In addition, the pattern of missing data, which is common in most trials, is another important consideration on the model selection. In cases where heterogeneity is of significant interest, the GLMM could be the better choice. The choice of modeling method should depend on the scientific questions and the validity of the underlying assumptions. The GLMM is more complicated and informative than the GEE approach by providing the estimation of the variance components, which are otherwise treated as nuisance parameters in GEE. 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. 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 the other is the marginal or population mean model using the generalized estimating equations (GEE) approach. Two modeling approaches are commonly used for the individual-level analyses of CRTs with the consideration of clustering. 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 however, only the individual-level analyses enable the adjustment of the participant characteristics to minimize the selection bias. Any statistical test ignoring the non-independence of participants within clusters will underestimate the variances of the intervention effects and consequently inflate the type I error rates. Although typically the ICC is small ( ρ < 0.05) and not known when a trial is planned, the adjustment for ICC is necessary for a valid statistical analysis at the subject level. Because the clusters are formed not at random but rather through some connections among their members, a positive intraclass correlation (ICC, denoted as ρ) among observations in the same cluster is expected. CRTs are distinct from other randomized controlled trials in that the identifiable clusters of subjects/participants such as medical practices, hospital wards, schools, or communities, rather than individuals, are randomly assigned to different intervention conditions. We 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.Ĭluster-randomized trials (CRTs), also called group-randomized trials, are widely used in the evaluation of interventions in health services research. 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. 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. 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. Our 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 results are also illustrated using a real CRT dataset. 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 small sample performances of five DDF approximations for the F test are compared and contrasted under CRT frameworks with simulations. 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. The most challenging issue for the approximate Wald F test is the estimation of the denominator degrees of freedom (DDF). F tests are commonly used in the generalized linear mixed model (GLMM) to test intervention effects in CRTs. 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.
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