A hazard ratio is the effect measure from survival analysis, also called time-to-event analysis. It compares the hazard, the instantaneous rate at which an event occurs, in a treatment group against a control group, averaged across the follow-up period. A hazard ratio of 1 means the event happens at the same rate in both groups; below 1 means the event happens more slowly in the treatment group, often a benefit; above 1 means it happens faster. Unlike a risk ratio, it accounts for when events happen and for participants whose outcome is unknown at the end of follow-up.
Why time-to-event data need their own measure
In a trial that follows patients for years, two things complicate a simple count of events. First, timing matters: a treatment that delays death by two years is valuable even if everyone eventually dies, and a measure that ignores timing would miss this entirely. Second, censoring is unavoidable: some participants are still event-free when the study ends, others withdraw or are lost, and their follow-up is incomplete rather than a known non-event. A plain ratio of binary event proportions cannot handle either feature properly. Survival methods, summarised in a Kaplan-Meier curve and quantified by the hazard ratio, were built exactly to use all the follow-up time each person contributes and to handle censored observations without discarding them.
The Cox proportional hazards model
The hazard ratio almost always comes from a Cox proportional hazards model, the standard regression for time-to-event outcomes. Its great convenience is that it estimates the ratio of hazards between groups without specifying the shape of the underlying baseline hazard over time; it is semi-parametric, modelling how covariates multiply the hazard while leaving the baseline rate unspecified. The model can also adjust for confounders, producing an adjusted hazard ratio that is the time-to-event counterpart of an adjusted odds ratio. The coefficient the model estimates is on the log scale, and exponentiating it gives the hazard ratio, exactly the same log-then-exponentiate logic used for ratio measures in any pooled quantitative synthesis. A hazard ratio is reported with a confidence interval, and if that interval includes 1 the difference is not statistically significant.
Interpreting values above and below 1
Read the hazard ratio as a relative rate. A hazard ratio of 0.70 means the treatment group experiences the event at 70 percent of the rate of the control group at any given moment, a 30 percent lower hazard. A hazard ratio of 1.5 means the event occurs at one and a half times the rate, a 50 percent higher hazard. A hazard ratio of 2.0 means the rate is doubled. The crucial subtlety is that a hazard ratio is a ratio of rates, not of probabilities, and not of survival times. A hazard ratio of 0.50 does not mean patients live twice as long, nor that half as many have the event; it means the moment-to-moment rate of the event is halved. Translating it into absolute terms, such as a difference in median survival or in the proportion event-free at five years, requires the baseline risk, in the same way that a relative measure must be combined with a baseline to give an absolute reduction in risk.
The proportional hazards assumption
The Cox model rests on one big assumption: proportional hazards, meaning the ratio of hazards between the groups stays constant over the whole follow-up. The single reported hazard ratio is an average over time, and it is only a faithful summary if that ratio really is stable. When the assumption fails, for example if a surgical treatment carries early harm but later benefit so the curves cross, a single hazard ratio averages a changing effect into one potentially misleading number. The assumption is checked by inspecting whether Kaplan-Meier curves diverge proportionally, by plotting log-minus-log survival, or by testing Schoenfeld residuals. When it does not hold, analysts report effects in time intervals, use a different model, or summarise with a measure such as the restricted mean survival time instead. Flagging a violated proportional hazards assumption is part of a proper appraisal of how an analysis was conducted.
Why a hazard ratio is not a risk ratio
The two are routinely conflated, but they are different quantities. A risk ratio compares the cumulative probability of the event by a fixed time point, ignoring when within that window events occurred and unable to use censored follow-up. A hazard ratio compares the instantaneous rate across the whole period and uses every participant’s follow-up time. They coincide only when the event is rare and follow-up short; otherwise they diverge, and a hazard ratio is generally further from 1 than the corresponding risk ratio. This matters for synthesis: you cannot pool a hazard ratio together with a risk ratio in the same meta-analysis, just as you cannot mix odds ratios and risk ratios, a pitfall covered in our guide to choosing a single effect measure to combine. You can sanity-check the related binary ratios for the same data with our odds ratio and risk ratio calculator.
How hazard ratios are pooled in meta-analysis
Hazard ratios are combined like any ratio measure: on the log scale, with each study weighted by inverse variance, and the pooled result exponentiated back. The practical difficulty is obtaining the variance, because many papers report only the hazard ratio and its confidence interval rather than the standard error directly. The standard error of the log hazard ratio is recovered from the interval, taking the difference of the logged confidence limits and dividing by 2 times 1.96. Where even that is missing, established methods estimate the log hazard ratio and its variance from the number of events, the number at risk, or summary statistics read off a published Kaplan-Meier curve, though these reconstructions add uncertainty that belongs in a sensitivity analysis of the pooled estimate. Differences in follow-up length and in whether the proportional hazards assumption held across studies are common drivers of heterogeneity between time-to-event studies. Handled with these checks in place, a pooled hazard ratio is a powerful summary of how a treatment changes the rate of an event over time.