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

Calculate the regional consistency probability (RCP) for restricted mean survival time (RMST) endpoints 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). The treatment effect is expressed as the difference in RMST at a prespecified truncation time $\tau^*$: $\delta = \mu(\tau^*) - \mu_0(\tau^*)$, where $\mu_0(\tau^*)$ is the historical control RMST at $\tau^*$.

The regional RMST estimator is defined as the area under the Kaplan-Meier curve up to $\tau^*$: $\hat{\mu}_j(\tau^*) = \int_0^{\tau^*} \hat{S}_j(t)\,dt$.

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 benefit across all regions. Evaluates whether all regional RMST estimates exceed the historical control RMST: $Pr[\hat{\mu}_j > \mu_0 \text{ for all } j]$.

Two calculation approaches are available:

"formula": Normal approximation based on exponential event times and exponential dropout. When $\tau^* \leq t_f$ (the most common setting in practice), the administrative censoring survival function equals 1 on $[0, \tau^*]$ and the variance integral has a closed-form solution. When $\tau^* > t_f$, the integral is evaluated numerically via integrate. Method 1 uses a two-block decomposition (Region 1 vs regions 2..J combined), valid for $J \geq 2$. Method 2 supports $J \geq 2$ regions.
"simulation": Monte Carlo simulation. The RMST for each simulation replicate is computed as the exact area under the Kaplan-Meier step function (sum of rectangles). Supports $J \geq 2$ regions.

Usage

rcp1armRMST(
  lambda,
  tau_star,
  mu0,
  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.
tau_star: Numeric scalar. Truncation time for RMST calculation. Must be positive and no greater than the total study duration $\tau = t_a + t_f$.
mu0: Numeric scalar. Historical control RMST at tau_star, i.e., $\mu_0(\tau^*) = \int_0^{\tau^*} S_0(t)\,dt$. Must be positive and less than tau_star.
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 normal approximation 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 "rcp1armRMST", which is a list containing:

approach: Calculation approach used ("formula" or "simulation").
formula_type: For approach = "formula": either "closed-form" (when $\tau^* \leq t_f$) or "numerical-integration" (when $\tau^* > t_f$). NULL for simulation.
nsim: Number of Monte Carlo iterations (NULL for "formula" approach).
lambda: True hazard rate under the alternative hypothesis.
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.
tau_star: Truncation time for RMST.
mu0: Historical control RMST at tau_star.
mu_est: True RMST under the alternative: $\mu(\tau^*) = (1 - e^{-\lambda \tau^*}) / \lambda$.
Method1: RCP using Method 1 (effect retention).
Method2: RCP using Method 2 (all regions positive).

Details

Variance formula. The asymptotic variance of the RMST estimator $\hat{\mu}_j(\tau^*)$ for a single arm with $n$ subjects is: $$ \mathrm{Var}(\hat{\mu}_j(\tau^*)) \approx \frac{1}{n} \int_0^{\tau^*} \frac{\left(e^{-\lambda t} - e^{-\lambda \tau^*}\right)^2} {\lambda \cdot e^{-\lambda t} \cdot G(t)} \,dt, $$ where $G(t) = P(\mathrm{observed\ time} \geq t)$ is the overall censoring survival function: $$ G(t) = e^{-\lambda_d t} \times G_a(t), \quad G_a(t) = \begin{cases} 1 & 0 \leq t \leq t_f, \\ (\tau-t)/t_a & t_f < t \leq \tau. \end{cases} $$ Closed-form variance ($\tau^* \leq t_f$). When $\tau^* \leq t_f$, $G_a(t) = 1$ on $[0, \tau^*]$ and the integrand reduces to $e^{\lambda_d t}(1 - e^{-\lambda(\tau^*-t)})^2 / \lambda$. Expanding and integrating term by term: $$ v(\tau^*) = \frac{1}{\lambda}\Bigl[ A(\lambda_d) - 2 e^{-\lambda\tau^*} A(\lambda_d + \lambda) + e^{-2\lambda\tau^*} A(\lambda_d + 2\lambda) \Bigr], $$ where $A(r) = (e^{r\tau^*} - 1)/r$ for $r > 0$ and $A(0) = \tau^*$. When there is no dropout ($\lambda_d = 0$) this further reduces to: $$ v(\tau^*) = \tau^* - \frac{2(1-e^{-\lambda\tau^*})}{\lambda} + \frac{1-e^{-2\lambda\tau^*}}{2\lambda}. $$

References

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 (tau_star <= t_f) with N = 100
lam   <- log(2) / 10
lam0  <- log(2) / 5
tstar <- 8
mu0_val <- (1 - exp(-lam0 * tstar)) / lam0

result1 <- rcp1armRMST(
  lambda         = lam,
  tau_star       = tstar,
  mu0            = mu0_val,
  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 : Restricted Mean Survival Time (RMST)
#> 
#>    Approach       : Closed-Form Solution
#>    True Hazard    : lambda  = 0.069315
#>    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
#>    Trunc. Time    : tau*    = 8.00
#>    Control RMST   : mu0     = 4.8339
#>    True RMST      : mu_est  = 6.1408
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.7767
#>    Method 2 (All Regions > mu0)    : 0.9284
#> 

# Example 2: Monte Carlo simulation with N = 100
result2 <- rcp1armRMST(
  lambda         = lam,
  tau_star       = tstar,
  mu0            = mu0_val,
  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 : Restricted Mean Survival Time (RMST)
#> 
#>    Approach       : Simulation-Based (nsim = 10000)
#>    True Hazard    : lambda  = 0.069315
#>    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
#>    Trunc. Time    : tau*    = 8.00
#>    Control RMST   : mu0     = 4.8339
#>    True RMST      : mu_est  = 6.1408
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.7920
#>    Method 2 (All Regions > mu0)    : 0.9335
#>