Beatrix E. Beisner, Pedro R. Peres-Neto, Eva S. Lindström, Allain Barnett, and Maria Lorena Longhi. 2006. The role of environmental and spatial processes in structuring lake communities from bacteria to fish. Ecology 87:2993–2999.

Appendix C. Adjusted redundancy statistic and variation partitioning in redundancy analysis.

The methods described here follow Peres-Neto et al. (2006).

The formulation of the redundancy statistic R²X|Y applied to species data matrices is calculated as follows:

where represents the matrix of predicted values for the species distribution matrix and T denotes matrix transpose ; note that this is identical to calculating predicted values for individual multiple regressions of each column of Y on X; X cent=(I-P)X and Y cent=(I-P)Y are matrices X and Y centered by their means (i.e., column means = 0). I is an (n × n) identity matrix and P is a (n × n) matrix with all elements = 1/n; n refers to the number of sampling units. In the case of matrix X, it makes no difference whether we center or standardize the data (column means=0 and column variances =1). This is not the case for Y, which is only centered in order to keep the variances of the different species unmodified (i.e., species abundances are kept proportional and standardizing Y would transform all species to equal variances).

Adjusted values R²X|Yadj for the redundancy statistic can be estimated as follows (Peres-Neto et al. 2006):

where p is the number of predictors, n is the number of sample sites and R²X|Y is the sample estimation of the population R²X|Y.

Variation partitioning for RDA using two sets of predictors (E: environment and S: spatial predictors, i.e., eigenvector maps) is straightforward as it is based on simply three canonical analyses. The first one uses both sets of predictors [E,S], the second one only E and the last one only S. Adjusted fractions of variations are calculated as follows:

Note that fractions are redundancy statistics based on appropriate combinations of the matrices involved, via canonical analysis. They actually represent squared semi-partial multiple correlations since the denominator is always the total variation in matrix Y, given by trace(Y^TY).

LITERATURE CITED

Peres-Neto, P. R., P. Legendre, S. Dray, and D. Borcard. 2006. Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology, in press.