Reformulating the hazard ratio to enhance communication with clinical investigators

Background: Clinical trials with time to event outcomes are often designed utilizing the Cox [1] proportional hazard model with a hazard ratio parameter Δ.

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 θ. Clinically interesting differences between the treatment arms are easier for researchers to quantify in terms of θ 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 θ is illustrated through a Lymphoma clinical trial example.

Results: The Cox proportional hazard model with parameter θ is shown to be equivalent to the Cox proportional hazard model with a hazard ratio parameter Δ. A table of typical hazard ratios Δ is presented with their equivalent θ values. In the appendix the mathematical derivations are developed and an unbiased estimate of θ 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 θ and the hazard ratio Δ 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 θ 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. Clinical Trials 2008; 5: 248—252. http://ctj.sagepub.com