Clarifying Selection Bias in Cluster Randomized Trials: Estimands and Estimation

In cluster randomized trials (CRTs), patients are typically recruited after clusters are randomized, and the recruiters and patients are not blinded to the assignment. This leads to differential recruitment process and systematic differences in baseline characteristics of the recruited patients between intervention and control arms, inducing post-randomization selection bias. We rigorously define the causal estimands in the presence of post-randomization confounding. We elucidate the conditions under which standard covariate adjustment methods can validly estimate these estimands. We discuss the additional data and assumptions necessary for estimating the causal effects when such conditions are not met. Adopting the principal stratification framework in causal inference, we clarify there are two intention-to-treat (ITT) causal estimands in CRTs: one for the overall population and one for the recruited population. We derive the analytical formula of the two estimands in terms of principal-stratum-specific causal effects. We assess the empirical performance of two common covariate adjustment methods, multivariate regression and propensity score weighting, under different data generating processes. When treatment effects are heterogeneous across principal strata, the ITT effect on the overall population differs from the ITT effect on the recruited population. A naive ITT analysis of the recruited sample leads to biased estimate of both ITT effects. In the presence of post-randomization selection and without additional data on the non-recruited subjects, the ITT effect on the recruited population is estimable only when the treatment effects are homogenous between principal strata, and the ITT effect on the overall population is generally not estimable. The extent to which covariate adjustment can remove selection bias depends on the degree of effect heterogeneity across principal strata.