Figure 11.5 CCA ordination biplot of the artificial data in Table 11.3; the numerical results of the analysis are in Table 11.5. (a) Biplot representing the species (dashed arrows), sites (dots, with site identifiers which also correspond to depths in m) and environmental variables (full arrow for depth, triangles for the three binary substrate variables). (b) Ranking of the species along a quantitative environmental variable (depth in the present case) is inferred by projecting the species onto the arrow representing that variable.

Scaling type 2 (from Subsection 9.4.1) was used, in this example, to emphasize the relationships among species. As a result, the species (matrix F) are at the centroids of the sites (matrix V) in Fig. 11.5a and distances among species approximate their chi-square distances. Species 3 and 4 characterize the sites with coral substrate, whereas species 5 and 6 indicate the sites with "other substrate". Species 1 and 2, which occupy an intermediate position between the sites with coral and other substrate, are not well represented in the biplot of canonical axes I and II; axis III is needed to adequately represent the variance of these species. Among the ubiquitous species 7 to 9, two are well represented in the subspace of canonical axes I and II; their arrows fall near the middle of the area encompassing the three types of substrate. The sites are not perfectly ordered along the depth vector; the ordering of sites along this variable mainly reflects the difference in species composition between the shallow sandy sites (1, 2 and 3) and the other sites.

Figure 11.5b shows how to infer the ranking of species along a quantitative environmental variable. Depth is used in this example. The graphical method simply consists in projecting (at right angle) the species onto the arrow representing that variable. This gives an approximation of the weighted averages of the species with respect to environmental variables. Ecologists like to interpret this ranking as representing the niche optima for the species under study. It is important to realize that three rather strong assumptions have to be made when attempting such an interpretation:

• that the various species have unimodal distributions along the environmental variable of interest (subsection 9.4.4);

• that the species distributions are under environmental control (Whittaker, 1956; Bray & Curtis, 1957), so that the mode of each species is at its optimum along the various environmental variables; and

• that the gradient under study is long enough to allow each species to go from some less-than-optimum low frequency to its high-frequency optimum, and back to some past-optimum low frequency.

In the data of the present example (Table 11.2), only species 1, 3 and 5 were constructed to approximately correspond to these criteria. Species 7, which may also look like it has a unimodal distribution, has actually been constructed using a pseudorandom number generator.

To investigate the similarities among sites or the relationships among species after controlling for the linear effects of depth and type of substrate, one could draw ordination biplots of the non-canonical axes in Table 11.5. These axes correspond to a correspondence analysis of the table of regression residuals, as in Fig. 11.2.

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