Multiplicity and Multiple Testing
Calculates the sample size needed when a trial is successful only if all primary endpoints achieve statistical significance. The joint power is evaluated from a multivariate normal model with a common correlation between endpoints, matching the methodology used in this method for multiple must-win endpoints.
This calculator implements the must-win/co-primary endpoint formulation of the multiple-primary-endpoint sample-size problem: the trial is considered successful only when every prespecified endpoint is statistically significant in the anticipated direction. In this setting, multiplicity does not require an alpha adjustment because requiring all endpoints to succeed is an intersection-union decision; instead, the important design issue is maintaining the desired overall power.
the relevant methodological literature describes this design as a multiple must-win endpoint design and calculates joint power from a multivariate normal distribution. The same framework is used here for a two-group study with equal allocation, standardized endpoint effects, a common pairwise correlation, and a known/asymptotic variance z-test.
For endpoint j, let δj be its standardized treatment effect and let n be the number of participants per treatment group. Under equal allocation, the mean of the corresponding standardized test statistic under the alternative is
For a two-sided test at level α, the critical value is z1-α/2. The joint power is the probability that all endpoint-specific test statistics exceed their corresponding critical boundary in the anticipated direction. When all endpoint correlations are equal to ρ, the correlation matrix has 1 on the diagonal and ρ in every off-diagonal position.
The calculator evaluates that multivariate-normal probability numerically using the equivalent one-dimensional common-factor representation of an equicorrelated normal distribution. It then searches over integer values of n until the requested overall power is reached.
With one endpoint, 90% power means a 10% probability of missing the effect. With several must-win endpoints, the trial can fail when any one endpoint misses significance. Consequently, the marginal power required for each endpoint must generally be higher than the desired overall power. Positive correlation between endpoints reduces this penalty because successful and unsuccessful results tend to occur together.
The calculator also reports an inflation factor relative to a single-endpoint calculation based on the smallest standardized effect size. This provides an intuitive measure of how much the multiple-endpoint requirement increases enrollment beyond the endpoint that would drive a conventional single-endpoint design.
The calculation is based on the asymptotic z-test formulation. It is therefore most directly applicable to continuous, binary, or other endpoints that can reasonably be represented by standardized asymptotically normal test statistics. The common-correlation assumption is a simplifying design assumption; when endpoint-specific correlations differ materially, a full covariance matrix or simulation-based calculation may be preferable.
Published worked example: 8 must-win endpoints, two-sided α = 0.05, target overall power = 0.90, equal allocation, known variance, and endpoint correlation ρ = 0.3.
The standardized effects are 0.65217 for one endpoint, 0.40825 for one endpoint, and 0.55 for six endpoints.
Expected result: 137 participants per group, 274 total, with approximately 90.27% joint power.
The implementation reproduces this Published worked example. For the same inputs, the numerical multivariate-normal calculation gives 137 participants per group and approximately 0.9026 joint power, matching the relevant methodological literature after rounding.
Julious, S. A., & McIntyre, N. E. (2012). Sample sizes for trials involving multiple correlated must-win comparisons. Pharmaceutical Statistics, 11(2), 177–185. DOI: 10.1002/pst.515.
the software, LLC. (2025). this method Sample Size Software: Tests for Two Groups with Multiple Must-Win Endpoints, Chapter 904. the relevant methodological literature. The procedure describes joint power using a multivariate normal distribution and assumes a common correlation between endpoints.
Sozu, T., Sugimoto, T., & Hamasaki, T. (2010). Sample size determination in clinical trials with multiple co-primary binary endpoints. Statistics in Medicine, 29(21), 2169–2179. DOI: 10.1002/sim.3972.