Tests of Hypotheses in Overdispersed Poisson Regression and other Quasi-Likelihood Models

Abstract

Test statistics for evaluating the significance of added variables in a regression equation are developed for mixed Poisson models, where the structural parameter φ that determines the mean/variance relationship var(μ; φ) = μ + φ · μ 2 is estimated by the method of moments and the regression coefficients are estimated by quasi-likelihood. The formulas presented for test statistics and related estimating equations are applicable generally to quasi-likelihood models specified by an arbitrary mean value function μ(x; β), together with a variance function V(μ; φ) that contains one or more unknown parameters. Two versions of the Wald and score tests are investigated—one calculated from the usual model-based covariance matrix whose validity depends on correct specification of the variance function, and another using an “empirical” covariance matrix that has a more general asymptotic justification. Monte Carlo simulations demonstrate that the quasi-likelihood/method of moments (QL/M) procedures yield approximately unbiased estimates of regression coefficients and their standard errors and that model-based Wald, score, and deviance tests approximate the nominal size at the 5% level for moderate sample sizes. The simpler Poisson analysis also produces approximately unbiased regression coefficients, even though the overdispersion is not accounted for. Although tests and standard errors based on Poisson theory are seriously in error in the presence of overdispersion, the empirical standard errors and especially the empirical score test obtained in conjunction with the Poisson analysis perform reasonably well with overdispersed data provided the sample size is sufficiently large. They perform less well in small samples than do the model-based QL/M procedures, probably because of the lack of precision in the empirical variances. These methods have important applications in epidemiology, toxicology, and related areas.