To the Editor

Abstract

In their article, Barraclough et al.1Barraclough H Simms L Govindan R What a clinician ought to know: Hazard ratios.J Thorac Oncol. 2011; : 978-982Abstract Full Text Full Text PDF PubMed Scopus (43) Google Scholar provide important insights into the interpretation of hazard ratio (HR) estimates from Cox models and Kaplan–Meier curves from clinical trials. Unfortunately, the authors provide a particular interpretation of the HR, which can distort both statistical and clinical interpretations. Specifically, the authors interpret (see ref.,1Barraclough H Simms L Govindan R What a clinician ought to know: Hazard ratios.J Thorac Oncol. 2011; : 978-982Abstract Full Text Full Text PDF PubMed Scopus (43) Google Scholar p. 981, Box 2) a 0.75 HR for overall survival (OS), comparing treatment E (experimental) versus C (control), as either (1) a 25% lower risk of death (via 100 × (1 − HR)%, denoted 1 − HR), or (2) a 33% increase in the survival time (via 100 × (1/HR − 1)%, denoted 1/HR − 1). The interpretation provided by (1) is appropriate and is standard throughout the statistical and medical literature. The interpretation provided by (2), however, is not sound and can cause considerable miscommunication of study results. Statement (2) suggests that treatment E extends the survival times of patients after treatment E by 33% compared with the survival times of patients after treatment C. For example, 1-year OS under treatment C is extended to 1.33 years under treatment E, 2-year OS extended to 2.66 years, and so on. The degree of improvement in OS times or probabilities cannot by itself be summarized by a single value as survival differences will vary across time. Although the HR is generally considered the most important comprehensive summary of survival comparisons, its interpretation should not be taken out of context. The authors’ interpretation assumes that survival times follow an exponential distribution. In this case, when both treatment groups follow exponential distributions, then the ratio of medians, me/mc (say), is equal to 1/HR. The exponential model, however, should not be the basis for general interpretation. For example, suppose that treatment C survival times follow a Weibull model with shape parameter ν and scale parameter θ (cf. ref.2Kalbfleisch J Prentice R The Statistical Analysis of Failure Time Data. John Wiley & Sons, Hoboken, NJ2002Crossref Scopus (3697) Google Scholar), and that the hazard function for treatment E is equal to three-fourths the hazard function for treatment C. We note that the exponential model is a special case of the Weibull model with shape parameter ν=1. Then HR=0.75 and the ratio of medians is equal to 1/HR=1.33 if and only if ν=1. However, if ν=0.25, for example, then the ratio of medians is equal to 3.16, and if ν=4, then the ratio is 1.07. These examples illustrate the pitfalls of interpreting 1/HR as the “increase in survival time.” Here we clarify that for HR=0.75, 1 − HR = 0.25 means that “treatment E reduces the risk of death by 25% relative to treatment C,” whereas 1/HR − 1 = 0.33 means that “treatment C increases the risk of death by 33% relative to treatment E.” In other words, converting the interpretation from 1 − HR to 1/HR − 1 simply changes the reference group; from E versus C to C versus E. This can be seen from the definition of the HR. Let λe denote the hazard of death for treatment E, and λc denote the hazard of death for treatment C. If the ratio of hazards r(t)=λe(t)/λc(t) does not depend on time t then the proportional hazards assumption holds. Denoting this ratio by HR we can interpret HR = 0.75 (without reference to time) as “patients on treatment E have a 25% reduced risk of death relative to treatment C.” If we switch the interpretation to be in terms of C relative to E then this is λc(t)/λe(t)=1/HR, which means that the hazard for C is (1/0.75) or 133% the hazard of E. Note that [λc(t)–λe(t)]/λe(t)=1/HR − 1, where λc(t)–λe(t) represents how much treatment C increases the hazard of death compared with E. Then 1/HR − 1 = 33% is the percentage increase in the hazard of death for treatment C relative to treatment E. Because survival probabilities (OS curves) are an explicit mathematical expression of the hazard function, that “treatment E reduces the risk (hazard) of death” already directly translates into prolonged OS for E relative to C. As we design trials to assess whether an experimental regimen prolongs OS relative to a control, the interpretation given by 1 − HR (E versus C) is what is needed.