[Durgical](https://durgical.com/en/) › [DurgiPedia](https://durgical.com/en/durgipedia/) › [Epidemiology](https://durgical.com/en/epidemiology/) › [Survival analysis](https://durgical.com/en/epidemiology/areas/survival/) › Hazard ratio and the Cox model

# 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.

Updated Nov 5, 2026

## Key points

- HR = hazard in the exposed or treated group / hazard in the reference group.
- HR = 1: no difference. HR 1: faster (harm).
- If the 95% CI of the HR includes 1, the difference is not significant at α = 0.05.
- Cox model: gives HRs adjusted for several variables; it assumes proportional hazards (curves that do not cross).
- HR vs RR: the RR compares the cumulative share with events at a fixed time; the HR compares rates at each instant and uses time and censoring.
- An HR of 0.7 does not mean ‘30% fewer deaths by the end’ but a 30% lower event rate at every moment.

## 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.

## Memory aid

‘The HR measures the pace, not the finish line’: it compares how fast events arrive, not how many there are at the end.

## 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.

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## Connections

## Related 5

- [Log-rank test](https://durgical.com/en/epidemiology/log-rank-test/)
- [Kaplan-Meier curve](https://durgical.com/en/epidemiology/kaplan-meier-curve/)
- [Relative risk](https://durgical.com/en/epidemiology/relative-risk/)
- [Logistic regression](https://durgical.com/en/epidemiology/logistic-regression/)
- [Survival analysis and censoring](https://durgical.com/en/epidemiology/survival-analysis-and-censoring/)

[← All of epidemiology](https://durgical.com/en/epidemiology/)

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](https://www.itl.nist.gov/div898/handbook/apr/section1/apr167.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); [Failure (or hazard) rate](https://www.itl.nist.gov/div898/handbook/apr/section1/apr123.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); [Chapter 6: Choosing effect measures and computing estimates of effect](https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06) (Cochrane, 2024).

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Representative model, for educational purposes only.

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