MANOVA

MANOVA, or multivariate analysis of variance, is a statistical technique used to assess whether mean vectors of two or more dependent variables differ across levels of one or more categorical independent variables. It generalizes analysis of variance (ANOVA) by considering multiple correlated outcomes simultaneously, reducing the risk of inflating type I error and providing a multivariate view of group differences.

In the typical model, there are g groups and p dependent variables. Observations form p-dimensional vectors within each group, assumed to follow a multivariate normal distribution with group-specific mean vectors and covariance matrices. The null hypothesis states that all group mean vectors are equal (mu1 = mu2 = ... = mug). The alternative posits that at least one mean vector differs. The test relies on the between-group and within-group SSCP (sum of squares and cross-products) matrices to produce statistics.

Common test statistics include Wilks' lambda, Pillai's trace, Hotelling's T-squared, and Roy's largest root. These statistics assess the overall multivariate difference among groups. A significant result indicates that the dependent variables, taken together, differ across groups. When significant, researchers typically follow with univariate ANOVAs on each dependent variable with appropriate adjustments, or use discriminant analysis for further interpretation.

Assumptions and design variants are key considerations. MANOVA assumes independence of observations, multivariate normality within groups, and homogeneity of covariance matrices across groups. Box's M test can assess covariance equality but is sensitive to non-normality. Designs include one-way MANOVA, factorial MANOVA, repeated-measures MANOVA, and MANCOVA when covariates are included. Effect sizes (e.g., partial eta squared) help quantify the strength of multivariate effects.