hazard ratio calculator

What is a hazard ratio?

A hazard ratio (HR) compares the instantaneous event rate between two groups over time. In survival analysis, it is commonly used to describe whether one group experiences an event (such as death, relapse, device failure, or recovery) faster or slower than another group.

• HR = 1: no difference in hazard between groups.
• HR < 1: treatment group has a lower hazard (protective effect).
• HR > 1: treatment group has a higher hazard (harmful effect).

How this calculator works

This tool estimates the hazard ratio by using event counts and total person-time in each group. That produces an incidence rate ratio, which is often used as an approximation to a hazard ratio when detailed time-to-event model outputs are unavailable.

Formulas used

Let:

• E_T = treatment events
• PT_T = treatment person-time
• E_C = control events
• PT_C = control person-time

Then:

• Rate_T = E_T / PT_T
• Rate_C = E_C / PT_C
• HR ≈ Rate_T / Rate_C

Confidence interval (log method):

• SE(log(HR)) ≈ √(1/E_T + 1/E_C)
• CI = exp(log(HR) ± z × SE)

How to use the hazard ratio calculator

1. Enter event counts for treatment and control groups.
2. Enter person-time for each group (same units in both groups, such as person-years).
3. Choose the confidence level (90%, 95%, or 99%).
4. Click Calculate Hazard Ratio.
5. Interpret the HR and confidence interval together, not HR alone.

Interpreting your output

Point estimate

The point estimate tells you the relative hazard magnitude. For example, HR = 0.70 means the treatment group hazard is about 30% lower than control.

Confidence interval

The confidence interval reflects uncertainty. If the 95% CI includes 1.00, the observed difference may not be statistically distinguishable from no effect at the 0.05 level.

Rate difference

This calculator also reports an absolute rate difference (treatment minus control), which gives practical context beyond a relative measure.

Worked example

Suppose a trial reports:

• Treatment: 28 events over 520 person-years
• Control: 41 events over 505 person-years

The event rate is lower in treatment, and the estimated HR will be below 1, indicating reduced hazard in the treatment arm.

Hazard ratio vs relative risk vs odds ratio

• Hazard ratio: compares event rates over continuous time and handles censoring in proper survival models.
• Relative risk: compares cumulative risk over a fixed interval.
• Odds ratio: compares odds, often used in logistic regression and case-control settings.

These are related but not interchangeable, especially when event rates are high or follow-up time varies.

Important assumptions and limitations

• This is an approximation, not a replacement for a full Cox proportional hazards model.
• Person-time units must match across groups.
• Very small event counts produce wide and unstable confidence intervals.
• If either group has zero events, a continuity correction may be needed to avoid infinite estimates.
• Potential confounding is not adjusted in this simple calculator.

When to use a full survival model instead

Use Cox regression or parametric survival models when you need covariate adjustment, formal proportional hazards testing, time-varying effects, stratification, competing risks, or publication-grade inference.

Frequently asked questions

Can this calculator replace Cox regression?

No. It is useful for quick estimation from summarized data but does not model individual-level time-to-event outcomes.

What if my event count is zero in one arm?

You can enable continuity correction. The estimate will still be sensitive, so interpret with caution.

What person-time units should I use?

Any unit is fine (person-days, person-months, person-years), as long as both groups use the same unit.

Bottom line

A hazard ratio calculator is great for rapid interpretation and back-of-the-envelope analysis. For critical clinical or scientific conclusions, always confirm with a full survival analysis and domain-appropriate sensitivity checks.