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Targeted Learning on Variable Importance Measure for Heterogeneous Treatment Effect

Haodong Li, Alan E Hubbard, Oliver J Hines, Andrea M Storås, Kajsa Kvist, Mark van der Laan

TL;DR

This article proposes a new targeted maximum likelihood estimator (TMLE) for a treatment effect variable importance measure, in the form of the difference of the variances of conditional average treatment effect, and applies it to data from a randomized clinical trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults.

Abstract

Quantifying the heterogeneity of treatment effect is important for understanding how a commercial product or medical treatment affects different population subgroups. While much of treatment effect heterogeneity analysis focuses on the conditional average treatment effect, an alternative parameter that captures treatment effect heterogeneity is the variance of treatment effect across different covariate groups. One can also derive variable importance parameters that measure (and rank) how much of treatment effect heterogeneity is explained by a targeted subset of covariates. In this article, we propose a new targeted maximum likelihood estimator (TMLE) for a treatment effect variable importance measure, in the form of the difference of the variances of conditional average treatment effect. This TMLE is a pure plug-in estimator that consists of two steps: 1) the initial estimation of relevant components to plug in and 2) an iterative updating step to optimize the bias-variance tradeoff. Simulation results show that the proposed estimator has competitive performance in terms of lower bias and better confidence interval coverage compared to a simple substitution estimator and an estimating equation estimator. We apply these methods to data from a randomized clinical trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults. We find that the estimating equation estimator and the proposed TMLE suggest similar rankings of variable importance. The application of this method also demonstrates the advantage of a plug-in estimator, which always respects the global constraints on the data distribution and that the estimand is a particular function of the distribution.

Targeted Learning on Variable Importance Measure for Heterogeneous Treatment Effect

TL;DR

This article proposes a new targeted maximum likelihood estimator (TMLE) for a treatment effect variable importance measure, in the form of the difference of the variances of conditional average treatment effect, and applies it to data from a randomized clinical trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults.

Abstract

Quantifying the heterogeneity of treatment effect is important for understanding how a commercial product or medical treatment affects different population subgroups. While much of treatment effect heterogeneity analysis focuses on the conditional average treatment effect, an alternative parameter that captures treatment effect heterogeneity is the variance of treatment effect across different covariate groups. One can also derive variable importance parameters that measure (and rank) how much of treatment effect heterogeneity is explained by a targeted subset of covariates. In this article, we propose a new targeted maximum likelihood estimator (TMLE) for a treatment effect variable importance measure, in the form of the difference of the variances of conditional average treatment effect. This TMLE is a pure plug-in estimator that consists of two steps: 1) the initial estimation of relevant components to plug in and 2) an iterative updating step to optimize the bias-variance tradeoff. Simulation results show that the proposed estimator has competitive performance in terms of lower bias and better confidence interval coverage compared to a simple substitution estimator and an estimating equation estimator. We apply these methods to data from a randomized clinical trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults. We find that the estimating equation estimator and the proposed TMLE suggest similar rankings of variable importance. The application of this method also demonstrates the advantage of a plug-in estimator, which always respects the global constraints on the data distribution and that the estimand is a particular function of the distribution.
Paper Structure (29 sections, 1 theorem, 21 equations, 8 figures, 1 algorithm)

This paper contains 29 sections, 1 theorem, 21 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Estimation Problem
  3. Data and Model
  4. Target Causal Parameters
  5. Identification Assumptions
  6. Target Statistical Parameters
  7. Variance of Treatment Effect
  8. Treatment Effect Variable Importance Measure
  9. Scaled Treatment Effect Variable Importance Measure
  10. TMLE for Treatment Effect VIM
  11. Overview of TMLE
  12. Initial Estimation of $\Psi_2(\tilde{Q}_0)$
  13. Machine Learning Algorithms
  14. CATE Metalearners
  15. Estimation of the Nuisance Functions
  16. ...and 14 more sections

Key Result

Theorem 1

Consider $O \sim{\mathbb{P}_0} \in \mathcal{M}$. Let $\Psi_2(\tilde{Q}_n^*): \mathcal{M} \rightarrow \mathbb{R}$ be defined by $\Psi_2(\tilde{Q}_n^*) \equiv \frac{1}{n}\sum_{i=1}^n [\gamma_{s,n}^*(w_i) - (\tau_{s,n}^*(w_i))^2]$. Let $D^*_{\Psi_2, \mathbb{P}_0}$ and $D^*_{\Psi_2, \mathbb{P}_n^*}$ be

Figures (8)

  • Figure 1: Main performance metrics of the estimators of the VTE parameter (Squares circles and crosses represent EE, SS and TMLE respectively. For the two metrics in the first row, the lower the value, the better. For the two coverage metrics on the second row, the closer to the 0.95 reference line, the better.)
  • Figure 2: Main performance metrics of the estimators of the Unscaled VIM parameter
  • Figure 3: Main performance metrics of the estimators of the Scaled VIM parameter
  • Figure 4: EE (left) and TMLE (right) VIMa results from the ACTG175 study with S-learner
  • Figure 5: Main performance metrics of the estimators of the VTE parameter (DR-learner)
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • proof