• Biserial correlation;
  • Intersection-union test;
  • Latent variable;
  • Multiplicity;
  • Multivariate normal

In the field of pharmaceutical drug development, there have been extensive discussions on the establishment of statistically significant results that demonstrate the efficacy of a new treatment with multiple co-primary endpoints. When designing a clinical trial with such multiple co-primary endpoints, it is critical to determine the appropriate sample size for indicating the statistical significance of all the co-primary endpoints with preserving the desired overall power because the type II error rate increases with the number of co-primary endpoints. We consider overall power functions and sample size determinations with multiple co-primary endpoints that consist of mixed continuous and binary variables, and provide numerical examples to illustrate the behavior of the overall power functions and sample sizes. In formulating the problem, we assume that response variables follow a multivariate normal distribution, where binary variables are observed in a dichotomized normal distribution with a certain point of dichotomy. Numerical examples show that the sample size decreases as the correlation increases when the individual powers of each endpoint are approximately and mutually equal.