Self-separated and self-connected models for mediator and outcome missingness in mediation analysis

1. Problem Statement

Mediation analysis is widely used to decompose causal effects into direct and indirect pathways. However, a significant challenge arises when both the mediator ($M$) and the outcome ($Y$) contain missing data.

• Limitations of Standard Approaches: Common methods like Multiple Imputation (MI) or Full Information Maximum Likelihood (FIML) typically assume data are Missing At Random (MAR) or Missing Completely At Random (MCAR). In mediation contexts, these assumptions are often violated because missingness in $M$ or $Y$ may depend on the unobserved values of $M$ or $Y$ themselves (Missing Not At Random, or MNAR).
• The Gap: While recent literature has developed identification strategies for MNAR using "shadow variables" (SVs) or conditional independence assumptions, these methods have not been systematically synthesized or extended specifically for the complex setting where both the mediator and outcome are subject to missingness.
• Goal: The paper aims to expand the range of identified models for mediator and outcome missingness, moving beyond MAR to MNAR settings, and to provide a unified theoretical framework for identification and model falsifiability.

2. Methodology and Framework

The authors utilize a Directed Acyclic Graph (DAG) framework within the potential outcomes setting, assuming Sequential Ignorability of Assignment and Mediator Assignment (SIAA). They categorize missingness mechanisms into two broad classes based on the relationship between a variable and its missingness indicator ($R_M, R_Y$):

A. Temporal Ordering

The authors explicitly define four temporal orderings for the measurement of variables and their missingness indicators, which dictates the structure of the DAGs:

1. In-time: $M, R_M, Y, R_Y$ (Most common; $M$ measured before $Y$).
2. Delayed: $M, Y, R_M, R_Y$ (Both measured retrospectively).
3. Unordered: $M, Y, R_M, R_Y$ (Missingness indicators occur simultaneously or order is unknown).
4. Reverse: $M, Y, R_Y, R_M$ (Outcome missingness precedes mediator missingness).

B. Class 1: Self-Separated Missingness Models

In these models, a variable and its missingness indicator do not influence each other, nor do they share unobserved common causes. Identification is achieved via conditional independence assumptions.

• Key Models (S1–S5):
  • S1 (Sequentially Ignorable Missingness - SIM): The least restrictive model for the "in-time" setting. It assumes $R_M \perp\!\!\perp (M, Y) | X, A$ and $R_Y \perp\!\!\perp Y | X, A, M, R_M$. It generalizes MAR but allows for specific dependencies.
  • S2–S5: Variations that relax assumptions for different temporal orders (e.g., unordered or reverse). These often require specific separations (e.g., $R_Y \perp\!\!\perp M$) to achieve identification.
• Technique: The authors introduce tilting functions ($h, k$) to map observed data densities to full-data densities. They show that many of these models are falsifiable (they imply testable conditions on the observed data distribution), unlike standard MAR.

C. Class 2: Self-Connected Missingness Models with Shadow Variables

In these models, a variable influences its own missingness (or shares unobserved causes with it), making standard conditional independence insufficient. Identification relies on Shadow Variables (SVs).

• Shadow Variable Definition: A variable $Z$ is an SV for $V$ if $Z \perp\!\!\perp R_V | V$ (independent of missingness given the variable) but $Z \not\perp\!\!\perp V$ (dependent on the variable).
• Theoretical Innovation: The authors synthesize and extend SV theory using an integral equation based on the odds of missingness:
  $$\text{odds}(R_V=0 | Z) = E[\text{odds}(R_V=0 | V) | Z, R_V=1]$$
  This equation allows the recovery of the missingness mechanism if a completeness condition holds (i.e., $Z$ has sufficient variation relative to $V$).
• Extensions:
  1. Built-in SVs: Using $M$ or $Y$ themselves as SVs for each other (Models Z1–Z5).
  2. External SVs: Using auxiliary variables (downstream, midstream, or upstream in the causal chain) or covariates ($X$) as SVs (Models D1–D4, M1–M4, U4, C1–C4).
  3. Missing SVs: The theory is extended to handle cases where the shadow variable itself is missing, provided specific separation assumptions (mSV) are met.

3. Key Contributions

1. Systematic Taxonomy of Identified Models:

   • The paper provides a comprehensive catalog of identified models for the joint missingness of mediators and outcomes, categorized by temporal order and missingness mechanism (self-separated vs. self-connected).
   • It clarifies that Sequentially Ignorable Missingness (SIM) is a more flexible and often more realistic alternative to joint MAR, as it allows for dependencies between missingness indicators while maintaining identification.

2. Synthesis and Extension of Shadow Variable Theory:

   • Unified Theory: The authors present a simple, unified SV theory that ties together existing results (e.g., from Zuo et al., Miao et al.) and accommodates missingness in the SV itself.
   • Two-Variable Extension: They extend SV theory to handle two variables of interest ($M$ and $Y$) simultaneously, deriving three new external SV models (Combined, Parallel, and Sequential) that minimize restrictions on the core missingness model.
   • Odds-Tilting Techniques: They introduce specific "odds-tilting" techniques to solve the integral equations required for identification, offering a practical toolkit for deriving tilting functions.

3. Model Falsifiability:

   • A major contribution is the derivation of testable implications for many of the proposed models. Unlike standard MAR, which is untestable, many MNAR models (e.g., S3, Z1, A2) imply specific constraints on the observed data distribution (e.g., specific ratios of missingness probabilities). This allows researchers to empirically screen out incorrect models.

4. Practical Application Guidelines:

   • The paper uses two motivating examples (School Intervention and Job Training) to demonstrate how to select models based on subject-matter knowledge, data availability, and the plausibility of conditional independence assumptions.
   • It emphasizes the importance of collecting auxiliary data (e.g., baseline measures, alternative sources like parent reports, or follow-up data) to serve as shadow variables when self-connected missingness is suspected.

4. Key Results

• Identification Conditions: The paper provides explicit identification formulas (tilting functions) for all proposed models. These formulas express the full-data densities $P(M|X,A)$ and $P(Y|X,A,M)$ in terms of observed data densities and identified response probabilities.
• Completeness Requirements: The authors detail the necessary completeness conditions for each model. For instance, in models where $M$ and $Y$ serve as mutual SVs (Model Z1), both must have sufficient variation (e.g., same number of categories if discrete).
• Handling Missing SVs: The paper proves that identifiability is preserved even if the shadow variable is missing, provided the missingness of the SV satisfies specific separation assumptions (e.g., the SV's missingness does not share unobserved causes with the main variables' missingness).
• Trade-offs: The analysis reveals a fundamental trade-off: models that allow for self-connected missingness (MNAR) generally require stronger restrictions elsewhere (e.g., no unobserved common causes between missingness indicators, or specific causal structures for SVs).

5. Significance

• Methodological Advancement: This work bridges the gap between theoretical missing data literature and the practical needs of mediation analysis. It moves the field beyond the restrictive MAR assumption, which is often untenable for sensitive variables (e.g., drug use, earnings).
• Robustness and Validation: By providing testable implications, the paper empowers researchers to validate their assumptions against data, reducing the risk of biased causal estimates due to untestable MNAR assumptions.
• Study Design Implications: The paper highlights that handling MNAR in mediation analysis is not just a post-hoc statistical fix but a design issue. It argues for the proactive collection of auxiliary variables (shadow variables) and careful consideration of temporal measurement orders during the study planning phase.
• Generalizability: While focused on mediation, the extended shadow variable theory and odds-tilting techniques are applicable to any setting involving joint missingness in multiple variables, offering a broader impact on causal inference methodology.

In conclusion, Nguyen et al. provide a rigorous, structured framework for identifying causal mediation effects in the presence of complex, non-ignorable missing data, offering both theoretical depth and practical guidance for researchers.