Apologies for the almost text-book like question.
I have a 2x2 design with fixed categories and a continuous response variable.
If the variances are equal between groups (Bartlett test) and residuals are normally distributed (Shapiro test), ok I can do standard ANOVA.
Try transforming the data (e.g: arcsin(sqrt), or log(), or even rank()). If transformed data is homoscedastic & normal residues, do normal ANOVA.
One option: Kruskal test (tells you whether any means differ between groups) followed by many pairs of wilcox tests (to identify which means differ). If all are significant, all factors (and interactions are significant).
Another option: Use the bootstrap approach (permuting residuals) outlined here: Is there an equivalent to Kruskal Wallis one-way test for a two-way model?
Is this correct?