Extending the Weighted Generalized Score Statistic for Comparison of Correlated Means

The generalized score (GS) statistic is widely used to test hypotheses about mean model parameters in the generalized estimating equations (GEE) framework. However, when comparing predictive values of two diagnostic tests in a paired study design, or comparing correlated proportions between two unequally sized groups with both paired and independent outcomes, GS has been shown neither to adequately control type I error nor to reduce to the score statistic under independence. Weighting the residuals in empirical variance estimation by the ratio of the two groups’ sample sizes produces a weighted generalized score (WGS) statistic that has been shown to resolve these issues and is now used in the diagnostic testing literature. Potential improvements from weighting in more general uses of GS have not previously been investigated.This dissertation extends the WGS method in several ways. Formulas are derived to extend the WGS statistic for paired and/or independent data from two binary proportions to two means in a quasi-likelihood model with any suitable link and variance functions, assuming finite fourth moments. The asymptotic convergence of WGS to the chi-square distribution in these general cases is proven. Finite-sample type I error rates are compared between GS and WGS, for which purpose the variance of the test statistic denominator (i.e., the variance of the empirical variance estimator) is proposed as an analytic heuristic. New weights are derived to optimize the variance-of-the-denominator criterion for approximate type I error control. Simulation results verify that the heuristically optimal weights achieve type I error rates closer to the nominal alpha level than GS or WGS for combinations of correlation and sample size where either GS or WGS demonstrates poor control.