Small sample correction for the variance of GEE estimators

When clustered multinomial responses are fit using the generalized logistic link, Morel (1989) introduced a small sample correction in the Taylor series based estimator of the covariance matrix of the parameter estimates. The correction reduces the bias of the Type I error rates in small samples and guarantees positive definiteness of the estimated variance-covariance matrix. It is well known that small sample bias in the use of the Delta method persists in any application of the Generalized Estimating Equations (GEE) methodology. In this article, we extend the correction originally suggested for the generalized logistic link, to other link functions and distributions, when parameters are estimated by GEE. In a Monte Carlo study with correlated data generated under different sampling schemes, the small sample correction has been shown to be effective in reducing the Type I error rates when the number of clusters is relatively small.