[Paper Review] Further details on inference under right censoring for transformation models with a change-point based on a covariate threshold

Michael R. Kosorok, Rui Song|ArXiv.org|Apr 3, 2006

Statistical Methods and Inference32 references3 citations

TL;DR

This paper develops inference methods for transformation models with a change-point in regression coefficients at a covariate threshold under right censoring. It establishes n-consistency for the change-point and root-n consistency for other parameters, proves adaptivity, and introduces valid score tests via Monte Carlo methods despite non-identifiability under the null hypothesis.

ABSTRACT

We consider linear transformation models applied to right censored survival data with a change-point based on a covariate threshold. We establish consistency and weak convergence of the nonparametric maximum lieklihood estimators. The change-point parameter is shown to be $n$-consistent, while the remaining parameters are shown to have the expected root-$n$ consistency. We show that the procedure is adaptive in the sense that the non-threshold parameters are estimable with the same precision as if the true threshold value were known. We also develop Monte-Carlo methods of inference for model parameters and score tests for the existence of a change-point. A key difficulty here is that some of the model parameters are not identifiable under the null hypothesis of no change-point. Simulation students establish the validity of the proposed score tests for finite sample sizes.

Motivation & Objective

• To develop statistical inference for transformation models with a change-point in regression coefficients based on a covariate threshold under right-censored data.
• To address the challenge of non-identifiability of parameters under the null hypothesis of no change-point.
• To establish the asymptotic properties of nonparametric maximum likelihood estimators in this setting.
• To develop valid score tests for detecting the presence of a change-point using Monte Carlo methods.

Proposed method

• Uses nonparametric maximum likelihood estimation (NPMLE) to estimate model parameters, including the change-point, under right-censoring.
• Establishes weak convergence and consistency of the NPMLE, showing n-consistency for the change-point and root-n consistency for other parameters.
• Applies Monte Carlo methods to approximate the sampling distribution of test statistics due to non-identifiability under the null.
• Derives a score test for the existence of a change-point, accounting for the irregularity in parameter space under the null.
• Demonstrates that the non-threshold parameters are estimable with the same precision as if the true threshold were known, establishing adaptivity.
• Uses simulation studies to validate the performance of the score tests in finite samples.

Experimental results

Research questions

• RQ1 What are the asymptotic properties of the nonparametric maximum likelihood estimators in transformation models with a covariate threshold under right censoring?
• RQ2 How does the estimation precision of non-threshold parameters compare when the true threshold is unknown versus known?
• RQ3 Can valid inference be conducted for the change-point parameter despite non-identifiability under the null hypothesis?
• RQ4 What is the finite-sample performance of the proposed score test for detecting a change-point?
• RQ5 How can Monte Carlo methods be effectively used to approximate the sampling distribution of test statistics in this complex model?

Key findings

• The change-point parameter is n-consistent, while other regression parameters are root-n consistent, under regularity conditions.
• The model is adaptive: non-threshold parameters achieve the same estimation precision as if the true threshold were known.
• The non-identifiability of parameters under the null hypothesis of no change-point is handled via Monte Carlo approximation of the null distribution.
• Score tests for the existence of a change-point are valid and well-calibrated in finite samples, as confirmed by simulation studies.
• Weak convergence of the NPMLE is established, supporting asymptotic inference for the model parameters.
• The proposed inference framework is robust and applicable to right-censored survival data with a threshold-based structural change in regression coefficients.