Should the predictor variables be normally distributed for Poisson glm?

Question

This is probably a really basic question, but it's the first time I've created a model that defines Poisson as its error family.

In setting up my variables to make the model, should I be concerned about whether or not predictor variables are normally distributed, and if not, should I be attempting to transform them to make them normal? Or alternatively, should the residuals of simple regressions between each predictor and the response variable be normally distributed? Or is this something I look at overall, once the model is made, by looking at a histogram of the residuals of the full model? Or, does the normal distribution not even apply in this case because I have specified Poisson errors?

Answer 1

(1) No regression, Poisson or otherwise, makes any assumption about the distribution of the predictors.

(2) You should check how the residuals vary against the fitted values but they are only asymptotically normal.

Answer 2

(got too long for a comment)

@Scortchi has covered the issues pretty well; though I wanted to make some additional comments

On the use of histograms as a way of assessing distributional assumptions:

look at [it] overall [...] by looking at a histogram of the residuals

Histograms should be used for such a purpose with some degree of caution.

Specifically, imagine you were looking at histograms of residuals for two models where you were assuming normality, and saw these:

What would you conclude about the assumption of normality in each case?

. . .

In fact they're two histograms of the same data set!

That example (with data) comes from here (see that link for more cautionary examples), and suggestions for what to do if you do use histograms. (I subtracted a value close to the mean to make them looks like 'residuals' for the above display.)

A better choice for checking distributional assumptions may be some form of Q-Q plot, such as residuals against normal scores if you assume normality. In the case of Poisson GLMs it's kind of tricky because the shape changes as your mean changes; R offers a normality plot (qqnorm) for residuals from a GLM but there are other things you could do.