Multivariate Analysis of Variance

Multivariate analysis of variance (MANOVA) is an extension of univariate analysis of variance (ANOVA) in which the independent variable is some combination of group membership but there is more than one dependent variable. MANOVA is often used either when the researcher has correlated dependent variables or, instead of a repeated-measures ANOVA, to avoid the sphericity assumption. While MANOVA has the advantage of providing a single, more powerful test of multiple dependent variables, it can be difficult to interpret the results.

For example, a researcher might have a large data set of information from a high school about their former students. Each student can be described using a combination of two factors: gender (male or female) and whether they graduated from high school (yes or no). The researcher wishes to analyze and make decisions about the statistical significance of the main effects and interaction of the factors using a simultaneous combination of interval predictor variables such as GPA, attendance, degree of participation in various extracurricular activities (e.g., band, athletics), weekly amount of screen time, and family income.

Put in a broader statistical context, MANOVA is a special case of canonical correlation and is closely related to discriminant function analysis (DFA). DFA predicts group membership based on multiple interval measures and can be used after a MANOVA to assist in the interpretation of the results.

This entry explains MANOVA by first reviewing the underlying theory of univariate ANOVA and then demonstrating how MANOVA extends ANOVA by using the simplest case of two [Page 1120]dependent variables. After the rationale of the analysis is understood, it can be extended to more than two dependent variables but is difficult to present visually. In that case, matrix algebra provides a shorthand method of mathematically presenting the analysis.

Univariate ANOVA

In univariate ANOVA, the independent variable is some combination of group membership and a single interval-dependent variable. The data can be visualized as separate histograms for each group, as seen in Figure 1, with four groups of 20 observations each.

Figure 1 Histogram of four groups

The ratio of the variability between the means of the groups relative to the variability within groups is fundamental to ANOVA. This is done by modeling the sampling distribution of each group with a normal curve model, assuming that both the separate sample means estimate µ and σ is equal in all groups and estimated by a formula using a weighted mean of the sample variances. The assumption of identical within-group variability is called the homogeneity of variance assumption. The model of the previous data is illustrated in Figure 2.

Figure 2 Normal curve model of four groups

From this model, two estimates of σ2 are computed. The first, mean square between (MSB), uses the variability of the means, and the second, mean square within (MSW), uses the estimate of combined variability within the groups. A computed statistic, called F, is the ratio of the two variance estimates:

$$F={\text{MS}}_{\text{B}}/{\text{MS}}_{\text{W}}.$$

The distribution of the F statistic is known, given the assumptions of the model are correct. If the computed F ratio is large relative to what would be expected by chance, then real effects can be inferred; that is, the means of the groups are significantly different from each other. The between variability can be partitioned using contrasts to account for the structure of group membership, with separate main effects, interactions, and nested main effects, among others, being tested using the ANOVA procedure.

...