Plot Regional Consistency Probability for Single-Arm MRCT (Binary Endpoint)

Description

Generate a faceted plot of Regional Consistency Probability (RCP) as a function of the regional allocation proportion f_1 for binary endpoints. Formula and simulation results are shown together for both Method 1 and Method 2. Facet columns correspond to total sample sizes specified in N_vec.

Regional sample sizes are allocated as: N_{j1} = \lfloor N \times f_1 \rfloor and N_{j2} = \cdots = N_{jJ} = (N - N_{j1}) / (J - 1).

Usage

plot_rcp1armBinary(
  p = 0.5,
  p0 = 0.2,
  PI = 0.5,
  N_vec = c(20, 40, 100),
  J = 3,
  f1_seq = seq(0.1, 0.9, by = 0.1),
  nsim = 10000,
  seed = 1,
  base_size = 28
)

Arguments

Value

A ggplot2 object.

Examples

p <- plot_rcp1armBinary(
  p     = 0.5,
  p0    = 0.2,
  PI    = 0.5,
  N_vec = c(20, 40, 100),
  J     = 3
)
print(p)

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

Description

Generate a faceted plot of Regional Consistency Probability (RCP) as a function of the regional allocation proportion f_1 for continuous endpoints. Formula and simulation results are shown together for both Method 1 and Method 2. Facet columns correspond to total sample sizes specified in N_vec.

Regional sample sizes are allocated as: N_{j1} = \lfloor N \times f_1 \rfloor and N_{j2} = \cdots = N_{jJ} = (N - N_{j1}) / (J - 1).

Usage

plot_rcp1armContinuous(
  mu = 0.5,
  mu0 = 0.1,
  sd = 1,
  PI = 0.5,
  N_vec = c(20, 40, 100),
  J = 3,
  f1_seq = seq(0.1, 0.9, by = 0.1),
  nsim = 10000,
  seed = 1,
  base_size = 28
)

Arguments

Value

A ggplot2 object.

Examples

p <- plot_rcp1armContinuous(
  mu  = 0.5,
  mu0 = 0.1,
  sd  = 1,
  PI  = 0.5,
  N_vec = c(20, 40, 100),
  J   = 3
)
print(p)

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

Description

Generate a faceted plot of Regional Consistency Probability (RCP) as a function of the regional allocation proportion f_1 for count endpoints (negative binomial model). Formula and simulation results are shown for Method 1 (log-RR and linear-RR scales) and Method 2. Facet rows correspond to the two Method 1 scales (\log(RR) and 1 - RR), and facet columns correspond to total sample sizes specified in N_vec.

Regional sample sizes are allocated as: N_{j1} = \lfloor N \times f_1 \rfloor and N_{j2} = \cdots = N_{jJ} = (N - N_{j1}) / (J - 1).

Usage

plot_rcp1armCount(
  lambda = 2,
  lambda0 = 3,
  dispersion = 1,
  PI = 0.5,
  N_vec = c(20, 40, 100),
  J = 3,
  f1_seq = seq(0.1, 0.9, by = 0.1),
  nsim = 10000,
  seed = 1,
  base_size = 28
)

Arguments

Value

A ggplot2 object.

Examples

p <- plot_rcp1armCount(
  lambda     = 2,
  lambda0    = 3,
  dispersion = 1,
  PI         = 0.5,
  N_vec      = c(20, 40, 100),
  J          = 3
)
print(p)

Plot Regional Consistency Probability for Single-Arm MRCT (Hazard Ratio Endpoint)

Description

Generate a faceted plot of Regional Consistency Probability (RCP) as a function of the regional allocation proportion f_1 for hazard ratio endpoints. Formula and simulation results are shown for Method 1 (log-HR and linear-HR scales) and Method 2. Facet rows correspond to the two Method 1 scales (\log(HR) and 1 - HR), and facet columns correspond to total sample sizes specified in N_vec.

Regional sample sizes are allocated as: N_{j1} = \lfloor N \times f_1 \rfloor and N_{j2} = \cdots = N_{jJ} = (N - N_{j1}) / (J - 1).

Usage

plot_rcp1armHazardRatio(
  lambda = log(2)/10,
  lambda0 = log(2)/5,
  t_a = 3,
  t_f = 10,
  lambda_dropout = NULL,
  PI = 0.5,
  N_vec = c(20, 40, 100),
  J = 3,
  f1_seq = seq(0.1, 0.9, by = 0.1),
  nsim = 10000,
  seed = 1,
  base_size = 28
)

Arguments

Value

A ggplot2 object.

Examples

p <- plot_rcp1armHazardRatio(
  lambda  = log(2) / 10,
  lambda0 = log(2) / 5,
  t_a     = 3,
  t_f     = 10,
  PI      = 0.5,
  N_vec   = c(20, 40, 100),
  J       = 3
)
print(p)

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

Description

Generate a faceted plot of Regional Consistency Probability (RCP) as a function of the regional allocation proportion f_1 for milestone survival endpoints. Formula and simulation results are shown together for both Method 1 and Method 2. Facet columns correspond to total sample sizes specified in N_vec.

Regional sample sizes are allocated as: N_{j1} = \lfloor N \times f_1 \rfloor and N_{j2} = \cdots = N_{jJ} = (N - N_{j1}) / (J - 1).

Usage

plot_rcp1armMilestoneSurvival(
  lambda = log(2)/10,
  t_eval = 8,
  S0 = exp(-log(2) * 8/5),
  t_a = 3,
  t_f = 10,
  lambda_dropout = NULL,
  PI = 0.5,
  N_vec = c(20, 40, 100),
  J = 3,
  f1_seq = seq(0.1, 0.9, by = 0.1),
  nsim = 10000,
  seed = 1,
  base_size = 28
)

Arguments

Value

A ggplot2 object.

Examples

p <- plot_rcp1armMilestoneSurvival(
  lambda = log(2) / 10,
  t_eval = 8,
  S0     = exp(-log(2) * 8 / 5),
  t_a    = 3,
  t_f    = 10,
  PI     = 0.5,
  N_vec  = c(20, 40, 100),
  J      = 3
)
print(p)

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

Description

Generate a faceted plot of Regional Consistency Probability (RCP) as a function of the regional allocation proportion f_1 for restricted mean survival time (RMST) endpoints. Formula and simulation results are shown together for both Method 1 and Method 2. Facet columns correspond to total sample sizes specified in N_vec.

Regional sample sizes are allocated as: N_{j1} = \lfloor N \times f_1 \rfloor and N_{j2} = \cdots = N_{jJ} = (N - N_{j1}) / (J - 1).

Usage

plot_rcp1armRMST(
  lambda = log(2)/10,
  tau_star = 8,
  mu0 = (1 - exp(-log(2)/5 * 8))/(log(2)/5),
  t_a = 3,
  t_f = 10,
  lambda_dropout = NULL,
  PI = 0.5,
  N_vec = c(20, 40, 100),
  J = 3,
  f1_seq = seq(0.1, 0.9, by = 0.1),
  nsim = 10000,
  seed = 1,
  base_size = 28
)

Arguments

Value

A ggplot2 object.

Examples

lam0    <- log(2) / 5
tstar   <- 8
mu0_val <- (1 - exp(-lam0 * tstar)) / lam0

p <- plot_rcp1armRMST(
  lambda   = log(2) / 10,
  tau_star = tstar,
  mu0      = mu0_val,
  t_a      = 3,
  t_f      = 10,
  PI       = 0.5,
  N_vec    = c(20, 40, 100),
  J        = 3
)
print(p)

Print Method for rcp1armBinary Objects

Description

Print Method for rcp1armBinary Objects

Usage

## S3 method for class 'rcp1armBinary'
print(x, ...)

Arguments

Value

Invisibly returns the input object x.

Print Method for rcp1armContinuous Objects

Description

Print Method for rcp1armContinuous Objects

Usage

## S3 method for class 'rcp1armContinuous'
print(x, ...)

Arguments

Value

Invisibly returns the input object x.

Print Method for rcp1armCount Objects

Description

Print Method for rcp1armCount Objects

Usage

## S3 method for class 'rcp1armCount'
print(x, ...)

Arguments

Value

Invisibly returns the input object x.

Print Method for rcp1armHazardRatio Objects

Description

Print Method for rcp1armHazardRatio Objects

Usage

## S3 method for class 'rcp1armHazardRatio'
print(x, ...)

Arguments

Value

Invisibly returns the input object x.

Print Method for rcp1armMilestoneSurvival Objects

Description

Print Method for rcp1armMilestoneSurvival Objects

Usage

## S3 method for class 'rcp1armMilestoneSurvival'
print(x, ...)

Arguments

Value

Invisibly returns the input object x.

Print Method for rcp1armRMST Objects

Description

Print Method for rcp1armRMST Objects

Usage

## S3 method for class 'rcp1armRMST'
print(x, ...)

Arguments

Value

Invisibly returns the input object x.

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

Description

Calculate the regional consistency probability (RCP) for binary 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{p}_1 - p_0) > \pi \times (\hat{p} - p_0)].

• Method 2: Simultaneous positivity across all regions. Evaluates whether all regional response rates exceed the null value: Pr[\hat{p}_j > p_0 \text{ for all } j].

Two calculation approaches are available:

• "formula": Exact closed-form solution via full enumeration of the 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

rcp1armBinary(
  p,
  p0,
  Nj,
  PI = 0.5,
  approach = "formula",
  nsim = 10000,
  seed = 1
)

Arguments

Value

An object of class "rcp1armBinary", which is a list containing:

approach

Calculation approach used ("formula" or "simulation").

nsim

Number of Monte Carlo iterations (NULL for "formula" approach).

p

True response rate under the alternative hypothesis.

p0

Null hypothesis response rate.

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: Exact solution with N = 100, Region 1 has 10 subjects
result1 <- rcp1armBinary(
  p  = 0.5,
  p0 = 0.2,
  Nj = c(10, 90),
  PI = 0.5,
  approach = "formula"
)
print(result1)

# Example 2: Monte Carlo simulation with N = 100, Region 1 has 10 subjects
result2 <- rcp1armBinary(
  p    = 0.5,
  p0   = 0.2,
  Nj   = c(10, 90),
  PI   = 0.5,
  approach = "simulation",
  nsim = 10000,
  seed = 1
)
print(result2)

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

Description

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

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)

# 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 (Count Endpoint)

Description

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

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)

# 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 (Time-to-Event Endpoint)

Description

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

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)

# 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 (Milestone Survival Endpoint)

Description

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

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}.

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).

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)

# 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 (RMST Endpoint)

Description

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

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}.

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).

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)

# 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)