Pearson statistics, goodness of fit, and overdispersion in generalised linear models

The Pearson statistic is commonly used for assessing goodness of fit in generalised linear models. However when data are sparse, asymptotic results based on the chi square distribution may not be valid. McCullagh (1985) recommended conditioning on the parameter estimates and obtained approximations to the first three conditional moments of Pearson’s statistic for generalised linear models with canonical link functions. This paper presents a generalisation of these results to non-canonical models, derived in Farrington (1995). A first order linear correction term to the Pearson statistic is defined which induces local orthogonality with the regression parameters, and leads to substantial simplifications in the expressions for the first three conditional and unconditional moments. Expressions are given for Poisson, binomial, gamma and inverse Gaussian models. The power of the modified statistic to detect overdispersion is assessed, and the methods are applied to adjusting the bias of the dispersion parameter estimate in exponential family models.