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Efficient Targeted Maximum Likelihood Estimators for Two-Phase Design Problems

Sky Qiu, Susan Gruber, Pamela A. Shaw, Brian D. Williamson, Mark J. van der Laan

Abstract

In a typical two-phase design, a random sample is drawn from the target population in phase 1, during which only a subset of variables is collected. In phase 2, a subsample of the phase-1 cohort is selected, and additional variables are measured. This setting induces a coarsened data structure on the data from the second phase. We assume coarsening at random, that is, the phase-2 sampling mechanism depends only on variables fully observed. We review existing estimators, including the generalized raking estimator and the inverse probability of censoring weighted targeted maximum likelihood estimation (IPCW-TMLE) along with its extensions that also target the phase-2 sampling mechanism to improve efficiency. We further introduce a new class of estimators constructed within the TMLE framework that are asymptotically equivalent.

Efficient Targeted Maximum Likelihood Estimators for Two-Phase Design Problems

Abstract

In a typical two-phase design, a random sample is drawn from the target population in phase 1, during which only a subset of variables is collected. In phase 2, a subsample of the phase-1 cohort is selected, and additional variables are measured. This setting induces a coarsened data structure on the data from the second phase. We assume coarsening at random, that is, the phase-2 sampling mechanism depends only on variables fully observed. We review existing estimators, including the generalized raking estimator and the inverse probability of censoring weighted targeted maximum likelihood estimation (IPCW-TMLE) along with its extensions that also target the phase-2 sampling mechanism to improve efficiency. We further introduce a new class of estimators constructed within the TMLE framework that are asymptotically equivalent.
Paper Structure (31 sections, 3 theorems, 88 equations, 1 figure, 8 tables, 3 algorithms)

This paper contains 31 sections, 3 theorems, 88 equations, 1 figure, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and terminology
  4. The efficient influence curve
  5. Targeted maximum likelihood estimation
  6. The exact remainder term
  7. Review of existing estimators for two-phase sampling designs
  8. Inverse probability of censoring weighted targeted maximum likelihood estimator (IPCW-TMLE)
  9. IPCW-TMLE with targeted phase-2 sampling mechanism
  10. Generalized raking (GR)
  11. Two estimators that rely on a slight rearrangement of the A-IPCW representation of the EIC
  12. Efficient estimating-equation (EEE) estimator
  13. Quasi-TMLE
  14. A TMLE using an alternative representation of the target parameter
  15. Analyzing the exact remainder terms
  16. ...and 16 more sections

Key Result

Lemma 5.1

Define The canonical gradient of the parameter $\Psi:\mathcal{M}\rightarrow\mathbb{R}$ at $P$, where is given by where

Figures (1)

  • Figure 1: Wald-type 95% confidence interval oracle coverage of raking, IPCW-TMLE, and IPCW-TMLE with targeting of $\Pi$. Oracle coverage is defined as the proportion of Monte-Carlo runs (out of 1,000) where the 95% confidence interval computed using the empirical standard error covers the true causal estimand.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 5.1
  • Remark 5
  • Lemma 6.1
  • Lemma 6.2
  • proof
  • proof