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

Guanglei Hong, Xu Qin, Zhengyan Xu, Fan Yang

TL;DR

This work tackles the problem of identifying the cumulative average treatment effect (ATE) of a two-phase intervention in multisite randomized trials when noncompliance arises after Phase I and posttreatment confounding may bias outcomes. The authors introduce MS2T-IV, a multisite two-phase treatment IV framework that uses Phase I randomization as the instrument, accommodates a posttreatment mediator $V$, and allows for non-additive effects, thereby relaxing the exclusion restriction and sequential ignorability assumptions. Identification hinges on within-site ignorability, within-site zero covariance, and between-site independence, yielding the cumulant δ_{ATE} = γ_1 + γ_2 + γ_3 + θ_V α_1; estimation proceeds via site-level ITT models (Stage 1) and a between-site regression (Stage 2), with covariance-adjusted and bootstrap BCa inference. Simulations show MS2T-IV reduces bias and improves mean-squared error relative to naive or IPTW approaches, though inference with proper bootstrap is sometimes conservative; the method applied to Project STAR estimates a positive two-year small-class effect on Grade 1 outcomes, illustrating practical utility for evaluating multisite two-phase programs. Overall, the paper advances causal inference for multi-phase interventions by delivering a robust, less assumption-reliant approach to estimate cumulative ATEs in the presence of noncompliance and posttreatment confounding.

Abstract

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.

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

TL;DR

This work tackles the problem of identifying the cumulative average treatment effect (ATE) of a two-phase intervention in multisite randomized trials when noncompliance arises after Phase I and posttreatment confounding may bias outcomes. The authors introduce MS2T-IV, a multisite two-phase treatment IV framework that uses Phase I randomization as the instrument, accommodates a posttreatment mediator , and allows for non-additive effects, thereby relaxing the exclusion restriction and sequential ignorability assumptions. Identification hinges on within-site ignorability, within-site zero covariance, and between-site independence, yielding the cumulant δ_{ATE} = γ_1 + γ_2 + γ_3 + θ_V α_1; estimation proceeds via site-level ITT models (Stage 1) and a between-site regression (Stage 2), with covariance-adjusted and bootstrap BCa inference. Simulations show MS2T-IV reduces bias and improves mean-squared error relative to naive or IPTW approaches, though inference with proper bootstrap is sometimes conservative; the method applied to Project STAR estimates a positive two-year small-class effect on Grade 1 outcomes, illustrating practical utility for evaluating multisite two-phase programs. Overall, the paper advances causal inference for multi-phase interventions by delivering a robust, less assumption-reliant approach to estimate cumulative ATEs in the presence of noncompliance and posttreatment confounding.

Abstract

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.

Paper Structure

This paper contains 31 sections, 1 theorem, 37 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Relevance
  3. Methodological Challenges
  4. Constraints of Existing Causal Inference Strategies
  5. Proposed Multisite Two-Phase Treatment IV Strategy
  6. Identification and Estimation of the Cumulative ATE via the MS2T-IV Strategy
  7. Notation and the Causal Estimand
  8. Theoretical Model
  9. Identification Assumptions
  10. Reduced-Form Models for the ITT Effects
  11. The Distinction Between the Cumulative ATE and the ITT Effect on the Outcome
  12. Estimation and Statistical Inference
  13. Simulations
  14. Research Questions for the Simulations
  15. Data Generation
  16. ...and 16 more sections

Key Result

Theorem 1

Under the theoretical model (eq1) and when Assumptions ass1, ass2, and ass3 hold, the cumulative ATE is the sum of four components as shown in equation (eq6):

Figures (1)

  • Figure 1: Directed Acyclic Graph for Two-Phase Treatment Sequences with Noncompliance. $Z$ and $D$ respectively denote the Phase I treatment and Phase II treatment receipt. $V$ and $Y$ respectively stand for the outcome by the end of Phase I and that by the end of Phase II. $X$ and $U$ denote the observed and unobserved vectors of baseline covariates, respectively.

Theorems & Definitions (1)

  • Theorem 1