Open Access
2023 Nonparametric inference under a monotone hazard ratio order
Yujian Wu, Ted Westling
Author Affiliations +
Electron. J. Statist. 17(2): 3181-3225 (2023). DOI: 10.1214/23-EJS2173


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.

Funding Statement

The authors gratefully acknowledge support from the University of Massachusetts Amherst Department of Mathematics and Statistics startup fund (TW) and NSF Award 2113171 (TW).


The authors are grateful for thoughtful feedback from Anna Liu, John Staudenmayer, and Marco Carone.


Download Citation

Yujian Wu. Ted Westling. "Nonparametric inference under a monotone hazard ratio order." Electron. J. Statist. 17 (2) 3181 - 3225, 2023.


Received: 1 August 2022; Published: 2023
First available in Project Euclid: 20 November 2023

Digital Object Identifier: 10.1214/23-EJS2173

Primary: 62Gxx , 62Nxx
Secondary: 62Mxx

Keywords: hazard ratio , nonparametric inference , Shape constrained estimation , Stochastic orders , Survival analysis , time-varying treatment

Vol.17 • No. 2 • 2023
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