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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. 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 t {displaystyle t} and t + Δ t {displaystyle t+Delta t} , given that he has survived up to time t {displaystyle t} , divided by Δ t {displaystyle Delta t} , as Δ t {displaystyle Delta t} approaches zero. 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 that treat the log of the HR as a function of a baseline hazard h 0 ( t ) {displaystyle h_{0}(t)} and a linear combination of explanatory variables: Such models are generally classed proportional hazards regression models; the best known being the Cox semiparametric proportional hazards model, and the exponential, Gompertz and Weibull parametric models. For two groups that differ only in treatment condition, the ratio of the hazard functions is given by e β {displaystyle e^{eta }} , where β {displaystyle eta } is the estimate of treatment effect derived from the regression model. This 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, is given by holding everything else constant, i.e. assuming proportionality of the hazard functions.

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