Quantifying the average of the time-varying hazard ratio via a class of transformations.

The hazard ratio derived from the Cox model is a commonly used summary statistic to quantify a treatment effect with a time-to-event outcome. The proportional hazards assumption of the Cox model, however, is frequently violated in practice and many alternative models have been proposed in the statistical literature. Unfortunately, the regression coefficients obtained from different models are often not directly comparable. To overcome this problem, we propose a family of weighted hazard ratio measures that are based on the marginal survival curves or marginal hazard functions, and can be estimated using readily available output from various modeling approaches. The proposed transformation family includes the transformations considered by Schemper et al. (Statist Med 28:2473-2489, 2009) as special cases. In addition, we propose a novel estimate of the weighted hazard ratio based on the maximum departure from the null hypothesis within the transformation family, and develop a Kolmogorov[Formula: see text]Smirnov type of test statistic based on this estimate. Simulation studies show that when the hazard functions of two groups either converge or diverge, this new estimate yields a more powerful test than tests based on the individual transformations recommended in Schemper et al. (Statist Med 28:2473-2489, 2009), with a similar magnitude of power loss when the hazards cross. The proposed estimates and test statistics are applied to a colorectal cancer clinical trial.