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Doubly robust inference with censoring unbiased transformations

Oliver Lunding Sandqvist

Abstract

This paper extends doubly robust censoring unbiased transformations to a broad class of censored data structures under the assumption of coarsening at random and positivity. This includes the classic survival and competing risks setting, but also encompasses multiple events. A doubly robust representation for the conditional bias of the transformed data is derived. This leads to rate double robustness and oracle efficiency properties for estimating conditional expectations when combined with cross-fitting and linear smoothers. Simulation studies demonstrate favourable performance of the proposed method relative to existing approaches. An application of the methods to a regression discontinuity design with censored data illustrates its practical utility.

Doubly robust inference with censoring unbiased transformations

Abstract

This paper extends doubly robust censoring unbiased transformations to a broad class of censored data structures under the assumption of coarsening at random and positivity. This includes the classic survival and competing risks setting, but also encompasses multiple events. A doubly robust representation for the conditional bias of the transformed data is derived. This leads to rate double robustness and oracle efficiency properties for estimating conditional expectations when combined with cross-fitting and linear smoothers. Simulation studies demonstrate favourable performance of the proposed method relative to existing approaches. An application of the methods to a regression discontinuity design with censored data illustrates its practical utility.

Paper Structure

This paper contains 15 sections, 4 theorems, 27 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Doubly robust censoring unbiased transformation
  3. Transformation
  4. Asymptotics and inference
  5. Sample splitting estimator
  6. Cross-fitted estimator
  7. Simulations
  8. Data-generating process
  9. Estimators
  10. Results
  11. Data application
  12. Background
  13. Model and results
  14. Competing interests
  15. Acknowledgements

Key Result

Theorem 1

(Doubly robust censoring unbiased transformation.) Let $\mathbb{P}_1$ and $\mathbb{P}_2$ be two probability measures, which may be thought of as candidate measures for $\mathbb{P}$. Let When $\mathbb{P}$ and $\mathbb{P}_1$ satisfy Assumption assumption:CAR and assumption:positivity it holds that Furthermore, it holds that $\mathrm{Var}[Y^\ast_{\mathbb{P},\mathbb{P}}(C,X^C)\mid W] \geq \mathrm{Va

Figures (5)

  • Figure 1: The irreversible illness-death model for the process $Z$. Transitions from state $j$ to state $k$ has the transition hazard $\mu_{jk}$.
  • Figure 2: Left Panel: Fitted HAL estimates and actual hazards at specific input values indicated at the top right corner for a single simulation. Right Panel: Estimators and true value of $\mathbb{E}[Y(X) \mid W]$ as a function of $W$ for a single simulation.
  • Figure 3: Violin plot of the $L^2([-4,4],\lambda)$ error for different estimators and values of $n$ with Mean $\pm$ Standard deviation indicated as a point range using 500 simulations.
  • Figure 4: Left Panel: Histogram of estimates at the point $W=-1$ for the doubly robust pseudo-outcomes with censoring estimated by HAL and a misspecified parametric family. The Gaussian approximation is obtained from the oracle pseudo-values and the dashed line is the true value of the estimand. Based on 500 simulations of size $n=30\,000$. Right Panel: The empirical coverages of the $99\%$, $95\%$, and $90\%$ confidence intervals using a Gaussian approximation with standard errors obtained from lprobust using HAL-estimated and oracle doubly robust pseudo-outcomes. Nominal values are shown with dashed lines.
  • Figure 5: Left Panel: Binned-means over the pseudo-outcomes for $Y$ and EMA thresholds for income in Wave 3. Middle Panel: Binned-means of the pseudo-outcomes for $Y$, estimate of the conditional mean, 95% confidence intervals, and EMA thresholds for Wave 3 incomes around the threshold for receiving high EMA. Right Panel: The same as the middle panel with pseudo-outcomes for $A$.

Theorems & Definitions (17)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 1
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 1
  • Remark 6
  • Remark 7
  • ...and 7 more