Multivariate Analysis of Variance

Multivariate Analysis Of Variance

Multivariate analysis of variance (MANOVA) is a statistical test procedure for comparing multivariate (population) means of several groups. Unlike ANOVA, it uses the variance-covariance between variables in testing the statistical significance of the mean differences.

It is a generalized form of univariate analysis of variance (ANOVA). It is used when there are two or more dependent variables. It helps to answer : 1. do changes in the independent variable(s) have significant effects on the dependent variables; 2. what are the interactions among the dependent variables and 3. among the independent variables. Essentially, MANOVA takes scores from the multiple dependent variable and creates a single dependent variable giving the ability to test for the above effects. Statistical reports however will provide individual p-values for each dependent variable, indicating whether differences and interactions are statistically significant.

Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices appear. The diagonal entries are the same kinds of sums of squares that appear in univariate ANOVA. The off-diagonal entries are corresponding sums of products. Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution.

Analogous to ANOVA, MANOVA is based on the product of model variance matrix, and inverse of the error variance matrix, or . The hypothesis that implies that the product . Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product, but there is no unique choice owing to the multi-dimensional nature of the alternative hypothesis.

The most common statistics are summaries based on the roots (or eigenvalues) of the matrix:

• Samuel Stanley Wilks' distributed as lambda (Λ)
• the Pillai-M. S. Bartlett trace,
• the Lawley-Hotelling trace,
• Roy's greatest root (also called Roy's largest root),

Discussion continues over the merits of each, though the greatest root leads only to a bound on significance which is not generally of practical interest. A further complication is that the distribution of these statistics under the null hypothesis is not straightforward and can only be approximated except in a few low-dimensional cases. The best-known approximation for Wilks' lambda was derived by C. R. Rao.

In the case of two groups, all the statistics are equivalent and the test reduces to Hotelling's T-square.