Hello Nazli,

Here are three options:

1. Run the usual mixed anova, but select bootstrap/resampling estimates for parameters and their CIs. Doing so frees you from distributional assumptions.

2. Look at "hybrid" methods: Two such approaches are due to Conover and Iman, and Puri and Sen. The steps are: (a) rank the scores; (b) run the usual parametric test; then (c) apply the significance criterion (these are a bit different; Conover and Iman say, just use the computed p-values, whereas Puri and Sen call for the computation involving explained variance tested as a chi--square...references below).

3. Consider the aligned ranks transform method. Here's a link to get you started: https://rcompanion.org/handbook/F_16.html

Good luck with your work.

References

Conover & Iman: https://www.jstor.org/stable/2683975

Puri & Sen: Article Puri, M. L., and P. K. Sen: Nonparametric Methods in Multiva...

There is a non-parametric function of the R package nparLD for your design: f1.ld.f2. See the R script in attachment. It includes this non-parametric test.

Outliers can be a problem and you should eliminate them or depotentiating by a rank or log transform, as for non-normality, forget it ! It is not a real problem (see attached file)