Contents
Program for transformation of frequency data
User's notes
Five transformations for frequency data, including species abundances, have been described by Legendre and Gallagher (2001).
The program (distribution: see below) converts a matrix of species abundances in such a way that the Euclidean distance among rows of the transformed matrix is equal to one of the following distances among rows of the original data matrix:
 Chord distance
 Chisquare metric
 Chisquare distance
 Distance between species profiles
 Hellinger distance
Example Input file: Frequency data for 3 species (columns) at 3 sites (rows), from Legendre and Legendre (1998, p. 457):
10 10 20 10 15 10 15 5 5Output file: Transformation such that Euclidean distances computed among rows of transformed data are equal to Chord distances among the original sites:
0.40825 0.40825 0.81650 0.48507 0.72761 0.48507 0.90453 0.30151 0.30151Output file: Transformation such that Euclidean distances computed among rows of transformed data are equal to Chisquare distances among the original sites:
0.42258 0.45644 0.84515 0.48295 0.78246 0.48295 1.01419 0.36515 0.33806Output file: Transformation such that Euclidean distances computed among rows of transformed data are equal to Hellinger distances among the original sites:
0.50000 0.50000 0.70711 0.53452 0.65465 0.53452 0.77460 0.44721 0.44721
Program Distribution
Updated March 22, 2001A preprint version of the article is Acrobat document. Computer programs to carry out these transformations are available from the following WWWeb sites:
 Fortran source code and compiled versions, written by P. Legendre

Transformations Macintosh version
 Fortran source code
 Compiled versions of the programs for Macintosh computers using 680x0, 680x0 + FPU, or PowerPC processor
 Program documentation, in Adobe Acrobat format
 Transformations 32bit DOS version

Transformations Macintosh version
 Matlab code, written by E. Gallagher
Program for biplot species scores
User's notes
After distancebased redundancy analysis (dbRDA, Legendre and Anderson, 1999) using BrayCurtis (or other) distance, species scores may be obtained by computing correlations between the species and the fitted site scores, also called "site scores that are linear combinations of environmental variables" or "sample scores that are linear combinations of environmental variables".
The correlations have to be weighted as follows before being used to draw the species as arrows in biplots:
SpeciesScore(jk) = r(jk)*s(j)/s(k)where
r(jk) = correlation between species j and fitted site score vector k,In standard RDA, the lengths of the species score vectors are 1 in distance biplots and they are equal to the square roots of the corresponding canonical eigenvalues in correlation biplots (Legendre and Legendre 1998), just as in PCA. Different programs may scale the site scores in different ways, however, as can be seen by comparing the three examples below. This may result in species scores that are larger than the original ones by a constant, which does not change the interpretation of biplots. The RDA scalings implemented in different programs are described in Legendre and Legendre (1998, pp. 585586). This program offers the option of scaling the species scores as in standard RDA, or as in program CANOCO versions 3 or 4. Compared to standard RDA, described in Legendre & Legendre (1998), the standard deviations of the site scores differ by a constant in CANOCO, so that the site scores also differ by the following constants:
s(j) is the standard deviation of species j,
and s(k) is the standard deviation of fitted site score vector k.
n / sqrt(totinert) in CANOCO 3.10where n is the sumber of objects (e.g., sites), p is the number of variables (species), and 'totinert' is the total inertia in the species matrix. Input file 1: Species data text file. Rectangular table of species presenceabsence or frequency data where the rows correspond to objects (e.g., sites) and the columns to species. There are no row (= site) or column (= species) identifiers. Data are separated by spaces or tabs. It is recommended to add a carriage return at the end of the last row of data. Input file 2: Text file of "fitted site scores", or "sample scores that are linear combinations of environmental variables", where rows correspond to objects (e.g., sites) and columns to canonical eigenvalues. This table is copied from the output file of Canoco or similar program into an Ascii (i.e., text) file. There are no row (= site) or column (= species) identifiers. Data are separated by spaces or tabs. The program checks that the number of rows is the same in input files 1 and 2. Add a carriage return at the end of the last row of site scores. Output file: Table of species scores for biplots. The rows correspond to the species and the columns to the canonical eigenvalues. This file may be added to the Species Scores and Site Scores (or Fitted Site Scores) tables to produce biplots. An automatic biplot drawing procedure is available in The R Package, version 4.0 (Casgrain and Legendre, 1999). The examples that follow were all scaled to obtain distance biplots. Program SpeciesScores can also compute biplot species scores from "fitted site scores" scaled to obtain correlation biplots. In all cases, users of CANOCO should request scalings for covariancebased scores in CANOCO 3.1 (negativenumber options), or without posttransformation in CANOCO 4.
sqrt(n*p) / sqrt(totinert) in CANOCO 4
Example 1 Coral reef fish data from Table 11.3 of Legendre and Legendre (1998, p. 590). There are 10 sites (rows) and 6 species (columns). Input file 1: Species file (text file; no row or column identifiers).
1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 11 4 0 0 8 1 11 5 17 7 0 0 9 6 0 0 6 2 9 7 13 10 0 0 7 8 0 0 4 3 7 9 10 13 0 0 5 10 0 0 2 4Input file 2: fitted site scores from program RdaCca. Scaling 1 was used here to obtain a distance biplot. The columns correspond to the three canonical eigenvalues. (Text file; no row or column identifiers).
6.79498 5.49498 2.24897 6.96197 5.91719 0.63774 7.12895 6.33941 0.97349 3.55205 6.52301 4.39356 12.69996 0.24686 3.17159 3.88603 5.67858 1.17109 12.36599 1.09129 0.05088 4.22000 4.83415 2.05138 12.03201 1.93572 3.27335 4.55398 3.98972 5.27384Output file: Species scores for biplots, computed by program SpeciesScores. The rows correspond to the species and the columns to the canonical eigenvalues. This table is identical to the table of species scores computed by program RdaCca and reported in Table 11.4 of Legendre and Legendre (1998, p. 591).
0.30127 0.64624 0.39939 0.20038 0.47265 0.74458 0.74098 0.16813 0.25689 0.55013 0.16841 0.26114 0.11588 0.50594 0.29319 0.06292 0.21535 0.25679
Example 2 Input file 1: Same species file as above. Input file 2: fitted site scores from program CANOCO 3.1. Scaling 1 was used to obtain a distance biplot with covariancebased scores. The columns correspond to the three canonical eigenvalues. (Text file; no row or column identifiers).
0.6741 0.5452 0.2231 0.6907 0.5870 0.0633 0.7073 0.6289 0.0966 0.3524 0.6471 0.4359 1.2600 0.0245 0.3147 0.3855 0.5634 0.1162 1.2268 0.1083 0.0050 0.4187 0.4796 0.2035 1.1937 0.1920 0.3247 0.4518 0.3958 0.5232Output file: Species scores for biplots. The rows correspond to species and the columns to canonical eigenvalues.
0.95271 2.04359 1.26323 0.63365 1.49471 2.35440 2.34316 0.53170 0.81282 1.73965 0.53257 0.82547 0.36643 1.59994 0.92714 0.19897 0.68102 0.81207
Example 3 Input file 1: Same species file as above. Input file 2: fitted site scores from program CANOCO 4. Biplot scores emphasizing intersample distances, without posttransformation (scaling 1), were computed to obtain a distance biplot. The columns correspond to the three canonical eigenvalues. (Text file).
0.6741 0.5452 0.2231 0.6907 0.5870 0.0633 0.7073 0.6289 0.0966 0.3524 0.6471 0.4359 1.2600 0.0245 0.3147 0.3855 0.5634 0.1162 1.2268 0.1083 0.0050 0.4187 0.4796 0.2035 1.1937 0.1920 0.3247 0.4518 0.3958 0.5232Output file: Species scores for biplots. The rows correspond to species and the columns to canonical eigenvalues.
0.73797 1.58296 0.97849 0.49082 1.15779 1.82371 1.81501 0.41186 0.62961 1.34753 0.41253 0.63941 0.28384 1.23931 0.71816 0.15412 0.52751 0.62902
Distribution A computer program to compute the biplot species scores is available from the following WWWeb site:
Fortran source code and compiled versions, written by P. Legendre:

Species Scores Macintosh version
 Fortran source code
 Compiled versions of the programs for Macintosh computers using 680x0, 680x0 + FPU, or PowerPC processor
 Program documentation, in Adobe Acrobat format
 Species Scores 32bit DOS version
References
Casgrain, P. and P. Legendre. 2001. The R Package for multivariate and spatial analysis, version 4.0  User's manual. Departement de sciences biologiques, Universite de Montreal. Available from the Web site http://numericalecology.com/ . Legendre, P. & M. J. Anderson. 1999. Distancebased redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecological Monographs 69 (1): 124. Legendre, P. and E. Gallagher. 2001. Ecologically meaningful transformations for ordination of species data. Oecologia 129: 271280. (Reprint available, © 2001 "SpringerVerlag". The original publication is available on http://link.springer.de/) Legendre, P. & Legendre, L. 1998. Numerical Ecology, 2nd English edition. Elsevier Science BV, Amsterdam. xv + 853 pages.