Consider a simple randomized experiment with 4 treatments(no control) where the outcome of interest is difference between pre-treatment and post-treatment count. I want to conclude 4 treatments does not differ from each other. Standard approach is longitudinal Poisson mixed effect model. Suppose I am ignorant of that model. I want to do ANOVA on difference with treatments as factors. Let me call this ANOVA analysis model $M$.
Q: When does $M$ check out with ANOVA diagnostics(residuals, VIF and etc)? It seems very unlikely that the model $M$ will check out with all ANOVA diagnostics. Is this possibly due to treatment being categorical with 4 levels and non continuous? I think that might be the reason by jiggling the distribution a bit.
Q: If a model checks out with diagnostics, is it valid? It is not appropriate as the difference must be discrete. However, if it is good first order approximation, it is not a bad approximation.