Response to Discussions of "Causal and Counterfactual Views of Missing Data Models"

TL;DR

This paper addresses identifiability of complete-data distributions under MNAR by leveraging graphical models to connect causal inference and missing data. By reframing identifiability as identifying the joint distribution over counterfactuals $L^{(1)}$ and missingness $R$, and deriving a counterfactual $g$-formula, it shows nonparametric identification under Markov restrictions of an m-DAG whenever $p(R=1\mid L^{(1)})$ is identifiable from the observed data, with $p(l^{(1)}) = p(l,R=1)/p(R=1\mid l^{(1)})$. Key contributions include clarifying two notions of nonparametric identification, cataloging identification results for various m-DAGs, examining relationships to SWIGs and instrumental/ shadow-variable strategies, and addressing estimation and validation challenges. The work provides a conceptual bridge between causal and missing-data theory, avoids rank-preservation assumptions, and outlines a roadmap to integrate graphical identification, auxiliary information, censoring-based interventions, and robust sensitivity analysis for practical MNAR analysis.

Abstract

We are grateful to the discussants, Levis and Kennedy [2025], Luo and Geng [2025], Wang and van der Laan [2025], and Yang and Kim [2025], for their thoughtful comments on our paper (Nabi et al., 2025). In this rejoinder, we summarize our main contributions and respond to each discussion in turn.

Figures (4)

• Figure 1: (a) Ignorable treatment model with SWIGs; (b) Its MCAR analogue, with a single relevant SWIG; (c) Conditionally ignorable treatment model with SWIGs; (d) Its MAR analogue, again with only one relevant SWIG.
• Figure 2: (a) A possible SWIG representation of a self-censoring missing data model; (b) Stitching SWIGs into an observed-data model produces a cycle; (c) A SWIG that draws a distinction between the full data variable $L^{(1)}$, viewed as an unobserved confounder $U$ with extra restrictions, and the observed variable $L$ under an intervention where $R$ is set to $1$; (d) The full data graph equating the full data variable $L^{(1)}$ with an unobserved confounder $U$ with special structure.
• Figure 3: (a) The block-parallel MNAR model, labeling missing variables as unmeasured for the purposes of constructing a SWIG; (b) The corresponding SWIG obtained from (a).
• Figure 4: (a) Extension of Levis and Kennedy's example to have two missing variables. (b) Graph obtained after fixing $R_A$ and $R_Y$ in parallel. (c) SWIG obtained by splitting the treatment variable $A$ after already having fixed $R_A$ and $R_Y$.