Canonical analysis of principal coordinates: a useful method of constrained ordination for ecology

A flexible method is needed for constrained ordination on the basis of any distance or dissimilarity measure, which will display a cloud of multivariate points by reference to a specific a priori hypothesis. We suggest the use of principal coordinate analysis (PCO, metric MDS), followed by either a canonical discriminant analysis (CDA, when the hypothesis concerns groups) or a canonical correlation analysis (CCorA, when the hypothesis concerns relationships with environmental or other variables), to provide a flexible and meaningful constrained ordination of ecological species abundance data. Called "CAP" for "Canonical Analysis of Principal coordinates," this method will allow a constrained ordination to be done on the basis of any distance or dissimilarity measure. We describe CAP in detail, including how it can uncover patterns that are masked in an unconstrained MDS ordination. Canonical tests using permutations are also given, and we show how the method can be used (1) to place a new observation into the canonical space using only interpoint dissimilarities, (2) to classify observations and obtain misclassification or residual errors, and (3) to correlate the original variables with patterns on canonical plots. Misclassification error or residual error is used to obtain a non-arbitrary decision concerning the appropriate dimensionality of the response data cloud (number of PCO axes) for the ensuing canonical analysis. We suggest that a CAP ordination and an unconstrained ordination, such as MDS, together will provide important information for meaningful multivariate analyses of ecological data by reference to explicit a priori hypotheses.