Summary Cluster randomized trials often exhibit a three-level structure with participants nested in subclusters such as health care providers, and subclusters nested in clusters such as clinics. While the average treatment effect has been the primary focus in planning three-level randomized trials, interest is growing in understanding whether the treatment effect varies among prespecified patient subpopulations, such as those defined by demographics or baseline clinical characteristics. In this article, we derive novel analytical design formulas based on the asymptotic covariance matrix for powering confirmatory analyses of treatment effect heterogeneity in three-level trials, that are broadly applicable to the evaluation of cluster-level, subcluster-level, and participant-level effect modifiers and to designs where randomization can be carried out at any level. We characterize a nested exchangeable correlation structure for both the effect modifier and the outcome conditional on the effect modifier, and generate new insights from a study design perspective for conducting analyses of treatment effect heterogeneity based on a linear mixed analysis of covariance model. A simulation study is conducted to validate our new methods and two real-world trial examples are used for illustrations.

Background In cluster randomized trials, patients are typically recruited after clusters are randomized, and the recruiters and patients may not be blinded to the assignment. This often leads to differential recruitment and consequently 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. Methods 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 analytical formula of the two estimands in terms of principal-stratum-specific causal effects. Furthermore, using simulation studies, we assess the empirical performance of the multivariable regression adjustment method under different data generating processes leading to selection bias. Results When treatment effects are heterogeneous across principal strata, the average treatment effect on the overall population generally differs from the average treatment effect on the recruited population. A naïve intention-to-treat analysis of the recruited sample leads to biased estimates of both average treatment effects. In the presence of post-randomization selection and without additional data on the non-recruited subjects, the average treatment effect on the recruited population is estimable only when the treatment effects are homogeneous between principal strata, and the average treatment 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. Conclusion There is a need and opportunity to improve the analysis of cluster randomized trials that are subject to post-randomization selection bias. For studies prone to selection bias, it is important to explicitly specify the target population that the causal estimands are defined on and adopt design and estimation strategies accordingly. To draw valid inferences about treatment effects, investigators should (1) assess the possibility of heterogeneous treatment effects, and (2) consider collecting data on covariates that are predictive of the recruitment process, and on the non-recruited population from external sources such as electronic health records.

Motivated by a suicide prevention trial with hierarchical treatment allocation (cluster‐level and individual‐level treatments), we address the sample size requirements for testing the treatment effects as well as their interaction. We assume a linear mixed model, within which two types of treatment effect estimands (controlled effect and marginal effect) are defined. For each null hypothesis corresponding to an estimand, we derive sample size formulas based on large‐sample z‐approximation, and provide finite‐sample modifications based on a t‐approximation. We relax the equal cluster size assumption and express the sample size formulas as functions of the mean and coefficient of variation of cluster sizes. We show that the sample size requirement for testing the controlled effect of the cluster‐level treatment is more sensitive to cluster size variability than that for testing the controlled effect of the individual‐level treatment; the same observation holds for testing the marginal effects. In addition, we show that the sample size for testing the interaction effect is proportional to that for testing the controlled or the marginal effect of the individual‐level treatment. We conduct extensive simulations to validate the proposed sample size formulas, and find the empirical power agrees well with the predicted power for each test. Furthermore, the t‐approximations often provide better control of type I error rate with a small number of clusters. Finally, we illustrate our sample size formulas to design the motivating suicide prevention factorial trial. The proposed methods are implemented in the R package H2x2Factorial.

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Background: Intracardiac catheter ablation for atrial fibrillation with pulmonary vein isolation may result in Takotsubo syndrome (TS), but the frequency, predisposing factors (age, sex, mental health disorders), and outcomes are currently unknown. This study sought to assess the frequency, predisposing factors, and outcomes of subjects who underwent intracardiac catheter ablation for atrial fibrillation with pulmonary vein isolation and were diagnosed with TS.Methods: This was a retrospective observational cohort study utilizing TriNetX® electronic health record (EHR) data. We included subjects aged older than 18 years who underwent intracardiac catheter ablation for atrial fibrillation with pulmonary vein isolation. The study population was divided into two groups (no TS diagnostic code presence and TS diagnostic code presence). We analyzed the distributions of age, sex, race, diagnostic codes, common terminology procedures (CPT), and vasoactive medication codes and examined mortality rate within 30 days. Results:We included 69,116 subjects. Of these, 27 (0.04%) had a TS diagnostic code, the cohort was comprised mostly of females [17 (63.0%)], and 1 (3.7%) death within 30 days was reported. There were no significant differences in age and frequency of mental health disorders between those patients in TS and non-TS cohorts. Adjusting for age, sex, race, ethnicity, patient regionality, and mental health disorder diagnostic code, those patients who developed TS had a significantly higher odds of dying in 30 days after catheter ablation compared to those without TS (OR = 15.97, 95% CI: 2.10-121.55, p = .007).Conclusions: Approximately 0.04% of subjects who underwent intracardiac catheter ablation of atrial fibrillation by pulmonary vein isolation had a subsequent diagnostic code of TS. Further study is needed to determine whether there are predisposing factors associated with the development of TS in subjects who undergo catheter ablation of atrial fibrillation by pulmonary vein isolation.

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