Assumptions Underlying Multivariate ANOVA with One Between-Subjects Factor

• Level of measurement. Each criterion variable should be assessed on an interval or ratio level of measurement. The predictor variable should be a nominal-level variable (i.e., a categorical variable).

• Independent observations. Across participants, a given observation should not be dependent on any other observation in any group. It is acceptable for the various criterion variables to be correlated with one another; however, a given participant’s score on any criterion variable should not be affected by any other participant’s score on any criterion variable. (For a more detailed explanation of this assumption, see Chapter 8, “t Tests: Independent Samples and Paired Samples.”)

• Random sampling. Scores on the criterion variables should represent a random sample drawn from the populations of interest.

• Multivariate normality. In each group, scores on the various criterion variables should follow a multivariate normal distribution. Under conditions normally encountered in social science research, violations of this assumption have a small effect on the Type I error rate (i.e., the probability of incorrectly rejecting a true null hypothesis). On the other hand, when the data are platykurtic (form a relatively flat distribution), the power of the test might be significantly attenuated. (The power of the test is the probability of correctly rejecting a false null hypothesis.) Platykurtic distributions can be transformed to better approximate normality (see Stevens, 2002; or Tabachnick & Fidell, 2001).

• Homogeneity of covariance matrices. In the population, the criterion-variable covariance matrix for a given group should equal the covariance matrix for each remaining group. This is the multivariate extension of the “homogeneity of variance” assumption in univariate ANOVA. To illustrate, consider a simple example with just two groups and three criterion variables (V1, V2, and V3). To satisfy the homogeneity assumptions, the variance of V1 in group 1 must equal the variance of V1 in group 2. The same must be true for the variances of V2 and V3. In addition, the covariance between V1 and V2 in group 1 must equal the covariance between V1 and V2 in group 2. The same must be true for the remaining covariances (between V1 and V3 and between V2 and V3). It becomes evident that the number of corresponding elements that must be equal increases dramatically as the number of groups increases and/or as the number of criterion variables increases. For this reason, the homogeneity of covariance assumption is rarely satisfied in real-world research. Fortunately, the Type I error rate associated with MANOVA is relatively robust against typical violations of this assumption so long as the sample sizes are equal. The power of the test, however, tends to be attenuated when the homogeneity assumption is violated.