Goodness of fit of generalized linear models to sparse data

We derive approximations to the first three moments of the conditional distribution of the deviance statistic, for testing the goodness of fit of generalized linear models with non-canonical links, by using an estimating equations approach, for data that are extensive but sparse. A supplementary estimating equation is proposed from which the modified deviance statistic is obtained. An application of a modified deviance statistic is shown to binomial and Poisson data. We also conduct a performance study of the modified Pearson statistic derived by Farrington and the modified deviance statistic derived in this paper, in terms of size and power, through a small scale simulation experiment. Both statistics are shown to perform well in terms of size. The deviance statistic, however, shows an advantage of power. Two examples are given.