Regional Consistency Probability for Single-Arm MRCT (Milestone Survival Endpoint)

Calculate the regional consistency probability (RCP) for milestone survival 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 at a prespecified evaluation time $t_{\mathrm{eval}}$ is expressed as the difference in survival rates: $\delta = S(t_{\mathrm{eval}}) - S_0(t_{\mathrm{eval}})$, where $S_0$ is the historical control survival rate at $t_{\mathrm{eval}}$.

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: $Pr[(\hat{S}_1(t) - S_0(t)) > \pi \times (\hat{S}(t) - S_0(t))]$.
Method 2: Simultaneous benefit across all regions. Evaluates whether all regional Kaplan-Meier estimates exceed the historical control: $Pr[\hat{S}_j(t) > S_0(t) \text{ for all } j]$.

Two calculation approaches are available:

"formula": Closed-form or semi-analytical solution based on the asymptotic variance of the Kaplan-Meier estimator derived from Greenwood's formula. When $t_{\mathrm{eval}} \leq t_f$, the administrative censoring survival function $G_a(t) = 1$ and the variance integral has a closed-form solution. When $t_{\mathrm{eval}} > t_f$, the integral is evaluated numerically via integrate. Method 1 uses a two-block decomposition (Region 1 vs regions 2..J combined). Method 2 supports $J \geq 2$ regions.
"simulation": Monte Carlo simulation with vectorized Kaplan-Meier estimation. Supports $J \geq 2$ regions.

Usage

rcp1armMilestoneSurvival(
  lambda,
  t_eval,
  S0,
  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.
t_eval: Numeric scalar. Evaluation time point for the milestone survival probability. Must be positive.
S0: Numeric scalar. Historical control survival rate at t_eval. Must be in $(0, 1]$.
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 (or semi-analytical) 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 "rcp1armMilestoneSurvival", which is a list containing:

approach: Calculation approach used ("formula" or "simulation").
formula_type: For approach = "formula": either "closed-form" (when $t_{\mathrm{eval}} \leq t_f$) or "numerical-integration" (when $t_{\mathrm{eval}} > t_f$). 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.
eval_time: Milestone evaluation time point.
S0: Historical control survival rate at eval_time.
S_est: True survival rate under the alternative at eval_time: $S(t) = e^{-\lambda t}$.
Method1: RCP using Method 1 (effect retention).
Method2: RCP using Method 2 (all regions positive).

Details

Variance formula. The asymptotic variance of the Kaplan-Meier estimator $\hat{S}_j(t)$ for a single arm with $N_j$ subjects is derived from Greenwood's formula: $$ \mathrm{Var}[\hat{S}_j(t)] \approx \frac{S^2(t)}{N_j} \int_0^t \frac{\lambda(u)}{S(u) \cdot G(u)} \, du, $$ where $G(u) = e^{-\lambda_d u} \cdot G_a(u)$ is the overall censoring survival function, and $$ G_a(u) = \begin{cases} 1 & 0 \leq u \leq t_f, \\ (\tau - u)/t_a & t_f < u \leq \tau. \end{cases} $$ Under the exponential model $S(u) = e^{-\lambda u}$, this simplifies to: $$ \mathrm{Var}[\hat{S}_j(t)] \approx \frac{e^{-2\lambda t}}{N_j} \int_0^t \frac{\lambda \, e^{(\lambda + \lambda_d) u}}{G_a(u)} \, du. $$

Closed-form solution ($t_{\mathrm{eval}} \leq t_f$). When $t_{\mathrm{eval}} \leq t_f$, $G_a(u) = 1$ throughout $[0, t_{\mathrm{eval}}]$, and the integral reduces to: $$ \int_0^t \lambda \, e^{(\lambda + \lambda_d) u} \, du = \frac{\lambda}{\lambda + \lambda_d} \bigl( e^{(\lambda + \lambda_d) t} - 1 \bigr). $$ Therefore: $$ \mathrm{Var}[\hat{S}_j(t)] = \frac{e^{-2\lambda t}}{N_j} \cdot \frac{\lambda}{\lambda + \lambda_d} \bigl( e^{(\lambda + \lambda_d) t} - 1 \bigr). $$ When there is no dropout ($\lambda_d = 0$), this further simplifies to the binomial variance: $$ \mathrm{Var}[\hat{S}_j(t)] = \frac{S(t)(1 - S(t))}{N_j}. $$

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 (t_eval <= t_f) with N = 100
result1 <- rcp1armMilestoneSurvival(
  lambda         = log(2) / 10,
  t_eval         = 8,
  S0             = exp(-log(2) * 8 / 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 : Milestone Survival
#> 
#>    Approach       : Closed-Form Solution (Greenwood)
#>    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
#>    Eval Time      : t_eval  = 8.00
#>    Control Surv   : S0      = 0.3299
#>    True Surv      : S_est   = 0.5743
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.7918
#>    Method 2 (All Regions > S0)     : 0.9410
#> 

# Example 2: Monte Carlo simulation with N = 100
result2 <- rcp1armMilestoneSurvival(
  lambda         = log(2) / 10,
  t_eval         = 8,
  S0             = exp(-log(2) * 8 / 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 : Milestone Survival
#> 
#>    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
#>    Eval Time      : t_eval  = 8.00
#>    Control Surv   : S0      = 0.3299
#>    True Surv      : S_est   = 0.5743
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.7922
#>    Method 2 (All Regions > S0)     : 0.9224
#>