Assumptions Underlying the One-Way ANOVA with One Repeated-Measures Factor

Assumptions for the Multivariate Test

• Level of measurement. Repeated measures designs are so named because they normally involve obtaining repeated measures on some criterion variable from a single sample of participants. This criterion variable should be assessed on an interval- or ratio-level of measurement. The predictor variable should be a nominal-level variable (a categorical variable) that typically codes “time,” “trial,” “treatment,” or some similar construct.

• 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. This is another way of saying, for example, that it is acceptable for participants’ scores in condition 1 to be correlated with their scores in condition 2 and condition 3.

• 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 very small effect on the Type I error rate (i.e., the probability of incorrectly rejecting a true null hypothesis).

Assumptions for the Univariate Test

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

• Sphericity. In order to understand the sphericity assumption, it is necessary to first understand the nature of the difference variables that are created in performing a repeated-measures ANOVA. Assume that a study is conducted in which each participant provides scores on the criterion variable under each of three conditions: the variable V1 includes scores from condition 1; V2 includes scores from condition 2; and V3 includes scores from condition 3.

  It is possible to create a difference variable (called D1) by subtracting participants’ scores on V2 from their score on V1:

  D1 = V1 - V2

  Similarly, it is possible to create a separate difference variable by subtracting participants’ scores on V3 from their score on V2:

  D2 = V2 - V3

  Difference variables are created in this way by subtracting participants’ scores observed in adjacent conditions (e.g., V1 – V2, V2 – V3). In a given study, the number of difference variables created will be equal to k–1, where k = the number of conditions. The present study included three conditions and therefore will include two difference variables (D1 and D2).

  It is now possible to compute the variances of each of these difference variables as well as the covariances between the difference variables. You can arrange these values in a variance-covariance matrix. You review this matrix to determine whether its corresponding matrix in the population demonstrates sphericity.

  Two conditions must be satisfied for a variance-covariance matrix to demonstrate sphericity. First, each variance on the diagonal of the matrix should be equal to every other variance on the diagonal. In the present case, this means that the variance for D1 should equal the variance for D2. Second, each covariance off of the diagonal should equal zero. (This is analogous to saying that the correlations between the difference variables should be zero.) In the present case, this means that the covariance between D1 and D2 should be equal to zero.

  PROC GLM performs a test for sphericity by requesting the PRINTE option in the REPEATED statement. When this test indicates that the sphericity assumption is satisfied, you may interpret the univariate test. When the test indicates that the sphericity assumption is not satisfied (as will often be the case), the situation becomes more complicated. The options available under these circumstances are discussed in detail in the section “Further Notes on Repeated-Measures Analysis,” earlier in this chapter.