Improved hypothesis testing for coefficients in generalized estimating equations with small samples of clusters

The sandwich standard error estimator is commonly used for making inferences about parameter estimates found as solutions to generalized estimating equations (GEE) for clustered data. The sandwich tends to underestimate the variability in the parameter estimates when the number of clusters is small, and reference distributions commonly used for hypothesis testing poorly approximate the distribution of Wald test statistics. Consequently, tests have greater than nominal type I error rates. We propose tests that use bias-reduced linearization, BRL, to adjust the sandwich estimator and Satterthwaite or saddlepoint approximations for the reference distribution of resulting Wald t-tests. We conducted a large simulation study of tests using a variety of estimators (traditional sandwich, BRL, Mancl and DeRouen's BC estimator, and a modification of an estimator proposed by Kott) and approximations to reference distributions under diverse settings that varied the distribution of the explanatory variables, the values of coefficients, and the degree of intra-cluster correlation (ICC). Our new method generally worked well, providing accurate estimates of the variability of fitted coefficients and tests with near-nominal type I error rates when the ICC is small. Our method works less well when the ICC is large, but it continues to out-perform the traditional sandwich and other alternatives.