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Quantile regression under dependent censoring with unknown association

Myrthe D'Haen, Ingrid Van Keilegom, Anneleen Verhasselt

TL;DR

This work tackles quantile regression under dependent censoring by introducing a fully parametric copula model for $(T,C)|X$ with identifiable association. The survival time $T|X$ is modeled through an Enriched Asymmetric Laplace distribution enriched with Laguerre polynomials, while the censoring time $C|X$ follows a parametric margin; the joint distribution is linked via a copula with $h$-functions to yield the observable likelihood for $(Y,\Delta|X)$. The authors establish identifiability, consistency, and asymptotic normality of the estimators, and develop a practical maximum-likelihood-type estimation procedure that selects Laguerre degrees via AIC and exploits a shift relation to obtain multiple quantiles from a single optimization. Simulations and a real-data liver transplant application show that accounting for dependent censoring materially alters predicted quantiles and improves inference compared to models that ignore dependence. The framework offers a robust, distributional perspective on survival quantiles under complex censoring and opens avenues for extensions to more flexible margins and copula structures with covariate-dependent associations.

Abstract

The study of survival data often requires taking proper care of the censoring mechanism that prohibits complete observation of the data. Under right censoring, only the first occurring event is observed: either the event of interest, or a competing event like withdrawal of a subject from the study. The corresponding identifiability difficulties led many authors to imposing (conditional) independence or a fully known dependence between survival and censoring times, both of which are not always realistic. However, recent results in survival literature showed that parametric copula models allow identification of all model parameters, including the association parameter, under appropriately chosen marginal distributions. The present paper is the first one to apply such models in a quantile regression context, hence benefiting from its well-known advantages in terms of e.g. robustness and richer inference results. The parametric copula is supplemented with a likewise parametric, yet flexible, enriched asymmetric Laplace distribution for the survival times conditional on the covariates. Its asymmetric Laplace basis provides its close connection to quantiles, while the extension with Laguerre orthogonal polynomials ensures sufficient flexibility for increasing polynomial degrees. The distributional flavour of the quantile regression presented, comes with advantages of both theoretical and computational nature. All model parameters are proven to be identifiable, consistent, and asymptotically normal. Finally, performance of the model and of the proposed estimation procedure is assessed through extensive simulation studies as well as an application on liver transplant data.

Quantile regression under dependent censoring with unknown association

TL;DR

This work tackles quantile regression under dependent censoring by introducing a fully parametric copula model for with identifiable association. The survival time is modeled through an Enriched Asymmetric Laplace distribution enriched with Laguerre polynomials, while the censoring time follows a parametric margin; the joint distribution is linked via a copula with -functions to yield the observable likelihood for . The authors establish identifiability, consistency, and asymptotic normality of the estimators, and develop a practical maximum-likelihood-type estimation procedure that selects Laguerre degrees via AIC and exploits a shift relation to obtain multiple quantiles from a single optimization. Simulations and a real-data liver transplant application show that accounting for dependent censoring materially alters predicted quantiles and improves inference compared to models that ignore dependence. The framework offers a robust, distributional perspective on survival quantiles under complex censoring and opens avenues for extensions to more flexible margins and copula structures with covariate-dependent associations.

Abstract

The study of survival data often requires taking proper care of the censoring mechanism that prohibits complete observation of the data. Under right censoring, only the first occurring event is observed: either the event of interest, or a competing event like withdrawal of a subject from the study. The corresponding identifiability difficulties led many authors to imposing (conditional) independence or a fully known dependence between survival and censoring times, both of which are not always realistic. However, recent results in survival literature showed that parametric copula models allow identification of all model parameters, including the association parameter, under appropriately chosen marginal distributions. The present paper is the first one to apply such models in a quantile regression context, hence benefiting from its well-known advantages in terms of e.g. robustness and richer inference results. The parametric copula is supplemented with a likewise parametric, yet flexible, enriched asymmetric Laplace distribution for the survival times conditional on the covariates. Its asymmetric Laplace basis provides its close connection to quantiles, while the extension with Laguerre orthogonal polynomials ensures sufficient flexibility for increasing polynomial degrees. The distributional flavour of the quantile regression presented, comes with advantages of both theoretical and computational nature. All model parameters are proven to be identifiable, consistent, and asymptotically normal. Finally, performance of the model and of the proposed estimation procedure is assessed through extensive simulation studies as well as an application on liver transplant data.
Paper Structure (39 sections, 10 theorems, 50 equations, 2 figures, 13 tables)

This paper contains 39 sections, 10 theorems, 50 equations, 2 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Related literature
  3. Copula models for dependent censoring
  4. Quantile regression under censoring
  5. The model and notation
  6. Model formulation
  7. Parameter status of the linear quantile level $\lambda$
  8. Identifiability of the regression coefficient $\beta$: intuition
  9. Implications: the parameter status of $\lambda$
  10. Implications: (numerical) multiple quantile modelling
  11. Identifiability
  12. Assumptions and identifiability statements
  13. Discussion on the assumptions
  14. Asymptotics
  15. Assumptions and notation
  16. ...and 24 more sections

Key Result

Theorem 4.1

Suppose that assumptions ass:idf:posdef-ass:idf:S2 are satisfied and suppose that two candidate sets of parameters $\pi_1 = (\theta_1, \theta_{T_1}, \theta_{C_1})$ and $\pi_2 = (\theta_2, \theta_{T_2}, \theta_{C_2})$ yield the same contribution $\ell(y,x,\delta; \pi_1) = \ell(y,x,\delta; \pi_2)$ to

Figures (2)

  • Figure 1: Shape of the quantile curves depending on the form of $\sigma(X; \gamma)$, for Case 1 with $\gamma_1 = 0$, i.e. homoscedasticity (first panel), for $\gamma_0 = 0$ (second), $\gamma_0 \neq 0$ and $\gamma_1 \neq 0$ (third); and the nonlinear-$\sigma(\cdot)$ Case 2 (fourth panel).
  • Figure 2: Time to death (in days) while waiting for a liver transplant ($T$) or censoring due to receiving a transplant ($C$), versus the UKELD score. With a logarithmic scale for $T$, Figure (b) on the right shows a rather homoscedastic scenario and a linear trend can be observed.

Theorems & Definitions (16)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.1: Generalisation of \ref{['ass:idf:S1']}
  • Remark 4.2: Generalisation of \ref{['ass:idf:LC']}: dependence on $X$
  • Theorem 5.1
  • Theorem 5.2
  • Corollary 5.1
  • ...and 6 more