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Clinical Trials, Vol. 5, No. 3, 248-252 (2008)
DOI: 10.1177/1740774508091452

Reformulating the hazard ratio to enhance communication with clinical investigators

Barry K Moser

Cancer and Leukemia Group B Statistical Center, Duke University Medical Center, Cancer Center Biostatistics and Information Systems, Durham, NC, 27710 USA, moser004{at}mc.duke.edu

Melinda H McCann

Department of Statistics, Oklahoma State University, Stillwater, OK 74074, USA

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 {theta} 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. Clinical Trials 2008; 5: 248—252. http://ctj.sagepub.com


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