Partial Canonical Correspondence Analysis



Figure 1: An illustration of "partialling out" the influence of a set of variables (W) from a model. a) Both the explanatory variable(s) in matrices X and Y explain a portion of the variation in the response data (Y). b) After the partialling out the effect of W (which may be a single variable or a set of variables), only the variance in the response variables (Y) which can be exclusively explained by the variance in one set of explanatory variables (X) is retained.

Partial canonical correspondence analysis (pCCA) is an extension of CCA wherein the influence of a set of variables stored in an additional matrix can be controlled for. The concept is related to partial correlation.

This is particularly useful when one wishes to control for a set of variables whose influence is known or at least anticipated and which are not of immediate interest. Examples include geographic distance, latitudinal temperature gradients, or depth-dependent photogradients.

Controlling for the effect of different sampling or measurement times or locations between samples is also possible. It must be determined whether time and/or space are best represented by dummy variables or appropriately transformed quantitative variables. As CCA is a method tuned to represent centroids (multivariate means) of data sets, if the influence of time and/or space is restricted to shifting these centroids then its effect can be well controlled for. However, if time/space effects show more complex influence or interact with other explanatory or control variables, these factors cannot be controlled for by pCCA.

The method can also be applied to examine the effect of a single variable in a matrix of explanatory variables using pCCA, while controlling for the other variables. This is done by placing all other explanatory variables in a matrix of control variables. Their effects may then be partialled out. A single canonical axis and eigenvalue will be generated which express the variation that the variable of interest is responsible for.