Nonparametric Estimation of the Generalized Propensity Score Based on the Highly Adaptive Lasso

Abstract

Continuous treatment variables have posed a significant challenge for causal inference, both in the formulation and identification of causal effects and in their robust estimation. Traditionally, focus has been placed on techniques applicable to binary or categorical treatments with few levels, settings allowing the application of propensity score-based methodology with relative ease. Efforts to accommodate continuous treatments introduced the generalized propensity score (the conditional density of treatment given covariates), a nuisance parameter required for the estimation of scientifically informative parameters like the causal dose-response curve and the causal effects of stochastic interventions that shift the value of treatment received (modified treatment policies). Unfortunately, the vast majority of generalized propensity score estimators impose restrictive modeling assumptions, sharply limiting the real-world applicability of classical and doubly robust estimators alike. We present several novel estimators of the generalized propensity score, all based on the highly adaptive lasso, a recently developed nonparametric regression function demonstrated to achieve a convergence rate suitably fast for the formulation of functional parameter estimators with desirable properties, including asymptotic linearity and variance converging to the nonparametric efficiency bound. Using a class of causal effect estimands tailored to modified treatment policies, we demonstrate the construction of nonparametric-efficient inverse probability weighted estimators in which the highly adaptive lasso generalized propensity score estimator is undersmoothed. Through numerical experiments, we compare the relative performance of our efficient inverse probability weighted estimators to variants of doubly robust estimators utilizing either our nonparametric generalized propensity score estimators or adaptations of common but restrictive semiparametric alternatives.