Obviously, the response data live on 5 points. And in our case, the responses are fairly skewed to the right, with many residents finding these core roles important.

I have graphically investigated the relationship between Perception (5 point Likert Scale) and my two independent variables (School, Residency Program). I have used mean plots, also considered mosaic plots. I have a sense of what is going on qualitatively but want estimated covariate effects, p-values, etc...corresponding to the impact of School, Residency and their interaction on self-reported importance.

Ideally, I could run this as a 2-way ANOVA (or linear regression model). However, diagnostic investigation of residuals suggests an issue of non-normality.

I considered the proportional odds logistic regression model next; however, in many instances the assumption of "proportional odds" was not satisfied.

From there I moved on to the multinomial logistic regression model; however, given the distribution of the response data (skewed to the right), very little information exists in cells {1}, {2}, so estimated covariate effects (relative to these response levels) appear to be estimated without a great deal of precision.

Next, I considered non-parametric tests...extensions of Kruskal Wallis test from one-way layouts to two-way (and higher order) layouts. I found the following: http://www.tandfonline.com/doi/abs/10.1207/s15327906mbr1503_4#.Ud1uCdpzY3E However, I could not find a statistical package to implement such a method. Does anyone know of one? I wouldn't expect these tests to elucidate the size of the covariate effect; however, they may at least be able to derive a valid overall p-value (something like the global F-test in regression/ANOVA)...or perhaps even more valuable LRT style p-values corresponding to each covariate effect (i.e. School, Program, School*Program).

I imagine this is a fairly common problem in applied research settings...just wonder what other people have done in the past?

Thanks Chris