Estimate an approximate hazard ratio from same-horizon treatment and control event counts, with confidence interval, event risks, and absolute risk difference.
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Same-horizon approximation
Use event counts measured against the same endpoint and the same follow-up window. This page estimates an
approximate hazard ratio from cumulative event probabilities under an exponential-survival /
proportional-hazards assumption.
This calculator needs at least one event and one non-event in each group. If you have censoring, variable
follow-up, or model output from a Cox regression, use the published hazard ratio directly instead of this
grouped-count approximation.
Enter treatment and control event counts to estimate an approximate hazard ratio.
Hazard ratio calculator: estimate an approximate HR from grouped event counts
This hazard ratio calculator estimates an approximate hazard ratio when you only have grouped event counts for treatment and control measured at the same follow-up horizon. It is designed for quick study interpretation and teaching, showing the approximate HR, 95% confidence interval, event-risk context, and absolute risk difference without pretending to replace a full survival-analysis workflow.
What this calculator actually estimates
A true hazard ratio compares instantaneous event rates over time among people still at risk, which is why it is usually estimated from survival-analysis software such as a Cox proportional hazards model. If all you have is grouped event-count data at a shared time horizon, you do not have the full time-to-event history needed to reproduce that model exactly.
This page therefore labels the result as an approximation. It converts the observed cumulative event probabilities in the treatment and control groups into cumulative hazards, then compares those values under an exponential-survival / proportional-hazards assumption. That is materially better than dividing treatment risk by control risk, which would only give a relative risk rather than a hazard ratio style estimate.
Interpret the headline cautiously. An approximate HR below 1 suggests the treatment arm accumulated events more slowly than the control arm across the same horizon; above 1 suggests faster accumulation; a value near 1 suggests little separation between the groups. The effect size still needs clinical or study-design context before you decide whether it is important.
Approx. HR ≈ ln(1 - P_t) / ln(1 - P_c)
Same-horizon approximation from cumulative event risk, where P_t and P_c are treatment and control event probabilities measured at the same follow-up time.
ARD = P_c - P_t
Absolute risk difference shown alongside the approximate HR so the relative effect is not read without an absolute-risk frame.
Why hazard ratio is not the same as relative risk
Relative risk compares cumulative probabilities at a fixed time point: risk in treatment divided by risk in control. Hazard ratio compares event intensity over time among those still event-free. Those are related ideas, but they are not interchangeable.
That difference matters because hazard ratios are often misread as simple risk ratios. A study might report HR 0.70, but that does not automatically mean a 30% lower cumulative risk at every time point. The exact relationship depends on baseline risk, follow-up length, censoring, and whether the proportional-hazards assumption is reasonable.
This is why the page shows both the grouped event risks and the approximate hazard ratio. The risk numbers answer the practical question, 'What share of each group experienced the event by this horizon?' The approximate HR answers the more technical question, 'If the hazard were roughly constant in relative terms across the horizon, what hazard ratio would these grouped risks imply?'
How to read the confidence interval and significance flag
The confidence interval is built on the log scale around the approximate hazard ratio. If the interval stays entirely below 1, the grouped data points toward a lower hazard in treatment. If it stays entirely above 1, the grouped data points toward a higher hazard in treatment. If the interval crosses 1, the grouped-count approximation does not clearly separate the groups at the conventional 5% threshold.
That still does not answer whether the finding is clinically meaningful. A narrow interval around HR 0.95 may be statistically convincing but practically modest, while a wider interval around HR 0.60 may suggest a potentially important effect that the available grouped data is too limited to pin down precisely.
Confidence intervals on this page should be read as rough inferential context, not as a substitute for the interval reported by the original trial or study analysis. Published survival models can differ because they account for censoring, covariate adjustment, stratification, or non-constant hazards over time.
Worked example and best-use cases
Suppose 20 of 200 people in the treatment arm and 40 of 200 in the control arm experienced the event by the same follow-up point. The raw risks are 10% versus 20%. A simple risk ratio would be 0.50, but the same-horizon hazard approximation is about 0.47 because it translates the event probabilities through the survival-to-hazard relationship before taking the ratio.
That makes this page useful when you are reading a study summary, teaching the difference between relative risk and hazard ratio, or sense-checking whether grouped event data is directionally consistent with a reported hazard ratio. It is also useful when you want to pair the relative estimate with an absolute risk difference rather than quoting the HR in isolation.
It is not the right tool when you have individual time-to-event records, censoring times, Kaplan-Meier curves, adjusted model output, or strong evidence that hazards are not proportional. In those settings, a dedicated survival-analysis package is the appropriate route and the published study HR should take precedence over any web approximation.
Frequently asked questions
How is hazard ratio different from relative risk?
Relative risk compares cumulative event probabilities at a fixed time point, while hazard ratio compares event intensity over time among people still at risk. They can look similar when events are uncommon and the follow-up horizon is shared, but they are not interchangeable labels. This page makes that distinction explicit by showing the grouped event risks and then deriving an approximate hazard ratio from the log-survival relationship instead of dividing one risk by the other.
What does it mean when the confidence interval includes 1?
When the 95% confidence interval includes 1, the grouped-count approximation does not clearly separate treatment from control at the usual 5% threshold. In practical terms, the available data is still compatible with no clear difference between the groups. That does not prove the treatment has no effect; it means the approximation is not precise enough to rule out a null result with confidence.
What does a hazard ratio below 1 mean?
A hazard ratio below 1 suggests the treatment group accumulated events more slowly than the control group over the period being studied. For example, an approximate HR of 0.70 points to a lower hazard in treatment than control, but it does not by itself prove that the cumulative risk is exactly 30% lower at every horizon. Always read the absolute event risks, confidence interval, and study context alongside the HR.
Can I calculate a true hazard ratio from simple event counts alone?
Not usually. A true hazard ratio normally comes from time-to-event data that records when events happened and which observations were censored. If you only have grouped event counts by one follow-up point, you can estimate an approximate HR under additional assumptions, but you cannot fully reproduce a Cox-model hazard ratio from those counts alone.
