Abstract
BackgroundSmall number of clusters and large variation of cluster sizes commonly exist in cluster-randomized trials (CRTs) and are often the critical factors affecting the validity and efficiency of statistical analyses. F tests are commonly used in the generalized linear mixed model (GLMM) to test intervention effects in CRTs. The most challenging issue for the approximate Wald F test is the estimation of the denominator degrees of freedom (DDF). Some DDF approximation methods have been proposed, but their small sample performances in analysing binary outcomes in CRTs with few heterogeneous clusters are not well studied.MethodsThe small sample performances of five DDF approximations for the F test are compared and contrasted under CRT frameworks with simulations. Specifically, we illustrate how the intraclass correlation (ICC), sample size, and the variation of cluster sizes affect the type I error and statistical power when different DDF approximation methods in GLMM are used to test intervention effect in CRTs with binary outcomes. The results are also illustrated using a real CRT dataset.ResultsOur simulation results suggest that the Between-Within method maintains the nominal type I error rates even when the total number of clusters is as low as 10 and is robust to the variation of the cluster sizes. The Residual and Containment methods have inflated type I error rates when the cluster number is small (<30) and the inflation becomes more severe with increased variation in cluster sizes. In contrast, the Satterthwaite and Kenward-Roger methods can provide tests with very conservative type I error rates when the total cluster number is small (<30) and the conservativeness becomes more severe as variation in cluster sizes increases. Our simulations also suggest that the Between-Within method is statistically more powerful than the Satterthwaite or Kenward-Roger method in analysing CRTs with heterogeneous cluster sizes, especially when the cluster number is small.ConclusionWe conclude that the Between-Within denominator degrees of freedom approximation method for F tests should be recommended when the GLMM is used in analysing CRTs with binary outcomes and few heterogeneous clusters, due to its type I error properties and relatively higher power.
Highlights
Small number of clusters and large variation of cluster sizes commonly exist in cluster-randomized trials (CRTs) and are often the critical factors affecting the validity and efficiency of statistical analyses
CRTs can be analyzed at the cluster level, by deriving summary statistics for each cluster, or at the individual level using the data for each participant in each cluster [1]; only the individual-level analyses enable the adjustment of the participant characteristics to minimize the selection bias
We compare the small sample performance of five denominator degrees of freedom (DDF) approximation methods in generalized linear mixed model (GLMM) to test the null hypothesis of intervention effect under the framework of CRTs with binary outcomes
Summary
Small number of clusters and large variation of cluster sizes commonly exist in cluster-randomized trials (CRTs) and are often the critical factors affecting the validity and efficiency of statistical analyses. One is the random effects model or generalized linear mixed model (GLMM), which incorporates random effects to reflect the correlation among observations of same cluster [6]; the other is the marginal or population mean model using the generalized estimating equations (GEE) approach [7]. These two modeling methods should provide similar results if both models are correctly specified and their underlying assumptions hold well, while the interpretation of the fixed effects estimates is a little different [8]. The GLMM is valid under both missing completely at random (MCAR) and missing at random (MAR), while the GEE approach is valid only under MCAR even though some imputation strategies have been proposed for valid GEE inference under MAR [8]
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