Small sample performance of bias‐corrected sandwich estimators for cluster‐randomized trials with binary outcomes

Abstract

SUMMARY The sandwich estimator in generalized estimating equations (GEE) approach underestimates the true variance in small samples and consequently results in inflated type I error rates in hypothesis testing. This fact limits the application of the GEE in cluster-randomized trials (CRTs) with few clusters. Under various CRT scenarios with correlated binary outcomes, we evaluate the small sample properties of the GEE Wald tests using bias-corrected sandwich estimators. Our results suggest that the GEE Wald z test should be avoided in the analyses of CRTs with few clusters even when bias-corrected sandwich estimators are used. With t-distribution approximation, the Kauermann and Carroll (KC)-correction can keep the test size to nominal levels even when the number of clusters is as low as 10, and is robust to the moderate variation of the cluster sizes. However, in cases with large variations in cluster sizes, the Fay and Graubard (FG)-correction should be used instead. Furthermore, we derive a formula to calculate the power and minimum total number of clusters one needs using the t test and KC-correction for the CRTs with binary outcomes. The power levels as predicted by the proposed formula agree well with the empirical powers from the simulations. The proposed methods are illustrated using real CRT data. We conclude that with appropriate control of type I error rates under small sample sizes, we recommend the use of GEE approach in CRTs with binary outcomes due to fewer assumptions and robustness to the misspecification of the covariance structure.

Covariance estimators for Generalized Estimating Equations (GEE) in longitudinal analysis with small samples

“…V WL is a combined variance estimator suggested by Wang and Long [16], which considered both the strength of V PAN and V MD for pooling information from all subjects and reducing the bias of the estimate for . The estimator is below (18) where (19) This estimator was confirmed to perform as well as or better than V PAN and V M , but the two additional assumptions specified above also need to be satisfied.…”

“…As indicated in our simulation study, in general, ttests based on the variance estimator V ML perform robustly well across different set-ups. In particular, the degrees of freedom for t-statistic are more accurately approximated as compared to Li and Redden [18]. However, there are still several limitations for this work.…”

“…The test statistic for t-test is , which is the same as the Wald test statistic with the degrees of freedom [16,20]. This satterthwaite-type degrees of freedom approximation incorporates the variability of the variance estimator, and thus performs better compared to depending only on the number of clusters in Li and Redden [18]. The outperformance of t-test over the Wald test was identified in the settings with small sample [16,29].…”

Continuation of non-essential medications in actively dying hospitalised patients

“…We reviewed methods proposed by several investigators 25–28. Based on Li and Redden29 we employed the correction factor proposed by Kauermann and Carroll 28…”

Best (but oft-forgotten) practices: designing, analyzing, and reporting cluster randomized controlled trials

“…The validity of inferences from a given analysis is dependent on sufficiently large K (per the central limit theorem) or under the usual parametric modeling assumptions, including the normality of residuals. Again, there are special issues when the number of clusters is small (e.g., ,10), and not all statistical methods will perform well in such circumstances (21,22).…”