Identification and estimation of the conditional average treatment effect with nonignorable missing covariates, treatment, and outcome

1. Problem Statement

In observational studies, researchers aim to estimate the Conditional Average Treatment Effect (CATE), defined as $\tau(x) = E[Y(1) - Y(0) | X=x]$, to understand heterogeneous treatment effects. However, data is often incomplete, with covariates ($X$), treatment ($T$), and outcomes ($Y$) subject to missingness.

The core challenge addressed is Missing Not At Random (MNAR), where the probability of missingness depends on the unobserved values themselves (e.g., individuals with low earnings are less likely to report them).

• Limitations of Standard Methods: Standard approaches like Complete Case Analysis (CCA), Multiple Imputation (MI), and Inverse Probability Weighting (IPW) rely on Missing At Random (MAR) or Missing Completely At Random (MCAR) assumptions. Under MNAR, these methods yield biased estimates.
• Gap in Literature: While some work exists on identifying the Average Treatment Effect (ATE) under MNAR, it often requires restrictive assumptions (e.g., missingness only in covariates, or no self-censoring in outcomes). Furthermore, recovering the full joint distribution (required for ATE) is often impossible under general MNAR, yet the CATE might still be identifiable.
• Specific Context: The paper focuses on prospective studies where $X$ and $T$ are measured before $Y$. This temporal ordering implies that $Y$ cannot influence the missingness of $X$ or $T$, but $X$ and $T$ can influence the missingness of $Y$, and $Y$ can influence its own missingness (self-censoring).

2. Methodology

2.1 Causal Framework and Assumptions

The authors adopt the potential outcomes framework with SUTVA, Strong Ignorability, and Positivity. They define three specific MNAR mechanisms (Assumptions 1–3) by restricting the Directed Acyclic Graph (DAG) of missingness. These assumptions allow general MNAR for $X$ and $T$ but impose specific restrictions on the outcome missingness indicator $R_Y$:

• Assumption 1 (Outcome-Independent): $R_Y \perp Y \mid (X, T, R_X, R_T)$. The outcome does not influence its own missingness (no self-censoring).
• Assumption 2 (Treatment-Independent): $R_Y \perp T \mid (X, Y, R_X, R_T)$. Outcome missingness depends on $Y$ (allowing self-censoring) but is independent of treatment $T$ conditional on $X$ and $Y$.
• Assumption 3 (Covariate-Independent): $R_Y \perp X \mid (T, Y, R_X, R_T)$. Outcome missingness depends on $Y$ but is independent of covariates $X$ conditional on $T$ and $Y$.

2.2 Identification Results

The paper establishes nonparametric identification of the CATE under these assumptions:

• Under Assumption 1: The CATE is identifiable using only complete cases ($R_X=R_T=R_Y=1$) because $P(Y|X,T)$ is recoverable from complete cases.
• Under Assumptions 2 & 3: Identification requires a completeness condition.
  • For Assumption 2, $T$ must act as a "shadow variable" (associated with $Y$ but not directly affecting $R_Y$ given $X, Y$). The conditional distribution $P(Y|X,T)$ is identified if the joint distribution of $(Y, R_Y=1)$ is complete in $T$.
  • For Assumption 3, $X$ acts as the shadow variable. Identification holds if the distribution is complete in $X$.
  • Special Case: If $Y \perp T \mid X$ (null effect stratum), the CATE is identified as 0 even without the completeness condition under Assumption 2.

Key Insight: The CATE depends only on the conditional distribution $P(Y|X,T)$, not the full joint distribution of $(X,T,Y)$. Therefore, CATE can be identified under weaker MNAR conditions where the ATE (requiring the full joint distribution) is not.

2.3 Estimation Strategies

The authors propose two estimators for the MNAR settings (Assumptions 2 & 3):

1. Nonparametric Series 2SLS Estimator:

   • Solves a Fredholm integral equation to recover the response odds function $\zeta(x,y) = P(R_Y=0|...)/P(R_Y=1|...)$.
   • Uses a sieve approximation (basis functions) and quadratic regularization to handle the ill-posed nature of the integral equation.
   • Constructs selection-corrected weights to estimate $E(Y|X,T)$ on complete cases.

2. Parametric EM Estimator:

   • Assumes parametric models for the outcome $P_\beta(Y|X,T)$ and the missingness mechanism $\pi_\lambda(R_Y|...)$.
   • Uses the Expectation-Maximization (EM) algorithm with Fractional Imputation (FI) to handle latent outcomes.
   • More stable than the nonparametric approach but relies on correct model specification.

3. Key Contributions

1. Novel Identification Theory: Provides the first nonparametric identification results for CATE under general multivariate MNAR mechanisms where $X$, $T$, and $Y$ are all partially observed, including outcome self-censoring.
2. Distinction between CATE and ATE: Demonstrates that CATE is identifiable under weaker conditions than ATE because it does not require recovering the joint distribution of covariates and treatment.
3. Flexible Framework: Develops a framework that accommodates self-censoring in outcomes while allowing general MNAR in covariates and treatment, provided specific independence restrictions hold.
4. Sensitivity Analysis Framework: Proposes a single-parameter sensitivity analysis approach. By reintroducing the excluded edge in the DAG (e.g., allowing $Y \to R_Y$ when it was assumed absent), the authors quantify how robust the CATE estimates are to violations of the baseline assumptions.
5. Practical Estimators: Develops both flexible nonparametric and stable parametric estimators, complete with inference procedures (bootstrap).

4. Results

4.1 Simulation Studies

• Setup: Simulations with $N=1000$ covering binary/continuous combinations of $X, T, Y$ under Assumptions 1–3.
• Findings:
  • Under Assumption 1, standard Complete Case Analysis (CCA) is unbiased.
  • Under Assumptions 2 & 3, CCA and standard MI methods exhibit substantial bias.
  • The proposed Nonparametric (NP) and Parametric (Para) estimators remain nearly unbiased when the completeness condition holds.
  • When the completeness condition fails (e.g., binary $T$ under Assumption 2), the NP estimator shows bias (due to underdetermined integral equations), while the Parametric estimator remains unbiased (as parametric models can satisfy completeness even with discrete variables).
  • Para (full) (modeling the full joint distribution) yields biased results, confirming that modeling non-identifiable components introduces error.

4.2 Application: National Job Corps Study (NJCS)

• Context: Estimating the effect of obtaining an educational credential ($T$) on weekly earnings ($Y$) using baseline covariates ($X$). Data has high missingness rates (up to 21%) in $X, T,$ and $Y$.
• Analysis:
  • Applied a two-part model (probability of earnings > 0 and magnitude of earnings) to handle zero-inflated earnings.
  • Main Finding: All estimators (CCA, MI, NP, Para) under Assumptions 1–3 consistently show a positive and statistically significant effect of credential attainment on earnings for the reference profile.
  • Sensitivity Analysis: The authors varied the sensitivity parameter $\delta$ (representing the strength of the violation of the MNAR assumption) over $[-2, 2]$. The estimated positive effect remained robust across all three assumptions and the entire sensitivity range.

5. Significance and Conclusion

This paper provides a critical methodological advance for causal inference in the presence of complex missing data.

• Policy Relevance: It offers a rigorous way to estimate heterogeneous treatment effects in real-world observational studies (like the NJCS) where data is often MNAR due to the sensitivity of variables (e.g., earnings, arrest history).
• Robustness: By showing that CATE can be identified even when the full joint distribution is not, the paper expands the scope of valid causal inference in settings previously considered intractable.
• Practical Guidance: The authors recommend using the parametric estimator as the primary analysis due to its stability and ease of sensitivity analysis, using the nonparametric estimator as a robustness check. The proposed sensitivity analysis framework allows researchers to quantify how much their conclusions depend on untestable missingness assumptions.

The work bridges the gap between theoretical identification under MNAR and practical estimation, providing tools for researchers to draw reliable conclusions from incomplete, non-ignorable data.