When does the ID algorithm fail?

The ID algorithm solves the problem of identification of interventional distributions of the form p(Y | do(a)) in graphical causal models, and has been formulated in a number of ways [12, 9, 6]. The ID algorithm is sound (outputs the correct functional of the observed data distribution whenever p(Y | do(a)) is identified in the causal model represented by the input graph), and complete (explicitly flags as a failure any input p(Y | do(a)) whenever this distribution is not identified in the causal model represented by the input graph). The reference [9] provides a result, the so called "hedge criterion" (Corollary 3), which aims to give a graphical characterization of situations when the ID algorithm fails to identify its input in terms of a structure in the input graph called the hedge. While the ID algorithm is, indeed, a sound and complete algorithm, and the hedge structure does arise whenever the input distribution is not identified, Corollary 3 presented in [9] is incorrect as stated. In this note, I outline the modern presentation of the ID algorithm, discuss a simple counterexample to Corollary 3, and provide a number of graphical characterizations of the ID algorithm failing to identify its input distribution.Comment: arXiv admin note: substantial text overlap with arXiv:2108.0681

Counterfactual Graphical Models for Longitudinal Mediation Analysis with Unobserved Confounding

Questions concerning mediated causal effects are of great interest in psychology, cognitive science, medicine, social science, public health, and many other disciplines. For instance, about 60% of recent papers published in leading journals in social psychology contain at least one mediation test (Rucker, Preacher, Tormala, & Petty, 2011). Standard parametric approaches to mediation analysis employ regression models, and either the "difference method" (Judd & Kenny, 1981), more common in epidemiology, or the "product method" (Baron & Kenny, 1986), more common in the social sciences. In this paper we first discuss a known, but perhaps often unappreciated fact: that these parametric approaches are a special case of a general counterfactual framework for reasoning about causality first described by Neyman (1923), and Rubin (1974), and linked to causal graphical models by J. Robins (1986), and Pearl (2000). We then show a number of advantages of this framework. First, it makes the strong assumptions underlying mediation analysis explicit. Second, it avoids a number of problems present in the product and difference methods, such as biased estimates of effects in certain cases. Finally, we show the generality of this framework by proving a novel result which allows mediation analysis to be applied to longitudinal settings with unobserved confounders.Comment: To appear in the 2012 Rumelhart prize special issue of Cognitive Science honoring Judea Pear

Semiparametric theory for causal mediation analysis: Efficiency bounds, multiple robustness and sensitivity analysis

While estimation of the marginal (total) causal effect of a point exposure on an outcome is arguably the most common objective of experimental and observational studies in the health and social sciences, in recent years, investigators have also become increasingly interested in mediation analysis. Specifically, upon evaluating the total effect of the exposure, investigators routinely wish to make inferences about the direct or indirect pathways of the effect of the exposure, through a mediator variable or not, that occurs subsequently to the exposure and prior to the outcome. Although powerful semiparametric methodologies have been developed to analyze observational studies that produce double robust and highly efficient estimates of the marginal total causal effect, similar methods for mediation analysis are currently lacking. Thus, this paper develops a general semiparametric framework for obtaining inferences about so-called marginal natural direct and indirect causal effects, while appropriately accounting for a large number of pre-exposure confounding factors for the exposure and the mediator variables. Our analytic framework is particularly appealing, because it gives new insights on issues of efficiency and robustness in the context of mediation analysis. In particular, we propose new multiply robust locally efficient estimators of the marginal natural indirect and direct causal effects, and develop a novel double robust sensitivity analysis framework for the assumption of ignorability of the mediator variable.Comment: Published in at http://dx.doi.org/10.1214/12-AOS990 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Semiparametric Causal Sufficient Dimension Reduction Of High Dimensional Treatments

Cause-effect relationships are typically evaluated by comparing the outcome responses to binary treatment values, representing two arms of a hypothetical randomized controlled trial. However, in certain applications, treatments of interest are continuous and high dimensional. For example, understanding the causal relationship between severity of radiation therapy, represented by a high dimensional vector of radiation exposure values and post-treatment side effects is a problem of clinical interest in radiation oncology. An appropriate strategy for making interpretable causal conclusions is to reduce the dimension of treatment. If individual elements of a high dimensional treatment vector weakly affect the outcome, but the overall relationship between the treatment variable and the outcome is strong, careless approaches to dimension reduction may not preserve this relationship. Moreover, methods developed for regression problems do not transfer in a straightforward way to causal inference due to confounding complications between the treatment and outcome. In this paper, we use semiparametric inference theory for structural models to give a general approach to causal sufficient dimension reduction of a high dimensional treatment such that the cause-effect relationship between the treatment and outcome is preserved. We illustrate the utility of our proposal through simulations and a real data application in radiation oncology