Nonparametric inference under a monotone hazard ratio order

TL;DR

This paper introduces a nonparametric method for estimating and making inferences about a monotone hazard ratio function in survival analysis, addressing limitations of Cox regression when the proportional hazards assumption is violated.

Contribution

It proposes a new estimator for the monotone hazard ratio function, proves its asymptotic properties, and develops valid confidence intervals, with applications to clinical survival data.

Findings

Estimator converges to a mean-zero limit distribution.

Constructed asymptotically valid confidence intervals.

Numerical studies demonstrate good finite-sample performance.

Abstract

The ratio of the hazard functions of two populations or two strata of a single population plays an important role in time-to-event analysis. Cox regression is commonly used to estimate the hazard ratio under the assumption that it is constant in time, which is known as the proportional hazards assumption. However, this assumption is often violated in practice, and when it is violated, the parameter estimated by Cox regression is difficult to interpret. The hazard ratio can be estimated in a nonparametric manner using smoothing, but smoothing-based estimators are sensitive to the selection of tuning parameters, and it is often difficult to perform valid inference with such estimators. In some cases, it is known that the hazard ratio function is monotone. In this article, we demonstrate that monotonicity of the hazard ratio function defines an invariant stochastic order, and we study the properties of this order. Furthermore, we introduce an estimator of the hazard ratio function under a monotonicity constraint. We demonstrate that our estimator converges in distribution to a mean-zero limit, and we use this result to construct asymptotically valid confidence intervals. Finally, we conduct numerical studies to assess the finite-sample behavior of our estimator, and we use our methods to estimate the hazard ratio of progression-free survival in pulmonary adenocarcinoma patients treated with Gefitinib or carboplatin-paclitaxel.