Multivariate Analysis of Variance (MANOVA)

Purpose

When populations are compared, they are generally compared with respect to multiple outcomes or response measures. For example, varying the levels of a vitamin dosage (e.g., 500, 1,000, 1,500, or 2,000 IU) may be the grouping variable under investigation, and the consequences of dosage variation with respect to several outcome measures (e.g., Y1 = diastolic blood pressure, Y2 = systolic blood pressure, Y3 = heart rate, Y4 = anxiety, Y5 = mood) may be of interest. Multiple outcomes are often observed because no single outcome measure can adequately capture the intended construct(s) of interest. For example, measures Y1, Y2, and Y3 may be indicators of physical health while Y4 and Y5 may be indicators of psychological health. Both physical and psychological health are latent constructs that cannot be adequately assessed by any single indicator. However, by combining several indicators, an estimate of a construct can be provided. The purpose of MANOVA is to determine the best combination of indicators to estimate one or more constructs that maximize group differences. Measures are combined by multiplying (i.e., weighting) individual indicators by constants (e.g., Z = b1Y1 + b2Y2 + … + b5Y5) to create a composite (i.e., Z) of the outcome measures. The analysis and interpretation of the composites that define the group differences is one of the primary advantages of the multivariate model compared with the univariate model. Other important advantages of the multivariate model include a reduction in the risk of Type I errors and more sensitive (powerful) group comparisons.

Hypothesis Tested

The set of means on the outcome measures within each group is called a mean centroid. MANOVA tests the hypothesis that the populations, represented by the groups, have identical centroids: H0: μ1 = μ2 = … = μj (j = 1, 2, …, j), where μj = [μj1, μj2, …, μjp [.]T = transpose of vector, μjm = mean of population j (j = 1, …, J) for outcome measure m (m = 1, …, p). Using the vitamin dosage example introduced in the previous section, suppose the mean scores for the five outcome measures observed for the group receiving 500 IU of the vitamin equaled 5,

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