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Cluster randomized trials (CRTs) are studies in which clusters of subjects are randomized to different trial arms. Due to the nature of outcomes within the same cluster to be correlated, generalized estimating equations (GEE) are growing as a popular choice for the analysis of data arising from CRTs. In the past, research has shown that analyses using GEE could result in liberal inference due to the use of the empirical sandwich covariance matrix estimator, which can yield negatively biased standard error estimates when the number of clusters is not large. Many techniques have been presented to correct this negative bias; however, use of these corrections can still result in biased standard error estimates and thus test sizes that are not consistently at their nominal level. Therefore, there is a need for an improved correction such that nominal type I error rates will consistently result. In this manuscript, we study the use of recently developed corrections for empirical standard error estimation and the use of a combination of two popular corrections. In an extensive simulation study, we found that nominal type I error rates can be consistently attained when using an average of two popular corrections developed by Mancl and DeRouen (, Biometrics 57, 126-134) and Kauermann and Carroll (, Journal of the American Statistical Association 96, 1387-1396). Therefore, use of this new correction was found to notably outperform the use of previously recommended corrections.

Generalized estimating equations (GEE) are commonly used for the marginal analysis of correlated data, although the quadratic inference function (QIF) approach is an alternative that is increasing in popularity. This method optimally combines distinct sets of unbiased estimating equations that are based upon a working correlation structure, therefore asymptotically increasing or maintaining estimation efficiency relative to GEE. However, in finite samples, additional estimation variability arises when combining these sets of estimating equations, and therefore the QIF approach is not guaranteed to work as well as GEE. Furthermore, estimation efficiency can be improved for both analysis methods by accurate modeling of the correlation structure. Our goal is to improve parameter estimation, relative to existing methods, by simultaneously selecting a working correlation structure and choosing between GEE and two versions of the QIF approach. To do this, we propose the use of a criterion based upon the trace of the empirical covariance matrix (TECM). To make GEE and both QIF versions directly comparable for any given working correlation structure, the proposed TECM utilizes a penalty to account for the finite-sample variance inflation that can occur with either version of the QIF approach. Via a simulation study and in application to a longitudinal study, we show that penalizing the variance inflation that occurs with the QIF approach is necessary and that the proposed criterion works very well.

Group randomized trials (GRTs) randomize groups, or clusters, of people to intervention or control arms. To test for the effectiveness of the intervention when subject-level outcomes are binary, and while fitting a marginal model that adjusts for cluster-level covariates and utilizes a logistic link, we develop a pseudo-Wald statistic to improve inference. Alternative Wald statistics could employ bias-corrected empirical sandwich standard error estimates, which have received limited attention in the GRT literature despite their broad utility and applicability in our settings of interest. The test could also be carried out using popular approaches based upon cluster-level summary outcomes. A simulation study covering a variety of realistic GRT settings is used to compare the accuracy of these methods in terms of producing nominal test sizes. Tests based upon the pseudo-Wald statistic and a cluster-level summary approach utilizing the natural log of observed cluster-level odds worked best. Due to weighting, some popular cluster-level summary approaches were found to lead to invalid inference in many settings. Finally, although use of bias-corrected empirical sandwich standard error estimates did not consistently result in nominal sizes, they did work well, thus supporting the applicability of marginal models in GRT settings.