The positivity assumption in causal mediation analyses? Checked!

Technical Summary: The Positivity Assumption in Causal Mediation Analyses

Problem Statement
Causal mediation analysis, which seeks to decompose the total effect of an exposure on an outcome into direct and indirect pathways, relies on strong identification assumptions. While the assumption of exchangeability (no unmeasured confounding) has received extensive attention, the positivity assumption is frequently overlooked in mediation settings. Positivity violations occur when specific subgroups defined by covariates have zero or near-zero probability of receiving a specific exposure or mediator value.

In mediation analysis, positivity is more complex than in standard total effect analysis. It requires non-zero probabilities for both the exposure and the mediator, and the specific form of the assumption depends on the causal estimand of interest (e.g., Controlled Direct Effect, Natural Effects, or Interventional Effects). Currently, there is a lack of robust, assumption-free tools to verify these positivity conditions in practice. Existing methods, such as checking propensity scores, are sensitive to model specification and may fail to identify the specific subgroups causing violations.

Methodology: The dePoRT Algorithm
The authors propose dePoRT (decomposition Positivity Regression Trees), an extension of the Positivity Regression Trees (PoRT) algorithm originally developed by Danelian et al. (2023) for total effect analysis. dePoRT is designed to verify practical positivity violations without requiring parametric assumptions about the data-generating process.

The algorithm operates through the following logic:

1. Tree Construction: It builds a sequence of regression or classification trees where the exposure or mediator is the dependent variable and covariates are independent variables. Unlike parametric models, trees learn non-linear and non-monotone associations directly from the data.
2. Subgroup Identification: The algorithm iteratively splits the data into subgroups (nodes). It "reads" these nodes to identify subgroups where the probability of the exposure or mediator falls below a pre-specified threshold ($\beta$).
3. Hyperparameters:
   • $\beta$: A bound for the minimum probability (e.g., Gruber's bound $\beta = 5/(\sqrt{n} \cdot \ln(n))$) below which a subgroup is considered to violate positivity.
   • $\alpha$: A threshold for the minimum subgroup size (e.g., 1% of the sample) to filter out violations in negligible subgroups that may not impact estimation.
   • $\gamma$: A stopping parameter limiting the number of covariates considered simultaneously in a tree to control complexity.
4. Adaptation for Mediation:
   • Controlled Direct Effects (CDE): Requires checking positivity for the exposure ($A$) and the mediator ($M$) conditional on covariates ($C$) within specific exposure strata.
   • Natural Effects: Involves nested potential outcomes ($Y_{a, M_{a^*}}$). The authors introduce a "reverse engineering" approach: first, identify subgroups where the mediator distribution in one exposure group ($A=a$) lacks support; second, verify if the mediator value exists in the counterfactual exposure group ($A=a^*$) for that same subgroup. If the value is absent in the counterfactual group, the violation is structural; if present, it is a violation of the specific estimand.
   • Interventional Effects: Similar to natural effects but accounts for intermediate confounders ($L$) influenced by the exposure. The support check involves verifying the mediator's prevalence conditional on both covariates ($C$) and the intermediate confounder ($L$).

Key Contributions

1. Algorithmic Extension: The paper extends the PoRT framework to the specific and complex positivity requirements of causal mediation analysis, covering Controlled Direct, Natural, and Interventional effects.
2. Tool Development: The authors implemented dePoRT in the R package port and provided a reproducible notebook, making these checks accessible to applied researchers.
3. Empirical Illustration: The method is applied to a dataset from the International Dating Violence Study to investigate the effect of childhood exposure to violence on dating violence, mediated by religiosity and post-traumatic stress symptoms (PTSS).
   • The analysis demonstrated that while exposure positivity held, mediator positivity was violated for specific subgroups (e.g., students with specific parental education levels and religious involvement).
   • The application showed how dePoRT can guide the choice of estimand; for instance, one decomposition of natural effects might be identifiable while another is not due to differing positivity violations.
   • In the presence of an intermediate confounder (PTSS), the authors illustrated how to check positivity for interventional effects, identifying violations that would bias natural effect estimates.

Results and Findings
In the motivating example, the application of dePoRT revealed:

• Total Effect: No positivity violations were found for the exposure.
• Controlled Direct Effect: 14 subgroups violated the mediator positivity assumption. The authors concluded these were likely random (practical) violations rather than structural, allowing for estimation using the iterated g-formula.
• Natural Effects: The analysis identified that the decomposition involving $E(Y_{0, M_1})$ had fewer positivity violations than the alternative ($E(Y_{1, M_0})$), suggesting the former is a more robust choice for this dataset.
• Interventional Effects: When accounting for PTSS, the method identified specific violations for interventional direct and indirect effects, reinforcing the conclusion that religiosity does not mediate the relationship in this context.

Significance and Claims
The authors claim that dePoRT fills a critical gap in the causal mediation toolkit by providing a model-free method to verify positivity assumptions. They emphasize that:

• Robustness: Unlike propensity score-based checks, dePoRT does not rely on parametric model specifications, reducing the risk of model-dependent false negatives or positives.
• Decision Support: The tool helps researchers determine which causal estimands are identifiable with their specific data. If positivity is violated for a preferred estimand (e.g., a specific natural effect decomposition), researchers can pivot to an alternative estimand (e.g., an interventional effect or a different decomposition) that is supported by the data.
• Transparency: By explicitly identifying subgroups where assumptions fail, dePoRT promotes transparency in causal inference, allowing for qualitative interpretation of whether violations are structural (requiring a redefinition of the target population) or random (addressable via extrapolation or alternative estimators).

The paper concludes modestly, noting that dePoRT is not a "silver bullet." Its validity still depends on the correct identification of confounders, the absence of measurement error, and the appropriate selection of hyperparameters. However, it represents a significant step toward more rigorous and transparent causal mediation analysis.