# When distributions are non normal and heteroscedastic, is it preferred to use ordinal logistic regression or permutation tests?

I am currently conducting statistical tests on my two independent samples (both with more than 1500 entries each). The sample sizes are no equal. My response variables are interval as well as quasi interval (Likert scale) variables. My two independent variables are nominal.

Normality test I have checked these on normality with a Shapiro Wilks test and several visualizations (ggq plots and histograms). However, all of them were non-normal.

Homogeneity test Afterwards, I have run a Levene test for each variable. The results also indicated that the variances are heterogeneous.

I tried to transform my data with the log transformation and box-cox but I haven't had luck yet.

Therefore, I tried to find alternatives. I have run the white adjusted two-way ANOVA as well as the one way Welch ANOVA. However, to back up my results, I would like to conduct a test that actually meets all my assumptions. Therefore, I wanted to use the Kruskal-Wallis test but read that this test is not robust when the data is heteroscedastic. On this site, I have read several recommendations including the permutation test and the ordinal logistic regression. My question is now which one of these tests is actually preferred in the nature of these distributions?

Any help is appreciated.

• With hetroscedasticity, I'd worry about exchangeability, required for permutation test. Jul 3, 2020 at 8:16
• You need exchangeability under the null; you won't have it under the alternative. As you don't know whether the null is true, you can't necessarily decide whether you would have had it under the null simply by looking at the data (not that you should be looking at the data to decide what test to use anyway). Typically exchangeability under the null is an assumption, not a result of peering at the data. It may well be that the spread and the mean change together (for example, consider a Gamma GLM, where differences in mean are due to differences in the scale ... ctd Jul 3, 2020 at 8:43
• ctd ... it would be perfectly okay to use a Kruskal-Wallis in that situation (though less efficient than the Gamma GLM if its assumptions hold), even though it's heteroskedastic under the alternative (and so you'd expect the spread of the data in each group to differ, since you anticipate that the means will differ). Jul 3, 2020 at 8:44
• @Glen_b, I see. Then a permutation test is not an option. Is the ordinal logistic regression more efficient than the Kruskal Wallis? I found a code in R with which I could calculate it very easy (clm from the ordinal package). I am just not sure whether I need to train my dataset beforehand or can I apply it directly on my dataset? Jul 3, 2020 at 9:12
• I don't see how you came to the conclusion that anything I said indicates a permutation test is not an option. Note that my comment is in reply to Bruce's comment. His comment suggests there may be a problem; I was indicating that there might not be a problem, and that you couldn't necessarily tell from looking at the data (and also that you probably shouldn't be looking at your data to choose your assumptions). Jul 3, 2020 at 9:15