I would like to compare the three groups, normally what an ANOVA would do. However, the data are distributed much more like Poisson than Normal (i.e. heavily weighted on slow rates but with a fair number of large rates - occasionally things go quite quickly). Clearly, these are not count data and it seems my only option is to use a non-parametric KW test.

As much as possible, I would prefer to use some kind of regression analysis and account for the lack of normality by matching the proper error distribution, which will provide more information (p-values for each group). Is there any way that I can use a regression to compare the groups but incorporate a Poisson distribution, for example?

Thanks a lot for any comments on this.