Bounds for causal mediation effects

Several frameworks have been proposed for studying causal mediation analysis. What these frameworks have in common is that they all make assumptions for point identifications that can be violated even when treatment is randomized. When a causal effect is not point-identified, one can sometimes derive bounds, i.e. a range of possible values that are consistent with the observed data. In this work, we study causal bounds for mediation effects under both the natural effects framework and the separable effects framework. In particular, we show that when there are unmeasured confounders for the intermediate variables(s) the sharp symbolic bounds on separable (in)direct effect coincide with existing bounds for natural (in)direct effects in the analogous setting. We compare these bounds to valid bounds for the natural direct effects when only the cross-world independence assumption does not hold. Furthermore, we demonstrate the use and compare the results of the bounds on data from a trial investigating the effect of peanut consumption on the development of peanut allergy in infants through specific pathways of measured immunological biomarkers.

💡 Research Summary

This paper addresses the problem of causal mediation analysis when point identification of direct and indirect effects is impossible due to violations of the cross‑world independence assumption, even in randomized experiments. The authors focus on two major frameworks: the traditional natural (in)direct effects (NDE/NIE) introduced by Robins, Greenland, and Pearl, and the newer separable (or interventionist) (in)direct effects (SDE/SIE) proposed by Robins, Richardson, and Stensrud. While natural effects rely on cross‑world counterfactual independence—an assumption that cannot be empirically verified and is often broken by unmeasured mediator‑outcome confounders—the separable effects require only “single‑world” independence assumptions that are, at least in principle, testable in a four‑arm trial where the two components of treatment can be intervened on separately.

The authors first clarify the distinction between single‑world and cross‑world causal models using directed acyclic graphs (DAGs) and single‑world intervention graphs (SWIGs). They show that the NPSEM‑IE model imposes the strong cross‑world independence, whereas the FFRCISTG model imposes only the weaker single‑world no‑confounding condition. This conceptual groundwork sets the stage for the main methodological contributions.

The paper studies two settings. Setting I is a simple mediation structure with a binary exposure A, binary mediator M, binary outcome Y, and an unmeasured confounder U that affects both M and Y. The exposure is conceptually split into two binary components, A_M (affecting Y only through M) and A_Y (affecting Y directly). Under this expansion, the separable direct effect (SDE) and separable indirect effect (SIE) are defined as differences in the probability of Y when the two components are set to different values. The authors demonstrate that, when the NPSEM‑IE holds, these definitions coincide exactly with the natural direct and indirect effects (NDE and NIE). Consequently, the two frameworks are mathematically equivalent in this setting.

To obtain sharp (i.e., attainable) bounds when U is unmeasured, the authors apply the linear‑programming based symbolic bounding method of Sachs et al. (2023), which they have previously implemented in the R package causaloptim. They prove that the resulting bounds for SDE and SIE are identical to the bounds previously derived for NDE and NIE by Sjölander (2009) under the same randomization‑only assumption. Thus, when only treatment randomization is assumed, the best possible bounds for natural and separable effects are the same.

The paper then turns to the case where only the cross‑world independence assumption fails, but there is no unmeasured mediator‑outcome confounding. In this scenario, earlier work by Robins and Richardson (2010) provided valid but not sharp bounds for the NDE. The authors compare those bounds to the Fréchet bounds and show that the earlier bounds are indeed valid but can be improved. They derive new sharp bounds for the NDE using the same linear‑programming approach, demonstrating that the previous bounds are a special (non‑sharp) case.

Setting II extends the analysis to a more complex structure with two sequential mediators (L and M) and allows for post‑treatment confounding. Three graphical variants are considered: (a) L is affected only by A_M, (b) L is affected only by A_Y, and (c) L is affected by a third component A_L. For each variant the authors specify the necessary dismissible‑component assumptions (A0.I, A1.I, etc.) that guarantee identification of the separable effects under the FFRCISTG model. They then derive sharp symbolic bounds for the corresponding natural and separable direct and indirect effects, showing how the bounds change with the placement of L in the causal graph.

A substantial empirical illustration uses data from a randomized trial of early peanut consumption in infants, where the outcome is peanut allergy and the mediators are immunological biomarkers (e.g., specific IgE, cytokine levels). The authors fit the required conditional probability models, compute the bounds for SDE, SIE, NDE, and NIE, and compare them to point estimates obtained under the (often untenable) cross‑world assumption. The bounded estimates are wider, reflecting the uncertainty due to possible violations of the cross‑world independence, yet they still provide useful information about the magnitude and direction of direct versus indirect pathways.

In the discussion, the authors acknowledge limitations: the current bounds are derived for binary variables and simple mediation structures; extending to continuous mediators, multiple outcomes, or high‑dimensional settings will require further methodological development. They also note that incorporating external information (e.g., Bayesian priors) could tighten the bounds, and that future work might explore optimal experimental designs that directly target the width of the bounds.

Overall, the paper makes three major contributions: (1) it establishes a formal equivalence between the sharp bounds for natural and separable effects under randomization‑only assumptions; (2) it clarifies when existing bounds are valid but not sharp and provides improved sharp bounds for the cross‑world violation case; (3) it demonstrates the practical applicability of these bounds in a real clinical trial, offering a robust alternative to point‑identification when key assumptions are doubtful. This work advances causal mediation analysis by providing a unified, theoretically sound, and implementable framework for bounding direct and indirect effects in the presence of unmeasured confounding or cross‑world assumption violations.