Non-parametric ANOVA with 2 predictors

Question

I fitted the same linear mixed model with changing response variables:

lmer(Response ~ categorical predictor 1 * categorical predictor 2 + (1|random effect1/random effect2)

(one predictor has 2 levels the other 3)

Afterwards I used the anova to check for signficant effects of the 2 predictors. Model validation tells me for one response, that the variances are not homogenous. Now I was thinking about the next steps. I thought about using a Kruskal-Wallis test, but that does not work with my 2 predictors in one line. Is it correct to do one Kruskal-test for the one predictor and an other for the other one? I am pretty new to statistics and it is really difficult when things are not following a typical recipe. So sorry if the question is a bit stupide.

I have read this question: General approach for non-parametric two-way ANOVA but this is for only one predictor

Answer 1

If all of your predictor variables are nominal, or factor, variables, another option for a nonparametric analysis is Aligned Ranks Transformation Analysis. In a good implementation (ARTool in R), it can handle the inclusion of random effects.

Considering your lmer model, if it's only the heteroscedasticity that's bothering you, you might look here to see if there is another approach that may be applicable: Alternatives to one-way ANOVA for heteroskedastic data

Answer 2

The generalization of the Wilcoxon and Kruskal-Wallis tests is the semiparametric proportional odds ordinal logistic model. While maintaining excellent power it will allow for any number of covariates. For clustered data random effects are easy to add to this model. See the Bayesian modeling package brms and frequentist R packages mixor and ordinal. If you have longitudinal data random effects are unlikely to find the resulting serial correlation pattern and you might consider instead a Markov model as detailed here. An introduction to the proportional odds model is in the Nonparametrics chapter of BBR.