twoCoprimary (version 1.0.0)

ss2MixedCountContinuous: Sample Size Calculation for Two Co-Primary Endpoints: One Count and One Continuous

Description

Determines the sample size for a two-arm superiority trial with two co-primary endpoints where one is a count (negative binomial) and one is continuous (normal), to achieve a specified power at a given significance level.

Usage

ss2MixedCountContinuous(
  r1,
  r2,
  nu,
  t,
  mu1,
  mu2,
  sd,
  r,
  rho1,
  rho2,
  alpha,
  beta
)

Value

A data frame with the following columns:

r1, r2

Count rates

nu

Dispersion parameter

t

Follow-up time

mu1, mu2

Means for continuous endpoint

sd

Standard deviation for continuous endpoint

r

Allocation ratio

rho1, rho2

Correlations

alpha

One-sided significance level

beta

Type II error rate

n1

Required sample size for group 1

n2

Required sample size for group 2

N

Total sample size (n1 + n2)

Arguments

r1

Mean count rate in group 1 for the count endpoint

r2

Mean count rate in group 2 for the count endpoint

nu

Dispersion parameter for the negative binomial distribution (nu > 0). Smaller values indicate greater overdispersion

t

Follow-up time period

mu1

Mean for group 1 for the continuous endpoint

mu2

Mean for group 2 for the continuous endpoint

sd

Common standard deviation for the continuous endpoint

r

Allocation ratio n1/n2 where n1 is sample size for group 1

rho1

Correlation between count and continuous endpoints in group 1

rho2

Correlation between count and continuous endpoints in group 2

alpha

One-sided significance level (typically 0.025 or 0.05)

beta

Type II error rate (typically 0.1 or 0.2). Power = 1 - beta

Details

This function implements the sample size calculation for mixed count-continuous co-primary endpoints following the methodology in Homma and Yoshida (2024).

The sequential search algorithm (Homma and Yoshida 2025, Algorithm 1) is used:

Step 1: Initialize with sample sizes from single endpoint formulas.

Step 2: Use sequential search:

• Calculate power at initial sample size

• If power >= target: decrease n2 until power < target, then add 1 back

• If power < target: increase n2 until power >= target

Step 3: Return final sample sizes.

Negative Binomial Distribution: The count endpoint follows a negative binomial distribution NB(lambda, nu) where:

• lambda = r * t is the mean count

• nu is the dispersion parameter

• Variance = lambda + lambda^2 / nu

Correlation: The correlations rho1 and rho2 must satisfy feasibility constraints that depend on the parameters. Use corrbound2MixedCountContinuous to check valid correlation bounds.

References

Homma, G., & Yoshida, T. (2024). Sample size calculation for count and continuous multiple co-primary endpoints. Pharmaceutical Statistics, 23(3), 372-388.

Homma, G., & Yoshida, T. (2025). Exact power and sample size in clinical trials with two co-primary binary endpoints. Statistical Methods in Medical Research, 34(1), 1-19.

Examples

# Sample size calculation for count and continuous endpoints
ss2MixedCountContinuous(
  r1 = 1.0,
  r2 = 1.25,
  nu = 0.8,
  t = 1,
  mu1 = -50,
  mu2 = 0,
  sd = 250,
  r = 1,
  rho1 = 0.4,
  rho2 = 0.4,
  alpha = 0.025,
  beta = 0.2
)

# With different dispersion parameter (more overdispersion)
ss2MixedCountContinuous(
  r1 = 1.0,
  r2 = 1.25,
  nu = 0.5,
  t = 1,
  mu1 = -50,
  mu2 = 0,
  sd = 250,
  r = 1,
  rho1 = 0.4,
  rho2 = 0.4,
  alpha = 0.025,
  beta = 0.2
)