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Incorporating Auxiliary Variables to Improve the Efficiency of Time-Varying Treatment Effect Estimation

Jieru Shi, Zhenke Wu, Walter Dempsey

TL;DR

This paper tackles the efficiency of estimating time-varying treatment effects in micro-randomized trials by introducing A2-WCLS, which leverages pre- and post-treatment auxiliary variables under a Neyman orthogonality framework to avoid bias and reduce variance. The method provides a principled way to incorporate rich longitudinal covariates while keeping nuisance parameters low-dimensional, and it establishes local efficiency gains when the causal model is correctly specified. Through comprehensive simulations and an application to the Intern Health Study, A2-WCLS demonstrates substantial improvements in estimation precision for both time-specific and lagged effects, with practical benefits for mobile health interventions. The work offers practical guidelines, theoretical guarantees, and open avenues for extensions such as non-linear centering and adaptive trial designs.

Abstract

Contextual sensing and delivery of digital interventions to improve health outcomes have gained significant traction in behavioral and psychiatric studies. Micro-randomized trials (MRTs) are a common experimental design for obtaining data-driven evidence on the effectiveness of digital interventions where each individual is repeatedly randomized to receive treatments over numerous time points. Throughout the study, individual characteristics and contextual factors around randomization are collected, with some prespecified as moderators for assessing time-varying causal effect moderation. However, many additional measurements beyond these moderators often go underutilized. Some of these may influence treatment randomization or known to strongly moderate the treatment effect. Incorporating such auxiliary information into the estimation procedure can reduce chance imbalances and improve asymptotic estimation efficiency. In this work, we propose a method to adjust for auxiliary variables in consistently estimating time-varying intervention effects. The approach can also be extended to include post-treatment auxiliary variables when evaluating lagged treatment effects. Under specific conditions, local efficiency gains are guaranteed. We demonstrate the method's utility through simulation studies and an analysis of data from the Intern Health Study (NeCamp et al., 2020).

Incorporating Auxiliary Variables to Improve the Efficiency of Time-Varying Treatment Effect Estimation

TL;DR

This paper tackles the efficiency of estimating time-varying treatment effects in micro-randomized trials by introducing A2-WCLS, which leverages pre- and post-treatment auxiliary variables under a Neyman orthogonality framework to avoid bias and reduce variance. The method provides a principled way to incorporate rich longitudinal covariates while keeping nuisance parameters low-dimensional, and it establishes local efficiency gains when the causal model is correctly specified. Through comprehensive simulations and an application to the Intern Health Study, A2-WCLS demonstrates substantial improvements in estimation precision for both time-specific and lagged effects, with practical benefits for mobile health interventions. The work offers practical guidelines, theoretical guarantees, and open avenues for extensions such as non-linear centering and adaptive trial designs.

Abstract

Contextual sensing and delivery of digital interventions to improve health outcomes have gained significant traction in behavioral and psychiatric studies. Micro-randomized trials (MRTs) are a common experimental design for obtaining data-driven evidence on the effectiveness of digital interventions where each individual is repeatedly randomized to receive treatments over numerous time points. Throughout the study, individual characteristics and contextual factors around randomization are collected, with some prespecified as moderators for assessing time-varying causal effect moderation. However, many additional measurements beyond these moderators often go underutilized. Some of these may influence treatment randomization or known to strongly moderate the treatment effect. Incorporating such auxiliary information into the estimation procedure can reduce chance imbalances and improve asymptotic estimation efficiency. In this work, we propose a method to adjust for auxiliary variables in consistently estimating time-varying intervention effects. The approach can also be extended to include post-treatment auxiliary variables when evaluating lagged treatment effects. Under specific conditions, local efficiency gains are guaranteed. We demonstrate the method's utility through simulation studies and an analysis of data from the Intern Health Study (NeCamp et al., 2020).
Paper Structure (65 sections, 7 theorems, 111 equations, 3 figures, 10 tables)

This paper contains 65 sections, 7 theorems, 111 equations, 3 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Main Contributions.
  3. Preliminaries
  4. Causal Excursion Effects and Existing Inferential Methods
  5. Literature Review: Covariate Adjustment
  6. Estimand and Inferential Method
  7. Time-Specific Causal Excursion Effect Estimation
  8. The Unadjusted and the WCLS Estimator
  9. Proposed and More Efficient Estimator
  10. Auxiliary Variable Adjusted WCLS (A2-WCLS) Estimation
  11. Pre-treatment Auxiliary Variable Adjustment
  12. Post-Treatment Auxiliary Variable Adjustment
  13. Theoretical Results
  14. Efficiency Comparison of the Proposed Estimator
  15. Estimating the centering functions
  16. ...and 50 more sections

Key Result

Lemma 3.2

Denote $\beta_{1,t}^\star = \Sigma(g_t(H_t))^{-1}\mathbb{E}[Y_{t+1}(A_t-p_t)\tilde{g}_t(H_t)]\in \mathbb{R}^{d}$, where $\Sigma(g_t(H_t)) = \mathbb{E}[ \tilde{g}_t(H_t) \tilde{g}_t(H_t)^\top] \in \mathbb{R}^{d\times d}$. The difference in the asymptotic variance between the WCLS estimator $\hat{\bet

Figures (3)

  • Figure 1: Causal effect estimates with the 95% confidence interval, and standard errors
  • Figure 2: Causal effect estimates with the 95% confidence interval.
  • Figure 3: Relative efficiency density curve over $M=1000$ replicates. (Left) Fixing total time points $T=50$ and varying sample size $N$.(Right) Fixing sample size $N=250$ and varying total time points $T$.

Theorems & Definitions (8)

  • Lemma 3.2: Asymptotic Variance Comparison between $\hat{\beta}_{0,t}^{\text{WCLS}}$ and $\hat{\beta}_{0,t}^{\text{U}}$
  • Lemma 3.3
  • Example 3.5: Fully marginal causal effect estimation
  • Lemma 3.6
  • Lemma 3.7
  • Theorem 4.3
  • Proposition 4.4
  • Lemma 4.5