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A Sensitivity Analysis of the Surrogate Index Approach for Estimating Long-Term Treatment Effects

Yanqin Fan, Carlos A. Manzanares, Hyeonseok Park, Yuan Qi

Abstract

This paper develops a sensitivity analysis of the surrogacy assumption for the surrogate index approach in Athey et al. [2025b]. We introduce "Weighted Surrogate Indices (WSIs)," the analog of the surrogate index under the surrogacy assumption. We show that under comparability, the ATE on WSI identifies the ATE on the long-term outcome when a copula of the treatment and the long-term outcome conditional on baseline covariates and surrogates is known. When the copula is unknown, we establish the identified set of the ATE on the long-term outcome. Furthermore, we construct debiased estimators of the ATE for any given copula and develop asymptotically valid inference in both point-identified and partially identified cases. Using data from a poverty alleviation program in Pakistan, we demonstrate the importance of sensitivity checks as well as the usefulness of our approach.

A Sensitivity Analysis of the Surrogate Index Approach for Estimating Long-Term Treatment Effects

Abstract

This paper develops a sensitivity analysis of the surrogacy assumption for the surrogate index approach in Athey et al. [2025b]. We introduce "Weighted Surrogate Indices (WSIs)," the analog of the surrogate index under the surrogacy assumption. We show that under comparability, the ATE on WSI identifies the ATE on the long-term outcome when a copula of the treatment and the long-term outcome conditional on baseline covariates and surrogates is known. When the copula is unknown, we establish the identified set of the ATE on the long-term outcome. Furthermore, we construct debiased estimators of the ATE for any given copula and develop asymptotically valid inference in both point-identified and partially identified cases. Using data from a poverty alleviation program in Pakistan, we demonstrate the importance of sensitivity checks as well as the usefulness of our approach.
Paper Structure (33 sections, 16 theorems, 181 equations, 4 figures, 1 algorithm)

This paper contains 33 sections, 16 theorems, 181 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup and Identification Under Surrogacy Assumption
  3. Point Identification Without Surrogacy Assumption
  4. Weighted Surrogate Indices
  5. Identification of ATE
  6. Surrogacy Bias---Stochastically Monotone Copulas
  7. A Numerical Illustration
  8. Worst-Case Bounds, Debiased Estimation, and Inference
  9. Comparison with Lemma 1 in athey2025surrogate
  10. Debiased Estimation
  11. Asymptotic Theory
  12. Debiased Estimation and Inference---General Copula
  13. Orthogonal Moment Function and Debiased Estimator
  14. Asymptotic Theory
  15. Empirical Application
  16. ...and 18 more sections

Key Result

Lemma 2.1

Under assumption: Random Sample-assumption:Comparability, $\tau$ is identified as

Figures (4)

  • Figure 1: Relationship between Kendall's tau $(\varrho_K)$ and $\tau$ given $\rho\in\{0.1,0.5,0.9\}$ for the Gaussian, Clayton, Gumbel, and Frank copulas.
  • Figure 2: Relationship between $\hat{\tau}_{C_\vartheta}$ and Kendall's tau using the Frank copula for Pakistan.
  • Figure 3: Relationship between $\hat{\tau}_{C_\vartheta}$ and Kendall's tau using the Plackett copula for Pakistan.
  • Figure 4: Local sensitivity analysis near the surrogacy benchmark (Kendall's tau $=0$).

Theorems & Definitions (36)

  • Remark 2.1
  • Definition 2.1: Surrogate Index, athey2025surrogate
  • Lemma 2.1: athey2025surrogate
  • Remark 2.2
  • Example 3.1: Families of Copulas
  • Definition 3.1: Weighted Surrogate Indices
  • Remark 3.1
  • Example 3.2: AVaR and WSIs for Upper and Lower Bound Copulas
  • Proposition 3.1
  • Theorem 3.1: Known Sub-copula
  • ...and 26 more