Hazard ratio and the Cox model
- Measure
The HR compares how fast events happen in two groups. HR = 0.5: at any moment, the event occurs at half the pace.
What it is
The hazard is the rate at which the event occurs at a given moment among those still at risk. The hazard ratio (HR) is the ratio of those rates between two groups, and the Cox proportional hazards model estimates it adjusted for other variables. It assumes the HR stays constant over time (proportional hazards). It is the usual effect measure in trials with time-to-event outcomes.
Formula and reading
- With a constant hazard, it is estimated as events / person-time.
- HR = hazard in group A / hazard in group B.
- Reading: (1 − HR) × 100 = relative reduction in the event rate; with HR = 0.6, 40% lower.
Worked example
- Data: a 2-year trial. Treatment: 30 deaths over 1 500 person-months. Control: 50 deaths over 1 250 person-months.
- Assuming a constant hazard, the treatment hazard = 30 / 1 500 = 0.02 per person-month.
- Control hazard = 50 / 1 250 = 0.04 per person-month.
- HR = 0.02 / 0.04 = 0.50.
- A Cox model adjusted for age and stage gives HR = 0.55, 95% CI 0.35 to 0.86.
- The CI excludes 1, so the reduction is significant at α = 0.05.
- Reading: at every point of follow-up, treated patients die at about half the pace of controls, at the same age and stage.
Markers
- HR
- Cox
- 95% CI
How to read it
- In the abstracts of oncology and cardiovascular trials, the HR with its CI is the headline efficacy figure.
- For decisions, translate it into absolute measures: the survival difference at a given time, or the NNT.
Common pitfalls
- Reading the HR as a relative risk of cumulative events at the end of the study.
- Not checking the proportional-hazards assumption: if the curves cross, a single HR summarises the effect poorly.
- Forgetting that the HR does not say how much time is gained; the median or the gap between curves shows that.
Test yourself
Think of the answer, then tap the question to see it.
What does an HR of 0.75 for mortality mean?
That at every point of follow-up the treated group dies at a 25% lower rate than the reference group.
What does the Cox model assume?
Proportional hazards: the ratio of the hazards between groups is constant over time.
How does the HR differ from the RR?
The RR compares cumulative proportions with events at a fixed time; the HR compares instantaneous rates and uses time to event and censoring.
An HR of 0.82 has a 95% CI of 0.64 to 1.05. Is it significant?
Not at α = 0.05: the interval includes 1.
Connections
Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.
Sources: Proportional hazards model (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); Failure (or hazard) rate (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); Chapter 6: Choosing effect measures and computing estimates of effect (Cochrane, 2024).
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Representative model, for educational purposes only.