Canonical Analysis Applied to Multivariate Analysis of Variance

In this paper it is shown how a generalized form of canonical analysis can be useful to reveal which parametric functions of a MANOVA model, for instance treatment contrasts or combinations of observed variables, are responsible for rejection of a general linear hypothesis on these functions. For the decomposition in successive canonical terms the choice of a matrix norm is crucial. It is shown that the norm derived from the standardization of the least squares estimators of the parametric functions involved is equal to the Lawley–Hotelling statistic for testing the hypothesis under investigation. Thus, some useful interpretations based on canonical variates can be given in terms of the contributions of the various parametric functions to the overall test statistic or to statistics relevant to specific subhypotheses. Corresponding to these possibilities for interpretation, three different types of biplot are proposed. As an example, an agricultural block design experiment is thoroughly analyzed.