Selective machine learning of doubly robust functionals

While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, selection of possibly high-dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we propose a selective machine learning framework for making inferences about a finite-dimensional functional defined on a semiparametric model, when the latter admits a doubly robust estimating function and several candidate machine learning algorithms are available for estimating the nuisance parameters. We introduce two new selection criteria for bias reduction in estimating the functional of interest, each based on a novel definition of pseudo-risk for the functional that embodies the double robustness property and thus is used to select the pair of learners that is nearest to fulfilling this property. We establish an oracle property for a multi-fold cross-validation version of the new selection criteria which states that our empirical criteria perform nearly as well as an oracle with a priori knowledge of the pseudo-risk for each pair of candidate learners. We also describe a smooth approximation to the selection criteria which allows for valid post-selection inference. Finally, we apply the approach to model selection of a semiparametric estimator of average treatment effect given an ensemble of candidate machine learners to account for confounding in an observational study

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

A Note on the Control Function Approach with an Instrumental Variable and a Binary Outcome

Unobserved confounding is a well known threat to causal inference in non-experimental studies. The instrumental variable design can under certain conditions be used to recover an unbiased estimator of a treatment effect even if unobserved confounding cannot be ruled out with certainty. For continuous outcomes, two stage least squares is the most common instrumental variable estimator used in epidemiologic applications. For a rare binary outcome, an analogous linear-logistic two-stage procedure can be used. Alternatively, a control function approach is sometimes used which entails entering the residual from the first stage linear model as a covariate in a second stage logistic regression of the outcome on the treatment. Both strategies for binary response have previously formally been justified only for continuous exposure, which has impeded widespread use of the approach outside of this setting. In this note, we consider the important setting of binary exposure in the context of a binary outcome. We provide an alternative motivation for the control function approach which is appropriate for binary exposure, thus establishing simple conditions under which the approach may be used for instrumental variable estimation when the outcome is rare. In the proposed approach, the first stage regression involves a logistic model of the exposure conditional on the instrumental variable, and the second stage regression is a logistic regression of the outcome on the exposure adjusting for the first stage residual. In the event of a non-rare outcome, we recommend replacing the second stage logistic model with a risk ratio regression