I was wondering, in relation to model selection for some count data I am looking at using poisson GLMs, would a high residual deviance (measure of dispersion) warrant switching to a quasi-poisson model before or after dropping unimportant explanatory variables?
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The question is not quite clear. In general if you are doing model selection you need to account for overdispersion during the model selection process (i.e. 'before' dropping unimportant explanatory variables); otherwise you will tend to overfit the model. So depending on the question the answer would be:
I would strongly caution you about model selection. If you drop "unimportant" explanatory variables and then make inferences based on the reduced model you will be making a mistake. (There are valid reasons to do model selection, but you have to be careful.) See Frank Harrell's book on Regression Modeling Strategies for a clear statement, or http://www.stata.com/support/faqs/stat/stepwise.html for a concise version.