Bias from cluster-specific time trajectories in cluster-randomized stepped wedge trials

Cluster-randomized stepped-wedge trials (SWT's) randomize each cluster to the time at which it crosses over from control to intervention. Commonly used models for SWT's assume that all clusters share a common time trajectory, meaning that the effect of time is the same across all clusters. But, the time effects may vary by cluster, resulting in cluster-specific time trajectories (CSTT's). This paper explores the bias induced in treatment effect estimatees is CSTT's exist but are ignored. A formula for estimating the bias is derived. To correct for the bias, two models with less restrictive assumptions are proposed that allow time trajectories to be cluster-specific, estimated using polynomials. It is shown using simulations that the bias formula is accurate, and that the proposed CSTT models reduce the bias. The bias seems particularly problematic when CSTT patterns are nonlinear. These methods are applied to two cluster-randomized SWT datasets.