Regional Consistency Probability for Single-Arm MRCT (Time-to-Event Endpoint)

Calculate the regional consistency probability (RCP) for time-to-event endpoints using the hazard ratio (HR) in single-arm multi-regional clinical trials (MRCTs) using the Effect Retention Approach (ERA).

Event times are modelled by the exponential distribution with hazard rate \(\lambda\) (treatment) relative to a known historical control hazard \(\lambda_0\). The treatment effect is expressed as a hazard ratio: \(HR = \lambda / \lambda_0 < 1\) (benefit). Two effect scales are considered:

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

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-HR: \(\log(\widehat{HR}_1) < \pi \times \log(\widehat{HR})\); Linear-HR: \((1 - \widehat{HR}_1) > \pi \times (1 - \widehat{HR})\).
Method 2: Simultaneous benefit across all regions. Evaluates whether all regional hazard ratios are below 1: \(\widehat{HR}_j < 1\) for all \(j\). (Equivalent for both log-HR and linear-HR scales.)

Two calculation approaches are available:

"formula": Closed-form solution based on normal approximation for \(\log(\widehat{HR})\) and the delta method for the linear-HR scale (Hayashi and Itoh 2018). 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 using individual patient data with person-years estimation of the hazard rate. Supports \(J \geq 2\) regions.

Usage

rcp1armHazardRatio(
  lambda,
  lambda0,
  Nj,
  t_a,
  t_f,
  lambda_dropout = NULL,
  PI = 0.5,
  approach = "formula",
  nsim = 10000,
  seed = 1
)

Arguments

lambda: Numeric scalar. True hazard rate under the alternative hypothesis. Must be positive. Under exponential distribution, median survival = \(\log(2) / \lambda\).
lambda0: Numeric scalar. Known hazard rate for the historical control (null hypothesis reference value). 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.
t_a: Numeric scalar. Accrual period (patient enrolment duration). Must be positive.
t_f: Numeric scalar. Follow-up period (additional follow-up after accrual ends). Must be positive.
lambda_dropout: Numeric scalar or NULL. Dropout hazard rate. If NULL (default), no dropout is assumed. If specified, dropout times follow an exponential distribution with rate lambda_dropout.
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 "rcp1armHazardRatio", which is a list containing:

approach: Calculation approach used ("formula" or "simulation").
nsim: Number of Monte Carlo iterations (NULL for "formula" approach).
lambda: True hazard rate under the alternative hypothesis.
lambda0: Historical control hazard rate.
Nj: Sample sizes for each region.
t_a: Accrual period.
t_f: Follow-up period.
tau: Total study duration (\(\tau = t_a + t_f\)).
lambda_dropout: Dropout hazard rate (NA if NULL).
PI: Effect retention threshold.
Method1_logHR: RCP using Method 1 (log-HR scale).
Method1_linearHR: RCP using Method 1 (linear-HR scale).
Method2: RCP using Method 2 (all regions show benefit; identical for log-HR and linear-HR scales).

References

Hayashi N, Itoh Y (2017). A re-examination of Japanese sample size calculation for multi-regional clinical trial evaluating survival endpoint. Japanese Journal of Biometrics, 38(2): 79–92.

Wu J (2015). Sample size calculation for the one-sample log-rank test. Pharmaceutical Statistics, 14(1): 26–33.

Examples

# Example 1: Closed-form solution with N = 100, Region 1 has 10 subjects
result1 <- rcp1armHazardRatio(
  lambda         = log(2) / 10,
  lambda0        = log(2) / 5,
  Nj             = c(10, 90),
  t_a            = 3,
  t_f            = 10,
  lambda_dropout = NULL,
  PI             = 0.5,
  approach       = "formula"
)
print(result1)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Time-to-Event (Hazard Ratio)
#> 
#>    Approach       : Closed-Form Solution
#>    True Hazard    : lambda  = 0.069315
#>    Control Hazard : lambda0 = 0.138629
#>    Sample Size    : Nj      = (10, 90)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = NA
#>    Threshold      : PI      = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-HR based    : 0.8007
#>       Linear-HR based : 0.8358
#>    Method 2 (All Regions Show Benefit):
#>       HR < 1          : 0.9478
#> 

# Example 2: Monte Carlo simulation with N = 100, Region 1 has 10 subjects
result2 <- rcp1armHazardRatio(
  lambda         = log(2) / 10,
  lambda0        = log(2) / 5,
  Nj             = c(10, 90),
  t_a            = 3,
  t_f            = 10,
  lambda_dropout = NULL,
  PI             = 0.5,
  approach       = "simulation",
  nsim           = 10000,
  seed           = 1
)
print(result2)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Time-to-Event (Hazard Ratio)
#> 
#>    Approach       : Simulation-Based (nsim = 10000)
#>    True Hazard    : lambda  = 0.069315
#>    Control Hazard : lambda0 = 0.138629
#>    Sample Size    : Nj      = (10, 90)
#>    Total Size     : N       = 100
#>    Accrual Period : t_a     = 3.00
#>    Follow-up      : t_f     = 10.00
#>    Study Duration : tau     = 13.00
#>    Dropout Hazard : lambda_d = NA
#>    Threshold      : PI      = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-HR based    : 0.8067
#>       Linear-HR based : 0.8444
#>    Method 2 (All Regions Show Benefit):
#>       HR < 1          : 0.9562
#>