On the Graphical Rules for Recovering the Average Treatment Effect Under Selection Bias

Selection bias is a major obstacle toward valid causal inference in epidemiology. Over the past decade, several graphical rules based on causal diagrams have been proposed as the sufficient identification conditions for addressing selection bias and recovering causal effects. However, these simple graphical rules are typically coupled with specific identification strategies and estimators. In this article, we show two important cases of selection bias that cannot be addressed by these existing simple rules and their estimators: one case where selection is a descendant of a collider of the treatment and the outcome, and the other case where selection is affected by the mediator. To address selection bias and recover average treatment effect in these two cases, we propose an alternative set of graphical rules and construct identification formulas by the g-computation and the inverse probability weighting (IPW) methods based on single-world intervention graphs (SWIGs). We conduct simulation studies to verify the performance of the estimators when the traditional crude selected-sample analysis (i.e., complete-case analysis) yields erroneous conclusions contradictory to the truth.

💡 Research Summary

This paper addresses a critical gap in causal inference under selection bias by focusing on two problematic scenarios that existing graphical rules fail to handle. The first scenario involves a collider L that is a common effect of treatment A and outcome Y, with the selection indicator S being a descendant of L. The second scenario features L as a mediator on the causal pathway from A to Y, and L directly influences selection. Traditional rules such as the selection‑backdoor criterion, the generalized adjustment criterion, and the Mathur‑Shpitser rules all impose a “no conditioning on post‑treatment variables” restriction, which precludes identification of the average treatment effect (ATE) in these settings.

The authors adopt single‑world intervention graphs (SWIGs) to make explicit the relationships among potential outcomes Y(a), post‑treatment variables L(a), and selection S(a). They demonstrate that, when external information about the distribution of L in the unselected portion of the population is available, both g‑computation and inverse‑probability weighting (IPW) can be used to recover the ATE. The key identification conditions are:

1. C1 – Y(a) is independent of S(a) conditional on L(a) and any pre‑treatment covariates X. This blocks all back‑door paths between outcome and selection.
2. C2 – For g‑computation: Y(a) ⟂ A | L(a), S(a), X; for IPW: (Y(a), S(a)) ⟂ A | L(a), X. This ensures that treatment effects on outcome (and on selection for IPW) can be separated after conditioning.
3. C3 – L(a) ⟂ A | X, guaranteeing that the post‑treatment variable is not confounded with treatment once X is accounted for.

Under these conditions, Theorem 1 provides a g‑formula that integrates the conditional expectation of Y given (A, L, S, X) with the external distribution p(L|A, X). Theorem 2 presents an IPW estimator that weights observed outcomes by the inverse of the selection probability p(S=1|L, X, A) and the treatment propensity p(A|X). The authors note a crucial distinction: the IPW approach requires inclusion of all variables that affect selection (e.g., a pre‑treatment confounder X₃), whereas the g‑formula can sometimes omit such variables if L sufficiently blocks the selection pathway.

The paper extends these results to more complex DAGs that include observed confounders between treatment and post‑treatment variables, as well as unobserved confounders between L and Y. Corollaries illustrate that the identification strategies remain valid in these richer settings, emphasizing the flexibility of the SWIG‑based approach.

Simulation studies emulate a randomized clinical trial where A and Y are binary, and L plays either the collider or mediator role. Selection S depends on L, and only L is observed among non‑selected units. Analyses that rely solely on the selected sample (complete‑case analysis) produce severely biased ATE estimates, sometimes even reversing the sign of the true effect. In contrast, applying the proposed g‑computation or IPW estimators—leveraging external information on the distribution of L—yields unbiased estimates with correctly calibrated confidence intervals. The simulations also confirm that IPW estimators are sensitive to omission of selection‑related covariates (X₃), while g‑computation tolerates their exclusion when L blocks the relevant paths.

In conclusion, the authors make three substantive contributions: (1) they pinpoint specific forms of selection bias that lie outside the reach of existing graphical identification rules; (2) they formulate new SWIG‑based graphical conditions that enable identification of the ATE using post‑treatment variables together with external data; and (3) they provide concrete g‑computation and IPW estimators, along with variance‑estimation methods, that perform well in simulation. Their work underscores the importance of incorporating external information and carefully examining the causal structure of selection mechanisms, offering a robust toolkit for epidemiologists and other researchers confronting selection bias in observational studies.