# Hazard ratio

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The hazard ratio in survival analysis is the effect of an explanatory variable on the hazard or risk of an event.

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 $h_0(t)$ 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 $h_0(t)$, 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 $e^\beta$, 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.