Two-Phase Treatment with Noncompliance: Identifying the Cumulative Average Treatment Effect via Multisite Instrumental Variables

When evaluating a two-phase intervention, the cumulative average treatment effect (ATE) is often the primary causal estimand of interest. However, some individuals who do not respond well to the Phase I treatment may subsequently display noncompliant behaviors. At the same time, exposure to the Phase I treatment is expected to directly influence an individual’s potential outcomes, thereby violating the exclusion restriction. Building on an instrumental variable (IV) strategy for multisite trials, we clarify the conditions under which the cumulative ATE of a two-phase treatment can be identified by employing the random assignment of the Phase I treatment as the instrument. Our strategy relaxes both the conventional exclusion restriction and sequential ignorability assumptions. We assess the performance of the new strategy through simulation studies. Additionally, we reanalyze data from the Tennessee class size study, in which students and teachers were randomly assigned to either small or regular class types in kindergarten (Phase I) with noncompliance emerging in Grade 1 (Phase II). Applying our new strategy, we estimate the cumulative ATE of receiving two consecutive years of instruction in a small versus regular class.

💡 Research Summary

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This paper addresses the methodological challenge of estimating the cumulative average treatment effect (ATE) in interventions that consist of two phases, where the first phase is randomly assigned but the second phase experiences non‑compliance, and the first‑phase treatment itself may have a direct effect on the final outcome. Traditional approaches—intention‑to‑treat (ITT), inverse‑probability‑of‑treatment weighting (IPTW), and standard instrumental variable (IV) methods—are inadequate in this setting. ITT underestimates the cumulative effect when non‑compliance occurs; IPTW relies on the sequential ignorability assumption, which is often violated because post‑treatment variables (e.g., intermediate outcomes) influence both second‑phase treatment receipt and the final outcome; standard IV requires an exclusion restriction that the instrument (the random assignment in phase I) affects the outcome only through the treatment received, an assumption that fails when phase‑I exposure directly influences the outcome.

The authors propose a novel Multisite Two‑Phase Treatment Instrumental Variable (MS2T‑IV) strategy that exploits the multisite randomization design. For each site (k), let (Z_{ik}) denote the binary random assignment in phase I, (D_{ik}) the binary indicator of receiving the phase‑II treatment, (V_{ik}) the intermediate outcome measured at the end of phase I, and (Y_{ik}) the final outcome after phase II. Observed baseline covariates are (X_{ik}) and unobserved confounders are (U_{ik}). The key insight is to treat (V_{ik}) as a post‑treatment confounder that simultaneously affects (D_{ik}) and (Y_{ik}). The authors define site‑specific parameters: (\beta_{0k}=E