Package {ssMRCT}

Regional consistency and sample size for MRCTs

Description

Tools for multi-regional clinical trial (MRCT) design and regional treatment-effect estimation. CP denotes a consistency probability conditional on overall trial significance, not overall power or interval coverage.

Choosing a function

conProb, conProb2

Fixed-effects Method I CP for one or two trials.

regFrac, regFrac2

Regional fraction for a target Method I CP. The two-trial solver requires a symmetric design.

conProbII, conProbII2

Probability that all regional effects are nonnegative, for one or two trials.

ssCPnorm, ssCPbinary

Random-effects design for normal or binary endpoints.

ssCPsurvPH, ssCPsurvNPH

Random-effects design for survival endpoints.

CPlbd

Lower bound on random-effects Method I CP.

regATE_dr

Covariate-adjusted regional ATE from participant-level data.

Conventions

alpha is one-sided. Fixed-effects functions take power; random-effects functions take beta = 1 - power. Allocation ratios are treatment/control. Higher treatment effects indicate benefit; PH effects are negative log hazard ratios.

Analytical CP uses the continuous design. Returned integer sample sizes are recommendations, not a re-evaluation of CP after regional rounding. Binary Method II simulation instead uses the returned actual regional arm counts.

Author(s)

Maintainer: Kunhai Qing kunhaiq@stu.ecnu.edu.cn

Authors:

• Xinru Ren

• Jin Xu

References

Qing, Kunhai, Xinru Ren, Shuping Jiang, Ping Yang, Menggang Yu, and Jin Xu. 2025. “Regional Consistency Evaluation and Sample Size Calculation under Two MRCTs.” Statistics in Medicine 44 (25–27): e70306. doi:10.1002/sim.70306.

Ren, Xinru, and Jin Xu. 2026. “Consistency Assessment and Regional Sample Size Calculation for MRCT under Random Effects Model.” Pharmaceutical Statistics 25 (3): e70090. doi:10.1002/pst.70090.

Random-effects consistency bound: Method I

Description

Calculate the Method I CP lower bound from Ren and Xu (2026) and the regional normal-endpoint counts at its equality condition.

Usage

CPlbd(
  alpha,
  beta,
  tau_delta_r,
  delta,
  sigma0r,
  sigma1r,
  fr,
  pi = 0.5,
  randRatio = 1
)

Arguments

Details

Let z=z_{1-\alpha}+z_{1-\beta} and a=(\tau/\delta)^2z^2/2. If a<1, the function uses the bound in equation (13), with regional control count at equality

n_{0r}=\frac{\Omega_r}{(1/a-1)\tau^2}.

If a\ge1, it uses equation (14), and equality counts are NA. These counts describe the regional equality condition, not a complete multi-region allocation satisfying the overall design equation.

Value

A list with:

CPlbd

Scalar CP lower bound, without rounding.

n0r, n1r

Rounded regional control and treatment counts.

nr

Regional total, n0r + n1r.

Sample-size entries are numeric NA when the equality condition fails.

References

Ren, Xinru, and Jin Xu. 2026. “Consistency Assessment and Regional Sample Size Calculation for MRCT under Random Effects Model.” Pharmaceutical Statistics 25 (3): e70090. doi:10.1002/pst.70090.

See Also

ssCPnorm().

Examples

CPlbd(alpha = 0.05, beta = 0.2, tau_delta_r = 0.4, delta = 0.25,
      sigma0r = 1, sigma1r = 1, fr = c(0.1, 0.5))

Consistency probability for one MRCT: Method I

Description

Calculate the probability that a regional treatment effect preserves a specified fraction of the overall effect, conditional on overall significance, under a fixed-effects model.

Usage

conProb(
  alpha,
  power,
  pi = 0.5,
  rF,
  d,
  sigmaTrt,
  sigmaCtrl = sigmaTrt,
  randRatio = 1
)

Arguments

Details

With regional effect estimate D_r and overall estimate D,

\mathrm{CP} = \Pr(D_r \ge \pi D \mid T > z_{1-\alpha}).

The normal approximation is integrated numerically. CP depends on alpha, power, pi and rF; the other inputs determine sample size.

Value

A list with CP (conditional consistency probability) and N (recommended total sample size).

Sample sizes

CP uses the continuous design at the requested power. The recommended control count is rounded up first, followed by the treatment count. Returned counts do not re-evaluate CP after integer regional allocation.

References

Qing, Kunhai, Xinru Ren, Shuping Jiang, Ping Yang, Menggang Yu, and Jin Xu. 2025. “Regional Consistency Evaluation and Sample Size Calculation under Two MRCTs.” Statistics in Medicine 44 (25–27): e70306. doi:10.1002/sim.70306.

See Also

regFrac() to solve for a regional fraction; conProb2() for two trials.

Examples

conProb(alpha = 0.05, power = 0.8, rF = 0.271,
        d = 1, sigmaTrt = 4)

Pooled consistency probability for two MRCTs: Method I

Description

Calculate Method I consistency probability from pooled data, conditional on both independent trials being significant.

Usage

conProb2(
  alpha,
  power1,
  power2 = power1,
  pi = 0.5,
  rF1,
  rF2 = rF1,
  d1,
  d2 = d1,
  muCtrl1 = 0,
  muCtrl2 = muCtrl1,
  sigmaTrt1,
  sigmaCtrl1 = sigmaTrt1,
  sigmaTrt2 = sigmaTrt1,
  sigmaCtrl2 = sigmaTrt2,
  randRatio1 = 1,
  randRatio2 = randRatio1
)

Arguments

Details

Treatment and control outcomes are pooled separately using their sample sizes. Overall and regional pooling weights can differ.

\Pr(D_{r,\mathrm{pool}} \ge \pi D_{\mathrm{pool}} \mid T_1 > z_{1-\alpha}, T_2 > z_{1-\alpha}).

The conditional normal probability is integrated over both significant study results. Study 2 defaults to the corresponding study 1 inputs.

Value

A list with CP, N1 and N2: the conditional consistency probability and recommended total sample sizes in studies 1 and 2.

Sample sizes

CP uses the continuous design at the requested power. The recommended control count is rounded up first, followed by the treatment count. Returned counts do not re-evaluate CP after integer regional allocation.

References

Qing, Kunhai, Xinru Ren, Shuping Jiang, Ping Yang, Menggang Yu, and Jin Xu. 2025. “Regional Consistency Evaluation and Sample Size Calculation under Two MRCTs.” Statistics in Medicine 44 (25–27): e70306. doi:10.1002/sim.70306.

See Also

conProb(), regFrac2().

Examples

conProb2(alpha = 0.05, power1 = 0.8, power2 = 0.9,
         rF1 = 0.141, d1 = 1, sigmaTrt1 = 4)

Consistency probability for one MRCT: Method II

Description

Calculate Method II consistency probability: all regional effect estimates are nonnegative, conditional on overall significance.

Usage

conProbII(
  alpha,
  power,
  rF,
  d,
  sigmaTrt,
  sigmaCtrl = sigmaTrt,
  pTrt,
  pCtrl,
  randRatio = 1,
  responseType = c("continuous", "binary"),
  B = 1e+05
)

Arguments

Details

\mathrm{CP} = \Pr(D_r \ge 0 \text{ for all }r \mid T > z_{1-\alpha}).

Continuous endpoints use a joint normal probability, retaining dependence between regional and overall effects. Numerical integration can vary slightly with the random seed.

Value

A list with CP and total sample size N. For binary endpoints, N is the actual simulated total and the list also contains:

nTrt, nCtrl

Actual overall treatment and control counts.

nTrtReg, nCtrlReg

Actual regional treatment and control counts.

achievedPower

Simulated probability of the conditioning event.

MCSE

Monte Carlo standard error of CP.

successes

Number of replicates meeting the conditioning event.

consistentSuccesses

Number also meeting regional consistency.

B

Number of simulation replicates.

Sample sizes

CP uses the continuous design at the requested power. The recommended control count is rounded up first, followed by the treatment count. Returned counts do not re-evaluate CP after integer regional allocation.

Binary calculation

The function simulates regional binomial counts with separately rounded arm sizes. CP is the number of consistent overall successes divided by the number of overall successes. Zero regional effects count as consistent. Wald tests use the actual arm counts. With zero estimated variance, only a positive estimated effect rejects the null.

No overall successes gives NA for CP and its Monte Carlo standard error, with a warning. Set the random seed for reproducibility.

References

Qing, Kunhai, Xinru Ren, Shuping Jiang, Ping Yang, Menggang Yu, and Jin Xu. 2025. “Regional Consistency Evaluation and Sample Size Calculation under Two MRCTs.” Statistics in Medicine 44 (25–27): e70306. doi:10.1002/sim.70306.

See Also

conProbII2(), conProb().

Examples

set.seed(42)
conProbII(alpha = 0.05, power = 0.8, rF = rep(1/3, 3),
          d = 1, sigmaTrt = 4)
conProbII(alpha = 0.05, power = 0.8, rF = rep(1/3, 3),
          pTrt = 0.8, pCtrl = 0.7, responseType = "binary", B = 10000)

Pooled consistency probability for two MRCTs: Method II

Description

Calculate Method II consistency probability from pooled regional data, conditional on both independent trials being significant.

Usage

conProbII2(
  alpha,
  power1,
  power2 = power1,
  rF1,
  rF2 = rF1,
  d1,
  d2 = d1,
  sigmaTrt1,
  sigmaCtrl1 = sigmaTrt1,
  sigmaTrt2 = sigmaTrt1,
  sigmaCtrl2 = sigmaTrt2,
  pTrt1,
  pCtrl1,
  pTrt2 = pTrt1,
  pCtrl2 = pCtrl1,
  randRatio1 = 1,
  randRatio2 = randRatio1,
  responseType = c("continuous", "binary"),
  B = 1e+05,
  muCtrl1 = 0,
  muCtrl2 = muCtrl1
)

Arguments

Details

Consistency requires every pooled regional effect to be nonnegative. Treatment and control outcomes are pooled separately. The conditioning event is significance in both trials, not in a single pooled test. For continuous endpoints, a joint normal calculation accounts for regional/overall dependence and different control means. Set the seed for reproducible numerical integration or simulation.

Value

A list with CP, N1 and N2. For binary endpoints, N1 and N2 are the actual simulated totals and additional entries are:

nTrtReg1, nCtrlReg1

Regional arm counts in study 1.

nTrtReg2, nCtrlReg2

Regional arm counts in study 2.

achievedPower

Simulated probability of the conditioning event.

MCSE

Monte Carlo standard error of CP.

successes

Number of replicates meeting the conditioning event.

consistentSuccesses

Number also meeting regional consistency.

B

Number of simulation replicates.

Here achievedPower is the simulated probability that both studies succeed.

Sample sizes

CP uses the continuous design at the requested power. The recommended control count is rounded up first, followed by the treatment count. Returned counts do not re-evaluate CP after integer regional allocation.

Binary calculation

The function simulates regional binomial counts with separately rounded arm sizes. CP is the number of consistent overall successes divided by the number of overall successes. Zero regional effects count as consistent. Wald tests use the actual arm counts. With zero estimated variance, only a positive estimated effect rejects the null.

No overall successes gives NA for CP and its Monte Carlo standard error, with a warning. Set the random seed for reproducibility.

References

Qing, Kunhai, Xinru Ren, Shuping Jiang, Ping Yang, Menggang Yu, and Jin Xu. 2025. “Regional Consistency Evaluation and Sample Size Calculation under Two MRCTs.” Statistics in Medicine 44 (25–27): e70306. doi:10.1002/sim.70306.

See Also

conProbII(), conProb2().

Examples

set.seed(42)
conProbII2(alpha = 0.05, power1 = 0.8, rF1 = rep(1/3, 3),
           d1 = 1, sigmaTrt1 = 4)

Doubly robust estimation of a regional treatment effect

Description

Estimate the average treatment effect (ATE) in a target region using covariate information from other regions. Optionally screen source regions before borrowing information.

Usage

regATE_dr(
  dat,
  target_region = 1,
  compute_se = TRUE,
  region_wise = FALSE,
  p1 = NULL,
  p0 = NULL,
  pi = NULL,
  v1 = "const",
  v0 = "const",
  config = list()
)

Arguments

Details

Let D_i indicate target-region membership and \widehat q=n^{-1}\sum_i D_i. The estimator is

\widehat\tau = \frac{1}{n\widehat q}\sum_{i=1}^n \left[D_i\widehat\Delta(X_i)+\widehat W_{1i}\widehat U_{1i} -\widehat W_{0i}\widehat U_{0i}\right].

Here \widehat\Delta=\widehat m_1-\widehat m_0, \widehat U_t=Y-\widehat m_t(X), and the residual weights combine selection scores, treatment assignment and variance ratios. Outcome models are linear; estimated probability models are logistic.

Borrowing assumes conditional outcome-mean exchangeability across the regions retained. Double robustness concerns correct specification of either outcome models or both selection and treatment-assignment models; it does not remove the exchangeability assumption.

Value

A list of class regATE_dr with:

estimate

ATE using all candidate source regions.

se_eif, ci

Standard error and 95% Wald interval, or NULL if disabled.

remaining_regions

All candidate source-region labels.

config

Screening settings used.

With region_wise = TRUE, the list also contains:

check_results

Source labels, arm-specific p-values and pass indicators.

excluded_by_check

Excluded source labels.

region_wise_estimate

ATE after source screening.

region_wise_se, region_wise_ci

Its standard error and 95% interval.

region_wise_remaining_regions

Retained source labels.

Source screening

Each arm compares y ~ X with y ~ X + D_src + X:D_src. A source is excluded if either p-value is below check_p_threshold. Treatment probability vectors are subset with the retained rows. Selection probabilities are re-estimated after exclusions, because the population changes. If no sources remain, the estimator uses only the target region. The reported influence-function interval does not account for uncertainty from selecting the source regions.

Examples

set.seed(42)
dat <- data.frame(region = rep(1:3, each = 100),
                  trt = rep(c(0, 1), 150), x = rnorm(300))
dat$y <- dat$x + dat$trt + rnorm(300)
regATE_dr(dat, target_region = 1, p1 = 0.5, p0 = 0.5)

Regional fraction for one MRCT: Method I

Description

Find the smallest continuous regional fraction attaining a target consistency probability under the fixed-effects model.

Usage

regFrac(
  alpha,
  power,
  pi = 0.5,
  CP = 0.8,
  d = 1,
  sigmaTrt = 4,
  sigmaCtrl = sigmaTrt,
  randRatio = 1
)

Arguments

Details

Solves conProb(...)$CP = CP for rF. A target of one returns rF = 1. Fractions are not converted to integer regional arm counts.

Value

A list with rF (regional fraction) and N (recommended total count).

Sample sizes

CP uses the continuous design at the requested power. The recommended control count is rounded up first, followed by the treatment count. Returned counts do not re-evaluate CP after integer regional allocation.

See Also

conProb(), regFrac2().

Examples

regFrac(alpha = 0.05, power = 0.8, CP = 0.8)

Regional fraction for two symmetric MRCTs: Method I

Description

Solve for a common regional fraction attaining the target Method I consistency probability in two independent, symmetric trials.

Usage

regFrac2(alpha, power, pi = 0.5, CP = 0.8)

Arguments

Details

Both trials must have the same treatment effect, arm-specific variances, randomization ratio, power and regional fraction. The function solves for that common fraction. It does not optimize two unequal regional fractions. A target of one returns the full-region boundary.

Value

A numeric scalar: the common regional fraction for both trials.

References

Qing, Kunhai, Xinru Ren, Shuping Jiang, Ping Yang, Menggang Yu, and Jin Xu. 2025. “Regional Consistency Evaluation and Sample Size Calculation under Two MRCTs.” Statistics in Medicine 44 (25–27): e70306. doi:10.1002/sim.70306.

See Also

conProb2() for a general two-trial CP calculation.

Examples

fraction <- regFrac2(alpha = 0.05, power = 0.8)
conProb2(alpha = 0.05, power1 = 0.8, rF1 = fraction,
         d1 = 1, sigmaTrt1 = 4)

Random-effects design for a binary endpoint: Method I

Description

Calculate overall sample size and shrinkage-based Method I consistency probability from regional response probabilities.

Usage

ssCPbinary(alpha, beta, p0r, p1r, pi = 0.5, randRatio = 1, target, f)

Arguments

Details

Regional effects are p1r - p0r. Their unweighted mean and sample standard deviation define delta and tau; the mean must be positive. The sampling variance component is

\Omega_r = p_{1r}(1-p_{1r})/\ell+p_{0r}(1-p_{0r}).

This function uses a normal random-effects approximation, not binomial simulation. For fixed-effects Method II simulation, use conProbII().

Value

A list with:

n0, n1

Recommended overall control and treatment counts.

n

Total count, n0 + n1.

CP

Conditional consistency probability, without rounding.

Design assumptions

Regional inputs may be scalars or vectors of length length(f). The calculation uses a normal random-effects approximation and the sample-size equation in ssCPnorm(). Invalid inputs and designs with no finite solution raise errors. CP uses the continuous design; returned group counts are rounded recommendations.

References

Ren, Xinru, and Jin Xu. 2026. “Consistency Assessment and Regional Sample Size Calculation for MRCT under Random Effects Model.” Pharmaceutical Statistics 25 (3): e70090. doi:10.1002/pst.70090.

See Also

ssCPnorm(), CPlbd().

Examples

ssCPbinary(alpha = 0.025, beta = 0.2, p0r = 0.3,
           p1r = c(0.6, 0.55, 0.5), target = 1, f = rep(1/3, 3))

Random-effects design for a normal endpoint: Method I

Description

Calculate overall sample size and conditional Method I consistency probability for an empirical shrinkage regional estimator.

Usage

ssCPnorm(
  alpha,
  beta,
  delta,
  tau,
  sigma1r,
  sigma0r,
  pi = 0.5,
  randRatio = 1,
  target,
  f
)

Arguments

Details

Let \ell be randRatio, z = z_{1-\alpha}+z_{1-\beta} and \Omega_r = \sigma_{1r}^2/\ell+\sigma_{0r}^2. The continuous control count solves

\sum_r \left(\tau^2+\frac{\Omega_r}{n_0 f_r}\right)^{-1} = \frac{z^2}{\delta^2}.

A finite solution requires \tau z/\delta < \sqrt{R}, where R is the number of regions.

The target-region criterion is

\Pr(\widetilde D_r \ge \pi\widetilde D \mid T > z_{1-\alpha}).

It uses the shrinkage estimator, not the raw regional mean difference. At tau = 0, regional and overall shrinkage estimates coincide, so CP is one.

Value

A list with:

n0, n1

Recommended overall control and treatment counts.

n

Total count, n0 + n1.

CP

Conditional consistency probability, without rounding.

Design assumptions

Regional inputs may be scalars or vectors of length length(f). The calculation uses a normal random-effects approximation and the sample-size equation above. Invalid inputs and designs with no finite solution raise errors. CP uses the continuous design; returned group counts are rounded recommendations.

References

Ren, Xinru, and Jin Xu. 2026. “Consistency Assessment and Regional Sample Size Calculation for MRCT under Random Effects Model.” Pharmaceutical Statistics 25 (3): e70090. doi:10.1002/pst.70090.

See Also

ssCPbinary(), ssCPsurvPH(), ssCPsurvNPH(), CPlbd().

Examples

ssCPnorm(alpha = 0.025, beta = 0.2, delta = 0.25, tau = 0.1,
         sigma1r = 1, sigma0r = 1, target = 1, f = rep(1/3, 3))

Random-effects design using RMST: Method I

Description

Calculate sample size and shrinkage-based Method I consistency probability for a treatment-minus-control restricted mean survival time (RMST) difference, allowing non-proportional hazards.

Usage

ssCPsurvNPH(
  alpha,
  beta,
  lambda0a,
  lambda0b,
  lambda1a,
  lambda1b,
  breakpoint,
  shape0,
  scale0,
  shape1,
  scale1,
  PW = TRUE,
  unimin,
  unimax,
  pi = 0.5,
  randRatio = 1,
  target,
  f,
  eta
)

Arguments

Details

Supply only the survival parameters needed by PW. All regional model parameters can be scalars or vectors of length length(f). Use the same time units for eta, breakpoint, censoring limits and scales.

With PW = TRUE, hazards change once at breakpoint. With PW = FALSE, Weibull survival is S(t)=\exp\{-(t/\mathrm{scale})^{\mathrm{shape}}\}. Scale equals the reciprocal exponential rate only when shape is one. Event and censoring times are independent. Before unimin, censoring survival equals one.

Regional effects are RMST differences; their mean and sample standard deviation define delta and tau. The mean must be positive. RMST variance is integrated numerically, with \Omega_r=\sigma_{1r}^2/\ell+\sigma_{0r}^2.

Value

A list containing n0, n1, n and CP as in ssCPnorm(), plus:

Dr

Treatment-minus-control RMST difference in each region.

sigma2.1r, sigma2.0r

Treatment and control RMST variance coefficients. Divide each by its regional arm count to obtain the estimator variance.

Design assumptions

Regional inputs may be scalars or vectors of length length(f). The calculation uses a normal random-effects approximation and the sample-size equation in ssCPnorm(). Invalid inputs and designs with no finite solution raise errors. CP uses the continuous design; returned group counts are rounded recommendations.

References

Ren, Xinru, and Jin Xu. 2026. “Consistency Assessment and Regional Sample Size Calculation for MRCT under Random Effects Model.” Pharmaceutical Statistics 25 (3): e70090. doi:10.1002/pst.70090.

See Also

ssCPsurvPH(), ssCPnorm().

Examples

# Piecewise exponential survival
ssCPsurvNPH(alpha = 0.025, beta = 0.2,
            lambda0a = 0.07, lambda0b = 0.04,
            lambda1a = c(0.02, 0.03, 0.04), lambda1b = 0.04,
            breakpoint = 10, unimin = 0, unimax = 240,
            target = 1, f = rep(1/3, 3), eta = 80)

# Weibull survival: shape = 1 gives exponential survival
ssCPsurvNPH(alpha = 0.025, beta = 0.2, PW = FALSE,
            shape0 = 1, scale0 = 10, shape1 = 1, scale1 = 15,
            unimin = 20, unimax = 30, eta = 10,
            target = 1, f = rep(1/3, 3))

Random-effects design for a PH endpoint: Method I

Description

Calculate sample size and shrinkage-based Method I consistency probability for exponential survival with a common fixed follow-up time.

Usage

ssCPsurvPH(alpha, beta, lambda0r, hr, pi = 0.5, randRatio = 1, target, f, L)

Arguments

Details

All participants are followed for L time units. Administrative censoring at L is the only censoring mechanism. Regional effects are -log(hr); their mean and sample standard deviation define delta and tau. With event probabilities P_{kr}=1-\exp(-\lambda_{kr}L),

\Omega_r=\frac{(\ell+1)^2}{\ell(P_{0r}+\ell P_{1r})}.

For delayed effects or crossing survival curves, use ssCPsurvNPH().

Value

A list with:

n0, n1

Recommended overall control and treatment counts.

n

Total count, n0 + n1.

CP

Conditional consistency probability, without rounding.

Design assumptions

Regional inputs may be scalars or vectors of length length(f). The calculation uses a normal random-effects approximation and the sample-size equation in ssCPnorm(). Invalid inputs and designs with no finite solution raise errors. CP uses the continuous design; returned group counts are rounded recommendations.

References

Ren, Xinru, and Jin Xu. 2026. “Consistency Assessment and Regional Sample Size Calculation for MRCT under Random Effects Model.” Pharmaceutical Statistics 25 (3): e70090. doi:10.1002/pst.70090.

See Also

ssCPnorm(), ssCPsurvNPH().

Examples

ssCPsurvPH(alpha = 0.025, beta = 0.2, lambda0r = 0.05,
           hr = c(0.7, 0.6, 0.4), target = 1, f = rep(1/3, 3), L = 36)

Legacy simulation tables for the two-MRCT paper

Description

Internal helper for the historical simulation settings in Qing et al. (2025). This is not a general design or optimization interface.

Usage

table2MRCTs(numSimu, choice, seed = 1)

Arguments

Details

Tables 1–2 use one trial, tables 3–4 use two symmetric trials, and tables 5–6 use different allocation ratios and powers. Some regional fractions are fixed historical settings. Numerical results need not exactly reproduce published rounding or earlier software versions.

Value

A data frame of scenario inputs, recommended counts and empirical CP.

See Also

conProb(), conProb2(), regFrac2().