Hazard ratio

In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two levels of an explanatory variable. 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 risks and odds ratios in that RRs and ORs are cumulative over an entire study, using a defined endpoint, while HRs 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.

Regression models are used to obtain hazard ratios and their confidence intervals.

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 approaches 0.

where N(t) is the number at risk at the beginning of an interval. A hazard is the probability that a patient fails between ${\displaystyle t}$ and ${\displaystyle t+\Delta t}$, given that he has survived up to time ${\displaystyle t}$, divided by ${\displaystyle \Delta t}$, as ${\displaystyle \Delta t}$ approaches zero.

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