Reformulating the hazard ratio to enhance communication with clinical investigators

Abstract

Background: Clinical trials with time to event outcomes are often designed utilizing the Cox [1] proportional hazard model with a hazard ratio parameter Delta. Purpose: The purpose of this article is to demonstrate that a Cox proportional hazard model with a hazard ratio parameter is equivalent to a Cox proportional hazard model with a parameter equal to the probability that a patient given one treatment will have an event earlier than if the same patient were given a different treatment. This probability will subsequently be referred to as theta. Clinically interesting differences between the treatment arms are easier for researchers to quantify in terms of in situations where they have a difficult time with the hazard ratio, allowing better communication between the statistician and the researcher. Methods: The problem and its solution are demonstrated mathematically. The utility of the Cox proportional hazard model in terms of theta is illustrated through a Lymphoma clinical trial example. Results: The Cox proportional hazard model with parameter theta is shown to be equivalent to the Cox proportional hazard model with a hazard ratio parameter Delta. A table of typical hazard ratios Delta is presented with their equivalent theta values. In the appendix the mathematical derivations are developed and an unbiased estimate of theta is provided using Gehan's[2] generalization of the Wilcoxon statistic. Limitations: The equivalence of the Cox proportional hazard model in terms of the probability theta and the hazard ratio Delta is established only for continuous failure times with a single binary covariate. Conditions under which approximate equivalence holds with multiple covariates are discussed in the Appendix. Conclusions: The probability theta provides a natural parameterization for the Cox proportional hazard model, affords a tool to conceptualize treatment differences, and provides a method to improve communication between statisticians and researchers.