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.