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

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

Two evaluation methods are supported:

Method 1: Effect retention approach. Evaluates whether Region 1 retains at least a fraction PI of the overall treatment effect: \(Pr[(\hat{\mu}_1 - \mu_0) > \pi \times (\hat{\mu} - \mu_0)]\).
Method 2: Simultaneous positivity across all regions. Evaluates whether all regional estimates exceed the null value: \(Pr[\hat{\mu}_j > \mu_0 \text{ for all } j]\).

Two calculation approaches are available:

"formula": Closed-form analytical solution based on normal approximation. 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

rcp1armContinuous(
  mu,
  mu0,
  sd,
  Nj,
  PI = 0.5,
  approach = "formula",
  nsim = 10000,
  seed = 1
)

Arguments

mu: Numeric scalar. True mean under the alternative hypothesis.
mu0: Numeric scalar. Null hypothesis mean (baseline or historical control). The treatment effect is defined as \(\delta = \mu - \mu_0\).
sd: Numeric scalar. True standard deviation, assumed common across all regions. 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 closed-form 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 "rcp1armContinuous", which is a list containing:

approach: Calculation approach used ("formula" or "simulation").
nsim: Number of Monte Carlo iterations (NULL for "formula" approach).
mu: True mean under the alternative hypothesis.
mu0: Null hypothesis mean.
sd: Standard deviation.
Nj: Sample sizes for each region.
PI: Effect retention threshold.
Method1: RCP using Method 1 (effect retention).
Method2: RCP using Method 2 (all regions positive).

Examples

# Example 1: Closed-form solution with N = 100, Region 1 has 10 subjects
result1 <- rcp1armContinuous(
  mu  = 0.5,
  mu0 = 0.1,
  sd  = 1,
  Nj  = c(10, 90),
  PI  = 0.5,
  approach = "formula"
)
print(result1)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Continuous
#> 
#>    Approach    : Closed-Form Solution
#>    Target Mean : mu  = 0.5000
#>    Null Mean   : mu0 = 0.1000
#>    Std. Dev.   : sd  = 1.0000
#>    Sample Size : Nj  = (10, 90)
#>    Total Size  : N   = 100
#>    Threshold   : PI  = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.7446
#>    Method 2 (All Regions > mu0)    : 0.8970
#> 

# Example 2: Monte Carlo simulation with N = 100, Region 1 has 10 subjects
result2 <- rcp1armContinuous(
  mu   = 0.5,
  mu0  = 0.1,
  sd   = 1,
  Nj   = c(10, 90),
  PI   = 0.5,
  approach = "simulation",
  nsim = 10000,
  seed = 1
)
print(result2)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Continuous
#> 
#>    Approach    : Simulation-Based (nsim = 10000)
#>    Target Mean : mu  = 0.5000
#>    Null Mean   : mu0 = 0.1000
#>    Std. Dev.   : sd  = 1.0000
#>    Sample Size : Nj  = (10, 90)
#>    Total Size  : N   = 100
#>    Threshold   : PI  = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.7421
#>    Method 2 (All Regions > mu0)    : 0.8922
#>