Package 'ssMRCT' reference manual

Title:	Regional Consistency Evaluation and Sample Size Calculation for MRCTs
Description:	Calculate conditional regional consistency probabilities and sample sizes for multi-regional clinical trials under fixed-effects and random-effects models. Supports two independent trials, regional fraction selection under symmetric designs, and doubly robust regional treatment-effect estimation. Methods include Qing et al. (2025) <doi:10.1002/sim.70306> and Ren and Xu (2026) <doi:10.1002/pst.70090>.
Authors:	Kunhai Qing [aut, cre], Xinru Ren [aut], Jin Xu [aut]
Maintainer:	Kunhai Qing <[email protected]>
License:	MIT + file LICENSE
Version:	0.2.0
Built:	2026-11-12 13:16:37 UTC
Source:	https://github.com/cran/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 [email protected]

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.

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

Arguments

alpha

One-sided significance level, between 0 and 0.5.

power

Target overall power, with ⁠alpha < power < 1⁠.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

rF

Fraction of participants in the region of interest, in ⁠(0, 1]⁠.

d

Positive treatment-minus-control mean difference.

sigmaTrt

Positive outcome standard deviation in the treatment arm.

sigmaCtrl

Positive outcome standard deviation in the control arm.

randRatio

Treatment-to-control allocation ratio, strictly positive.

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

Examples

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

alpha

One-sided significance level, between 0 and 0.5.

power1, power2

Overall power in studies 1 and 2; each lies in ⁠(alpha, 1)⁠.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

rF1, rF2

Fractions in the target region in studies 1 and 2, in ⁠(0, 1]⁠.

d1, d2

Positive treatment-minus-control mean differences.

muCtrl1, muCtrl2

Control-arm means; needed when pooling weights differ.

sigmaTrt1, sigmaTrt2

Positive treatment-arm outcome standard deviations.

sigmaCtrl1, sigmaCtrl2

Positive control-arm outcome standard deviations.

randRatio1, randRatio2

Positive treatment-to-control allocation ratios.

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

References

Examples

conProb2(alpha = 0.05, power1 = 0.8, power2 = 0.9,
         rF1 = 0.141, d1 = 1, sigmaTrt1 = 4)
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
)
conProbII(
  alpha,
  power,
  rF,
  d,
  sigmaTrt,
  sigmaCtrl = sigmaTrt,
  pTrt,
  pCtrl,
  randRatio = 1,
  responseType = c("continuous", "binary"),
  B = 1e+05
)

Arguments

alpha

One-sided significance level, between 0 and 0.5.

power

Target overall power, with ⁠alpha < power < 1⁠.

rF

Positive fractions for all regions, summing to one.

d, sigmaTrt, sigmaCtrl

Continuous-endpoint effect and standard deviations; see conProb(). Ignored for binary endpoints.

pTrt, pCtrl

Treatment and control response probabilities for a binary endpoint, strictly between zero and one, with pTrt > pCtrl.

randRatio

Treatment-to-control allocation ratio, strictly positive.

responseType

Endpoint type: "continuous" or "binary".

B

Positive integer number of replicates; used only for binary endpoints.

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

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

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)
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
)
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

alpha

One-sided significance level, between 0 and 0.5.

power1, power2

Overall power in studies 1 and 2; each lies in ⁠(alpha, 1)⁠.

rF1, rF2

Positive regional fraction vectors, each summing to one and listing the same regions in the same order.

d1, d2

Positive continuous-endpoint effects; ignored for binary endpoints.

sigmaTrt1, sigmaTrt2

Positive treatment-arm outcome standard deviations.

sigmaCtrl1, sigmaCtrl2

Positive control-arm outcome standard deviations.

pTrt1, pTrt2

Treatment response probabilities for binary endpoints, strictly between zero and one.

pCtrl1, pCtrl2

Control response probabilities for binary endpoints, strictly between zero and the corresponding treatment probability.

randRatio1, randRatio2

Positive treatment-to-control allocation ratios.

responseType

Endpoint type: "continuous" or "binary".

B

Positive integer number of replicates; used only for binary endpoints.

muCtrl1, muCtrl2

Control means for continuous endpoints only.

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

Binary calculation

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

References

Examples

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

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

Arguments

alpha

One-sided significance level, between 0 and 0.5.

beta

Type II error probability; ⁠alpha < 1 - beta < 1⁠.

tau_delta_r

Positive ratio tau / delta.

delta

Positive mean of the regional treatment-effect distribution.

sigma0r, sigma1r

Positive control and treatment outcome standard deviations; scalars or vectors of length length(fr).

fr

Fractions of the regions of interest, each in ⁠(0, 1]⁠. Retained for compatibility and output sizing; counts are regional and are not divided by these fractions. The entries need not sum to one.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

randRatio

Treatment-to-control allocation ratio, strictly positive.

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.

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))
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))

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()
)
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

dat

A data frame with numeric outcome y, binary treatment trt (1 = treatment), region label region, and at least one numeric covariate. All other numeric columns are covariates. Values must be complete and finite; every region must contain both arms. Use valid R covariate names; D, D_src and resid_sq are reserved for internal use.

target_region

A single label identifying the target region in dat.

compute_se

Calculate the influence-function standard error and 95% interval.

region_wise

Screen each source region using separate treatment-arm and control-arm ANCOVA tests before refitting.

p1, p0

Treatment assignment probabilities in the target and source populations. Supply NULL to fit logistic models, a scalar constant, or a vector aligned with all rows of dat. Supplied values must be in ⁠(0, 1)⁠.

pi

Probability of membership in the target region, given covariates. Accepts NULL, a scalar or a row-aligned vector, as for p1. This is a selection probability, not the CP preservation fraction.

v1, v0

Target-to-source variance ratios in the treatment and control arms. Use a positive scalar, "const" for a marginal variance ratio, or "log_linear" for a model based on log squared residuals.

config

A named list with optional check_p_threshold in ⁠(0, 1)⁠ (default 0.05). Other entries are not supported.

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)

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

Arguments

alpha

One-sided significance level, between 0 and 0.5.

power

Target overall power, with ⁠alpha < power < 1⁠.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

CP

Target consistency probability, in ⁠(0.5, 1]⁠.

d

Positive treatment-minus-control mean difference.

sigmaTrt

Positive outcome standard deviation in the treatment arm.

sigmaCtrl

Positive outcome standard deviation in the control arm.

randRatio

Treatment-to-control allocation ratio, strictly positive.

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

Examples

regFrac(alpha = 0.05, power = 0.8, CP = 0.8)
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)
regFrac2(alpha, power, pi = 0.5, CP = 0.8)

Arguments

alpha

One-sided significance level, between 0 and 0.5.

power

Common overall power for both trials, with ⁠alpha < power < 1⁠.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

CP

Target consistency probability, in ⁠(0.5, 1]⁠.

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

Examples

fraction <- regFrac2(alpha = 0.05, power = 0.8)
conProb2(alpha = 0.05, power1 = 0.8, rF1 = fraction,
         d1 = 1, sigmaTrt1 = 4)
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)
ssCPbinary(alpha, beta, p0r, p1r, pi = 0.5, randRatio = 1, target, f)

Arguments

alpha

One-sided significance level, between 0 and 0.5.

beta

Type II error probability; ⁠alpha < 1 - beta < 1⁠.

p0r, p1r

Control and treatment response probabilities, strictly in ⁠(0, 1)⁠, as scalars or vectors with one value per region.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

randRatio

Treatment-to-control allocation ratio, strictly positive.

target

Index of the region of interest in f.

f

Positive regional fractions summing to one. At least two regions are required.

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.

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

Arguments

alpha

One-sided significance level, between 0 and 0.5.

beta

Type II error probability; ⁠alpha < 1 - beta < 1⁠.

delta

Positive mean of the regional treatment-effect distribution.

tau

Nonnegative standard deviation of regional treatment effects.

sigma1r, sigma0r

Positive treatment and control outcome standard deviations, as scalars or vectors with one value per region.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

randRatio

Treatment-to-control allocation ratio, strictly positive.

target

Index of the region of interest in f.

f

Positive regional fractions summing to one. At least two regions are required.

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.

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))
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
)
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

alpha

One-sided significance level, between 0 and 0.5.

beta

Type II error probability; ⁠alpha < 1 - beta < 1⁠.

lambda0a, lambda0b

Positive control hazard rates before and after breakpoint. Used only when PW = TRUE.

lambda1a, lambda1b

Positive treatment hazard rates before and after breakpoint. Used only when PW = TRUE.

breakpoint

Nonnegative, common hazard change time. Used only when PW = TRUE.

shape0, shape1

Positive Weibull shape parameters for control and treatment. Used only when PW = FALSE.

scale0, scale1

Positive Weibull scale parameters for control and treatment. Used only when PW = FALSE.

PW

Use piecewise exponential survival (TRUE) or Weibull survival (FALSE).

unimin, unimax

Uniform censoring limits, with ⁠0 <= unimin < unimax⁠.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

randRatio

Treatment-to-control allocation ratio, strictly positive.

target

Index of the region of interest in f.

f

Positive regional fractions summing to one. At least two regions are required.

eta

Positive common RMST horizon; must be below every unimax.

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

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.

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

Arguments

alpha

One-sided significance level, between 0 and 0.5.

beta

Type II error probability; ⁠alpha < 1 - beta < 1⁠.

lambda0r

Positive control hazard rates, as a scalar or regional vector.

hr

Positive treatment-to-control hazard ratios, as a scalar or regional vector. The mean of -log(hr) must be positive.

pi

Fraction of the overall effect to preserve; ⁠0 <= pi < 1⁠.

randRatio

Treatment-to-control allocation ratio, strictly positive.

target

Index of the region of interest in f.

f

Positive regional fractions summing to one. At least two regions are required.

L

Positive follow-up duration, in the same time units as the hazard rates.

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

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.

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)
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)

Package 'ssMRCT'

Help Index

Regional consistency and sample size for MRCTs

Description

Choosing a function

Conventions

Author(s)

References

Consistency probability for one MRCT: Method I

Description

Usage

Arguments

Details

Value

Sample sizes

References

See Also

Examples

Pooled consistency probability for two MRCTs: Method I

Description

Usage

Arguments

Details

Value

Sample sizes

References

See Also

Examples

Consistency probability for one MRCT: Method II

Description

Usage

Arguments

Details

Value

Sample sizes

Binary calculation

References

See Also

Examples

Pooled consistency probability for two MRCTs: Method II

Description

Usage

Arguments

Details

Value

Sample sizes

Binary calculation

References

See Also

Examples

Random-effects consistency bound: Method I

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Doubly robust estimation of a regional treatment effect

Description

Usage

Arguments

Details

Value

Source screening

Examples

Regional fraction for one MRCT: Method I

Description

Usage

Arguments

Details

Value

Sample sizes

See Also

Examples

Regional fraction for two symmetric MRCTs: Method I

Description

Usage

Arguments