The estimation of average hazard ratios by weighted Cox regression

Stat Med. 2009 Aug 30;28(19):2473-89. doi: 10.1002/sim.3623.


Often the effect of at least one of the prognostic factors in a Cox regression model changes over time, which violates the proportional hazards assumption of this model. As a consequence, the average hazard ratio for such a prognostic factor is under- or overestimated. While there are several methods to appropriately cope with non-proportional hazards, in particular by including parameters for time-dependent effects, weighted estimation in Cox regression is a parsimonious alternative without additional parameters. The methodology, which extends the weighted k-sample logrank tests of the Tarone-Ware scheme to models with multiple, binary and continuous covariates, has been introduced in the nineties of the last century and is further developed and re-evaluated in this contribution. The notion of an average hazard ratio is defined and its connection to the effect size measure P(X<Y) is emphasized. The suggested approach accomplishes estimation of intuitively interpretable average hazard ratios and provides tools for inference. A Monte Carlo study confirms the satisfactory performance. Advantages of the approach are exemplified by comparing standard and weighted analyses of an international lung cancer study. SAS and R programs facilitate application.

Publication types

  • Research Support, Non-U.S. Gov't

MeSH terms

  • Biometry
  • Humans
  • Kaplan-Meier Estimate
  • Lung Neoplasms / mortality
  • Lung Neoplasms / therapy
  • Monte Carlo Method
  • Prognosis
  • Proportional Hazards Models*
  • Survival Analysis