Nonparametric estimation of path-specific effects in the presence of nonignorable missing covariates

The path-specific effect (PSE) is of primary interest in mediation analysis when multiple intermediate variables are in the pathway from treatment to outcome, as it can isolate the specific effect through each mediator, thus mitigating potential bias arising from other intermediate variables serving as mediator-outcome confounders. However, estimation and inference of PSE become challenging in the presence of nonignorable missing covariates, a situation particularly common in studies involving sensitive individual information. This paper proposes a fully nonparametric methodology to address this challenge. We establish identification for PSE by expressing it as a function of observed data. By leveraging a shadow variable, we demonstrate that the associated nuisance functions can be uniquely determined through sequential optimization problems. Then, we propose a sieve-based regression imputation approach for estimation. We establish the large-sample theory for the proposed estimator and introduce an approach to make inferences for PSE. The proposed method is applied to the NHANES dataset to investigate the mediation roles of dyslipidemia and obesity in the pathway from Type 2 diabetes mellitus to cardiovascular disease.