# Non normal distribution for 4-way mixed ANOVA

I am analysing a study using a 4-way mixed ANOVA. It has one between-subjects factor and 3 within-subjects factors (with 2, 2, and 3 levels). My sample size is 63 - 29 in one group and 34 in the other.

I have 9 dependent variables, all which asked participants to choose their answer on a scale of 0 to 10. I ran normality tests on my data, and since it is a 2 x 2 x 2 x 3 ANOVA, I had to run 24 tests per dependent variable. The histograms from most of the data don't look very normal, but it is a particular problem from 2 groups of data, since participants were more likely to enter '0'. This means that transforming the data has minimal effect, since the value remains the same at '0'.

Since there are no non-parametric tests for mixed ANOVAs, what can I do?

Thanks for any help you can give me, I really appreciate it. I am really stuck and have spent so much time looking at different questions but have not been able to find my answer yet.

You do not need to run 24 separate normality tests since the assumption to be met in ANOVA is the normality of the residuals of the model and not the normality of the dependent variables in combinations of all factors levels.

This misunderstanding comes from the fact that the necessary normality of residuals is derived from the normality of the dependent variables in all factor combinations however this is a more strict assumption since the opposite is true only with the additional assumption of homogeneity among all factor combinations. It is also worth noting that SPSS does not offer the choice to save residuals in the simple 1-way ANOVA (Analyze > Compare Means > ANOVA) since they probably assume that when one factor is present, normality tests for the dependent in all levels of the factor is an easy thing to do (with Explore procedure for example).

So, you should compute and save the residuals and after check for normality with just one test of your choice.

Beware that the residuals are possible to be computed only after the model has been created so you should find how to save that residuals in a separate variable. (In case of SPSS, in the main ANOVA dialog of your choice, press Save button, in case of R residuals are computed and saved in the output of every lm command)

Hope this helps you.

• Thanks for your reply. I'm still confused though, because all of the stats books I have been reading (e.g. Andy Field) have said that you need to check for normality at each level. When I run a repeated measures test, SPSS computes the residuals for the 12 dependent variable levels. Is that what you want me to check for normality, the 12 of them? May 5, 2014 at 12:59
• You should make only one normality test for the variable RES_1 created by SPSS in your data set. Probably many books propose the normality of DV among groups because it is a clear separate step before main analysis, thus more user friendly for a statistics student, while testing directly the distribution of residuals is more technical think to do. May 5, 2014 at 13:27
• Thanks again for your reply. I ran the Q-Q plots of the residuals, and Res_1 has a distinct S shape, meaning it's not normal. What's my next step after that? Transform? Because I have floored data with many 0's, transformations don't seem to help. May 5, 2014 at 14:57
• Are you interested in interactions between factors? If this is not a crucial part of your analysis then you may run separate MANOVA (or Repeated Measures) and consider a correction in your type I error. May 5, 2014 at 17:20
• Yes, I am interested in interactions unfortunately. Is there an alternative if I was interested in interactions? May 5, 2014 at 19:44