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One-step TMLE for weighted average treatment effects

Yang Liu, Patrick Lopatto, Ivana Malenica

Abstract

We consider Targeted Maximum Likelihood Estimation (TMLE) of weighted average treatment effects (WATEs), a class of causal estimands that reweight the covariate distribution using a specified function of the propensity score. This class includes the average treatment effect and average treatment effect on the treated, as well as various overlap-based targets. We provide a comprehensive analysis of the one-step TMLE along the universal least favorable path for such parameters. Under explicit regularity conditions on the weight function and initialization, we show that the targeting procedure is well-defined, reaches a solution of the estimating equation in finite time, and yields an asymptotically efficient estimator. In particular, convergence of the targeting dynamics and control of the second-order remainder are derived from these conditions rather than imposed as separate assumptions on the output of the algorithm.

One-step TMLE for weighted average treatment effects

Abstract

We consider Targeted Maximum Likelihood Estimation (TMLE) of weighted average treatment effects (WATEs), a class of causal estimands that reweight the covariate distribution using a specified function of the propensity score. This class includes the average treatment effect and average treatment effect on the treated, as well as various overlap-based targets. We provide a comprehensive analysis of the one-step TMLE along the universal least favorable path for such parameters. Under explicit regularity conditions on the weight function and initialization, we show that the targeting procedure is well-defined, reaches a solution of the estimating equation in finite time, and yields an asymptotically efficient estimator. In particular, convergence of the targeting dynamics and control of the second-order remainder are derived from these conditions rather than imposed as separate assumptions on the output of the algorithm.

Paper Structure

This paper contains 25 sections, 31 theorems, 716 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Statistical Model
  4. Full Model.
  5. Fixed-Marginal Model.
  6. Weighted Average Treatment Effect
  7. Efficient Influence Functions
  8. Main Results
  9. Notation
  10. Assumptions
  11. Results
  12. Spline Estimation
  13. Background on TMLE
  14. Semiparametric Preliminaries
  15. Tangent Space Identifications
  16. ...and 10 more sections

Key Result

Proposition 2.4

Let $P\in\mathcal{M}_{\mathrm{full}}$ be such that $0<e(X)<1$ almost surely and $\Omega(P)>0$, and assume that $\lambda$ is differentiable on the essential range of $e(X)$. Define $\varphi$ as in e:aipw. The EIF of $\psi$ at $P$ relative to $\mathcal{M}_{\mathrm{full}}$ is where $\Omega(P)=\mathbb{E}[\lambda(e(X))]$. Moreover, $D_{\mathrm{full}}^* = D_q^* + D_e^* + D_X^*$, with The displayed for

Theorems & Definitions (66)

  • Remark 2.1
  • Remark 2.3
  • Proposition 2.4: Full-model EIF
  • Remark 2.5
  • Proposition 2.6: Restricted-model EIF
  • Lemma 2.7
  • Definition 3.6
  • Theorem 3.8
  • Theorem 3.9
  • Theorem 3.10
  • ...and 56 more