Set the Study Assumptions
Use standardized mean differences for two continuous co-primary endpoints. Required sample size uses equal allocation.
Example Data Table
| Scenario | Effect 1 | Effect 2 | Correlation | Groups | Planning use |
|---|---|---|---|---|---|
| Balanced moderate effects | 0.45 | 0.40 | 0.45 | 100 and 100 | Assess achieved joint power. |
| Lower second endpoint | 0.50 | 0.30 | 0.50 | Equal groups | Find the limiting sample size. |
| High positive correlation | 0.35 | 0.35 | 0.75 | Equal groups | Test correlation sensitivity. |
Formula Used
For endpoint j, the expected test statistic uses the standardized effect size and the two-group standard error factor.
The two-sided endpoint threshold is:
The joint design power is the bivariate normal probability that both shifted test statistics exceed the same critical value:
Here, Φ is the standard normal distribution function. The correlation is applied to the endpoint test statistics. The required-size search finds the smallest equal per-group size that reaches the requested joint power.
How to Use This Calculator
- Select achieved power to evaluate a planned allocation.
- Select required sample size to search for equal groups.
- Enter clinically meaningful standardized effects for both endpoints.
- Enter a justified test-statistic correlation and alpha level.
- Set the joint power target and maximum sample limit.
- Calculate, inspect endpoint powers, then test alternative assumptions.
- Export the result table or save the page through print.
Planning Co-Primary Endpoint Studies
Co-primary endpoints require success on every selected outcome. A positive result for one endpoint is not enough. This design fits studies where a treatment must deliver benefits across separate dimensions. However, the requirement raises the sample size challenge. The study must have enough information for both tests at the same time.
Power is the chance of meeting the planned significance rule when the assumed effects are real. Individual endpoint power is helpful, but it is not the final design target here. Joint power matters. It measures the probability that both endpoint tests cross their required thresholds. A design with eighty percent power for each independent endpoint has only about sixty four percent joint power.
The calculator uses standardized mean differences for two continuous endpoints. A standardized effect divides the anticipated mean difference by its standard deviation. This makes the two endpoint inputs comparable. Larger positive effects create larger expected test statistics. Larger samples also improve precision. A smaller effect needs more participants to reach the same power.
Correlation connects the two endpoint test statistics. Participants who improve on one measure may also improve on the other. Positive correlation often increases joint power compared with an independence assumption. Negative correlation can reduce it. The correct correlation should come from prior data, relevant publications, or an expert planning assumption. Do not use a convenient value without justification.
The alpha level controls the statistical threshold. A two-sided alpha of 0.05 is common. Each endpoint must pass its own threshold in the intended treatment direction. The calculator reports the critical normal value, marginal endpoint powers, and joint power. These outputs show why the least favorable endpoint often drives the design.
The achieved-power option evaluates a proposed allocation. Enter the treatment and control sample sizes, effect sizes, alpha, and correlation. The result appears above the form after submission. Review the endpoint powers before accepting the combined result. A high value for one endpoint cannot compensate for weak power on the other.
The required-sample option searches equal group sizes. It increases the per-group size until the selected joint-power target is reached. This is a planning aid, not a final protocol calculation. Allow additional participants for missing data, withdrawals, nonadherence, and analysis exclusions. Use realistic retention assumptions before setting recruitment goals.
This normal-approximation model is best for parallel two-arm studies with continuous outcomes. It assumes similar standardization choices, approximately normal test statistics, and a plausible correlation estimate. Clustered designs, repeated measurements, covariate adjustment, binary outcomes, survival outcomes, interim analyses, and unequal variances require methods designed for those conditions.
Document every assumption. State the endpoint definitions, clinically meaningful effects, standard deviations, alpha rule, correlation source, allocation, and target joint power. Run sensitivity checks across lower effects and different correlations. A robust design should remain useful when assumptions are slightly wrong. Statistical review is essential before any study decision. Careful planning makes evidence more credible, efficient, and interpretable.
Frequently Asked Questions
1. What are co-primary endpoints?
They are primary outcomes that must all meet the predefined success rule. A favorable result for only one endpoint does not establish overall study success.
2. Why is joint power lower than marginal power?
Joint power requires both endpoint tests to succeed together. Marginal power describes each test separately, so it does not fully represent the combined success requirement.
3. Which endpoint effect should be entered?
Enter the anticipated mean difference divided by the relevant standard deviation. Use effects that are clinically meaningful and supported by prior data whenever possible.
4. Does positive correlation always help?
For this all-endpoints success rule, positive correlation generally increases the chance that both tests succeed together. The assumed value still needs evidence and sensitivity testing.
5. Why use a two-sided alpha value?
Two-sided alpha is common when either direction of a treatment difference is statistically testable. This calculator then evaluates success in the anticipated beneficial direction.
6. Can I use unequal group sizes?
Yes. Achieved-power mode accepts different treatment and control sizes. Required-size mode intentionally searches equal groups for a clear and efficient planning baseline.
7. Does this model handle binary endpoints?
No. This page is designed for two continuous endpoints using a normal approximation. Binary, survival, repeated-measures, and cluster designs need specialized calculations.
8. Should dropout be added to the result?
Yes. Inflate the planned recruitment target after calculating analyzable participants. Use realistic assumptions for dropout, missing outcomes, ineligibility, and other losses.
9. Is alpha split across endpoints here?
No. Co-primary endpoints typically require each endpoint to meet its stated criterion. The calculator uses the entered two-sided alpha for each endpoint test.
10. What does the required-size search return?
It returns the smallest equal sample size per group whose computed joint power meets or exceeds the chosen target, subject to the maximum limit.
11. Is this result sufficient for a protocol?
No. A protocol needs endpoint definitions, analysis methods, assumptions, sensitivity checks, operational constraints, and statistical review beyond a single planning calculation.