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# Hazard ratio

The hazard ratio in survival analysis is the effect of an explanatory variable on the hazard or risk of an event. For a less technical definition than is provided here, consider hazard ratio to be an estimate of relative risk and see the explanation on that page.

### Additional recommended knowledge

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.

$h(t) = \lim_{\Delta t\rightarrow 0}\frac{\mathrm{observed \;events}(t)/N(t)}{\Delta t}$

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

The 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 which treat the log of the hazard rate 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 hazard rate over time), and include the Cox semi-parametric 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 hazard rate will differ by eβ, 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 hazard rate models have different formulations and the interpretation of the parameter estimates differs accordingly.

## References

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

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Hazard_ratio". A list of authors is available in Wikipedia.
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