Efficient Nonparametric Inference for Mediation Analysis with Nonignorable Missing Confounders

Mediation analysis is widely used for exploring treatment mechanisms; however, it faces challenges when nonignorable missing confounders are present. Efficient inference of mediation effects and the efficiency loss due to nonignorable missingness have been rarely studied in the literature because of the difficulties arising from the ill-posed inverse problem. In this paper, we propose a general shadow variable framework for identifying mediation effects, allowing shadow variables to be selected from either observed covariates or externally collected auxiliary data. We then propose a Sieve-based Iterative Outward (SIO) approach for estimation. We establish large-sample theory, particularly asymptotic normality, for the proposed estimator despite the ill-posedness of the problem. We show that our estimator is locally efficient and attains the semiparametric efficiency bound under certain conditions. Building on the efficient influence function, we explicitly quantify the efficiency loss attributable to missingness and propose a debiased machine learning approach for estimation and inference. We examine the finite-sample performance of the proposed approach using extensive simulation studies and showcase its practical applicability through an empirical analysis of CFPS data.

💡 Research Summary

This paper tackles the challenging problem of causal mediation analysis when pre‑treatment covariates (confounders) are missing not at random (MNAR). Existing mediation methods rely on the sequential ignorability assumption, which fails when the missingness of covariates depends on their unobserved values. The authors introduce a general “shadow variable” framework that allows identification of mediation effects even under MNAR. A shadow variable is a fully observed auxiliary variable that is associated with the missing covariate but independent of the missingness mechanism conditional on observed data. The framework permits shadow variables to be drawn either from existing covariates that satisfy certain independence conditions or from external auxiliary data sources.

Using this framework, the authors derive a stratified identification strategy. The population is partitioned into overlapping strata defined by missingness patterns. Within each stratum, a subset of the partially observed covariates becomes fully observed, enabling the construction of conditional expectations that are expressed as weighted averages. The weights are inverse response probabilities (the probability of being a complete case) which satisfy a Fredholm integral equation of the first kind. By solving these equations non‑parametrically, the key mediation functional θ = E