General approach for non-parametric two-way ANOVA 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. 
Otherwise: 


*

*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?
 A: The proportional odds ordinal logistic model is a generalization of the Wilcoxon and Kruskal-Wallis tests that extend to multiple covariates, interactions, etc.  It is a semiparametric method that only uses the ranks of Y.  It handles continuous Y, creating k-1 intercepts where k is the number of unique Y values.
A: This sounds good to me. There are some issues to consider though:


*

*In option 2, you need to make sure to correct the p-values in your wilcox tests for multiple hypothesis testing. The pairwise.wilcox.test function in R will do this for you. 

*In my experience, even though the bootstrap approach is very nice here, if other people in your field (e.g. paper reviewers) are unfamiliar with it you can draw a lot of criticism. 
It really depends on what is normal in your field, and what the purpose of the analysis is. If this work is for a paper, and practitioners in your field have a recipe for data analysis which does not match this one, it might be easier to justify using that approach (even if it's wrong). For example, in some fields the 'correct' procedure is just "Use ANOVA". No extra tests are performed, and the results are accepted as valid. ANOVA is reasonably robust to the violation of normality too, so in practice this approach (although overly simplistic) works out okay.
