August 15, 2024

In clinical trials that use time to event endpoints, a traditional measure has been using the hazard ratio derived from a Cox proportional hazard regression, but one must satisfy this assumption. Over time, measures that have relaxed this assumption have been developed. One of them is the average hazard ratio (AHR) where its core idea is to utilize a time-dependent weighting function that accounts for time variation. However, though it has been published in methodological research papers, the AHR has been rarely used in practice. To facilitate its application, the authors have published their approaches for sample size calculation of an AHR test.

One of the alternate measures is the restricted mean survival time (RMST) which describes the average event-free survival up to a specific time point (Royston and Parmar, 2013). The AHR has shown to stay interpretable under non-proportional hazard ratios. The RMST has had a sample size calculation package made in R, SSRMST (Horiguchi and Uno, 2017; Uno et al, 2015). The R package, coxphw, does have an implementation of AHR and does have a sample size calculation based on Schoenfeld’s formula for proportional hazards. The authors have instead provided guidance where AHR is the primary endpoint in a clinical trial. They designed a very simplistic sample size calculation based on assuming asymptotic normality of the AHR and a variance which they say was based on medical knowledge and the literature. It was obviously motivated as well by the Schoenfeld calculation.

In simulations they compared their test to the log-rank test, Schoenfeld test, asymptotic AHR, and simulation based AHR and found their new estimation for AHR appeared to be more robust to misspecifications and they also tested this in a real dataset analysis. They did admit in their discussion that they focused on one type of weight which could have had limitations. Their method still has seemed to work better under the assumption of non-proportional hazards than the usual log-rank test or Schoenfeld test.

Written by,

Usha Govindarajulu, PhD

**Keywords: **survival analysis, RMST, average hazard, Cox model, hazard ratio, sample size

**References**

Dormuth I, Pauly M, Rauch G, and Herrmann C (2024) Sample Size Calculation under Nonproportional Hazards Using Average Hazard Ratios. Biometrical Journal. https://doi.org/10.1002/bimj.202300271

Horiguchi, M., and H. Uno. 2017. *SSRMST: Sample Size Calculation Using Restricted Mean Survival Time*. R package version 0.1.1.

Royston, P., and M. K. Parmar. 2013. “Restricted Mean Survival Time: An Alternative to the Hazard Ratio for the Design and Analysis of Randomized Trials With a Time-to-Event Outcome.” *BMC Medical Research Methodology* 13, no. 1: 1–15.

Uno, H., J. Wittes, H. Fu, et al. 2015. “Alternatives to Hazard Ratios for Comparing the Efficacy or Safety of Therapies in Noninferiority Studies.” *Annals of Internal Medicine* 163, no. 2: 127–134.