Assessing Heterogeneous Causal Effects across Clusters in Partially Nested Designs

Intervention studies in psychology often have a partially nested design (PND): After individuals are assigned to study arms, individuals in a treatment arm are subsequently assigned to clusters (e.g., therapists/therapy-groups) to receive treatment, whereas individuals in a control arm are unclustered. Given the presence of clustering in the treatment arm, it can be of interest to examine heterogeneity of treatment effects across the clusters; but this is challenging in PNDs. First, in defining a causal effect of treatment for a specific cluster, it is unclear how the treatment and control outcomes should be compared, as the clustering is absent in the control arm. Although it may be tempting to compare outcomes between a specific cluster and the entire control arm, this crude comparison may not represent a causal effect even in PNDs with randomized treatment assignments, as the cluster assignment may be nonrandomized (elaborated in this study). In this study, we develop methods to define, identify, and estimate the causal effects of treatment across specific clusters in a PND where the treatment and/or cluster assignment may be nonrandomized. Using the principal stratification approach and potential outcomes framework, we define causal estimands for the cluster-specific treatment effects in two scenarios: (i) no-interference and (ii) within-cluster interference. We identify the effects under the principal ignorability assumption. For estimation, we provide a multiply-robust method that can protect against misspecification in a nuisance model and can incorporate machine learning methods in the nuisance model estimation. We evaluate the estimators’ performance through simulations, and illustrate the application using an empirical PND example.