Introduction to SingleArmMRCT

Background

Multi-regional clinical trials (MRCTs) are increasingly used in global drug development to allow simultaneous regulatory submissions across multiple regions. A key requirement for regional approval — particularly in Japan under the Japanese MHLW guidelines — is the demonstration of regional consistency: evidence that the treatment effect observed in a specific region (e.g., Japan) is consistent with the overall trial result.

Two widely used consistency evaluation methods, originally proposed under the Japanese guidelines, are:

Method 1 (Effect Retention Approach): Evaluates whether Region 1 retains at least a fraction \(\pi\) of the overall treatment effect.
Method 2 (Simultaneous Positivity Approach): Evaluates whether all regional estimates simultaneously show a positive effect in the direction of benefit.

These methods were originally developed for two-arm randomised controlled trials. However, single-arm trials are now common in oncology and rare disease settings, where historical control comparisons are standard. The SingleArmMRCT package extends Method 1 and Method 2 to the single-arm setting, in which the treatment effect is defined relative to a pre-specified historical control value.

Regional Consistency Probability

The Regional Consistency Probability (RCP) is defined as the probability that a consistency criterion is satisfied, evaluated under the assumed true parameter values at the trial design stage. A trial design is said to have adequate regional consistency if the RCP exceeds a pre-specified target (commonly 0.80).

Method 1: Effect Retention Approach

Let \(\theta\) denote the endpoint parameter for a given endpoint (e.g., mean, proportion, rate). Method 1 requires that Region 1 retains at least a fraction \(\pi\) of the overall treatment effect:

\[ \text{RCP}_1 = \Pr\!\left[\,(\hat{\theta}_1 - \theta_0) \geq \pi \times (\hat{\theta} - \theta_0)\,\right] \]

where \(\hat{\theta}_1\) is the treatment effect estimate for Region 1, \(\hat{\theta}\) is the overall pooled estimate, \(\theta_0\) is the null (historical control) value, and \(\pi \in [0, 1]\) is the pre-specified retention threshold (typically \(\pi = 0.5\)).

The consistency condition can be rewritten as \(D \geq 0\), where:

\[ D = \bigl(1 - \pi f_1\bigr)\,(\hat{\theta}_1 - \theta_0) - \pi(1 - f_1)\,(\hat{\theta}_{-1} - \theta_0) \]

with \(f_1 = N_1/N\) being the regional allocation fraction and \(\hat{\theta}_{-1}\) the pooled estimate for regions \(2, \ldots, J\) combined. Under the assumption of homogeneous treatment effects across regions, \(D\) follows a normal distribution with mean \((1-\pi)\delta\) and a variance that depends on the endpoint type, yielding a closed-form expression for \(\text{RCP}_1\), where \(\delta = \theta - \theta_0\) is the treatment effect.

For endpoints where a smaller value indicates benefit (e.g., hazard ratio, rate ratio), the inequality direction is reversed. See the endpoint-specific vignettes for exact formulae.

Method 2: Simultaneous Positivity Approach

Method 2 requires that all \(J\) regional estimates simultaneously demonstrate a positive effect. For endpoints where a larger value indicates benefit (continuous, binary, milestone survival, RMST):

\[ \text{RCP}_2 = \Pr\!\left[\,\hat{\theta}_j > \theta_0 \;\text{ for all } j = 1, \ldots, J\,\right] \]

For endpoints where a smaller value indicates benefit (hazard ratio, rate ratio):

\[ \text{RCP}_2 = \Pr\!\left[\,\hat{\theta}_j < \theta_0 \;\text{ for all } j = 1, \ldots, J\,\right] \]

Because regional estimators are independent across regions, \(\text{RCP}_2\) factorises as:

\[ \text{RCP}_2 = \prod_{j=1}^{J} \Pr\!\left[\,\hat{\theta}_j \text{ shows benefit}\,\right] \]

Package Structure

The package provides a pair of functions for each of six endpoint types.

Endpoint	Calculation function	Plot function
Continuous	rcp1armContinuous()	plot_rcp1armContinuous()
Binary	rcp1armBinary()	plot_rcp1armBinary()
Count (negative binomial)	rcp1armCount()	plot_rcp1armCount()
Time-to-event (hazard ratio)	rcp1armHazardRatio()	plot_rcp1armHazardRatio()
Milestone survival	rcp1armMilestoneSurvival()	plot_rcp1armMilestoneSurvival()
Restricted mean survival time (RMST)	rcp1armRMST()	plot_rcp1armRMST()

Each calculation function supports two approaches:

"formula": Closed-form or semi-analytical solution based on normal approximation. Computationally fast and, for binary and count endpoints, exact.
"simulation": Monte Carlo simulation. Serves as an independent numerical check of the formula results.

Common Parameters

All six calculation functions share the following parameters.

Parameter	Type	Default	Description
`Nj`	integer vector	—	Sample sizes for each region; length equals the number of regions \(J\)
`PI`	numeric	`0.5`	Effect retention threshold \(\pi\) for Method 1; must be in \([0, 1]\)
`approach`	character	`"formula"`	Calculation approach: `"formula"` or `"simulation"`
`nsim`	integer	`10000`	Number of Monte Carlo iterations; used only when `approach = "simulation"`
`seed`	integer	`1`	Random seed for reproducibility; used only when `approach = "simulation"`

Time-to-event endpoints (hazard ratio, milestone survival, RMST) additionally require the following trial design parameters.

Parameter	Type	Default	Description
`t_a`	numeric	—	Accrual period: duration over which patients are uniformly enrolled
`t_f`	numeric	—	Follow-up period: additional observation time after accrual closes; total study duration is \(\tau = t_a + t_f\)
`lambda_dropout`	numeric or `NULL`	`NULL`	Exponential dropout hazard rate; `NULL` assumes no dropout

Quick Start Example

The following example computes RCP for a continuous endpoint with the setting below:

Parameter	Value
Total sample size	\(N = 100\) (\(J = 2\) regions)
Region 1 allocation	\(N_1 = 10\) (\(f_1 = 10\%\))
True mean	\(\mu = 0.5\)
Historical control mean	\(\mu_0 = 0.1\) (mean difference \(\delta = 0.4\))
Standard deviation	\(\sigma = 1\)
Retention threshold	\(\pi = 0.5\)

Closed-form solution

result_formula <- rcp1armContinuous(
  mu       = 0.5,
  mu0      = 0.1,
  sd       = 1,
  Nj       = c(10, 90),
  PI       = 0.5,
  approach = "formula"
)
print(result_formula)
#> 
#> 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

Monte Carlo simulation

result_sim <- rcp1armContinuous(
  mu       = 0.5,
  mu0      = 0.1,
  sd       = 1,
  Nj       = c(10, 90),
  PI       = 0.5,
  approach = "simulation",
  nsim     = 10000,
  seed     = 1
)
print(result_sim)
#> 
#> 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

The closed-form and simulation results are in close agreement. The small difference is attributable to Monte Carlo sampling variation and diminishes as nsim increases.

Visualisation

Each endpoint type has a corresponding plot_rcp1arm*() function. These functions display RCP as a function of the regional allocation proportion \(f_1 = N_1/N\), with separate facets for different total sample sizes \(N\). Both Method 1 (blue) and Method 2 (yellow) are shown, with solid lines for the formula approach and dashed lines for simulation. The horizontal grey dashed line marks the commonly used design target of RCP \(= 0.80\).

The base_size argument controls font size: use the default (base_size = 28) for presentation slides, and a smaller value (e.g., base_size = 11) for documents and vignettes.

plot_rcp1armContinuous(
  mu        = 0.5,
  mu0       = 0.1,
  sd        = 1,
  PI        = 0.5,
  N_vec     = c(20, 40, 100),
  J         = 3,
  nsim      = 5000,
  seed      = 1,
  base_size = 11
)

Line plot of RCP versus regional allocation proportion f1 for a continuous endpoint, comparing Method 1 and Method 2 using formula and simulation approaches across sample sizes N = 20, 40, and 100

Several features are evident from the plot:

Method 1 (blue) increases with \(f_1\): as Region 1 becomes larger, its estimator \(\hat{\theta}_1\) becomes more precise, making the retention condition easier to satisfy.
Method 2 (yellow) is maximised when all regions have equal allocation \(f_1 = f_2 = \cdots = f_J = 1/J\), and decreases as \(f_1\) deviates from this balance, because unequal allocation reduces the marginal probability \(\Pr(\hat{\theta}_j \text{ shows benefit})\) for the smaller regions.
Both RCP values increase with total sample size \(N\), as expected.
The formula (solid) and simulation (dashed) curves are closely aligned, confirming the accuracy of the normal approximation.

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. https://doi.org/10.5691/jjb.38.79

Homma G (2024). Cautionary note on regional consistency evaluation in multiregional clinical trials with binary outcomes. Pharmaceutical Statistics, 23(3):385–398. https://doi.org/10.1002/pst.2358

Wu J (2015). Sample size calculation for the one-sample log-rank test. Pharmaceutical Statistics, 14(1): 26–33. https://doi.org/10.1002/pst.1654