Hazard Ratio
The hazard ratio is the ratio of the hazard rate — the instantaneous rate at which events occur among those still at risk — in one group to that in another. It is the relative measure of association for time-to-event (survival) data and the effect estimate produced by the Cox proportional-hazards model.
Definition
The hazard ratio is the ratio of the hazard (the instantaneous event rate among those still at risk) in one group to the hazard in a comparison group, typically assumed constant over follow-up time.
Scope
This entry covers the definition of the hazard and the hazard ratio, its null value of one, the proportional-hazards assumption it usually rests on, how it differs from a risk ratio, and what it does and does not say about timing of events. It treats the hazard ratio as a methodological measure, not as clinical guidance.
Key concepts
- Hazard (instantaneous event rate)
- Time-to-event data and censoring
- Null value of one
- Proportional-hazards assumption
- Cox regression
- Distinction from the risk ratio
Mechanisms
The hazard is the instantaneous rate at which the event of interest occurs among individuals who have survived event-free to a given moment. The hazard ratio divides the hazard in one group by that in another; a value of 1 means no association, above 1 a higher event rate, and below 1 a lower event rate. The Cox proportional-hazards model estimates this ratio while leaving the underlying baseline hazard unspecified, which is why it is so widely used; it rests on the proportional-hazards assumption that the ratio is constant over follow-up. Because it is built from event rates over time and accommodates censoring, the hazard ratio handles incomplete follow-up that a simple risk ratio cannot. It summarises the relative rate of events but does not by itself convey absolute risks or how long events are delayed, so it is interpreted alongside survival curves and absolute measures.
Clinical relevance
The hazard ratio is the standard summary of treatment or exposure effects in survival analyses, including many randomised trials with time-to-event endpoints, so reading it correctly is central to appraising that evidence. It quantifies a relative event rate over time and does not on its own indicate absolute benefit or the timing of events; it is a measure for interpreting research, not a directive for individual care.
Epidemiology
Hazard ratios are produced wherever time-to-event outcomes are analysed — survival in cohort studies and clinical trials, time to relapse, time to disease onset — and are the usual output of Cox regression for confounder adjustment. They are routinely pooled in meta-analyses of survival outcomes and are reported with survival curves and, ideally, absolute measures to convey impact.
History
The hazard ratio entered epidemiology and biostatistics through David Cox's 1972 paper introducing the proportional-hazards regression model, which made it possible to estimate relative event rates from censored survival data without specifying the baseline hazard. The Cox model became one of the most heavily used methods in medical research, and the hazard ratio became the default relative measure for time-to-event endpoints, prompting later clarifications of how it should and should not be interpreted.
Debates
- Interpreting the hazard ratio as a risk ratio
- A hazard ratio summarises relative event rates over follow-up and is not the same as a risk ratio or a statement about how much sooner events occur; reading it as a simple ratio of probabilities, or ignoring violations of the proportional-hazards assumption, can mislead.
Key figures
- David Cox
- Kenneth Rothman
- Sander Greenland
Related topics
Seminal works
- cox-1972
- spruance-2004
Frequently asked questions
- How does a hazard ratio differ from a relative risk?
- A relative risk compares the cumulative probability of an outcome over a fixed period, while a hazard ratio compares instantaneous event rates among those still at risk over follow-up; the hazard ratio handles censored survival data and usually assumes the ratio is constant over time.
- What is the proportional-hazards assumption?
- It is the assumption underlying the Cox model that the hazard ratio between groups stays constant throughout follow-up; when it is violated, a single hazard ratio can be a misleading summary and the survival curves should be examined directly.