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.