# Understanding repeated measure ANOVA assumptions for correct interpretation of SPSS output

I am investigating whether different reward conditions may affect task performance. I have data from a small study with two groups, each with n=20. I collected data on a task that involved performance in 3 different "reward" conditions. The task involved performance in each of the 3 conditions twice but in random order. I want to see if there is a mean difference in task performance for each group, in each of the different "reward" conditions.

• IV= Group type
• DV = mean measure of task performance across 3 conditions

I have output from a repeated measures ANOVA and access to the raw data set in SPSS but am unsure how to proceed. I haven't been able to find a step-by-step guide for this interpretation, as the Pallant text is somewhat limited. My particular issues are in the following areas:

1. Do I check the normality of each of my variables individually or within combinations of each of the levels of the IV? If it within combinations, how do I check that?
2. Do I check Mauchly's Test first? If it is violated, what does that mean? If it is not violated, what does that mean?
3. When is it okay to look at the multivariate tests tables, or the tests of within-subjects effects? I'm not sure when it is appropriate to use either (or both?)?
4. Is it always okay to look at the pairwise comparisons? It seems counterintuitive to do so if the multivariate or within-subjects effects don't indicate significance (ie P<0.05) but I am again unsure.
• You got some good responses here. If any of them helped you, please consider accepting one of them. It's what keeps people answering questions :) – ThomasH Nov 28 '12 at 18:03

1. Your dependent variables should be normal in each cell of between-subject design. You have 2 such cells: 2 groups, so normality should be in both groups. Also, variance-covariance matrix between your 3 DV should be same in the 2 groups. You could check normality by Shapiro-Wilk test or Kolmogorov-Smirnov (with Lilliefors correction) test in EXPLORE procedure. Variance-covariance homogeneity could be tested by Box's M test (found in Discriminant analysis). Note however that ANOVA is quite robust to violations to both assumptions.

2. Mauchly's test checks the so called sphericity assumption which is necessary for univariate approach to repeated measures ANOVA. This assumption requires that, roughly speaking, differences between your repeated measure DVs don't intercorrelate. If the assumption is violated you should disregard "Spericity assumed" in Tests of Within-Subjects Effects table - there found some corrections (such as Greenhouse-Geisser) instead.

3. While Tests of Within-Subjects Effects table reflects "univariate approach" in RM-ANOVA, Multivariate Tests table reflects "multivariate approach". These two are both useful and there's a little debate which is "better". Read a little here about them, a bit more here.

4. Usually one won't check pairwise tests if overall effect is non-significant, it has little sense.

• Since the test of the between-factor here is equivalent to a oneway ANOVA with the per-person averages over the within-factor, these averages need to be normal and have identical theoretical variances - not the original data. For the test of the within-factor, one needs to assume multivariate normality of the per-person data vectors. Of course, if this is the case, then their average is also normal. – caracal Sep 5 '11 at 13:16
• Did I understand you right, that if we take interest only in between-subject effect, DVs need not make mutivariate normal cloud, it's just their average variable that should be normal. If we take interest in within-subject effect, DVs have to make mutivariate normal cloud. – ttnphns Sep 6 '11 at 8:21
• Exactly, and the stricter assumptions for the test of the full split-plot model imply the assumptions for the test of only the between factor (multivariate normality $\rightarrow$ normality of per-person means, equality of theoretical covariance-matrices $\rightarrow$ equality of theoretical variances of per-person means). – caracal Sep 6 '11 at 10:39
• @ttnphns I've seen multiple references state that the normality should be in the within-subjects factor, not the between-. The within-subjects factor here is the reward condition. Here are two references where this is stated: stat.cmu.edu/~hseltman/309/Book/chapter14.pdf (pg. 11); google.com/… (pg. 4) – Meg Dec 11 '15 at 20:42

### General Resource on interpreting repeated measures ANOVA with SPSS

It sounds like you need a better general resource on repeated measures ANOVA. Here are a few web resources, but in general a search for "SPSS repeated measures ANOVA" will yield many useful options.

### 1. Checking normality

• From a practical perspective, tests of normality are often used to justify transformations. If you do apply a transformation, then you need to apply the same transformation to all cells of the design.
• A common way to assess normality using SPSS is to set up your model and save the residuals and then examine the distribution of residuals.

### 2. Value of Mauchly's test

• A common strategy is to look at Mauchly's test and if it is statistically significant, interpret either the univariate corrected tests or the multivariate tests.

### 3. Multivariate

• I think @ttnphns has summed this up well.

### 4. Pairwise comparisons

• I think @ttnphns has summed this up well.
• I would avoid the Field article, which has been somewhat carelessly put together and makes at least one definite error (mistaking Type I and Type II). – rolando2 Jun 5 '18 at 10:07