Multivariate Analysis of Variance

Multivariate analysis of variance tests whether group mean vectors differ across two or more groups when several response variables are measured together.

Definition

Multivariate analysis of variance is a method that tests the equality of mean vectors across groups by comparing between-group and within-group sum-of-squares-and-cross-products matrices using multivariate test statistics.

Scope

This topic covers the comparison of mean vectors, the two-sample Hotelling's T-squared test, the partition of the total sum-of-squares-and-cross-products matrix into hypothesis and error components, the multivariate test statistics built from their eigenvalues, and the advantages of a single multivariate test over separate univariate analyses of variance.

Core questions

• Do groups differ on a set of response variables considered jointly?
• How is the two-group comparison of mean vectors tested?
• How are the hypothesis and error cross-product matrices combined into a test?
• Why prefer a multivariate test to several univariate tests?

Key theories

Hotelling's T-squared

For comparing two mean vectors, Hotelling's T-squared generalizes the two-sample t statistic using the pooled covariance and Mahalanobis distance between the sample means, providing a single multivariate test.

Hypothesis and error matrices

The total cross-product matrix splits into between-group and within-group parts, and statistics such as Wilks's lambda and the Pillai trace are functions of the eigenvalues of their combination, giving the multivariate test of equal mean vectors.

Clinical relevance

Multivariate analysis of variance is used to compare groups across several correlated outcomes simultaneously, controlling the overall error rate and detecting differences in combinations of variables that univariate tests might miss.

History

The comparison of mean vectors developed from Hotelling's generalization of the t-test in the early 1930s and from Wilks's likelihood-ratio criterion, forming the multivariate analysis-of-variance framework that became standard in classical multivariate analysis.

Debates

Following up a significant MANOVA

How best to interpret a significant overall test, whether through univariate follow-ups, discriminant analysis, or examination of specific contrasts, is debated, since each approach answers a different question about where the difference lies.

Key figures

• Harold Hotelling
• Samuel Wilks
• S. N. Roy

Seminal works

• anderson2003
• johnson2007
• mardia1979

Frequently asked questions

Why use MANOVA instead of several ANOVAs?

MANOVA controls the overall error rate across outcomes and can detect group differences in combinations of correlated variables that separate univariate tests would miss.

What is Hotelling's T-squared?

It is the multivariate generalization of the two-sample t statistic, measuring the Mahalanobis distance between two sample mean vectors under a pooled covariance matrix.