Non-parametric Methods for Doubly Robust Estimation of Continuous Treatment Effects

Abstract

Summary Continuous treatments (e.g., doses) arise often in practice, but many available causal effect estimators are limited by either requiring parametric models for the effect curve, or by not allowing doubly robust covariate adjustment. We develop a novel kernel smoothing approach that requires only mild smoothness assumptions on the effect curve, and still allows for misspecification of either the treatment density or outcome regression. We derive asymptotic properties and give a procedure for data-driven bandwidth selection. The methods are illustrated via simulation and in a study of the effect of nurse staffing on hospital readmissions penalties.

Doubly Robust Additive Hazards Models to Estimate Effects of a Continuous Exposure on Survival

“…Briefly, we fit a linear regression (propensity score model) on X ij against a function of covariates V ij which explains the variation in the exposure. Assuming a normal distribution for the residuals N (0, σ 2 ) (this assumption can be relaxed using kernel density estimation 23 ), we calculated the conditional density of being exposed to the observed exposure $\widehat{f}(x_{ij}\mid {\mathrm{bold-italicv}}_{\mathrm{bold-italicij}};\widehat{\mathrm{bold\alpha }})=\frac{1}{\sqrt{2\pi {\widehat{\sigma }}^2}}exp(-\frac{{\stackrel{true^}{{\in }_{ij}}}^2}{2{\hat{\sigma }}^2})$, where $\widehat{\mathrm{bold-italic\alpha }}=\left({\hat{\sigma }}^2,\stackrel{true^}{{\in }_{ij}}\right)$ is the maximum likelihood estimators for the residual variance σ 2 and the residual of exposure for individual i in year j . Similarly, assuming a marginally normal distribution of the exposure (this assumption can be relaxed by marginalizing the kernel density estimation for the conditional distribution 23 ), we calculated the marginal density of the exposure using $\widehat{f}\left(x_{ij}\right)=\frac{1}{\sqrt{2\pi {{\hat{\sigma }}^{\prime }}^2}}exp(-\frac{{false(x_{ij}-\hat{\mu }false)}^2}{2{{\widehat{\sigma }}^{\prime }}^2})$, where μ̂ and σ̂ ′ 2 are the maximum likelihood estimators for the marginal mean and variance of the exposure.…”

Robust Causal Inference with Continuous Instruments Using the Local Instrumental Variable Curve

“…In particular, our cross‐validation approach could be used to select between working models whose complexity increases with the sample size, yielding doubly robust yet non‐parametric estimators of the LIV curve, in the same spirit as Robins and Rotnitzky () and Kennedy et al . ().…”

“…To estimate the instrument density π , we used a model that was previously used by Kennedy et al . (), in which the density depends on covariates through only the mean and variance functions but is otherwise flexible. Specifically this model assumes that Z = π 1 ( X )+ π 2 ( X ) ε , where ε satisfies and , the density f ε of ε is unspecified but smooth and ( π 1 , π 2 ) follow generalized additive models with identity and log‐links respectively.…”

“…In this section we derive the efficient influence function for the risk of a given candidate estimator and show how it can be used as a doubly robust loss function in the cross-validation framework that was developed by van der Laan and Dudoit (2003). In particular, our cross-validation approach could be used to select between working models whose complexity increases with the sample size, yielding doubly robust yet non-parametric estimators of the LIV curve, in the same spirit as Robins and Rotnitzky (2001) and Kennedy et al (2017).…”

Semiparametric Theory and Empirical Processes in Causal Inference

“…As mentioned above, Robins et al [43,45,67] considered semiparametric minimax estimation in settings where the parameter of interest is Euclidean, but root-n rates of convergence cannot be attained due to high-dimensional covariates. Estimation of functional effect parameters was considered by [12,21] in the context of continuous treatment effects; in such settings the target parameter is a non-pathwise differentiable curve, and root-n rates of convergence are again not possible. Inference for a non-regular parameter in an optimal treatment regime setting was considered by [23]; in this case nonregularity does not preclude the existence of root-n rate inference.…”