Centroids of sites with code "1" for BINARY environmental variables, in ordination diagram Coral 12.36599 1.09129 0.05088

Sand -6.96197 5.91719 -0.63774

Other subs. -4.05301 -5.2563 6 0.44014

the eigenanalysis (PCA in Fig. 11.2) has been conducted on matrix Y. Both the "site scores" (matrix F) and "fitted site scores" (matrix Z) may be used in RDA biplots.

Correlations of the environmental variables with the ordination vectors can be obtained in two forms: either with respect to the "site scores" (eq. 11.12) or with respect to the "fitted site scores" (eq. 11.13). The latter set of correlations is used to draw biplots containing the sites as well as the variables from Y and X (Fig. 11.3). There were three binary variables in Table 11.3. Each such variable may be represented by the centroid of the sites possessing state "1" for that variable (or else, the centroid of the sites possessing state "0"). These three variables are represented by both arrows (correlations) and symbols (centroids) in Fig. 11.3 to show the difference between these representations; in real-case studies, one chooses one of the representations.

The following question may arise when the effect of some environmental variables on the dependent variables Y is already well known (e.g. the effect of altitude on vegetation along a mountain slope, or the effect of depth on plankton assemblages): what would the residual ordination of sites (or the residual correlations among variables) be like if one could control for the linear effect of such well-known environmental variables? An approximate answer may be obtained by looking at the structure of the residuals obtained by regressing the original variables on the variables representing the well-known factors. With the present data set, for instance, one could examine the residual structure, after controlling for depth and substrate, by plotting ordination biplots of the non-canonical axes in Table 11.4. These axes correspond to a PCA of the table of residual values of the multiple regressions (Fig. 11.2).

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