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).

Package Structure

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

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.

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

Quick Start Example

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

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

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

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
)

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.

Further Reading

For endpoint-specific statistical models, derivations, and worked examples, see the companion vignettes:

• Non-survival endpoints: continuous, binary, and count (negative binomial) endpoints.
• Survival endpoints: hazard ratio, milestone survival probability, and RMST endpoints.

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