Empirical evaluation of statistical models for counts or rates

We consider methods for selecting the joint specification of the mean and variance functions in statistical models for rates or counts. Based on analyses of diagnosis-specific hospital discharge rates in Michigan, we show that a Poisson model with an extra variance component for the systematic variation is superior to several other probability models with regard to specification of the error structure. Further, the deviance residual appears superior to the Pearson residual. The proper specification of such variation is crucial for many types of analyses, such as identification of outliers and regression analyses designed to explain the systematic component of the variation.