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### Cox analysis of survival data with non-proportional hazard functions

### Cox analysis of survival data with non-proportional hazard functions

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Schemper, M. (1992): *The Statistician* 41, 455 - 465

The consequences of violated assumptions for Cox's proportional hazards model are discussed and current options to deal with non-proportionality in Cox's model are reviewed. An additional option for analysis is suggested, which produces weighted estimates of log hazard ratios, weighted at the time points where failures occur. The procedure amounts to generalizations of the tests by Breslow or Prentice for multiple covariates in the same manner that the proportional hazards model is a generalization of the log rank test by Mantel. Its advantages are representative estimates of average hazard ratios also for covariates with non-proportional and, in particular, converging hazard functions. The latter are often encountered in clinical applications. By means of an empirical study these average hazard ratios are shown to be very close to exact calculations of average hazard ratios as defined by Kalbfleisch and Prentice. Two examples illustrate the advantages of the weighted estimation and of other strategies for analysis with the Cox model in the presence of non-proportional hazards. Furthermore, with respect to checking proportionality, it is demonstrated how misleading the frequently used log-minus-log plots can be and that the lesser known Arjas plots seem to perform quite well.

Schemper, M. (1992): *The Statistician* 41, 455 - 465

The consequences of violated assumptions for Cox's proportional hazards model are discussed and current options to deal with non-proportionality in Cox's model are reviewed. An additional option for analysis is suggested, which produces weighted estimates of log hazard ratios, weighted at the time points where failures occur. The procedure amounts to generalizations of the tests by Breslow or Prentice for multiple covariates in the same manner that the proportional hazards model is a generalization of the log rank test by Mantel. Its advantages are representative estimates of average hazard ratios also for covariates with non-proportional and, in particular, converging hazard functions. The latter are often encountered in clinical applications. By means of an empirical study these average hazard ratios are shown to be very close to exact calculations of average hazard ratios as defined by Kalbfleisch and Prentice. Two examples illustrate the advantages of the weighted estimation and of other strategies for analysis with the Cox model in the presence of non-proportional hazards. Furthermore, with respect to checking proportionality, it is demonstrated how misleading the frequently used log-minus-log plots can be and that the lesser known Arjas plots seem to perform quite well.

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