Multivariate Analysis of Variance (MANOVA)

Multivariate analysis of variance, or MANOVA, is a data-modeling technique that is a powerful alternative to its univariate analysis of variance (ANOVA) counterpart. With traditional ANOVA, mean differences between groups on a single quantitative variable can be analyzed. For instance, consider a school psychologist who is interested in examining the average performance of Asian, Caucasian, and Hispanic children on a standardized test of intelligence that yields continuous scores on three subscales: Mathematics, Verbal, and Processing Speed. Adopting a univariate frame of mind, the psychologist could conduct three separate ANOVAs to determine if the means of the ethnic groups are equivalent on each of the three subscales. If the psychologist viewed the data from a multivariate standpoint, however, he or she could conduct a single MANOVA to examine differences between the Asian, Caucasian, and Hispanic children on linear combinations of the three intelligence subscales. In other words, he or she could combine the Mathematics, Verbal, and Processing Speed scores to form one or more multivariate composites on which the three ethnic groups could be compared. Perhaps a combination of high Mathematics, high Processing Speed, and low Verbal scores will provide the most effective composite for discriminating among the three groups? Perhaps the simple difference between the Mathematics and Verbal subscale scores will instead provide the most effective composite? Such questions are multivariate in nature because they address the interrelationships among the continuous measures of intellectual ability.

The primary strength of MANOVA is thus the capability to examine group differences on linear combinations of quantitative variables. For the sake of convenience, grouping variables will herein be referred to as independent variables, and the quantitative variables will be referred to as dependent variables. In the example above, ethnicity (with three levels: Asian, Caucasian, and Hispanic) is the independent variable, and the three intelligence subscales are the dependent variables. Mathematically speaking, there are no limits to the number of independent variables or dependent variables that can be included in the analysis, and the independent variables can be composed of two or more groups. When only one independent variable is included, the analysis is referred to as a one-way (or one-factor) MANOVA, and when two or more independent variables are included, the analysis is referred to as a factorial MANOVA. In the special case of one independent variable with two levels, the analysis is often referred to as a Hotelling's T, which is the multivariate generalization of the independent samples t test. Regardless[Page 670] of the study design, the goal of the MANOVA is to determine if the independent variable groups differ in their means on at least one linear combination of the dependent variables.