In cluster randomized trials, patients are recruited after clusters are randomized, and the recruiters and patients may not be blinded to the assignment. This often leads to differential recruitment and systematic differences in baseline characteristics of the recruited patients between intervention and control arms, inducing post-randomization selection bias. We aim to rigorously define causal estimands in the presence of selection bias. We elucidate the conditions under which standard covariate adjustment methods can validly estimate these estimands. We further discuss the additional data and assumptions necessary for estimating causal effects when such conditions are not met. Adopting the principal stratification framework in causal inference, we clarify there are two average treatment effect (ATE) estimands in cluster randomized trials: 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. Further, using simulation studies, we assess the empirical performance of the multivariable regression adjustment method under different data generating processes leading to selection bias. When treatment effects are heterogeneous across principal strata, the ATE on the overall population generally differs from the ATE on the recruited population. A naive intention-to-treat analysis of the recruited sample leads to biased estimates of both ATEs. In the presence of post-randomization selection and without additional data on the non-recruited subjects, the ATE on the recruited population is estimable only when the treatment effects are homogenous between principal strata, and the ATE 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.
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