# How to cope with non-normal ANOVA residuals?

IV1    IV2    IV3    IV4    RV
1      1      0.5    B      5.3
2      2      1      U      15.3
2      1      0.5    A      51.3
3      3      2      B      5.35


where:

• IV1 has 3 levels: 1, 2 and 0.5
• IV2 has 3 levels: 1, 2 and 0.5
• IV3 has 3 levels: 1, 2 and 3
• IV4 has 3 levels: B, U, R, S, A

All of them are categorical. The response variable is times measured in seconds.

Due to the non-normality of the data and the heteroscedasticity I cannot use Analysis of Variance (ANOVA). I have tried five data transformations to see if I can apply ANOVA (square-root, 2-power, e-power, log, ln). None of these fixes the data so that they pass either visual exploration or Levene and Kolmogorov's tests. Non-parametric test (distribution-free) like Kruskal-Wallis or Freidman cannot handle factor interaction as can ANOVA.

What can I do?

• What does a Q-Q plot of the ANOVA residuals look like? – Glen_b -Reinstate Monica Oct 29 '14 at 16:26
• – rica01 Oct 29 '14 at 20:37
• Thanks. That's an ... interesting plot. Was the response variable (the times) almost discrete (lots of observations with the same or nearly the same value) or heavily multimodal? – Glen_b -Reinstate Monica Oct 29 '14 at 21:16
• there are several time measures that repeat – rica01 Oct 30 '14 at 21:58
• I'm trying to find some way to account for the odd structure in residuals. Is it only a small percentage? How does that occur? It seems to argue against treating it as continuous. – Glen_b -Reinstate Monica Oct 30 '14 at 22:05

Most of the answer can be found here: What is the non-parametric equivalent of a two-way ANOVA that can handle interactions? Namely, you would use ordinal logistic regression. This is probably your best bet. The way it would work is that you convert the times to ranks and regress the ranks onto your covariates using OLR. Certainly if $L$ seconds were greater than $k$ seconds then this contains valid ordinal information. Depending on the software you use, you may not even have to convert the times explicitly into ranks (it would be done for you under the assumption that bigger numbers correspond to higher ranks). Bear in mind that the Kruskal-Wallis test is a special case of OLR, so any time KW might be valid OLR is as well. However, OLR is more flexible than KW, for example, it can handle factorial structure and interactions.