Dear Ben, sorry for not replying and thanking you for your answer before. I was relocating.
The reason that there's very little attention given to the distribution of the predictors is that in general the definition of standard statistical models such as GLMMs **does not say anything about the distribution of the predictors**. In particular, as far as I am aware your statement that "in classic regression my data would certainly invalidate the analysis" is not true -- at least if we're only talking about the distribution of the predictor. The main importance of the distribution of the predictor is that it affects the power of the test -- obviously if most of your predictor data are zeros, they won't give you very much information about how the response changes as a function of the response.
I went too far on the generalization of my thought. I was thinking on a related problem that was commonly addressed during my statistical training: that a few extreme values of the predictor accompanied by a very different response (considered outliers) could falsely imply a linear relationship. The point you raise is clear and simple (yet I had not thought about it).
I haven't read Sheater's book, but the purpose of transforming the predictor in this context is to take a response that is *not* log-odds-linear on the original scale of the predictor, but (e.g.) might be log-odds-linear when the predictor is on a log scale.
Yes, this is the way to express it.
Thus the transformation is *not* fixing a problem with the distribution of the predictor, but rather with the linearity of the response.
Even if not as crucial as the assumptions on the distribution of the response variable (or rather the model residuals), I beleive this is an important correction to come-up with an appropriate model and, therefore, study conclusions. I "discovered" this in Sheater?s book by looking for something else.
As always I'm happy to be corrected by others on the list ...
Thanks again for your helpful response. It?s so important to backup the level of science we (non-statistitians) are doing.
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