Assumptions Underlying Factorial ANOVA with Repeated-Measures Factors and Between-Subjects Factors

All of the statistical assumptions associated with a one-way ANOVA, repeated-measures design are also required for a factorial ANOVA with repeated-measures factors and between-subjects factors. In addition, the latter design requires a homogeneity of covariances assumption for the multivariate test and the homogeneity assumption as well as two symmetry conditions for the univariate test. This section reviews the assumptions for the one-way repeated measures ANOVA and introduces the new assumptions for the factorial ANOVA, mixed design.

Assumptions for the Multivariate Test

• Level of measurement. The criterion variable should be assessed on an interval or ratio level of measurement. The predictor variables should both be nominal-level variables (i.e., categorical variables). One predictor codes the within-subjects variable, and the second codes the between-subjects variable.

• Independent observations. A given participant’s score in any one condition should not be affected by any other participant’s score in any of the study’s conditions. However, it is acceptable for a given participant’s score in one condition to be dependent upon his or her own score in a different condition (under the within-subjects predictor variable).

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

• Multivariate normality. The measurements obtained from participants should follow a multivariate normal distribution. Under conditions normally encountered in social science research, violations of this assumption have only a small effect on the Type I error rate (i.e., the probability of incorrectly rejecting a true null hypothesis).

• Homogeneity of covariance matrices. In the population, the criterion-variable covariance matrix for a given group (under the between-subject’s predictor variable) should be equal to the covariance matrix for each of the remaining groups. This assumption was discussed in greater detail in Chapter 11, “Multivariate Analysis of Variance (MANOVA) with One Between-Subjects Factor.”

Assumptions for the Univariate Test

The univariate test requires all of the preceding assumptions as well as the following assumptions of sphericity and symmetry:

• Sphericity. Sphericity is a characteristic of a difference-variable covariance matrix that is obtained when performing a repeated-measures ANOVA. (The concept of sphericity was discussed in greater detail in Chapter 12.) Briefly, two conditions must be satisfied for the covariance matrix to demonstrate sphericity. First, each variance on the diagonal of the matrix should be equal to every other variance on the diagonal. Second, each covariance off the diagonal should equal zero. (This is analogous to saying that the correlations between the difference variables should be zero.) Remember that, in a study with a between-subjects factor, there is a separate difference-variable covariance matrix for each group under the between-subjects variable.

• Symmetry conditions. There are two symmetry conditions, and the first of these is the sphericity condition just described. The second condition is that the difference-variable covariance matrices obtained for the various groups (under the between-subjects factor) should be equal to one another.

For example, assume that a researcher has conducted a study that includes a repeated-measures factor with three conditions and a between-subjects factor with two conditions. Participants assigned to condition 1 under the between-subjects factor are designated as “group 1” and those assigned to condition 2 are designated as “group 2.” With this research design, one difference-variable covariance matrix is obtained for group 1 and a second for group 2. (The nature of these difference-variable covariance matrices was discussed in Chapter 12.) The symmetry conditions are met if both matrices demonstrate sphericity and each element in the matrix for group 1 is equal to its corresponding element in the matrix for group 2.