Modeling clustering and treatment effect heterogeneity in parallel and stepped-wedge cluster randomized trials

Cluster randomized trials are frequently used in health service evaluation. It is common practice to use an analysis model with a random effect to allow for clustering at the analysis stage. In designs where clusters are exposed to both control and treatment conditions, it may be of interest to examine treatment effect heterogeneity across clusters. In designs where clusters are not exposed to both control and treatment conditions, it can also be of interest to allow heterogeneity in the degree of clustering between arms. These two types of heterogeneity are related. It has been proposed in both parallel cluster trials, stepped-wedge, and other cross-over designs that this heterogeneity can be allowed for by incorporating additional random effect(s) into the model. Here, we show that the choice of model parameterization needs careful consideration as some parameterizations for additional heterogeneity induce unnecessary or implausible assumptions. We suggest more appropriate parameterizations, discuss their relative advantages, and demonstrate the implications of these model choices using a real example of a parallel cluster trial and a simulated stepped-wedge trial.