FAQ: When do we have to worry about a violation of sphericity?

Whenever you run a repeated measures design with more than 2 repeated measures (e.g. measuring a group of participants on the criterion three times each, at Time 1, Time 2, and Time 3), you need to worry about sphericity on all of your within-subjects effects.

So why don’t we need to worry about sphericity if we only have 2 repeated measures (e.g. measuring a group of participants on the criterion at just Time 1 and Time 2)? Remember what this assumption actually refers to: the population variances of the sets of difference scores should be equal. So for three time points, you’ll have three sets of difference scores: T1-T2, T1-T3, and T2-T3. When you only have two time points, there’s only one set of difference scores: T1-T2. It’s impossible to have a violation. The test we run to check this assumption is Mauchly’s W, which is actually technically a test of compound symmetry rather than sphericity (it works out well, though, since whenever we have compound symmetry, we can trust that sphericity is also okay). Mauchly’s compares the covariance matrix for the repeated measures to a smoothed out version of it, where all of the covariances are replaced by the average covariance, and all of the variances (down the diagonal) are replaced by the average variance. So basically, Mauchly’s tests whether all of the variances are equal, and whether all of the covariances are equal. Again, if there are only two repeated measures, it’s impossible to have a violation since there is only one covariance.

Note that this logic applies to contrasts as well as to F tests on main effects. When you run contrasts on the levels of your repeated measure, you’re only comparing two groups at a time (it’s possible that one or both of those groups is made up of observations pooled from other groups, but when you’re testing them it’s been simplified to just two groups). Since there are only two groups, it’s impossible to have a violation of sphericity or compound symmetry, so SPSS doesn’t provide corrections for tests with 1 df (i.e. tests with only two groups), such as contrasts.