision scientific notation (E Fields).

For each axis the eigenvalue is listed, along with the proportion of variance "explained" by that axis.  This proportion is simply the eigenvalue for that axis divided by the total variance, i.e. the sum of the diagonal of the cross-products matrix.  If you choose correlation for the cross-products matrix then the diagonal is occupied by ones and the total variance to be explained is equal to the number of columns (p)  in your main matrix.  In this case the proportion of variance explained by a given axis is the eigenvalue for that axis divided by p.

To help you decide how many components (axes) are worth interpreting, a broken-stick eigenvalue is provided (Jackson 1993).  If the broken-stick eigenvalue is less than the actual eigenvalue for an axis, then that axis contains more information than expected by chance and should be considered for interpretation.

The ordination scores are written into the result file and an output file for graphing, GRAPHROW.GPH.  A graph is given for each pair among the first three axes.  Overlays and correlations of environmental variables with sample scores on the first three axes can be performed.

PCA Setup

Principal Components Analysis (PCA)

Ordination


Reciprocal Averaging

Ordination | RA

Reciprocal averaging (Hill 1973); is also known as correspondence analysis.  It is performed in PC-ORD by selecting options in program DECORANA (Hill 1979a) adapted from the Cornell Ecology Program series.  Reciprocal averaging (RA) yields both normal and transpose ordinations automatically.

Graphical representations of the first three axes of the ordination are constructed by selecting Graph in the menu after ordination.

RA suffers several faults which make detract from its utility as an ordination method.  It gives reasonable representations of only one axis, the second axis introducing an arch into the first.  Subsequent axes introducing more complex polynomials of the first axis.  Points in the center are spread too much relative to the edges.  Outliers can have a strong effect.  In general RA should be avoided with community data sets that have more than one important axis of variation.

RA can also be represented as an eigenanalysis technique where compositional dissimilarity is measured using the chi-squared distance measure and samples are weighted by their totals (Chardy et al. 1976).  In theory RA results in an ordination space in which distances between sample points are proportional to their chi-squared distance values.

The effect of the chi-squared distance measure is to give high weight to species whose total abundance in the data matrix is low.  This can result in an overemphasis of the distinctness of samples containing several rare species (Faith et al. 1987; Minchin 1987a).  Unlike proportionate city-block distance measures, chi-squared distance does not have a constant maximal value for pairs of sample units with no species in common, but fluctuates according to variations in the abundance of species with high or low total abundances (Minchin 1987a).

RA Setup

RA Output

Ordination


RA Setup

Because a given data matrix has only one RA solution, there are few options to consider.  A simultaneous row and column weighting is performed as part of the analysis.  This double weighting is equivalent to a relativization that divides each element of the data matrix by the square root of the product of the relevant row and column totals.

Downweight Rare Species

You may choose to downweight rare species.  If many species have only a few occurrences in the data, the ordination may be improved by removing them (see Delete Columns with Fewer than N Nonzero Values) or downweighting them.  If Fmax is the frequency of the commonest species, then species rarer than Fmax/5 are downweighted in proportion to their frequency.  Species commoner than Fmax/5 are not downweighted.

Reciprocal Averaging (RA)

RA Output

Ordination


RA Output

The first part of the output is a record of the options you selected.  Then, for each axis the following information is given:

•	List of residuals.  The eigenvectors are found by iteration and the residuals indicate how close a given vector is to an eigenvector.  A vector is deemed to be an eigenvector if it has a residual of less than 10^-7.  If this value is not reached in 999 iterations, the current trial vector is taken to be the eigenvector and a warning is printed that the residual exceeds the desired tolerance.

•	Eigenvalue.  The relative sizes of the eigenvalues indicate the relative amounts of variance explained by each axis.

•	Following the axis-by-axis output, species and sample scores for each axis are printed.

•	Graphs and correlations with ordination axes are then available through Graph in the menu.

Reciprocal Averaging (RA)

RA Setup

Ordination


Weighted Averaging Ordination

Ordination | Weighted Averaging

The simplest yet often effective method of ordination is weighted averaging (Whittaker 1967).  This has been reinvented under many names at numerous times for various purposes.  By whatever name, the essential operation is the same:  a set of pre-assigned species weights (or weights for species groups) are used to calculate scores for sites (sample units).  The calculation is a weighted averaging for species or species groups actually present in a sample unit.

Several examples illustrate the many incarnations of this method:

Curtis and McIntosh (1951) assigned "climax adaptation values" to tree species in Wisconsin.  The climax adaptation values weight species on a continuum from pioneer species (weight=1) to climax species (weight=10).  These values are used as weights to calculate the "continuum index" for each sample unit.  The continuum index is a score on a gradient from early successional to late successional species composition.

LeBlanc and de Sloover (1970) constructed an "index of atmospheric purity" (IAP).  The IAP was calculated from presence of epiphytic lichen species.  Each species was assigned a weight indicating its degree of tolerance to air pollution.

The Federal Wetlands Manual (Federal Interagency Committee for Wetland Delineation 1989) uses five categories of plant tolerance to hydric soils to calculate scores for individual sites.  The purpose of these scores is to give a quantitative statement of the degree of dominance by wetland-demanding plants (obligate wetland plants, facultative wetland plants, facultative plants, facultative upland plants, and upland plants).  These categories are weighted 1 through 5, respectively (the "EI values").  The resulting score for each plot is called a "prevalence index," but we can also think of it as a one-dimensional ordination in species space.  If the prevalence index is less than three, the site is considered to have hydrophytic vegetation.

Aquatic invertebrates have been used to score lakes and streams with a variety of weighted averaging ordination techniques (although rarely explicitly recognized as such).  The Benthic Quality Index (BQI) weights the proportions of counts in a given taxon or indicator group by a series of weights from 1 to 5, according to pollution tolerance (Wiederholm 1980; Johnson 1998).  Similar approaches in Britain are the Biological Monitoring Working Party score (BMWP; Wright et al. 1993) and the Average Score Per Taxon (ASPT; Armitage et al. 1983).  The BMWP score is a weighted sum while the ASPT divides the weighted sum by the number of nonzero items contributing to the sum, converting the index to a weighted average and improving its performance.  The Index of Biotic Integrity (IBI) uses a weighted average of indicator values that attempt to incorporate sensitivity to several different stressors.  An IBI has been constructed using fish (Karr 1991) and invertebrates (Kerans and Karr 1994).

PC-ORD allows weighted averaging through either Ordination | Weighted Averaging or Modify Data | Multiply Matrices.  The community matrix is opened as the main matrix, with sample units as rows and species as columns. The species weights are opened as the second matrix, with species as rows and one or more set of weights as columns.  Historically a single set of weights (e.g. moisture preferences below) is used, but there is no reason why you cannot use more than one set of weights simultaneously.

If you use the Ordination | Weighted Averaging route, no options need to be set.  If you use Modify Data | Multiply matrices, the following options give the same result as Weighted Averaging ordination:

•	Multiplication type = Calculate AE and save it as a graph file

•	Matrices NOT normalized

•	Division after Multiplication = by species totals for weighted average.

An example is worked through below.  Assume we start with a main matrix A, containing abundances of p species in n sample units, for example a matrix of 2 plots x 3 species.

2	plots
3	species
	Q	Q	Q
	sp 1	sp 2	sp 3
plot A	4	1	0
plot B	0	2	4

Create a second matrix containing the weight (or indicator value) for each species with respect to a particular environmental gradient.  In this case we assigned weights to each species (based on prior knowledge), the weights indicating tolerance of wet to dry sites.  Sp 1 is a dry site indicator, and we assign it a weight of zero.  Sp 2 indicates medium moisture levels, and we assign it a weight of 50.  Sp 3 indicates the wettest sites and receives a weight of 100.  The second matrix thus looks like:

3	species
1	factor
	Q
	Moisture
Sp1	0
Sp2	50
Sp3	100

Now calculate a moisture score vi for each plot i:

Applying this equation to the example we have

This high value indicates that plot B is relatively wet, as we would have expected from the prevalence of the wet site indicator, species 3.  The same calculation for plot A yields a moisture score of 10, indicating a relatively dry site.  In this way, we have constructed a one-dimensional ordination of plots in species space.

The resulting matrix is substituted for the second matrix:

2	plots
1	factor
	Q
	Moisture
plot A	10
plot B	83.3

Although not part of most weighted averaging ordinations, the procedure can be reversed to refine the species weights.  A new set of species weights can be calculated from the site scores:

To do this in PC-ORD, repeat the procedure given above, but this time we are calculating new species weights from the plot weights.

Repeating this procedure back and forth, between calculating new species weights and new sample unit scores, we have reciprocal averaging ordination.  Actually, it's not quite that simple.  Attention must be paid to the scaling of the scores and calculating more than one axis.

You can calculate the variance explained by your ordination axes by evaluating the correspondence between distances in your ordination space and distances in the original unreduced species space.   We recommend an after-the-fact coefficient of determination between Relative Euclidean distance in the unreduced species space and Euclidean distance in the ordination space (Graph | Ordination | Statistics | Percent of Variance ).

Matrix multiplication

Ordination


Scores by Weighted Averaging

Historically ecologists used ordination methods based on correspondence analysis (RA, DCA, CCA) to produce ordinations with simultaneous plots of sample units and species, both represented as points in the same ordination space.  In version 4 of PC-ORD an option was added to the other ordination techniques (Bray-Curtis, NMS, and PCA)  to calculate scores for the column variables (usually species) by applying a single weighted averaging step.

Assume that the rows of the main matrix are sample units and the columns are species.  PC-ORD will calculate ordination scores for the species by weighted averaging.  Sample unit scores are used as weights to find the average position of each species along each ordination axis.  In contrast to Reciprocal Averaging, this weighted averaging step is applied only once, and the sample unit scores are not altered.

Plotting both variables and sample units on the same diagram can be considered a kind of biplot, using this word in a broad sense.  A biplot is simply a way of representing sample units and variables in a single diagram.  Exactly how this is done, and more precise, narrow definitions of biplots, vary among authors.  Gabriel (1971) coined the word "biplot" and applied the concept to Principal Components Analysis.  Categorical variables can also be represented on a biplot, a point representing a particular level (or category) of a variable (Benzécri et al. 1973; Gower & Hand 1996).

Gower and Hand (1996) represented continuous variables in ordination diagrams as additional axes.  These are usually straight, intersect the centroid of the point cloud, and at various angles to the original coordinate system for the sample units.  The biplot axes may also be curved or irregular in some applications.

The traditional biplots produced from PCA are different from the "scores by weighted averaging" offered by PC-ORD.  Traditional PCA biplots are based on the linear relationships of the variables to the sample unit scores.  In contrast, the weighted averaging method in PC-ORD is based on the average position of a species on each axis, and is thus more accommodating to non-linear unimodal species abundance curves along an ordination axis.  Of course one could argue that given the nonlinear responses of species to environmental gradients, one should not be using PCA in the first place!

Biplots were popularized in ecology through their adaptation to canonical correspondence analysis (ter Braak 1986).  These could be considered "triplots" because they plot the sample units as points, the variables in the species matrix as an additional set of points, and the environmental variables as vectors of various lengths radiating from the centroid.  As redefined by ter Braak (1986) and Jongman et al. (1995), biplots can be considered a special case of a joint plot where the positions of the tips of the vectors are calculated from the linear equations that define the ordination.  By their definition, biplots follow a certain set of rules (Jongman et al. 1987).  Both biplots and joint plots, however, are ordination diagrams that show two kinds of entities (often sample units and environmental variables).  Their method of constructing biplots is explained further under Canonical Correspondence Analysis.

Bray-Curtis Setup

NMS Setup

PCA Setup

Ordination


Managing Large Projects

By “large project” we mean large data matrices (several hundred or more rows or columns) or many sets of interrelated matrices.  These burden your system’s resources in one way or another.  We offer a few tips below to help ease, or at least understand, this burden.  Large projects may require special attention to the following topics:

•	Computational speed

•	Data management

•	Graphs

•	Memory

Getting Started


Computational speed

With the ability to address a large amount of memory you will find that computational speed becomes a limiting factor.  This is particularly true for some of the computationally intensive procedures, such as NMS and Monte Carlo tests.  These become extremely slow with large data sets.

If your computations are unbearably slow, you have only a few options:

•	Pare down the size of your data sets (consider deleting rare species or partitioning).

•	Use a faster computer.

•	Run the slow problems overnight.

Getting Started

Managing Large Projects


Data management

For building, archiving, error checking, and combining large data files we recommend using the PC-ORD compact format.

With multiple data sets drawing from the overlapping sets of species it is usually most efficient to build a large species dictionary (species file), to which all of the data sets refer.  This results in consistent use of names across data sets.  Furthermore, it allows a central way of handling nomenclatural changes.

For example, McCune maintains a species file with about a thousand epiphytic lichen species, all those known from North America.  He uses the species code numbers in this file for all of his data sets on epiphytic lichens.  Each data set is stored in compact format in relatively small plain text files, each file derived from a particular region, field crew, date, or combinations of those.

Error checking in large spreadsheets is awkward at best.  If you use compact  format a much more convenient method is available.  Generate plot by plot lists via Summary in PC-ORD.  These lists have code numbers, acronyms, and full species names, so they are very easy to use.  They are also nice to use if you are revisiting permanent plots, since you'll have a list for each plot.

Compact format files don't take much space.  You can enter and save the data with any precision you want.

When combining these files into a single data set, just append one file onto the next using a text editor.  There is no header information that needs to be edited.

Large data sets are frequently hierarchical in some way.  You can maintain the hierarchical structure of the data by defining each group of sample units as a "break group," so that each section of data starts with the key word "break", then the name for that group, then the data.

In summarizing the data and converting to spreadsheets you can choose to ignore the hierarchical structure or aggregate to a higher level.  For more information see Compact Format Data Summary.  See also suggestions for organizing your files and projects.

Getting Started

Managing Large Projects


Graphing large data sets

Ordination graphs of large data sets are typically messy and cluttered.  You can reduce the clutter by doing some or all of the following:

•	choose small font sizes (most systems will have “Small fonts” available as small as 2 pt.),

•	turn off labeling of the ordination points,

•	select small ordination symbols, and

•	plot only one set of points at a time.

If you want to label your ordination points but have a blob of overwritten labels in the center, then:

•	Arrange the peripheral labels to minimize overlap by dragging the labels (click and hold while dragging the mouse).

•	Eliminate labels from the densest parts of the graph (double click on a label, then press delete).

Plotting large data sets can be very slow.  You can speed up the process by turning off labeling – this is a bottleneck in the algorithm.

Getting Started

Managing Large Projects


Memory

With 16 mb of memory (RAM) or more, most users will find that use of large data sets are limited by computational speed, not memory.

Different procedures in PC-ORD require different amounts of memory.  The File | Memory Requirements utility provides a convenient way of sizing up your memory needs.  You can do this for the data set that is open or you can type in hypothetical values.

For more information on memory requirements, see Memory Use.

Getting Started

Managing Large Projects


Contents

The following Help Topics are available:

GETTING STARTED

Getting Started

Help for Old Friends

Examples

Managing Large Projects

YOU REALLY NEED TO KNOW THESE THINGS

File formats

Distance Measures

TOPICS CORRESPONDING TO MENU ITEMS

File Menu

Edit

Modify Data

Summary

Ordination

Graph Menu

Groups

Window

Options

Help

Acknowledgments

References

For Help on Help, Press F1


TOPICS CORRESPONDING TO MENU ITEMS

File

Edit

Modify Data

Summary

Ordination

Graph

Groups

Window

Options

Help