Chi-square Tests for Generalized Estimating Equations with Possibly Misspecified Weights

A difficulty associated with current hypothesis tests for generalized estimating equations is that, if the working correlation assumption is incorrect, then consistent estimates of the true covariance matrices of the observations within clusters are needed, either in the calculation of the test statistics, or in the evaluation of percentage points of these statistics. This amounts to estimating all the correlation coefficients within a cluster, which is undesirable particularly when the cluster size is relatively large compared with the number of independent clusters. In this paper we introduce a class of test statistics that do not require consistent estimates of the true correlations, and whose limiting distribution remains chi-square regardless of the true correlation structure. In the cases where the working correlation assumption is correct, these tests are asymptotically equivalent to the "working" version of Rao's score test and the Wald test, and are thus optimal in these situations. The new tests are invariant under changes of the coordinate system in the parameter space. The method is demonstrated on a seed data set.