Assessing the Causal Effect of Policies: An Example Using Stochastic Interventions

Abstract

Assessing the causal effect of an exposure often involves the definition of counterfactual outcomes in a hypothetical world in which the stochastic nature of the exposure is modified. Although stochastic interventions are a powerful tool to measure the causal effect of a realistic intervention that intends to alter the population distribution of an exposure, their importance to answer questions about plausible policy interventions has been obscured by the generalized use of deterministic interventions. In this article, we follow the approach described in Díaz and van der Laan (2012) to define and estimate the effect of an intervention that is expected to cause a truncation in the population distribution of the exposure. The observed data parameter that identifies the causal parameter of interest is established, as well as its efficient influence function under the non-parametric model. Inverse probability of treatment weighted (IPTW), augmented IPTW and targeted minimum loss-based estimators (TMLE) are proposed, their consistency and efficiency properties are determined. An extension to longitudinal data structures is presented and its use is demonstrated with a real data example.

Causal Mediation Analysis for Stochastic Interventions

“…the intervention fails to satisfy assumption 1 and integrals over the range of A cannot be computed by using the change‐of‐variable formula. This behaviour has been previously observed for other interventions that do not satisfy assumption 1 (Díaz and van der Laan, ). Even though this lemma implies that consistent estimation of g is required, the bias terms are still second order, so an estimator of g converging at rate n 1/4 or faster is sufficient, as we shall see in what follows.…”

“…Stochastic interventions are a generalization of this framework and are loosely defined as interventions which yield an exposure that is a random variable after conditioning on baseline variables. Estimation of total effects of stochastic interventions was first considered by Stock () and has been the subject of recent study (Robins et al ., ; Didelez et al ., ; Tian, ; Pearl, ; Taubman et al ., ; Stitelman et al ., ; Díaz and van der Laan, ; Dudík et al ., ; Haneuse and Rotnitzky, ; Young et al ., ). Particularly relevant to this work are the methods of Díaz and van der Laan (), Haneuse and Rotnitzky (), who defined total effects for modified treatment policies, and Kennedy (), who studied identification and estimation of the total effect of propensity score interventions that shift a binary exposure distribution.…”

“…Díaz and van der Laan () studied the effect of increasing the amount of leisure time physical activity in the elderly on subsequent all‐cause mortality. Díaz and van der Laan () studied the effect of a (hypothetical) policy that enforces pollution levels below a certain cut‐off point. Kennedy () showed that stochastic interventions can also be used in longitudinal studies to define and estimate total effects without reliance on the positivity assumption.…”

Causal Modeling in Environmental Health

“…In 2012, Padula et al (76) quantified the causal effects of exposure to traffic-related air pollution on birth weight using machine learning and targeted maximum likelihood estimation (TMLE), which, instead of minimizing variance or mean square errors, targets the maximum likelihood estimate but in a way that reduces bias. In 2013, Diaz & van der Laan (30) proposed to use inverse probability of treatment weighted (IPTW), augmented IPTW (a doubly robust estimator), TLME, and stochastic interventions to assess causality. In 2014, Mauderly and coauthors (70, 71) performed a multiple additive regression tree analysis in a multipollutant air quality study in rodents.…”

Targeted Data Adaptive Estimation of the Causal Dose–Response Curve

“…The main reason why consistent and efficient estimators of the causal dose-response curve (CDRC) for continuous treatments in the nonparametric model have not yet been developed is that it is not a pathwise differentiable parameter [10, chapter 3 and 5] and therefore cannot be estimated at a consistency rate of n À1=2 . Examples of pathwise differentiable parameters that measure the causal effect of a continuous exposure on an outcome of interest are given by the parameters defined in Díaz and van der Laan [11,12]. These approaches make use of stochastic interventions [13][14][15] as a means to define a counterfactual outcome in a post-intervened world, which compared to the expectation of the actual outcome defines the causal effect of an intervention.…”