### Department of Biostatistics Seminar/Workshop Series

# The estimation of average hazard ratios by Mantel-Haenszel estimators and by Cox regression

## Michael Schemper, PhD

### Professor of Clinical Biostatistics, Center for Medical Statistics, Informatics and Intelligent Systems, Medical University of Vienna, Austria

### Wednesday, April 6, 1:30-2:30pm, MRBIII Conference Room 1220

In the survival setting hazard ratios can be expected to change over time, either randomly or systematically. Therefore, the concept of an average hazard ratio (AHR) as introduced by Kalbfleisch and Prentice (Biometrika, 1981) for the two-sample case is of interest. In practice, two definitions of an AHR are of interest: AHRMH takes all times as equally important while AHROC weights hazard ratios at different times proportional to the numbers of individuals at risk. AHROC can alternatively be defined by P(T0<T1) / P(T0<T1), the odds of concordance, T0 and T1 denoting the survival times of two groups. It will be shown that under no censoring (or under censoring and proportional hazards) the standard and a weighted Mantel-Haenszel (MH) estimator estimate the underlying AHRMH and AHROC, respectively. Under censoring and non-proportional hazards an additional inverse probability of censoring weight (IPCW) is required to estimate the underlying AHRs. While the logrank test is the test corresponding to the standard MH-estimator, weighted logrank tests are available for weighted MH- estimators, extending the Tarone-Ware scheme of non-parametric two sample tests of survival outcomes. Empirical properties of the suggested tests and estimators are demonstrated by a Monte Carlo study. The differential role of the weighting functions considered is illustrated by a comparative analysis of four real data sets.

The concept of estimating an AHR has been extended to multiple, possibly continuous covariates within the framework of Cox regression (CR) by Schemper, Wakounig and Heinze (Statist. Med, 2009). The required modifications of the estimating equations for CR will be shown. Often the effect of at least one of the prognostic factors included in a CR changes over time, which violates the proportional hazards assumption of this model. As a consequence, the underlying AHR 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 (WCR) is a parsimonious alternative without additional parameters. The possible advantages of WCR over CR are demonstrated by means of a Monte Carlo study of efficiency and bias, and by comparative analyses of a big multicenter lung cancer study. Our empirical investigations permit us to recommend WCR, which is implemented in a SAS macro and in an R package available at: www.muw.ac.at/msi/biometrie/programs

This presentation is based on joint work by the author, by Georg Heinze, and by Samo Wakounig, who was supported by grant P18553-N13 of the Austrian Science Fund.