Multivariate Analysis of Variance

Multivariate analysis of variance (MANOVA) is a statistical technique used extensively in all types of research. It is the same thing as an analysis of variance (ANOVA), except that there is more than one dependent or response variable. The mathematical methods and assumptions of MANOVA are simply expansions of ANOVA from the univariate case to the multivariate case.

A situation in which MANOVA is appropriate is when a researcher conducts an experiment where several responses are measured on each experimental unit (subject) and experimental units have been randomly assigned to experimental conditions (treatments). For example, in a double-blind study systolic and diastolic blood pressures are measured on subjects who have been randomly assigned to one of two treatment groups. One group receives a new medication to treat high blood pressure, and the other group receives a placebo. The group that receives the placebo is considered a control group. The researcher wants to know whether the new medication is effective at lowering blood pressure.

There are several compelling reasons for conducting a MANOVA instead of an ANOVA. First, it is more efficient and economical in the long run to measure more than one response variable during the course of an experiment. If only one response is measured, there is the risk that another important response has been ignored. The measurement of several response variables provides a more thorough understanding of the nature of group differences given the response variables.

Another good reason to use MANOVA is that analyzing multiple responses simultaneously with a multivariate test is more powerful than analyzing the individual responses separately with multiple univariate tests. The chances of incorrectly rejecting the null hypothesis are inflated with multiple univariate tests, because the Type I error rate (a) increases with each additional test. For instance, the overall Type I error [Page 702]rate for two univariate tests each with a set at .05 is .10 (1 − (.95)2) rather than .05.

Finally, the correlation between the response variables is taken into account in a multivariate test. The result is that differences between groups that are not detected by multiple univariate analyses may become obvious. Figure 1 illustrates the hypothetical univariate distributions of two response variables, X1 and X2, for two study groups. The distributions appear to overlap such that no significant difference in means between groups is expected. In Figure 2, the multivariate distributions for the same response variables are illustrated with 95% confidence ellipses drawn about the group means. The figure shows that the two groups do not overlap as much as might be expected given the univariate distributions. Figure 3 illustrates the same multivariate distributions, except that the response variables are negatively correlated rather than positively correlated as in Figure 2. Given the degree to which the ellipses overlap in Figure 3, the null hypotheses may not be rejected.

When conducting a MANOVA, several assumptions are made about the data. When these assumptions do not hold, conclusions based on the analysis may be erroneous. The assumptions are as follows:

• The experimental units are random samples of target populations.
• The observations are independent of each other.
• The response variables are distributed multivariate normal. There is no test for multivariate normality commonly available. Generally, multivariate normality can be assumed when the individual variables are normally distributed; however, it is not guaranteed. Additionally, MANOVA is particularly sensitive to outliers so it is important to check for them prior to analysis. Outliers should be transformed or omitted from the analysis. Deviation from multivariate normality has less impact in larger samples.
• All populations have the same covariance matrix (homogeneity of covariance). This assumption is made because the error sums of squares are computed by adding the treatment sums of squares weighted by (ni − 1), where ni is the number of experimental units in each treatment. Otherwise, adding the treatment sums of squares would be inappropriate.

Figure 1 Univariate Distributions for Two Populations, Two Responses

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