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Abstract Poisson regression models are widely used in analyzing count data. This article develops tests for detecting extra-Poisson variation in such situations. The tests can be obtained as score tests against arbitrary mixed Poisson alternatives and are generalizations of tests of Fisher (1950) and Collings and Margolin (1985). Accurate approximations for computing significance levels are given, and the power of the tests against negative binomial alternatives is compared with those of the Pearson and deviance statistics. One way to test for extra-Poisson variation is to fit models that parametrically incorporate and then test for the absence of such variation within the models; for example, negative binomial models can be used in this way (Cameron and Trivedi 1986; Lawless 1987a). The tests in this article require only the Poisson model to be fitted. Two test statistics are developed that are motivated partly by a desire to have good distributional approximations for computing significance levels. Simulations suggest that one of the statistics should be satisfactory for testing extra-Poisson variation in most practical situations involving Poisson regression models.
Dean et al. (Thu,) studied this question.