CANCOR: A general least-squares program for univariate and multivariate analysis of variance and covariance

Abstract

Many psychologists conceive of canonical correlation as a statistical technique that relates two sets of continuous variables; whether one set is deemed the "predictor set" and the other the "criterion set" is often arbitrary. Psychologists are now aware of the applicability of least-squares or regression methods to conventional experimental designs, with the independent variables serving as predictors and the dependent measures as criteria. Cohen (1968), for example, points out that, if one constructs dummy variables (+1, 0, -1) representing each between-cell degree of freedom in an experimental design and regresses the dependent measure upon these predictors, then a multiple regression analog to univariate analysis of variance (ANOVA) is achieved. If, in addition to the discrete design predictors, the researcher includes one or more continuous variables in the predictor set, then a univariate analysis of covariance (ANCOVA) can be accomplished. Leastsquares techniques are also applicable to designs involving multiple dependent measures (Woodward & Overall, 1975). Just as multiple correlation is the least-squares analog to univariate ANOVA and ANCOVA, canonical correlation is the analog to multivariate analysis of variance (MANOVA) and covariance (MANCOVA) (Knapp, 1978). SinceANOVA is simplya special case of multiple regression (i.e., one which employs discrete design predictors), which in turn is a special case of canonical correlation (Le., one which has only one criterion), canonical correlation seems to be the most general of these techniques, the closest approximation to the general linear model. Insofar as a canonical correlation analysis is concerned, the only factors that distinguish the various techniques are the number of criteria and whether the predictors are continuous, discrete, or mixed. In sum, canonical correlation can be used to perform any univariate or multivariate analysis of variance, covariance, or regression. Although there are several canonical correlation programs available [e.g., SPSS's CANCORR (Nie, Hull,