Hazard ratio

In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two sets of explanatory variables. For example, in a drug study, the treated population may die at twice the rate per unit time as the control population. The hazard ratio would be 2, indicating higher hazard of death from the treatment. Or in another study, men receiving the same treatment may suffer a certain complication ten times more frequently per unit time than women, giving a hazard ratio of 10.

Hazard ratios differ from relative risk ratios in that the latter are cumulative over an entire study, using a defined endpoint, while the former represent instantaneous risk over the study time period, or some subset thereof. Hazard ratios suffer somewhat less from selection bias with respect to the endpoints chosen, and can indicate risks that happen before the endpoint.

Definition

The instantaneous hazard rate is the limit of the number of events per unit time divided by the number at risk as the time interval decreases.

$h(t) = \lim_{\Delta t\rightarrow 0}\frac{\mathrm{observed \;events \;in \; interval} [t, t+\Delta t]/N(t)}{\Delta t}$

where N(t) is the number at risk at the beginning of an interval.

Hazard ratio is the effect on this hazard rate of a difference, such as group membership (for example, treatment or control, male or female), as estimated by regression models that treat the log of the HR as a function of a baseline hazard h₀(t) and a linear combination of explanatory variables:

$\log h(t) = f(h_0(t),\alpha + \beta_1 X_1 + \cdots + \beta_k X_k).\,$

Such models are generally classed proportional hazards regression models (they differ in their treatment of h₀(t), the underlying pattern the HR over time), and include the Cox semiparametric proportional hazards model, and the exponential, Gompertz and Weibull parametric models.

For two individuals who differ only in the relevant membership (e.g., treatment vs. control), their predicted log-hazard will differ additively by the relevant parameter estimate, which is to say that their predicted HR will differ by e^β, i.e., multiplicatively by the anti-log of the estimate. Thus the estimate can be considered a hazard ratio, that is, the ratio between the predicted hazard for a member of one group and that for a member of the other group, holding everything else constant.

For a continuous explanatory variable, the same interpretation applies to a unit difference.

Other HR models have different formulations and the interpretation of the parameter estimates differs accordingly.

In simplified terms the hazard ratio is used to describe time-to-event in survival analysis. It is the ratio of the rate at which subjects in two groups are experiencing events where a slower rate suggests a longer time of event-free-survival. This type of analysis is frequently used to evaluate a drug's ability to prevent disease as a function of time.

See also

• Survival analysis
• Failure rate and Hazard rate
• Proportional hazards models
• Odds ratio
• Relative risk ratio

References

• Altman, DG. Practical Statistics for Medical Research. Chapman & Hall. London, 1991. ISBN 0-412-27630-5. pp383–4.

External links

• Motulsky, Harvey (1995). "Comparing Two Survival Curves". Intuitive Biostatistics|. Oxford University Press. ISBN 0-19-508607-4. http://www.graphpad.com/www/book/compsurv.htm. 
• Spruance, Spotswood L.; Reid, Julia E.; Grace, Michael; Samore,Matthew (August 2004). "Hazard Ratio in Clinical Trials". Antimicrobial Agents and Chemotherapy 48 (8): 2787–2792. doi:10.1128/AAC.48.8.2787-2792.2004. PMC 478551. PMID 15273082. http://aac.asm.org/cgi/content/full/48/8/2787. 
• Duerden, Martin. "What are Hazard Ratios". Sofi Aventis. http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/What_are_haz_ratios.pdf. Retrieved 11/2/2011.