Sample Size Calculation for a Single Count Endpoint (Negative Binomial)

Calculates the required sample size for a two-arm superiority trial with a single overdispersed count endpoint following a negative binomial distribution, as described in Homma and Yoshida (2024).

Usage

ss1Count(r1, r2, nu, t, r, alpha, beta)

Arguments

r1: Mean rate (events per unit time) for the treatment group
r2: Mean rate (events per unit time) for the control group
nu: Common dispersion parameter for the negative binomial distribution (nu > 0)
t: Common follow-up time period
r: Allocation ratio (treatment:control = r:1, where r > 0)
alpha: One-sided significance level (typically 0.025 or 0.05)
beta: Target type II error rate (typically 0.1 or 0.2)

Value

A data frame with the following columns:

r1: Mean rate for treatment group
r2: Mean rate for control group
nu: Dispersion parameter
t: Follow-up time
r: Allocation ratio
alpha: One-sided significance level
beta: Type II error rate
n1: Required sample size for treatment group
n2: Required sample size for control group
N: Total sample size (n2 + n1)

Details

The test statistic for the negative binomial rate ratio is: $$Z_1 = \frac{\hat{\beta}_1}{\sqrt{Var(\hat{\beta}_1)}}$$ where $\hat{\beta}_1 = \log(\bar{Y}_1) - \log(\bar{Y}_2)$ and the variance is: $$Var(\hat{\beta}_1) = \frac{1}{n_2}\left[\frac{1}{t}\left(\frac{1}{r_2} + \frac{1}{r \cdot r_1}\right) + \frac{1+r}{\nu \cdot r}\right]$$

This is equation (8) in Homma and Yoshida (2024).

References

Homma, G., & Yoshida, T. (2024). Sample size calculation in clinical trials with two co-primary endpoints including overdispersed count and continuous outcomes. Pharmaceutical Statistics, 23(1), 46-59.

Examples

# Sample size for count endpoint with nu = 0.8
ss1Count(r1 = 1.0, r2 = 1.25, nu = 0.8, t = 1, r = 1,
         alpha = 0.025, beta = 0.1)
#> 
#> Sample size calculation for single count endpoint
#> 
#>              n1 = 908
#>              n2 = 908
#>               N = 1816
#>            rate = 1, 1.25
#>              nu = 0.8
#>               t = 1
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.1
#> 

# Unbalanced design with 2:1 allocation
ss1Count(r1 = 1.0, r2 = 1.5, nu = 1.0, t = 1, r = 2,
         alpha = 0.025, beta = 0.2)
#> 
#> Sample size calculation for single count endpoint
#> 
#>              n1 = 256
#>              n2 = 128
#>               N = 384
#>            rate = 1, 1.5
#>              nu = 1
#>               t = 1
#>      allocation = 2
#>           alpha = 0.025
#>            beta = 0.2
#> 

# Higher dispersion
ss1Count(r1 = 1.5, r2 = 2.0, nu = 3.0, t = 1, r = 1,
         alpha = 0.025, beta = 0.1)
#> 
#> Sample size calculation for single count endpoint
#> 
#>              n1 = 233
#>              n2 = 233
#>               N = 466
#>            rate = 1.5, 2
#>              nu = 3
#>               t = 1
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.1
#>