Statistical Analysis for a Bayesian Hybrid Design

Description

This function performs the statistical analysis for a Bayesian Hybrid Design using a dynamic power prior approach.

Usage

Bayesian.Hybrid.Analysis(
  Yt = 20,
  nt = 40,
  Yc = 12,
  nc = 40,
  Ych = 73,
  nche = 40,
  nch = 234,
  sig = 0.9,
  credlev = 0.8,
  a0c = 0.001,
  b0c = 0.001,
  a0t = 0.001,
  b0t = 0.001,
  delta_threshold = 0.1
)

Arguments

Value

A list containing the following components:

• prob.pt.gt.pc: Probability of experimental arm having a better posterior response rate than control.

• median_hca: Posterior median response rate for hybrid control.

• CI_hca: Credible interval for median response rate for hybrid control.

• median_c: Posterior median response rate for current study control.

• CI_c: Credible interval for median response rate for current study control.

• median_t: Posterior median response rate for current study experimental arm.

• CI_t: Credible interval for median response rate for current experimental arm.

• delta.m: Posterior median response rate difference (experimental - control) based on hybrid design.

• delta.CI: Credible interval for response rate difference based on hybrid design.

• delta.m_trial: Posterior median response rate difference (experimental - control) based on current study only.

• delta.CI_trial: Credible interval for response rate difference based on current study only.

• conclusion: Statistical inference conclusion string.

Examples

# Note: This example relies on the internal package function 'borrow.wt'
Bayesian.Hybrid.Analysis(Yt=18, nt=40, Yc=13, nc=40, Ych=73,
                         nche=40, nch=234)

Bayesian Hybrid Design using Dynamic Power Prior Framework

Description

This function calculates the sample size and Bayesian hybrid design parameters using dynamic power prior framework.

Usage

DPP(
  pt,
  pc,
  pch,
  pc.calib,
  nch,
  nc.range = NULL,
  r,
  q,
  alpha = 0.1,
  power = 0.8,
  delta_threshold = 0.1,
  method = "Empirical Bayes",
  theta = 0.5,
  eta = 1,
  a0c = 0.001,
  b0c = 0.001,
  a0t = 0.001,
  b0t = 0.001,
  nsim = 1e+05,
  seed = NULL
)

Arguments

Value

An object with values

• alpha: nominal type I error rate

• power: The calculated power by simulation.

• type1err: empirical type I error rate.

• tau: The calibrated threshold for statistical significance.

• nt: sample size for experimental arm

• nc: sample size for control arm

• nche: maximum amount of borrowing in terms of number of subjects

• delta.bound: significance boundary of delta between the study experimental group and study control group

• mean.PMD: mean of posterior mean difference over nsim estimates

• sd.PMD: standard deviation of posterior mean difference over nsim estimates

• mean_pc_hca: a vector of nsim length. Storing the posterior means of pc based on hybrid control for nsim replications.

• mean_pc_c: a vector of nsim length. Storing the posterior means of pc based on study control for nsim replications.

Examples

o <- DPP(pt = 0.5, pc = 0.3, pch = 0.3, pc.calib = 0.3, nch = 200, nc.range = NULL,
         r = 2, q = 1, alpha = 0.1, power = 0.8,
         delta_threshold = 0.1,
         method = "Empirical Bayes", theta = 0.5, eta = 1,
         a0c = 0.001, b0c = 0.001, a0t = 0.001, b0t = 0.001,
         nsim = 1000, seed = 2000)
print(o)

Analysis of a Study Using Bayesian Hybrid Design using Dynamic Power Prior Framework

Description

This function Perform Analysis for a Study Using Bayesian Hybrid Design using Dynamic Power Prior Framework.

Usage

DPP.analysis(
  Yt = 39,
  nt = 60,
  Yc = 13,
  nc = 30,
  Ych = 90,
  nch = 200,
  nche = 30,
  a0c = 0.001,
  b0c = 0.001,
  a0t = 0.001,
  b0t = 0.001,
  delta_threshold = 0.1,
  method = "Empirical Bayes",
  theta = 0.5,
  eta = 1
)

Arguments

Value

An object of class list with values:

• w: Borrowing weight.

• phat_pt_larger_pc: Posterior probability P(ORR_trt > ORR_ctrl | data).

• apost_c_trial, bpost_c_trial: Parameters for the posterior Beta distribution of the concurrent control arm response rate.

• apost_c_hca, bpost_c_hca: Parameters for the posterior Beta distribution of the hybrid control arm response rate.

• apost_t, bpost_t: Parameters for the posterior Beta distribution of the experimental arm response rate.

• m.t: Posterior median response rate for the experimental arm.

• m.c: Posterior median response rate for the concurrent control arm.

• m.hca: Posterior median response rate for the hybrid control arm.

Examples

o <- DPP.analysis(Yt=39, nt=60, Yc=13, nc=30, Ych=90, nch=200, nche = 30,
a0c= 0.001, b0c= 0.001, a0t= 0.001, b0t= 0.001,
delta_threshold = 0.1, method = "Empirical Bayes",
theta = 0.5, eta = 1)
print(o)

Bayesian Hybrid Design

Description

This function calculates the expected effective sample size of the DPP.

Usage

EESS(
  pc,
  nc,
  pch,
  nche,
  nch,
  a0c = 0.001,
  b0c = 0.001,
  delta_threshold = 0.1,
  method = "Empirical Bayes",
  theta = 0.5,
  eta = 1
)

Arguments

Value

The expected effective sample size.

Examples

EESS(pc=0.3,nc=40,pch=0.3,nche=40,nch=180, a0c=0.001,b0c=0.001,
delta_threshold=0.1,method="Empirical Bayes", theta=0.5, eta=1)

Statistical Analysis Using Frequentist Method

Description

Performs a frequentist analysis of a two-arm study using current data only.

Usage

Frequentist.Analysis(Yt = 20, nt = 40, Yc = 12, nc = 40, conflev = 0.8)

Arguments

Value

An object of class list with the following values:

• pt: The response rate of the experimental arm.

• pc: The response rate of the control arm.

• delta: The difference in response rates (pt - pc).

• exactCI.c: A vector containing the Clopper-Pearson confidence interval for the control arm.

• exactCI.t: A vector containing the Clopper-Pearson confidence interval for the experimental arm.

• p.fisher: A one-sided p-value from Fisher's exact test, testing the alternative hypothesis that the experimental response rate is greater than the control response rate.

Examples

Frequentist.Analysis(Yt=18, nt=40, Yc=13, nc=40, conflev=0.8)

Calculating SAM priors

Description

This function is exactly from the SAMprior R package version 1.1.1. It is included to support the runSAM function in this package.

Usage

SAM_prior(if.prior, nf.prior, weight, ...)

## S3 method for class 'betaMix'
SAM_prior(if.prior, nf.prior, weight, ...)

## S3 method for class 'gammaMix'
SAM_prior(if.prior, nf.prior, weight, ...)

## S3 method for class 'normMix'
SAM_prior(if.prior, nf.prior, weight, ..., sigma)

Arguments

Details

The SAM_prior function is designed to display the SAM prior, given the informative prior (constructed from historical data), non-informative prior, and the mixture weight calculated using SAM_weight function (Yang, et al., 2023).

SAM prior is constructed by mixing an informative prior \pi_1(\theta), constructed based on historical data, with a non-informative prior \pi_0(\theta) using the mixture weight w determined by SAM_weight function to achieve the degree of prior-data conflict (Schmidli et al., 2015, Yang et al., 2023).

Let \theta and \theta_h denote the treatment effects associated with the current arm data D and historical data D_h, respectively. Let \delta denote the clinically significant difference such that if |\theta_h - \theta| \ge \delta, then \theta_h is regarded as clinically distinct from \theta, and it is therefore inappropriate to borrow any information from D_h. Consider two hypotheses:

H_0: \theta = \theta_h, ~ H_1: \theta = \theta_h + \delta ~ or ~ \theta = \theta_h - \delta.

H_0 represents that D_h and D are consistent (i.e., no prior-data conflict) and thus information borrowing is desirable, whereas H_1 represents that the treatment effect of D differs from D_h to such a degree that no information should be borrowed.

The SAM prior uses the likelihood ratio test (LRT) statistics R to quantify the degree of prior-data conflict and determine the extent of information borrowing.

R = P(D | H_0, \theta_h) / P(D | H_1, \theta_h) = P(D | \theta = \theta_h) / \max(P(D | \theta = \theta_h + \delta), P(D | \theta = \theta_h - \delta)) ,

where P(D | \cdot) denotes the likelihood function. An alternative Bayesian choice is the posterior probability ratio (PPR):

R = P(D | H_0, \theta_h) / P(D | H_1, \theta_h) = P(H_0) / P( H_1) \times BF,

where P(H_0) and P(H_1) is the prior probabilities of H_0 and H_1 being true. BF is the Bayes Factor that in this case is the same as the LRT.

The SAM prior, denoted as \pi_{sam}(\theta), is then defined as a mixture of an informative prior \pi_1(\theta), constructed based on D_h and a non-informative prior \pi_0(\theta):

\pi_{sam}(\theta) = w\pi_1(\theta) + (1-w)\pi_0(\theta),

where the mixture weight w is calculated as:

w = R / (1 + R).

As the level of prior-data conflict increases, the likelihood ratio R decreases, resulting in a decrease in the weight w assigned to the informative prior and thus a decrease in information borrowing. As a result, \pi_{sam}(\theta) is data-driven and has the ability to self-adapt the information borrowing based on the degree of prior-data conflict.

Value

Displays the SAM prior as a mixture of an informative prior (constructed based on the historical data) and a non-informative prior.

Methods (by class)

• SAM_prior(betaMix): The function calculates the SAM prior for beta mixture distribution. The default nf.prior is set to be mixbeta(c(1,1,1)) which represents a uniform prior Beta(1,1).

• SAM_prior(gammaMix): The function calculates the SAM prior for gamma mixture distribution. The default nf.prior is set to be mixgamma(c(1,0.001,0.001)) which represents a vague gamma prior Gamma(0.001,0.001).

• SAM_prior(normMix): The function calculates the SAM prior for normal mixture distribution. The default nf.prior is set to be mixnorm(c(1,summary(if.prior)['mean'], sigma)) which represents a unit-information prior.

References

Yang P, Zhao Y, Nie L, Vallejo J, Yuan Y. SAM: Self-adapting mixture prior to dynamically borrow information from historical data in clinical trials. Biometrics 2023; 00, 1–12. https://doi.org/10.1111/biom.13927

Schmidli H, Gsteiger S, Roychoudhury S, O'Hagan A, Spiegelhalter D, Neuenschwander B. Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics 2014; 70(4):1023-1032.

See Also

SAM_weight

Examples

set.seed(123)
## Examples for binary endpoints
## Suppose that the informative prior constructed based on historical data is
## beta(40, 60)
prior.historical <- RBesT::mixbeta(c(1, 40, 60))
## Data of the control arm
data.control     <- stats::rbinom(60, size = 1, prob = 0.42)
## Calculate the mixture weight of the SAM prior
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.15,      ## Clinically significant difference
                   data = data.control  ## Control arm data
                   )
## Assume beta(1,1) as the non-informative prior used for mixture
nf.prior  <- RBesT::mixbeta(nf.prior = c(1,1,1))
## Generate the SAM prior
SAM.prior <- SAM_prior(if.prior = prior.historical, ## Informative prior
                       nf.prior = nf.prior,         ## Non-informative prior
                       weight = wSAM                ## Mixture weight of the SAM prior
                       )
graphics::plot(SAM.prior)

## Examples for continuous endpoints
## Suppose that the informative prior constructed based on historical data is
## N(0, 3)
sigma      <- 3
prior.mean <- 0
prior.se   <- sigma/sqrt(100)
prior.historical <- RBesT::mixnorm(c(1, prior.mean, prior.se), sigma = sigma)
## Data of the control arm
data.control <- stats::rnorm(80, mean = 0, sd = sigma)
## Calculate the mixture weight of the SAM prior
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.2 * sigma,   ## Clinically significant difference
                   data = data.control     ## Control arm data
                   )
## Assume unit-information prior N(0,3) as the non-informative prior used
## for the mixture
nf.prior         <- RBesT::mixnorm(nf.prior = c(1,prior.mean, sigma),
                                   sigma = sigma)
## Generate the SAM prior
SAM.prior <- SAM_prior(if.prior = prior.historical, ## Informative prior
                       nf.prior = nf.prior,         ## Non-informative prior
                       weight = wSAM                ## Mixture weight of the SAM prior
                       )
graphics::plot(SAM.prior)

Calculating Mixture Weight of SAM Priors

Description

This function is exactly from the SAMprior R package version 1.1.1. It is included to support the runSAM function in this package.

Usage

SAM_weight(if.prior, theta.h, method.w, prior.odds, data, delta, ...)

## S3 method for class 'betaMix'
SAM_weight(if.prior, theta.h, method.w, prior.odds, data, delta, n, r, ...)

## S3 method for class 'normMix'
SAM_weight(
  if.prior,
  theta.h,
  method.w,
  prior.odds,
  data,
  delta,
  m,
  n,
  sigma,
  ...
)

## S3 method for class 'gammaMix'
SAM_weight(if.prior, theta.h, method.w, prior.odds, data, delta, u, w, ...)

Arguments

Details

The SAM_weight function is designed to calculate the mixture weight of the SAM priors according to the degree of prior-data conflicts (Yang, et al., 2023).

SAM prior is constructed by mixing an informative prior \pi_1(\theta), constructed based on historical data, with a non-informative prior \pi_0(\theta) using the mixture weight w determined by SAM_weight function to achieve the degree of prior-data conflict (Schmidli et al., 2015, Yang et al., 2023).

Let \theta and \theta_h denote the treatment effects associated with the current arm data D and historical data D_h, respectively. Let \delta denote the clinically significant difference such that if |\theta_h - \theta| \ge \delta, then \theta_h is regarded as clinically distinct from \theta, and it is therefore inappropriate to borrow any information from D_h. Consider two hypotheses:

H_0: \theta = \theta_h, ~ H_1: \theta = \theta_h + \delta ~ or ~ \theta = \theta_h - \delta.

H_0 represents that D_h and D are consistent (i.e., no prior-data conflict) and thus information borrowing is desirable, whereas H_1 represents that the treatment effect of D differs from D_h to such a degree that no information should be borrowed.

The SAM prior uses the likelihood ratio test (LRT) statistics R to quantify the degree of prior-data conflict and determine the extent of information borrowing.

R = P(D | H_0, \theta_h) / P(D | H_1, \theta_h) = P(D | \theta = \theta_h) / \max(P(D | \theta = \theta_h + \delta), P(D | \theta = \theta_h - \delta)) ,

where P(D | \cdot) denotes the likelihood function. An alternative Bayesian choice is the posterior probability ratio (PPR):

R = P(D | H_0, \theta_h) / P(D | H_1, \theta_h) = P(H_0) / P( H_1) \times BF,

where P(H_0) and P(H_1) is the prior probabilities of H_0 and H_1 being true. BF is the Bayes Factor that in this case is the same as the LRT.

The SAM prior, denoted as \pi_{sam}(\theta), is then defined as a mixture of an informative prior \pi_1(\theta), constructed based on D_h and a non-informative prior \pi_0(\theta):

\pi_{sam}(\theta) = w\pi_1(\theta) + (1-w)\pi_0(\theta),

where the mixture weight w is calculated as:

w = R / (1 + R).

As the level of prior-data conflict increases, the likelihood ratio R decreases, resulting in a decrease in the weight w assigned to the informative prior and thus a decrease in information borrowing. As a result, \pi_{sam}(\theta) is data-driven and has the ability to self-adapt the information borrowing based on the degree of prior-data conflict.

Value

The mixture weight of the SAM priors.

Methods (by class)

• SAM_weight(betaMix): The function calculates the mixture weight of SAM priors for beta mixture distribution. The input data can be patient-level data (i.e., a vector of 0 and 1 representing the response status of each patient) or summary statistics (i.e., the number of patients and the number of responses).

• SAM_weight(normMix): The function calculates the mixture weight of SAM priors for normal mixture distribution. The input data should be a vector of patient-level observations. The input data can be patient-level data (i.e., a vector of continuous response of each patient) or summary statistics (i.e., the mean estimate, number of subjects, and the standard deviation in the control arm).

• SAM_weight(gammaMix): The function calculates the mixture weight of SAM priors for gamma mixture distribution. The input data can be patient-level data (i.e., a matrix with the first row as the censoring indicator and the second row recording the observed time) or summary statistics (i.e., the number of uncensored observations u and total observed time w).

References

Yang P, Zhao Y, Nie L, Vallejo J, Yuan Y. SAM: Self-adapting mixture prior to dynamically borrow information from historical data in clinical trials. Biometrics 2023; 00, 1–12. https://doi.org/10.1111/biom.13927

Examples

set.seed(123)
## Examples for binary endpoints
## Example 1: no prior-data conflict
## Suppose that the informative prior is beta(40, 60)
prior.historical <- RBesT::mixbeta(c(1, 40, 60))
## Data of control arm
data.control     <- stats::rbinom(60, size = 1, prob = 0.42)
## Calculate the mixture weight of the SAM prior
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.15,      ## Clinically significant difference
                   data = data.control  ## Control arm data
                   )
print(wSAM)

## Example 2: with prior-data conflict (12 responses in 60 patients)
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.15,
                   method.w = 'PPR',
                   prior.odds = 1/9,
                   n = 60,
                   r = 12
                   )
print(wSAM)

## Examples for continuous endpoints
## Example 1: no prior-data conflict
## Suppose the informative prior is N(0, 3)
sigma      <- 3
prior.mean <- 0
prior.se   <- sigma/sqrt(100)
prior.historical <- RBesT::mixnorm(c(1, prior.mean, prior.se), sigma = sigma)
## Data of the control arm
data.control <- stats::rnorm(80, mean = 0, sd = sigma)
wSAM <- SAM_weight(if.prior = prior.historical,
                   delta = 0.3 * sigma,
                   data = data.control
                   )
print(wSAM)

Sample Size for Comparing Two Proportions

Description

Calculates the required sample size for a two-group comparison of proportions based on the method of Casagrande, Pike, and Smith (1978).

Usage

Two.Prop.Test.Sample.Size(p1, p2, alpha, beta, r)

Arguments

Value

A numeric vector of length two:

• The required sample size for Arm 1 ($n_1$).

• The total required sample size ($n_1 + n_2$).

See Also

https://www2.ccrb.cuhk.edu.hk/stat/proportion/Casagrande.htm

Examples

Two.Prop.Test.Sample.Size(p1=0.27, p2=0.47, alpha = 0.10, beta=0.2, r=2)

Calculate Borrowing Weights from Historical Data

Description

Calculates borrowing weights for a hybrid control arm using one of several dynamic borrowing methods.

Usage

borrow.wt(
  Yc,
  nc,
  Ych,
  nch,
  nche,
  a0c = 0.001,
  b0c = 0.001,
  delta_threshold = 0.1,
  method = "Empirical Bayes",
  theta = 0.5,
  eta = 1
)

Arguments

Details

The function implements the following methods:

• "Empirical Bayes": The weight is determined by maximizing the marginal likelihood of the current data.

• "Bayesian p": Similarity is measured by a Bayesian p-value comparing the posterior distributions.

• "Generalized BC": Uses the Generalized Bhattacharyya Coefficient. The standard BC is a special case with theta = 0.5.

• "JSD": Uses the Jensen-Shannon Divergence to measure similarity.

Value

A list containing three values:

• a: The global borrowing weight, calculated as nche / nch.

• wd: The dynamic borrowing weight, calculated based on the chosen method.

• w: The final overall borrowing weight, which is a product of a, wd, and an indicator for whether the response rates are sufficiently similar.

Examples

borrow.wt(Yc=12, nc=40, Ych=54, nch=180, nche=40, a0c=0.001,
b0c=0.001, delta_threshold=0.1, eta=1)

Calibration for Bayesian Hybrid Design

Description

This function calculates the threshold tau for calibration of Bayesian Hybrid Design. P(pt>pc|hybrid data)>tau is used to determine statistical significance.

Usage

calibration(
  nt,
  pc.calib,
  nc,
  pch,
  nche,
  nch,
  alpha = 0.1,
  a0c = 0.001,
  b0c = 0.001,
  a0t = 0.001,
  b0t = 0.001,
  delta_threshold = 0.1,
  method = "Empirical Bayes",
  theta = 0.5,
  eta = 1,
  datamat = NULL,
  w0 = NULL,
  nsim = 10000,
  seed = NULL
)

Arguments

Value

The scalar threshold for statistical significance that can control the type I error (1-sided)

Examples

  tau <- calibration(nt=40, pc.calib=0.3, nc=40,
                     pch=0.3, nche=40, nch=200,
                     alpha = 0.10,
                     a0c=0.001, b0c=0.001, a0t=0.001, b0t=0.001,
                     delta_threshold=0.1,
                     method="Empirical Bayes", theta=0.5, eta=1,
                     nsim = 1000, seed=2000) # nsim reduced for quick example

Clopper-Pearson Exact Confidence Interval for a Binomial Proportion

Description

Calculates the two-sided Clopper-Pearson exact confidence interval for a binomial proportion.

Usage

exactci(r, n, conflev)

Arguments

Value

A numeric vector of length two containing the lower and upper confidence limits.

Examples

exactci(r=4, n=20, conflev=0.95)

Explore Power Across Multiple Scenarios for a Bayesian Hybrid Design

Description

Evaluates statistical power and other design parameters across a grid of specified settings, using parallel computing for efficiency.

Usage

explore.power.DPP(
  method,
  pc,
  nc,
  pc.calib,
  q,
  delta,
  r,
  pch,
  nch,
  alpha = 0.1,
  a0c = 0.001,
  b0c = 0.001,
  a0t = 0.001,
  b0t = 0.001,
  delta_threshold = 0.1,
  theta = 0.5,
  eta = 1,
  nsim = 1e+05,
  seed = NULL,
  ncore = NULL
)

Arguments

Details

This function serves as a wrapper for power.DPP(). It uses expand.grid() to create a full factorial design from the vector-based input parameters (method, pc, nc, pc.calib, q). It then iterates through each scenario to calculate the power, type I error, and other metrics.

Value

A data.frame where each row corresponds to one scenario from the input grid. The columns include the input parameters and the following results:

method

The dynamic borrowing method used.

nt, nc, nche

Sample sizes for the experimental, current control, and effective historical control arms.

pt, pc, pc.calib, pch

Response rates used in the scenario.

q

The ratio of nche / nc.

tau

The calibrated threshold for statistical significance.

type1err

The simulated Type I error rate for the scenario.

power

The simulated statistical power for the scenario.

delta.boundary

The significance boundary for the difference between groups.

mean.PMD

The mean of the posterior mean difference over simulations.

sd.PMD

The standard deviation of the posterior mean difference.

See Also

power.DPP for the underlying single-scenario calculation.

Examples

# Run with 2 cores as an example; on CRAN, examples should use <= 2 cores.
Res1 <- explore.power.DPP(method=c("Empirical Bayes"),
                          pc=c(0.27, 0.37),
                          nc=23:25,
                          pc.calib = 0.27,
                          q = c(1, 1.5),
                          delta = 0.2,
                          r = 1,
                          pch=0.27, nch=500,
                          alpha=0.1,
                          nsim = 200, # Reduced for a quick example
                          seed=1000,
                          ncore=2)

Fisher's Exact Test for a 2x2 Contingency Table

Description

A wrapper for stats::fisher.test to conveniently compare response rates between two arms (experimental and control).

Usage

fisher(Yc, nc, Yt, nt, alternative = "greater")

Arguments

Value

An object of class htest as returned by stats::fisher.test.

See Also

fisher.test

Examples

fisher(Yc=12, nc=40, Yt=19, nt=40)

Calculate Rejection Boundary for Fisher's Exact Test

Description

For a given control group response rate and sample sizes, this function finds the smallest number of responders in the experimental arm (rt) that achieves statistical significance based on a one-sided Fisher's exact test.

Usage

fisher.bound(pc, nc, nt, alpha = 0.1)

Arguments

Value

A list containing the following components:

M

The 2x2 contingency table at the boundary.

p

The p-value corresponding to the boundary.

rc

The number of responders in the control arm, calculated as round(nc * pc).

nc

The sample size of the control arm.

rt

The smallest number of responders in the experimental arm that achieves a p-value <= alpha.

nt

The sample size of the experimental arm.

delta

The minimum detectable difference in response rates (rt/nt - rc/nc).

Examples

fisher.bound(pc=0.3, nc=40, nt=40, alpha=0.1)

Power Calculation by Fisher's Exact Test

Description

This function calculates the power by simulations using Fisher's exact test.

Usage

fisher.power(pt, nt, pc, nc, alpha = 0.1, nsim = 10000, seed = NULL)

Arguments

Value

A numeric scalar representing the calculated statistical power.

Examples

# nsim is reduced for a quick example run
fisher.power(pt = 0.5, nt = 40, pc = 0.3, nc = 40, nsim = 1000, seed = 2000)

Plot a DPP Object

Description

Plots the posterior distributions of the response rates for the control and experimental arms from a DPP object created by the DPP.analysis() function.

Usage

plotDPP(DPP)

Arguments

Value

No return value, called for side effects.

Examples

o <- DPP.analysis(Yt=39, nt=60, Yc=13, nc=30, Ych=90, nch=200, nche = 30,
                  a0c= 0.001, b0c= 0.001, a0t= 0.001, b0t= 0.001,
                  delta_threshold = 0.1, method = "Empirical Bayes",
                  theta = 0.5, eta = 1)

# Call the function using its defined name 'plotDPP'
plotDPP(DPP = o)

Plot Posterior Mean Difference (PMD) Distributions

Description

Plots the density of the posterior mean difference between the hybrid and concurrent control arms, based on an object from power.DPP(). Can compare one or two such objects on the same plot.

Usage

plotPMD(o, o2 = NULL)

Arguments

Value

Invisibly returns a list containing summary statistics. The structure of the list depends on whether o2 is provided.

• If o2 is NULL: A list with a scalar mean_PMD, a scalar sd_PMD, and a numeric vector CI95_PMD.

• If o2 is provided: A list with numeric vectors for mean_PMD and sd_PMD, and a matrix for CI95_PMD, with each row corresponding to an input object.

Examples

# nsim reduced for faster example
o1a <- power.DPP(pt=0.65, nt=60, pc=0.45, nc=30, pc.calib=0.45,
                 pch=0.45, nche=60, nch=200, alpha=0.1, nsim=1000)
o1b <- power.DPP(pt=0.65, nt=60, pc=0.45, nc=30, pc.calib=0.45,
                 pch=0.45, nche=30, nch=200, alpha=0.1, nsim=1000)

# Plot a single object
stats1 <- plotPMD(o=o1a)

# Plot two objects for comparison
stats2 <- plotPMD(o=o1a, o2=o1b)

Power Calculation for Bayesian Hybrid Design

Description

Calculates statistical power and other design parameters for a Bayesian Hybrid Design using a dynamic power prior approach, based on simulations.

Usage

power.DPP(
  pt,
  nt,
  pc,
  nc,
  pc.calib,
  pch,
  nche,
  nch,
  alpha = 0.1,
  tau = NULL,
  a0c = 0.001,
  b0c = 0.001,
  a0t = 0.001,
  b0t = 0.001,
  delta_threshold = 0.1,
  method = "Empirical Bayes",
  theta = 0.5,
  eta = 1,
  datamat = NULL,
  w0 = NULL,
  nsim = 1e+05,
  seed = NULL
)

Arguments

Value

A large list containing the power, the calibrated tau, all input parameters, and detailed simulation results such as:

power

The calculated statistical power.

tau

The calibrated significance threshold.

pc.PMD, pc.sd.PMD

The mean and standard deviation of the posterior mean difference between the hybrid and concurrent controls.

delta.bound

The minimum detectable difference in response rates.

phat_pt_larger_pc_all

A vector of posterior probabilities P(pt > pc | data) for each of the nsim simulations.

mean_hca, mean_c

Vectors of the posterior means for the hybrid and concurrent control arms for each simulation.

simulated.data

A matrix of the simulated response counts used.

w

A vector of the final borrowing weights used in each simulation.

...

and all input parameters.

Examples

o <- power.DPP(pt=0.5, nt=40, pc=0.3, nc=40, pc.calib = 0.3, pch=0.3,
               nche=40, nch=180, alpha=0.1, nsim = 1000, seed=2000) # nsim is reduced

Power Calculation for the Exact Binomial Test

Description

Calculates the power of a one-sample exact binomial test via simulation. This is used to compare a treatment response rate against a fixed historical benchmark.

Usage

power.binom.test(
  n = 20,
  p = 0.5,
  p0 = 0.3,
  alpha = 0.05,
  alternative = c("two.sided", "greater", "less"),
  nsim = 1e+05,
  seed = 2025
)

Arguments

Value

A numeric value representing the statistical power.

Examples

power.binom.test(n=110, p=0.4, p0=0.128)

Run a Calibrated Simulation Using SAM Priors

Description

Performs a two-step simulation.

1. Calibration Step: It first runs a simulation under the null hypothesis (using ⁠*.calib⁠ parameters) to determine the posterior probability thresholds (quantiles) needed to control the Type I Error Rate at the specified typeIER.cal level.

2. Main Simulation Step: It then uses these calibrated thresholds to run the main simulation (using parameters nc, nt, pc, pt) to evaluate final operating characteristics (e.g., Type I Error or Power) and bias.

Usage

runCalibratedSAM(
  nsim = 1000,
  pch,
  delta_threshold,
  nche.c,
  nc.calib,
  nt.calib,
  pc.calib,
  xt.cal = NULL,
  xc.cal = NULL,
  typeIER.cal = 0.1,
  nche,
  nc,
  nt,
  pc,
  pt,
  xt = NULL,
  xc = NULL,
  nf.prior = RBesT::mixbeta(c(1, 0.001, 0.001)),
  seed.hist = 1000,
  seed.gMAP = 2000,
  seed.SAM = 3000,
  seed.cal = 4000
)

Arguments

Details

This function wraps the core runSAM() function by adding a calibration layer. It derives historical data priors using RBesT::gMAP() and RBesT::automixfit() based on nche.c and pch.

The calibration step (runSAM with ...calib parameters) finds the (1 - typeIER.cal) quantiles for the posterior distributions (SAM, rMAP, Non-info) under the null.

The main simulation step (runSAM with main parameters) then calculates the proportion of simulations where the posterior probability exceeds these calibrated thresholds. This proportion represents the final Type I Error or Power.

Value

A list with the following components:

Sim_Result

A numeric vector (length 3) with the final simulation result (Type I Error or Power) for the "SAMprior", "MAP", and "Noninfo" methods.

pc.PMD

Posterior Mean Difference (Bias) for the three methods.

pc.PSDD

Posterior SD of the Difference (Bias) for the three methods.

pc.PM

A matrix (⁠3 x nsim⁠) of posterior means for the control rate for each simulation iteration.

Calibration_Thresholds

A matrix (1x3) containing the decision thresholds (quantiles) determined during the calibration step for the SAM, rMAP, and Non-info methods.

See Also

runSAM, gMAP, automixfit, mixbeta, decision2S

Examples

# Example uses functions from the RBesT package
library(RBesT)

sim_power_pc30 <- runCalibratedSAM(
   nsim = 100,
   pch = 0.30,
   delta_threshold = 0.1,
   nche.c = 180,
   nc.calib = 45,
   nt.calib = 45,
   pc.calib = 0.3,
   typeIER.cal = 0.10,
   nche = 180,
   nc = 45,
   nt = 45,
   pc = 0.30,
   pt = 0.50
)

print(sim_power_pc30$Sim_Result)

Generating Operating Characteristics of SAM Priors

Description

This function is modified based on the get_OC function from the SAMprior R package version 1.1.1.

Usage

runSAM(
  if.prior,
  theta.h,
  method.w,
  prior.odds,
  nf.prior,
  delta,
  n,
  n.t,
  decision,
  ntrial,
  if.MAP,
  weight,
  theta,
  theta.t,
  datamat = NULL,
  ...
)

## S3 method for class 'betaMix'
runSAM(
  if.prior,
  theta.h,
  method.w,
  prior.odds,
  nf.prior,
  delta,
  n,
  n.t,
  decision,
  ntrial,
  if.MAP,
  weight,
  theta,
  theta.t,
  datamat = NULL,
  ...
)

## S3 method for class 'normMix'
runSAM(
  if.prior,
  theta.h,
  method.w,
  prior.odds,
  nf.prior,
  delta,
  n,
  n.t,
  decision,
  ntrial,
  if.MAP,
  weight,
  theta,
  theta.t,
  datamat = NULL,
  ...,
  sigma
)

Arguments

Details

This function is modified based on the get_OC from the SAMprior R package version 1.1.1. The modification is made in order that the function can take simulated data (the datamat argument) from outside the function rather than simulating data within the funciton We used the same data for all methods for comparison in order to get a better comparison.

The runSAM function is designed to generate the operating characteristics of SAM priors (Yang, et al., 2023), including the relative bias, relative mean squared error, and type I error and power under a two-arm comparative trial design. As an option, the operating characteristic of robust MAP priors (Schmidli, et al., 2014) can also be generated for comparison.

The runSAM function is designed to generate the operating characteristics of SAM priors, including the relative bias, relative mean squared error, and type I error, and power under a two-arm comparative trial design. As an option, the operating characteristics of robust MAP priors (Schmidli, et al., 2014) can also be generated for comparison.

The relative bias is defined as the difference between the bias of a method and the bias of using a non-informative prior. The relative mean squared error is the difference between the mean squared error (MSE) of a method and the MES of using a non-informative prior.

To evaluate type I error and power, the determination of whether the treatment is superior to the control is calculated based on function decision2S.

Value

Returns a list result - dataframe that contains the relative bias, relative MSE, type I error, and power for both SAM priors, as well as robust MAP priors. Additionally, the mixture weight of the SAM prior is also displayed.

simulated.data - a matrix of two columns, first treatment, second control post_theta_t_list - list of all replication, posterior distribution of treatment group post_theta_c_list - list of all replication, posterior distribution of control group, non informative prior post_theta_c_SAM_list - list of all replication, posterior distribution of control group, SAM prior post_theta_c_MAP_list - list of all replication, posterior distribution of control group, MAP prior

Methods (by class)

• runSAM(betaMix): The function is designed to generate the operating characteristics of SAM priors for binary endpoints.

• runSAM(normMix): The function is designed to generate the operating characteristics of SAM priors for continuous endpoints.

References

Yang P, Zhao Y, Nie L, Vallejo J, Yuan Y. SAM: Self-adapting mixture prior to dynamically borrow information from historical data in clinical trials. Biometrics 2023; 00, 1–12. https://doi.org/10.1111/biom.13927

Schmidli H, Gsteiger S, Roychoudhury S, O'Hagan A, Spiegelhalter D, Neuenschwander B. Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics 2014; 70(4):1023-1032.

Examples

set.seed(123)
## Example of a binary endpoint
## Consider a randomized comparative trial designed to borrow information
## from historical data on the control. We assumed a non-informative prior
## beta(1, 1) and an informative prior beta(30, 50) after incorporating
## the historical data. The treatment is regarded as superior to the control
## if Pr(RR.t > RR.c | data) > 0.95, where RR.t and RR.c are response rates
##  of the treatment and control, respectively. The operating characteristics
##  were assessed under the scenarios of (RR.c, RR.t) = (0.3, 0.36) and (0.3, 0.56).
## OC <- runSAM(## Informative prior constructed based on historical data
##              if.prior = mixbeta(c(1, 30, 50)),
##              ## Non-informative prior used for constructing the SAM prior
##              nf.prior = mixbeta(c(1,1,1)),
##              delta    = 0.2,  ## Clinically significant difference
##              n = 35,          ## Sample size for the control arm
##              n.t = 70,        ## Sample size for the treatment arm
##              ## Decision rule to compare the whether treatment is superior
##              ## than the control
##              decision = decision2S(0.95, 0, lower.tail=FALSE),
##              ntrial   = 1000,  ## Number of trials simulated
##              ## Weight assigned to the informative component for MAP prior
##              weight = 0.5,
##              ## A vector of response rate for the control arm
##              theta    = c(0.3, 0.36),
##              ## A vector of response rate for the treatment arm
##              theta.t  = c(0.3, 0.56))
## OC

## Example of continuous endpoint
## Consider a randomized comparative trial designed to borrow information
## from historical data on the control. We assumed a non-informative prior
## N(0, 1e4) and an informative prior N(0.5, 2) after incorporating
## the historical data. The treatment is regarded as superior to the control
## if Pr(mean.t > mean.c | data) > 0.95, where mean.t and mean.c are mean
##  of the treatment and control, respectively. The operating characteristics
##  were assessed under the scenarios of (mean.c, mean.t) = (0.1, 0.1) and
## (0.5, 1.0).
sigma      <- 2
prior.mean <- 0.5
prior.se   <- sigma/sqrt(100)
## OC <- runSAM(## Informative prior constructed based on historical data
##              if.prior = mixnorm(c(1, prior.mean, prior.se)),
##              ## Non-informative prior used for constructing the SAM prior
##              nf.prior = mixnorm(c(1, 0, 1e4)),
##              delta    = 0.2 * sigma,  ## Clinically significant difference
##              n = 100,                 ## Sample size for the control arm
##              n.t = 200,               ## Sample size for the treatment arm
##              ## Decision rule to compare the whether treatment is superior
##              ## than the control
##              decision = decision2S(0.95, 0, lower.tail=FALSE),
##              ntrial   = 1000,  ## Number of trials simulated
##              ## A vector of mean for the control arm
##              theta    = c(0.1, 0.5),
##              ## A vector of mean for the treatment arm
##              theta.t  = c(0.1, 1.0),
##              sigma = sigma)
## OC

Support of Distributions

Description

Returns the support of a distribution. This function is adapted from the 'SAMprior' package (v1.1.1) to support the runSAM function.

Usage

mixlink(mix, x)

mixinvlink(mix, x)

mixJinv_orig(mix, x)

mixlJinv_orig(mix, x)

mixlJinv_link(mix, l)

is.dlink(x)

is.dlink_identity(x)

is.mixidentity_link(mix, l)

support(mix)

## Default S3 method:
support(mix)

## S3 method for class 'betaMix'
support(mix)

## S3 method for class 'gammaMix'
support(mix)

## S3 method for class 'normMix'
support(mix)

Arguments

Value

A numeric vector of length 2 containing the lower and upper bounds of the support for the specified mixture distribution. If the mixture type is unknown (default method), it returns the character string "Unknown mixture".