Study Design Inputs
Enter proportions as decimals. For example, enter 0.80 for 80% power.
Download Results
Calculate first, then export the result summary for your planning notes.
Example Data Table
| Input | Example value | Planning meaning |
|---|---|---|
| Hazard ratio | 0.70 | Expected 30% lower hazard in treatment. |
| Allocation | 0.50 | Equal recruitment into two groups. |
| Event probability | 0.45 | Forty-five events expected per 100 participants before losses. |
| Loss to follow-up | 0.10 | Simple planning reduction in observable events. |
| Covariate R² | 0.15 | Inflates the event requirement for covariate overlap. |
Formula Used
This page uses a Schoenfeld-style two-group event approximation for a Cox proportional hazards model.
D = (z1−α/2 + zpower)² ÷ [p(1−p)(ln HR)²]
Dadjusted = D ÷ (1 − R²)
N = ceil[Dadjusted ÷ {event probability × (1 − loss proportion)}]
Here, p is the treatment allocation proportion, HR is the assumed hazard ratio, and R² is the selected covariate-overlap inflation factor. The optional achieved-power estimate reverses the same approximation using the planned enrollment.
Method reference: Schoenfeld, D. A. (1983), Sample-size formula for the proportional-hazards regression model, Biometrics 39, 499–503.
How to Use This Calculator
- Choose the smallest hazard ratio that would be scientifically meaningful.
- Set the two-sided significance level and desired study power.
- Enter the planned fraction assigned to treatment or exposure.
- Estimate the event probability across the complete study period.
- Enter expected loss to follow-up as a decimal proportion.
- Add an R² inflation value when adjustment covariates overlap with exposure.
- Optionally enter a planned enrollment to review approximate achieved power.
- Use the required events and total enrollment for a fuller protocol review.
Planning Cox Proportional Hazards Studies
Events carry the information
Survival studies do not gain precision from enrollment alone. They gain precision when outcome events occur. A large cohort with few events can remain underpowered. A smaller cohort with frequent events may answer the study question sooner. This calculator starts with the event requirement. It then converts that requirement into an enrollment target using the expected event probability.
Choose a realistic hazard ratio
The hazard ratio describes the relative instantaneous event rate between two groups. A value below one suggests a lower event hazard for treatment. A value above one suggests a higher hazard. Ratios close to one are difficult to detect. Their logarithms are small. Therefore, the required event count increases quickly as the assumed effect approaches no difference.
Allocation affects efficiency
Equal allocation gives the largest value of p(1−p). It is usually the most statistically efficient arrangement. Unequal allocation can be justified for safety data, cost, ethics, or treatment availability. It also increases the required event total. Enter the actual treatment proportion rather than only the allocation ratio.
Convert events into participants carefully
The event probability is the share expected to experience the endpoint during the observed period. It depends on recruitment, follow-up, baseline risk, and endpoint definition. This page applies a simple loss adjustment by reducing that event probability. It is useful for early planning. It does not replace a detailed accrual and censoring model.
Understand the covariate adjustment
Adjustment variables can improve the analysis. Yet strong overlap between exposure and adjustment covariates reduces independent exposure information. The R² input applies a transparent inflation factor. Set it to zero when no such adjustment is intended. Use values supported by pilot data whenever possible. Avoid guessing a large R² without justification.
Check key assumptions
The calculation assumes two independent groups and a roughly constant hazard ratio over time. It uses a two-sided normal approximation. Nonproportional hazards, clustering, competing risks, repeated events, interim analyses, and complex stratification can change the design. Those settings often need specialized software or simulation. Review the final plan with a biostatistician before recruitment begins.
Use sensitivity scenarios
Run several plausible cases. Lower event rates increase enrollment. Higher dropout also increases enrollment. Smaller effects require many more events. Compare an optimistic, central, and conservative scenario. Select a target that remains credible when conditions are less favorable. This approach produces a more resilient protocol and a clearer resource plan.
Document assumptions
Document every assumption in the protocol. State endpoint clearly. Define time zero and the censoring rules. Explain how the event probability was estimated. Record the expected recruitment pace and follow-up length. Show why the selected hazard ratio matters clinically. Describe anticipated losses and any planned covariate adjustment. Include sensitivity tables for lower event rates and weaker effects. Keep the total enrollment target rounded upward. Allow room for data quality problems. Review eligibility criteria because they affect baseline risk. Confirm that endpoint adjudication is consistent across groups. Recalculate after major design changes. A transparent power plan helps reviewers understand feasibility. It protects study resources. It makes later interpretation more defensible.
Frequently Asked Questions
What does this calculator estimate?
It estimates required events and participants for a two-group Cox proportional hazards design. It can also estimate approximate achieved power when you enter a planned enrollment.
Why are events more important than participants?
The comparison is informed mainly by observed endpoint events. Enrollment without enough observed events provides limited information for detecting a hazard ratio.
Can the hazard ratio equal one?
No. A hazard ratio of one represents no assumed group difference. The required event count becomes undefined because there is no effect to detect.
What allocation should I use?
Use the proportion planned for treatment or exposure. Enter 0.50 for equal allocation. Unequal allocation is allowed, but it usually requires more events.
How should I estimate the event probability?
Use historical data, pilot results, registries, or credible published evidence. Match the estimate to the intended endpoint, recruitment period, and follow-up duration.
Does the loss input model censoring exactly?
No. It reduces the expected event probability with a simple planning adjustment. Use a detailed accrual and censoring model when losses vary over time.
What does the R² input represent?
It represents the selected inflation for exposure information shared with adjustment covariates. Higher R² values raise the event requirement through the factor 1 ÷ (1 − R²).
Is the test one-sided or two-sided?
This calculator uses a two-sided Type I error. Its critical value is z at 1 minus alpha divided by two.
Can I use this for observational research?
It can support preliminary planning. Observational studies may require stronger treatment of confounding, covariate overlap, missing data, and nonrandom allocation.
When is simulation preferable?
Simulation is preferable for nonproportional hazards, staggered recruitment, complex censoring, competing risks, clusters, adaptive monitoring, or multiple endpoints.
Should this result be used without review?
No. Use it as a transparent preliminary estimate. Careful planning protects participants, budgets, timelines, and scientific credibility.