Supplement 1. Software to compute nonlinear canonical analysis (program POLYNOMIAL RDACCA: source code, compiled versions for Macintosh and Windows program documentation, and example data files).

File List User Guide.pdf Polynomial_RDA-CCA_PC.zip Polynomial_RDA-CCA_Mac.sit Description These files are respectively the user's manual and two versions (PC and Mac) of the Polynomial RDA-CCA program used in our article to carry out linear and polynomial RDA and CCA. This program performs four forms of canonical analysis: linear or polynomial redundancy analysis (RDA) and linear or polynomial canonical correspondence analysis (CCA). Classical linear redundancy analysis (Rao, 1964) and canonical correspondence analysis (ter Braak, 1986, 1987) are computed using multiple linear regression followed by direct eigenanalysis of the matrix of fitted values. The method of calculation is described in Chapter 11 of Legendre and 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, 2001). The polynomial methods are based on the use of multiple polynomial 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. In 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. The program is also available on the web site of Pierre Legendre. See the User's Guide for more information. Press, W. H., B. P. Flanery, S. A. Teukolsky, and W. T. Vetterling. 1986. Numerical recipes - The art of scientific computing. Cambridge University Press, Cambridge, UK.