# 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
    )

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