Regional Consistency Probability for Single-Arm MRCT (Count Endpoint)

Calculate the regional consistency probability (RCP) for count (overdispersed) endpoints in single-arm multi-regional clinical trials (MRCTs) using the Effect Retention Approach (ERA).

Count data are modelled by the negative binomial distribution, and the treatment effect is expressed as a rate ratio (RR) relative to a historical control rate \(\lambda_0\). Two effect scales are considered:

Log-RR scale: \(\log(\widehat{RR}_j) = \log(\hat{\lambda}_j / \lambda_0)\).
Linear-RR scale: \(1 - \widehat{RR}_j = 1 - \hat{\lambda}_j / \lambda_0\).

Two evaluation methods are supported (for each scale):

Method 1: Effect retention approach. Evaluates whether Region 1 retains at least a fraction PI of the overall treatment effect. Log-RR: \(\log(\widehat{RR}_1) < \pi \times \log(\widehat{RR})\); Linear-RR: \((1 - \widehat{RR}_1) > \pi \times (1 - \widehat{RR})\).
Method 2: Simultaneous benefit across all regions. Evaluates whether all regional rate ratios are below 1: \(\widehat{RR}_j < 1\) for all \(j\). (Equivalent for both log-RR and linear-RR scales.)

Two calculation approaches are available:

"formula": Exact closed-form solution via full enumeration of the negative binomial joint distribution. Method 1 uses a two-block decomposition (Region 1 vs regions 2..J combined), which is valid for \(J \geq 2\). Method 2 supports \(J \geq 2\) regions.
"simulation": Monte Carlo simulation. Supports \(J \geq 2\) regions.

Usage

rcp1armCount(
  lambda,
  lambda0,
  dispersion,
  Nj,
  PI = 0.5,
  approach = "formula",
  nsim = 10000,
  seed = 1
)

Arguments

lambda: Numeric scalar. Expected count per patient under the alternative hypothesis. Must be positive.
lambda0: Numeric scalar. Expected count per patient under the historical control (null hypothesis reference value). Must be positive.
dispersion: Numeric scalar. Dispersion parameter (size) of the negative binomial distribution, assumed common across all regions. Smaller values indicate greater overdispersion. Must be positive.
Nj: Integer vector. Sample sizes for each region. For example, c(10, 90) indicates Region 1 has 10 subjects and Region 2 has 90 subjects. All elements must be positive integers.
PI: Numeric scalar. Prespecified effect retention threshold for Method 1. Typically \(\pi \geq 0.5\). Must be in \([0, 1]\). Default is 0.5.
approach: Character scalar. Calculation approach: "formula" for the exact solution or "simulation" for Monte Carlo simulation. Default is "formula".
nsim: Positive integer. Number of Monte Carlo iterations. Used only when approach = "simulation". Default is 10000.
seed: Non-negative integer. Random seed for reproducibility. Used only when approach = "simulation". Default is 1.

Value

An object of class "rcp1armCount", which is a list containing:

approach: Calculation approach used ("formula" or "simulation").
nsim: Number of Monte Carlo iterations (NULL for "formula" approach).
lambda: Expected count per patient under the alternative hypothesis.
lambda0: Expected count per patient under the historical control.
dispersion: Dispersion parameter.
Nj: Sample sizes for each region.
PI: Effect retention threshold.
Method1_logRR: RCP using Method 1 (log-RR scale).
Method1_linearRR: RCP using Method 1 (linear-RR scale).
Method2: RCP using Method 2 (all regions show benefit; identical for log-RR and linear-RR scales).

Examples

# Example 1: Exact solution with N = 100, Region 1 has 10 subjects
result1 <- rcp1armCount(
  lambda     = 2,
  lambda0    = 3,
  dispersion = 1,
  Nj         = c(10, 90),
  PI         = 0.5,
  approach   = "formula"
)
print(result1)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Count (Negative Binomial)
#> 
#>    Approach       : Exact Solution
#>    Expected Count : lambda     = 2.000000
#>    Control Count  : lambda0    = 3.000000
#>    Dispersion     : dispersion = 1.000000
#>    Sample Size    : Nj         = (10, 90)
#>    Total Size     : N          = 100
#>    Threshold      : PI         = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-RR based    : 0.7481
#>       Linear-RR based : 0.7675
#>    Method 2 (All Regions Show Benefit):
#>       RR < 1          : 0.8845
#> 

# Example 2: Monte Carlo simulation with N = 100, Region 1 has 10 subjects
result2 <- rcp1armCount(
  lambda     = 2,
  lambda0    = 3,
  dispersion = 1,
  Nj         = c(10, 90),
  PI         = 0.5,
  approach   = "simulation",
  nsim       = 10000,
  seed       = 1
)
print(result2)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Count (Negative Binomial)
#> 
#>    Approach       : Simulation-Based (nsim = 10000)
#>    Expected Count : lambda     = 2.000000
#>    Control Count  : lambda0    = 3.000000
#>    Dispersion     : dispersion = 1.000000
#>    Sample Size    : Nj         = (10, 90)
#>    Total Size     : N          = 100
#>    Threshold      : PI         = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-RR based    : 0.7448
#>       Linear-RR based : 0.7636
#>    Method 2 (All Regions Show Benefit):
#>       RR < 1          : 0.8810
#>