Abstract Diagnostic tools are proposed for assessing the dependence of extrabinomial or extra-Poisson variation on explanatory variables and for comparing several common models for overdispersion. These tools are based on tests for regression terms in the dispersion parameter of a generalized linear model, using double exponential family and “pseudolikelihood” formulations. Score tests do not require the full fitting of models for variation and lead to easy graphical and numerical procedures based on squared residuals (deviance or Pearson). Robust modifications of these are motivated by Levene's test in linear models. The diagnostic tools are intended primarily to ensure prudent modeling of the variance to make correct inferences about parameters in the mean.