On goodness‐of‐fit measures for Poisson regression models

Summary In this article, we study the statistical properties of the goodness‐of‐fit measure mpp proposed by (Eshima & Tabata 2007, Statistics & Probability Letters 77, 583–593) for generalised linear models. Focusing on the special case of Poisson regression using the canonical log link function, and assuming a random vector X of covariates, we obtain an explicit form for mpp that enables us to study its properties and construct a new estimator for the measure by utilising information about the shape of the covariate distribution. Simulations show that the newly proposed estimator for mpp exhibits better performance in terms of mean squared error than the simple unbiased covariance estimator, especially for larger absolute values of the slope coefficients. In contrast, it may be more unstable when the value of the slope coefficient is close to boundary of the domain of the moment generating function for the corresponding covariate. We illustrate the application of mpp on a data set of counts of complaints against doctors working in an emergency unit in hospital, in particular, showing how our proposed estimator can be efficiently computed across a series of candidate models.