Hazard Ratio Calculator
Calculate hazard ratio and 95% confidence interval from events and person-time for two groups.
Includes p-value and interpretation for survival analysis.
Hazard Ratio (HR)
The hazard ratio compares the rate of an event (death, relapse, failure) between two groups over time. It is the primary effect measure in survival analysis and clinical trials.
Formula
HR = (d₁ / T₁) / (d₂ / T₂)
Where:
- d₁, d₂ = number of events in Group 1 and Group 2
- T₁, T₂ = total person-time at risk in each group
This is equivalent to the ratio of two incidence rates.
Confidence Interval
95% CI = exp(ln(HR) ± 1.96 × SE)
SE = √(1/d₁ + 1/d₂)
A 95% CI that does not include 1.0 indicates statistical significance at p < 0.05.
Interpreting the Hazard Ratio
| HR | Meaning |
|---|---|
| HR = 1 | No difference between groups |
| HR < 1 | Group 1 has lower hazard (protective effect) |
| HR > 1 | Group 1 has higher hazard (increased risk) |
| HR = 0.5 | Group 1 has 50% lower event rate |
| HR = 2.0 | Group 1 has twice the event rate |
P-value
The p-value is calculated from the z-score: z = ln(HR) / SE Under the null hypothesis (HR = 1), z follows a standard normal distribution.
Common Applications
Clinical trials: drug vs. placebo survival curves. Epidemiology: exposed vs. unexposed cohort studies. Reliability engineering: failure rates between component designs.
Person-Time
Person-time accounts for participants who leave the study early. 100 participants followed for 2 years each = 200 person-years of observation.
The assumption underneath it all. A single hazard ratio only means something if the ratio stays roughly constant over the follow-up period. That is the proportional-hazards assumption, and it fails more often than reported. A surgical treatment can carry high early risk and lower risk later, so its hazard ratio starts above 1 and ends below it. Average those into one number and you get something near 1 that describes neither phase. If your survival curves cross, a single HR is the wrong summary and no confidence interval will tell you so.
Ratio versus difference. A hazard ratio of 0.5 sounds identical whether it halves a risk of one in ten or one in a million, and those are very different findings. This is why the calculator prints the underlying rates and their difference next to the ratio: the ratio tells you the relative effect, and only the difference tells you whether it is worth anything. Drug marketing has a long history of quoting the first and omitting the second.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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