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Nonparametric regression with dependent censoring or competing risks

Jia-Han Shih, Simon M. S. Lo, Ralf A. Wilke

Abstract

Single-index models or time-to-event models are frequently applied in empirical research. These models are non-identifiable in presence of unknown (dependent) censoring or competing risks and do not give informative results in empirical analysis unless rather strong, non-testable restrictions hold. Little is known, whether the known robustness properties of the single-index model carry over to models with dependent censoring or competing risks. This paper shows that the ratio of partial covariate effects on the margins is identifiable in nonparametric models with unknown dependent censoring or nonparametric competing risks models with nonparametric dependence structure, provided an exclusion restriction holds. Commonly used (semi)parametric models for the margin and independent censoring, such as Cox proportional hazards, accelerated failure time or proportional odds models, can be used to obtain relative covariate effects despite their misspecified censoring mechanism. Several nonparametric estimators for the general model are introduced and their numerical properties are studied.

Nonparametric regression with dependent censoring or competing risks

Abstract

Single-index models or time-to-event models are frequently applied in empirical research. These models are non-identifiable in presence of unknown (dependent) censoring or competing risks and do not give informative results in empirical analysis unless rather strong, non-testable restrictions hold. Little is known, whether the known robustness properties of the single-index model carry over to models with dependent censoring or competing risks. This paper shows that the ratio of partial covariate effects on the margins is identifiable in nonparametric models with unknown dependent censoring or nonparametric competing risks models with nonparametric dependence structure, provided an exclusion restriction holds. Commonly used (semi)parametric models for the margin and independent censoring, such as Cox proportional hazards, accelerated failure time or proportional odds models, can be used to obtain relative covariate effects despite their misspecified censoring mechanism. Several nonparametric estimators for the general model are introduced and their numerical properties are studied.
Paper Structure (15 sections, 6 theorems, 68 equations, 2 figures, 6 tables)

This paper contains 15 sections, 6 theorems, 68 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The model
  3. Identifiability of relative covariate effects
  4. Using the conditional expectation of the overall survival
  5. The exclusion restriction
  6. Examples
  7. Single-index models
  8. Non-single-index model
  9. Estimation
  10. Simulations
  11. Estimation of $\eta_{\pi}$, $\eta_m$ and $\eta$.
  12. Comparison with (semi)parametric single-index models.
  13. Empirical applications
  14. Analysis of unemployment
  15. Analysis of time to death

Key Result

Proposition 1

The relative covariate effects $\eta_\pi ( t,x,y )$ in (proportion) is identifiable.

Figures (2)

  • Figure S1: Simulation results for $N=50,000$, Gumbel with $\tau=0.1$ from 100 runs: true functions $\pi(t \mid \bar{x},\bar{y})$, $\Lambda(t \mid \bar{x},\bar{y})$, $\frac{\partial \pi(t\mid \bar{x},\bar{y})}{\partial z}$ for $z\in\{x,y\}$ and $\frac{\partial \Lambda(t\mid \bar{x},\bar{y})}{\partial z}$ for $z\in\{x,y\}$ in $t$ (solid black lines), mean of their nonparametric estimates (dashed gray lines) and 5'th and 95'th percentiles of the respective distributions (grey dots).
  • Figure S2: Simulation results for $N=50,000$ and $h=0.2$ under the Gumbel copula with $\tau=0.1$ from 100 runs: true $\eta_\pi (t,\bar{x},\bar{y}) = \beta_x/\beta_y = 1$ for all $t$ (solid black line), mean of $\hat{\eta}_\pi(t,\bar{x},\bar{y})$ and $\hat{\eta}_\Lambda(t,\bar{x},\bar{y})$ (dashed grey line) and 5th and 95th percentiles of the distribution of $\hat{\eta}_\pi(t,\bar{x},\bar{y})$ and $\hat{\eta}_\Lambda(t,\bar{x},\bar{y})$.

Theorems & Definitions (14)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Corollary 1
  • Example 1: Cox PH
  • ...and 4 more