## Linear and Polynomial RDA and CCA

Département de Sciences Biologiques
Université de Montréal
C.P. 6128, succursale Centre-ville
This program performs four forms of canonical analysis: linear or polynomial redundancy analysis (RDA) and linear or polynomial canonical correspondence analysis (CCA).

### Algorithms

In this program, classical linear redundancy analysis (Rao, 1964) and canonical correspondence analysis (ter Braak, 1986, 1987) are computed using multiple regressions followed by direct eigenanalysis of the matrix of fitted values. The method of calculation is described in Chapter 11 of Legendre & Legendre (1998). Polynomial RDA and CCA, which are generalizations of the linear forms, are implemented using a new approach proposed by Makarenkov and Legendre (1999, 2002). The polynomial methods are based on the use of polynomial multiple regression, during the first stage of RDA and CCA, instead of the multiple linear regression used in the linear forms. The explanatory variables are limited to their quadratic form in any term of the polynomial. The program produces the output required to draw biplot diagrams for linear and polynomial RDA or CCA. The explanatory variables can be represented in biplots in two different ways: (1) the individual terms of the polynomial equation can be represented as separate variables, or (2) one can choose to represent an explanatory variable using the multiple correlations (rescaled as required by the selected scaling method) of the canonical ordination axes against the linear and quadratic forms of the variable. A permutation procedure allows one to test the significance of the two models (linear and polynomial) and of the difference between them.

### Distribution of the program

• Macintosh version
• Fortran77 source code for the Macintosh Language Systems Fortran compiler (file Polynomial RDA_CCA.f). Users can modify the Parameter statement in the parameter file RDA_CCA-parameters.f (provided) which fixes the size (nmax) of the largest data matrices that can be analyzed. To become effective, any modification to the file of parameters requires recompiling the source-code file "Polynomial RDA_CCA.f".
• Compiled versions of the program for any Macintosh computer (fat binary)
• Program documentation, in Adobe Acrobat format
• Sample files
• 32-bit DOS version (The executable file is a Win32 "console" executable, not a DOS executable. Therefore it cannot run under plain DOS, nor in a DOS window under Windows 3.x, only in Windows 95/98 or Windows NT consoles)
• GNU Fortran source code for Mac OS X, Unix/Linux, or Windows (file PARAFIT.FOR), which can be compiled using a GNU FORTRAN77 compiler. Users can modify the Parameter statement in the parameter file RDPARAM.F (provided) which fixes the size (nmax) of the largest data matrices that can be analyzed. To become effective, any modification to the file of parameters requires recompiling the source-code file "RDA_CCA.F".
• Compiled version of the program for Win32 compatible computers
• Program documentation, in Adobe Acrobat format
• Sample files

### References:

Legendre, P. & Legendre, L. 1998. Numerical Ecology, 2nd English edition. Elsevier Science BV, Amsterdam. xv + 853 pages. Makarenkov, V., & Legendre, P. 1999. Une méthode d’analyse canonique non-linéaire et son application à des données biologiques. Mathématiques, informatique et sciences humaines, 37e année, n° 147: 135-147. Makarenkov, V., & Legendre, P. 2002. Nonlinear redundancy analysis and canonical correspondence analysis based on polynomial regression. Ecology 83(4): 1146-1161. Press, W. H., B. P. Flanery, S. A. Teukolsky & W. T. Vetterling. 1986. Numerical Recipes - The art of scientific computing. Cambridge Univ. Press, Cambridge. xx + 818 p. Rao, C. R. 1964. The use and interpretation of principal component analysis in applied research. Sankhya A 26: 329-358. 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. 1987. The analysis of vegetation-environment relationships by canonical correspondence analysis. Vegetatio 69: 69-77.