In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions characterised by two distinct levels of a treatment variable of interest. For example, in a clinical study of a drug, the treated population may die at twice the rate of the control population. The hazard ratio would be 2, indicating a higher hazard of death from the treatment. For example, a scientific paper might use an HR to state something such as: "Adequate COVID-19 vaccination status was associated with significantly decreased risk for the composite of severe COVID-19 or mortality with a[n] HR of 0.20 (95% CI, 0.17–0.22)." In essence, the hazard for the composite outcome was 80% lower among the vaccinated relative to those who were unvaccinated in the same study. So, for a hazardous outcome (e.g., severe disease or death), an HR below 1 indicates that the treatment (e.g., vaccination) is protective against the outcome of interest. In other cases, an HR greater than 1 indicates the treatment is favorable. For example, if the outcome is actually favorable (e.g., accepting a job offer to end a spell of unemployment), an HR greater than 1 indicates that seeking a job is favorable to not seeking one (if "treatment" is defined as seeking a job). Hazard ratios differ from relative risks (RRs) and odds ratios (ORs) 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.
Definition and derivation 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:
h ( t ) = lim Δ t → 0 observed events in interval [ t , t + Δ t ] / N ( t ) Δ t , {\displaystyle h(t)=\lim _{\Delta t\to 0}{\frac {{\text{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. A hazard is the probability that a patient fails between t {\displaystyle t} and t + Δ t {\displaystyle t+\Delta t} , given that they have 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 logarithm of the HR as a function of a baseline hazard h 0 ( t ) {\displaystyle h_{0}(t)} and a linear combination of explanatory variables:
log h ( t ) = f ( h 0 ( t ) , α + β 1 X 1 + ⋯ + β k X k ) . {\displaystyle \log h(t)=f{\big (}h_{0}(t),\alpha +\beta _{1}X_{1}+\cdots +\beta _{k}X_{k}{\big )}.}
Such models are generally classed proportional hazards regression models; the best known being the Cox 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^{\beta }} , where β {\displaystyle \beta } 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. 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.
Interpretation
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