RARtrials: Response-Adaptive Randomization in Clinical Trials

Description

RAR package is designed for simulating most popular response-adaptive randomization methods in the literature with comparisons of each treatment group to a control group under no delay and delayed scenarios. All the designs are based on one-sided tests with a choice from values of 'upper' and 'lower'. The general assumption is that binary outcomes follow a Binomial distribution, while continuous outcomes follow a normal distribution. Additionally, the number of patients accrued in the population follows a Poisson distribution and users can specify the enrollment rate of patients enrolled in the trial.

Author(s)

Maintainer: Chuyao Xu cxu870@aucklanduni.ac.nz

Authors:

• Thomas Lumley t.lumley@auckland.ac.nz [thesis advisor]

• Alain Vandal alain.vandal@auckland.ac.nz [thesis advisor]

Other contributors:

• Sofia S. Villar sofia.villar@mrc-bsu.cam.ac.uk [copyright holder]

• Kyle J. Wathen kylewathen@gmail.com [copyright holder]

See Also

Useful links:

• https://github.com/yayayaoyaoyao/RARtrials

Gittins Indices

Description

Gittins can provide Gittins indices for binary reward processes and normal reward processes with known and unknown variance for certain discount factors. Binary reward process can handle scenarios with up to 2000 participants in a trial, while normal reward process can handle scenarios with up to 10000 participants in a trial.

Usage

Gittins(Gittinstype, df)

Arguments

Details

Gittins indices for binary outcomes are generated from bmab_gi_multiple_ab function from gittins package with time horizon 100, 100, 100, 1000, 1000 for discount factor 0, 0.5, 0.7, 0.99 and 0.995 respectively. Gittins indices for continuous outcomes are obtained by linear extrapolation using Table 8.1 and Table 8.3 in (Gittins et al. 2011).

Value

A vector of Gittins indices for Gittinstype in 'UNKV' and 'KV'. A matrix of Gittins indices for Gittinstype in 'Binary'.

References

Gittins J, Glazebrook K, Weber R (2011). Multi-Armed Bandit Allocation Indices, 2nd Edition, volume 33. Hoboken,NJ:John Wiley & Sons. ISBN 9780470670026, doi:10.1002/9780470980033.ch8.

Examples

Gittins(Gittinstype='KV',df=0.5)

Gittins(Gittinstype='Binary',df=0.995)
Gittins(Gittinstype='UNKV',df=0.99)

Select au in Bayesian Response-Adaptive Randomization with a Control Group for Binary Endpoint

Description

brar_select_au_binary involves selecting au in Bayesian Response-Adaptive Randomization with a control group for binary outcomes with two to five arms. The conjugate prior distributions follow Beta (Beta(\alpha,\beta)) distributions and can be specified individually for each arm.

Usage

brar_select_au_binary(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  armn,
  h,
  N2,
  tp,
  armlabel,
  blocksize,
  alpha1 = 1,
  beta1 = 1,
  alpha2 = alpha1,
  beta2 = beta1,
  alpha3 = alpha1,
  beta3 = beta1,
  alpha4 = alpha1,
  beta4 = beta1,
  alpha5 = alpha1,
  beta5 = beta1,
  minstart,
  deltaa,
  tpp = 0,
  deltaa1,
  side,
  output = NULL,
  ...
)

Arguments

Details

This function generates a data set or a value in one iteration for selecting the appropriate au using Bayesian response-adaptive randomization with a control group under null hypotheses with no delay and delayed scenarios. The function can handle trials with up to 5 arms for binary outcomes. This function uses the formula \frac{Pr(p_k={\sf max}\{p_1,...,p_K\})^{tp}} {\sum_{k=1}^{K}{Pr(p_k={\sf max}\{p_1,...,p_K\})^{tp}}} with side equals to 'upper', and \frac{Pr(p_k={\sf min}\{p_1,...,p_K\})^{tp}} {\sum_{k=1}^{K}{Pr(p_k={\sf min}\{p_1,...,p_K\}){tp}}} with side equals to 'lower', utilizing available data at each step. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

A list of results generated from formula Pr(p_k>p_{{\sf control}}+\delta|{\sf data}_{t-1}) at each step. Note that before final stage of the trial, test statistics is calculated from deltaa, and test statistics is calculated from deltaa1 at the final stage.

References

Wathen J, Thall P (2017). “A simulation study of outcome adaptive randomization in multi-arm clinical trials.” Clinical Trials, 14, 174077451769230. doi:10.1177/1740774517692302.

Examples

#brar_select_au_binary with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days, where h1=c(0.2,0.4), tp=0.5,
#early futility stopping is set at -0.085, and the minimal effect size is 0.1.
set.seed(123)
stopbound1<-lapply(1:10,function(x){brar_select_au_binary(Pats=10,
nMax=50000,TimeToOutcome=expression(rnorm( length( vStartTime ),30, 3)),
enrollrate=0.1,N1=24,armn=2,h=c(0.3,0.3),N2=224,tp=0.5,armlabel=c(1,2),
blocksize=4,alpha1=1,beta1=1,alpha2=1,beta2=1,minstart=24,deltaa=-0.01,
tpp=0,deltaa1=0.1,side='upper')})
simf<-list()
for (xx in 1:10){
    if (any(stopbound1[[xx]][24:223,2]<0.01)){
      simf[[xx]]<-NA
   }  else{
      simf[[xx]]<-stopbound1[[xx]][224,2]
 }
}
simf1<-do.call(rbind,simf)
sum(is.na(simf1))/10  #1, achieve around 10% futility
sum(simf1>0.36,na.rm=TRUE)/10  #0.1
#the selected stopping boundary is 0.36 with an overall upper one-sided type I
#error of 0.1, based on 10 simulations. It is recommended to conduct more simulations 
#(i.e., 1000) to obtain an accurate au.

Select au in Bayesian Response-Adaptive Randomization with a Control Group for Continuous Endpoint with Known Variances

Description

brar_select_au_known_var involves selecting au in Bayesian Response-Adaptive Randomization with a control group for continuous endpoints with known variance in trials with two to five arms. The conjugate prior distributions follow Normal (N(mean,sd)) distributions and can be specified individually for each arm.

Usage

brar_select_au_known_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  armn,
  N2,
  tp,
  armlabel,
  blocksize,
  mean,
  sd,
  minstart,
  deltaa,
  tpp,
  deltaa1,
  mean10 = 0,
  mean20 = mean10,
  mean30 = mean10,
  mean40 = mean10,
  mean50 = mean10,
  sd10 = 1,
  sd20 = sd10,
  sd30 = sd10,
  sd40 = sd10,
  sd50 = sd10,
  n10 = 1,
  n20 = n10,
  n30 = n10,
  n40 = n10,
  n50 = n10,
  side,
  output = NULL,
  ...
)

Arguments

Details

This function generates a data set or a value in one iteration for selecting the appropriate au using Bayesian response-adaptive randomization with a control group under null hypotheses with no delay and delayed scenarios. The function can handle trials with up to 5 arms for continuous outcomes with known variances. This function uses the formula \frac{Pr(\mu_k={\sf max}\{\mu_1,...,\mu_K\})^{tp}} {\sum_{k=1}^{K}{Pr(\mu_k={\sf max}\{\mu_1,...,\mu_K\})^{tp}}} with side equals to 'upper', and \frac{Pr(\mu_k={\sf min}\{\mu_1,...,\mu_K\})^{tp}} {\sum_{k=1}^{K}{Pr(\mu_k={\sf min}\{\mu_1,...,\mu_K\}){tp}}} with side equals to 'lower', utilizing available data at each step. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

A list of results generated from formula Pr(\mu_k>\mu_{control}+\delta|data_{t-1}) at each step. Note that before final stage of the trial, test statistics is calculated from deltaa, and test statistics is calculated from deltaa1 at the final stage.

References

Wathen J, Thall P (2017). “A simulation study of outcome adaptive randomization in multi-arm clinical trials.” Clinical Trials, 14, 174077451769230. doi:10.1177/1740774517692302.

Examples

#brar_select_au_known_var with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days, where mean=c(8.9/100,8.74/100,8.74/100),
#sd=c(0.009,0.009,0.009), tp=0.5 and the minimal effect size is 0.
set.seed(789)
stopbound1<-lapply(1:10,function(x){
brar_select_au_known_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm(length( vStartTime ),30, 3)),enrollrate=0.1, N1=21,armn=3,
N2=189,tp=0.5,armlabel=c(1,2,3),blocksize=6,mean=c((8.9/100+8.74/100+8.74/100)/3,
(8.9/100+8.74/100+8.74/100)/3,(8.9/100+8.74/100+8.74/100)/3),
sd=c(0.009,0.009,0.009),minstart=21,deltaa=c(0,0.001),tpp=0,deltaa1=c(0,0),
mean10=0.09,mean20=0.09,mean30=0.09, sd10=0.01,sd20=0.01,sd30=0.01,n10=1,n20=1,
n30=1,side='lower')})
simf<-list()
simf1<-list()
for (xx in 1:10){
 if (any(stopbound1[[xx]][21:188,2]<0.01)){
      simf[[xx]]<-NA
   }  else{
      simf[[xx]]<-stopbound1[[xx]][189,2]
 }
 if (any(stopbound1[[xx]][21:188,3]<0.01)){
      simf1[[xx]]<-NA
   }  else{
      simf1[[xx]]<-stopbound1[[xx]][189,3]
 }
}
simf2<-do.call(rbind,simf)
sum(is.na(simf2)) #1, achieve around 10% futility
simf3<-do.call(rbind,simf1)
sum(is.na(simf3)) #1, achieve around 10% futility
stopbound1a<-cbind(simf2,simf3)
stopbound1a[is.na(stopbound1a)] <- 0
sum(stopbound1a[,1]>0.973 | stopbound1a[,2]>0.973)/10 #0.1
#the selected stopping boundary is 0.973 with an overall lower one-sided type I
#error of 0.1, based on 10 simulations. Because it is under the permutation null hypothesis,
#the selected deltaa should be an average of 0 and 0.001 which is 0.0005, although
#deltaa could be close to each other with larger simulation numbers.
#It is recommended to conduct more simulations (i.e., 1000) to obtain an accurate deltaa and au.
#As the simulation number increases, the choice of deltaa could be consistent for comparisons
#of each arm to the control.

Select au in Bayesian Response-Adaptive Randomization with a Control Group for Continuous Endpoint with Unknown Variances

Description

brar_select_au_unknown_var involves selecting au in Bayesian Response-Adaptive Randomization with a control group for continuous endpoints with unknown variance in trials with two to five arms. The conjugate prior distributions follow Normal-Inverse-Gamma (NIG) ((\mu,\sigma^2) \sim NIG({\sf mean}=m,{\sf variance}=V \times \sigma^2,{\sf shape}=a,{\sf rate}=b)) distributions and can be specified individually for each arm.

Usage

brar_select_au_unknown_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  armn,
  N2,
  tp,
  armlabel,
  blocksize,
  mean,
  sd,
  minstart,
  deltaa,
  tpp,
  deltaa1,
  side,
  output = NULL,
  V01,
  a01,
  b01,
  m01,
  V02 = V01,
  V03 = V01,
  V04 = V01,
  V05 = V01,
  a02 = a01,
  a03 = a01,
  a04 = a01,
  a05 = a01,
  b02 = b01,
  b03 = b01,
  b04 = b01,
  b05 = b01,
  m02 = m01,
  m03 = m01,
  m04 = m01,
  m05 = m01,
  ...
)

Arguments

Details

This function generates a data set or a value in one iteration for selecting the appropriate au using Bayesian response-adaptive randomization with a control group under null hypotheses with no delay and delayed scenarios. The function can handle trials with up to 5 arms for continuous outcomes with unknown variances. This function uses the formula \frac{Pr(\mu_k={\sf max}\{\mu_1,...,\mu_K\})^{tp}} {\sum_{k=1}^{K}{Pr(\mu_k={\sf max}\{\mu_1,...,\mu_K\})^{tp}}} with side equals to 'upper', and \frac{Pr(\mu_k={\sf min}\{\mu_1,...,\mu_K\})^{tp}} {\sum_{k=1}^{K}{Pr(\mu_k={\sf min}\{\mu_1,...,\mu_K\}){tp}}} with side equals to 'lower', utilizing available data at each step. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

A list of results generated from formula Pr(\mu_k>\mu_{{\sf control}}+\delta|{\sf data}_{t-1}) at each step. Note that before final stage of the trial, test statistics is calculated from deltaa, and test statistics is calculated from deltaa1 at the final stage.

References

Wathen J, Thall P (2017). “A simulation study of outcome adaptive randomization in multi-arm clinical trials.” Clinical Trials, 14, 174077451769230. doi:10.1177/1740774517692302.

Examples

#brar_select_au_unknown_var with delayed responses follow a normal distribution with
#a mean of 60 days and a standard deviation of 3 days, where 
#mean=c((9.1/100+8.74/100+8.74/100)/3,(9.1/100+8.74/100+8.74/100)/3,
#(9.1/100+8.74/100+8.74/100)/3),sd=c(0.009,0.009,0.009),tp=1 and 
#the minimal effect size is 0. All arms have the same prior distributions.
set.seed(789)
stopbound1<-lapply(1:5,function(x){
brar_select_au_unknown_var(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm(
length( vStartTime ),30, 3)), enrollrate=0.1, N1=48, armn=3, N2=480, tp=1,
armlabel=c(1,2,3), blocksize=6, mean=c((9.1/100+8.74/100+8.74/100)/3,
(9.1/100+8.74/100+8.74/100)/3,(9.1/100+8.74/100+8.74/100)/3) ,
sd=c(0.009,0.009,0.009),minstart=48, deltaa=c(-0.0003,-0.00035), tpp=0, 
deltaa1=c(0,0),V01=1/2,a01=0.3,m01=9/100,b01=0.00001,side='lower')
})

simf<-list()
simf1<-list()
for (xx in 1:5){
 if (any(stopbound1[[xx]][48:479,2]<0.01)){
      simf[[xx]]<-NA
   }  else{
      simf[[xx]]<-stopbound1[[xx]][480,2]
 }
 if (any(stopbound1[[xx]][48:479,3]<0.01)){
      simf1[[xx]]<-NA
   }  else{
      simf1[[xx]]<-stopbound1[[xx]][480,3]
 }
}
simf2<-do.call(rbind,simf)
sum(is.na(simf2)) #1, achieve around 20% futility
simf3<-do.call(rbind,simf1)
sum(is.na(simf3)) #1, achieve around 20% futility
stopbound1a<-cbind(simf2,simf3)
stopbound1a[is.na(stopbound1a)] <- 0
sum(stopbound1a[,1]>0.85 | stopbound1a[,2]>0.85)/5 #0.2
#the selected stopping boundary is 0.85 with an overall lower one-sided type 
#I error of 0.2, based on 5 simulations. Because it is under the permutation null hypothesis,
#the selected deltaa should be an average of -0.0003 and -0.00035 which is -0.000325.
#It is recommended to conduct more simulations (i.e., 1000)  
#to obtain an accurate deltaa and au. As the simulation number increases, the
#choice of deltaa could be consistent for comparisons of each arm to the control.

Convert parameters from a Normal-Inverse-Chi-Squared Distribution to a Normal-Inverse-Gamma Distribution

Description

Convert parameters from a Normal-Inverse-Chi-Squared distribution to a Normal-Inverse-Gamma distribution.

Usage

convert_chisq_to_gamma(cpar)

Arguments

Details

This function convert parameters from a Normal-Inverse-Chi-Squared ((\mu,\sigma^2) \sim NIX({\sf mean}=\mu,{\sf effective sample size}=\kappa,{\sf degrees of freedom}=\nu,{\sf variance}=\sigma^2/\kappa)) distribution to a Normal-Inverse-Gamma ((\mu,\sigma^2) \sim NIG({\sf mean}=m,{\sf variance}=V \times \sigma^2,{\sf shape}=a,{\sf rate}=b)) distribution.

Value

A list of parameters including m, V, a, b from a Normal-Inverse-Gamma distribution.

References

Murphy K (2007). “Conjugate Bayesian analysis of the Gaussian distribution.” University of British Columbia. https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf.

Examples

convert_chisq_to_gamma(list(mu=0.091,kappa=2,nu=1,sigsq=4e-05))

Convert parameters from a Normal-Inverse-Gamma Distribution to a Normal-Inverse-Chi-Squared Distribution

Description

Convert parameters from a Normal-Inverse-Gamma distribution to a Normal-Inverse-Chi-Squared distribution.

Usage

convert_gamma_to_chisq(gpar)

Arguments

Details

This function convert parameters from a Normal-Inverse-Gamma ((\mu,\sigma^2) \sim NIG({\sf mean}=m,{\sf variance}=V \times \sigma^2,{\sf shape}=a,{\sf rate}=b)) distribution to a Normal-Inverse-Chi-Squared ((\mu,\sigma^2) \sim NIX({\sf mean}=\mu,{\sf effective sample size}=\kappa,{\sf degrees of freedom}=\nu,{\sf variance}=\sigma^2/\kappa)) distribution.

Value

A list of parameters including mu, kappa, nu, sigsq from a Normal-Inverse-Chi-Squared distribution.

References

Murphy K (2007). “Conjugate Bayesian analysis of the Gaussian distribution.” University of British Columbia. https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf.

Examples

convert_gamma_to_chisq(list(V=1/2,a=0.5,m=9.1/100,b=0.00002))

Allocation Probabilities Using Doubly Adaptive Biased Coin Design with Maximal Power Strategy for Binary Endpoint

Description

dabcd_max_power can be used for doubly adaptive biased coin design with maximal power strategy for binary outcomes, targeting generalized Neyman allocation and generalized RSIHR allocation. The return of this function is a vector of allocation probabilities to each arm, with the pre-specified number of participants in the trial.

Usage

dabcd_max_power(NN, Ntotal1, armn, BB, type, dabcd = FALSE, gamma = 2)

Arguments

Details

The function simulates allocation probabilities for doubly adaptive biased coin design with maximal power strategy targeting generalized Neyman allocation with 2-5 arms which is provided in (Tymofyeyev et al. 2007) or generalized RSIHR allocation with 2-3 arms which is provided in (Jeon and Feifang 2010), with modifications for typos in (Sabo and Bello 2016). All of those methods are not smoothed. The output of this function is based on Hu \& Zhang's formula (Hu and Zhang 2004). With more than two armd the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction.

Value

A vector of allocation probabilities to each arm.

Author(s)

Chuyao Xu, Thomas Lumley, Alain Vandal

References

Hu F, Zhang L (2004). “Asymptotic Properties of Doubly Adaptive Biased Coin Designs for Multitreatment Clinical Trials.” The Annals of Statistics, 32(1), 268–301.

Tymofyeyev Y, Rosenberger WF, Hu F (2007). “Implementing Optimal Allocation in Sequential Binary Response Experiments.” Journal of the American Statistical Association, 102(477), 224-234. doi:10.1198/016214506000000906.

Jeon Y, Feifang H (2010). “Optimal Adaptive Designs for Binary Response Trials With Three Treatments.” Statistics in Biopharmaceutical Research, 2, 310-318. doi:10.1198/sbr.2009.0056.

Sabo R, Bello G (2016). “Optimal and lead-in adaptive allocation for binary outcomes: a comparison of Bayesian methodologies.” Communications in Statistics - Theory and Methods, 46.

Examples

dabcd_max_power(NN=c(54,67,85,63,70),Ntotal1=c(100,88,90,94,102),armn=5,BB=0.2, type='Neyman')
dabcd_max_power(NN=c(54,67,85,63),Ntotal1=c(100,88,90,94),armn=4,BB=0.2, type='Neyman')

Allocation Probabilities Using Doubly Adaptive Biased Coin Design with Minimal Variance Strategy for Binary Endpoint

Description

dabcd_min_var can be used for doubly adaptive biased coin design with minimal variance strategy for binary outcomes, targeting generalized Neyman allocation and generalized RSIHR allocation. The return of this function is a vector of allocation probabilities to each arm, with the pre-specified number of participants in the trial.

Usage

dabcd_min_var(NN, Ntotal1, armn, type, dabcd = FALSE, gamma = 2)

Arguments

Details

The function simulates allocation probabilities for doubly adaptive biased coin design with minimal variance strategy targeting generalized Neyman allocation and generalized RSIHR allocation with 2-5 arms. The output of this function is based on Hu \& Zhang's formula (Hu and Zhang 2004). With more than two armd the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction.

Value

A vector of allocation probabilities to each arm.

Author(s)

Chuyao Xu, Thomas Lumley, Alain Vandal

References

Hu F, Zhang L (2004). “Asymptotic Properties of Doubly Adaptive Biased Coin Designs for Multitreatment Clinical Trials.” The Annals of Statistics, 32(1), 268–301.

Examples

dabcd_min_var(NN=c(54,67,85,63,70),Ntotal1=c(100,88,90,94,102),armn=5, type='Neyman')
dabcd_min_var(NN=c(54,67,85,63),Ntotal1=c(100,88,90,94),armn=4,type='RSIHR')

Cut-off Value of the Forward-looking Gittins Index Rule in Binary Endpoint

Description

Function for simulating cut-off values at the final stage using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule for binary outcomes in trials with 2-5 arms. The conjugate prior distributions follow Beta (Beta(\alpha,\beta)) distributions and should be the same for each arm.

Usage

flgi_cut_off_binary(
  Gittinstype,
  df,
  gittins = NULL,
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  I0,
  K,
  noRuns2 = 100,
  Tsize,
  ptrue,
  block,
  rule,
  ztype
)

Arguments

Details

This function simulates trials using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule under both no delay and delayed scenarios to obtain cut-off values at the final stage, with control of type I error. The user is expected to run this function multiple times to determine a reasonable cut-off value for statistical inference. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

Value of Z test statistics for one trial.

References

Gittins J, Glazebrook K, Weber R (2011). Multi-Armed Bandit Allocation Indices, 2nd Edition, volume 33. Hoboken,NJ:John Wiley & Sons. ISBN 9780470670026, doi:10.1002/9780470980033.ch8.

Villar S, Wason J, Bowden J (2015). “Response-Adaptive Randomization for Multi-arm Clinical Trials Using the Forward Looking Gittins Index Rule.” Biometrics, 71. doi:10.1111/biom.12337.

Examples

#The forward-looking Gittins Index rule with delayed responses follow a normal distribution
#with a mean of 60 days and a standard deviation of 3 days
#One can run the following command 20000 times to obtain the selected cut-off value around 1.9991
#with an upper one-sided type I error 0.025

stopbound1<-lapply(1:20000,function(x){ flgi_cut_off_binary(Gittinstype='Binary',df=0.5,Pats=10,
nMax=50000,TimeToOutcome=expression(rnorm( length( vStartTime ),60, 3)),
enrollrate=0.9,I0= matrix(1,nrow=2,ncol=2),K=2,Tsize=992,ptrue=c(0.6,0.6),block=20,
rule='FLGI PM',ztype='unpooled')})
stopbound1a<-do.call(rbind,stopbound1)
sum(stopbound1a>(1.9991) )/20000
#The selected cut-off value is around 1.9991 with an overall lower one-sided type I
#error of 0.025, based on 20000 simulations.

Cut-off Value of the Forward-looking Gittins Index Rule in Continuous Endpoint with Known Variances

Description

Function for simulating cut-off values at the final stage using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule for continuous outcomes with known variance in trials with 2-5 arms. The conjugate prior distributions follow Normal (N({\sf mean},{\sf sd})) distributions and should be the same for each arm.

Usage

flgi_cut_off_known_var(
  Gittinstype,
  df,
  gittins = NULL,
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  K,
  noRuns2,
  Tsize,
  block,
  rule,
  prior_n,
  prior_mean,
  mean,
  sd,
  side
)

Arguments

Details

This function simulates trials using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule under both no delay and delayed scenarios to obtain cut-off values at the final stage, with control of type I error. The user is expected to run this function multiple times to determine a reasonable cut-off value for statistical inference. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

Value of Z test statistics for one trial.

References

Williamson SF, Villar S (2019). “A Response‐Adaptive Randomization Procedure for Multi‐Armed Clinical Trials with Normally Distributed Outcomes.” Biometrics, 76. doi:10.1111/biom.13119.

Examples

#The forward-looking Gittins Index rule with delayed responses follow a normal 
#distribution with a mean of 30 days and a standard deviation of 3 days
#One can run the following command 20000 times to obtain the selected cut-off
#value around -2.1725 with an overall lower one-sided type I error 0.025

stopbound1<-lapply(1:20000,function(x){ 
flgi_cut_off_known_var(Gittinstype='KV',df=0.995,Pats=10,nMax=50000,
TimeToOutcome=expression(rnorm( length( vStartTime ),30, 3)),enrollrate=0.5,
K=3,noRuns2=100,Tsize=852,block=20,rule='FLGI PM',prior_n=rep(1,3),
prior_mean=rep(9/100,3),mean=c(9.1/100,9.1/100,9.1/100),sd=c(0.009,0.009,0.009),
side='lower')})
stopbound1a<-do.call(rbind,stopbound1)
sum(stopbound1a<(-2.1725) )/20000
#The selected cut-off value is around -2.1725 with an overall lower one-sided 
#type I error of 0.025, based on 20000 simulations.



#One can run the following command 20000 times to obtain the selected cut-off 
#value around -2.075 with an overall lower one-sided type I error 0.025

stopbound1<-lapply(1:20000,function(x){
flgi_cut_off_known_var(Gittinstype='KV',df=0.995,Pats=10,nMax=50000,
TimeToOutcome=expression(rnorm( length( vStartTime ),30, 3)),enrollrate=0.1,
K=3,noRuns2=100,Tsize=852,block=20,rule='CFLGI',prior_n=rep(1,3),
prior_mean=rep(9/100,3),mean=c(9.1/100,9.1/100,9.1/100),sd=c(0.009,0.009,0.009),
side='lower')})
stopbound1a<-do.call(rbind,stopbound1)
sum(stopbound1a<(-2.075) )/20000
#The selected cut-off value is around -2.075 with an overall lower one-sided type I
#error of 0.025, based on 20000 simulations.

Cut-off Value of the Forward-looking Gittins Index rule in Continuous Endpoint with Unknown Variances

Description

Function for simulating cut-off values at the final stage using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule for continuous outcomes with known variance in trials with 2-5 arms. The prior distributions follow Normal-Inverse-Gamma (NIG) ((\mu,\sigma^2) \sim NIG({\sf mean}=m,{\sf variance}=V \times \sigma^2,{\sf shape}=a,{\sf rate}=b)) distributions and should be the same for each arm.

Usage

flgi_cut_off_unknown_var(
  Gittinstype,
  df,
  gittins = NULL,
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  K,
  noRuns2,
  Tsize,
  block,
  rule,
  prior_n,
  prior_mean1,
  prior_sd1,
  mean,
  sd,
  side
)

Arguments

Details

This function simulates trials using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule under both no delay and delayed scenarios to obtain cut-off values at the final stage, with control of type I error. The user is expected to run this function multiple times to determine a reasonable cut-off value for statistical inference. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

Value of T test statistics for one trial.

References

Williamson SF, Villar S (2019). “A Response‐Adaptive Randomization Procedure for Multi‐Armed Clinical Trials with Normally Distributed Outcomes.” Biometrics, 76. doi:10.1111/biom.13119.

Examples

#The forward-looking Gittins Index rule with delayed responses follow a normal 
#distribution with a mean of 60 days and a standard deviation of 3 days
#One can run the following command 20000 times to obtain the selected cut-off 
#value around -1.9298 with an overall lower one-sided type I error 0.025

stopbound1<-lapply(1:20000,function(x){ 
flgi_cut_off_unknown_var(Gittinstype='UNKV',df=0.5,Pats=10,nMax=50000,
TimeToOutcome=expression(rnorm( length( vStartTime ),60, 3)),enrollrate=0.9,
K=3,noRuns2=100,Tsize=852,block=20,rule='FLGI PM',prior_n=rep(2,3),
prior_mean1=rep(9/100,3),prior_sd1=rep(0.006324555,3),
mean=c(9.1/100,9.1/100,9.1/100),sd=c(0.009,0.009,0.009),side='lower')})
stopbound1a<-do.call(rbind,stopbound1)
sum(stopbound1a<(-1.9298) )/20000
#The selected cut-off value is around -1.9298 with an overall lower one-sided 
#type I error of 0.025, based on 20000 simulations.

Calculate the Futility Stopping Probability for Continuous Endpoint with Unknown Variances Using a Normal-Inverse-Chi-Squared Distribution

Description

Calculate the futility stopping probability in Bayesian response-adaptive randomization with a control group using the Thall \& Wathen method for continuous outcomes with unknown variances. The prior distributions follow Normal-Inverse-Chi-Squared (NIX) distributions and can be specified individually for each treatment group.

Usage

pgreater_NIX(par1, par2, delta = 0, side, ...)

Arguments

Details

This function calculates the results of Pr(\mu_k>\mu_{{\sf control}}+\delta|{\sf data}) for side equals to 'upper' and the results of Pr(\mu_{{\sf control}}>\mu_k+\delta|{\sf data}) for side equals to 'lower'. The result indicates the posterior probability of stopping a treatment group due to futility around 1\% in Bayesian response-adaptive randomization with a control arm using Thall \& Wathen method, with accumulated results during the conduct of trials. Parameters used in a Normal-Inverse-Gamma ((\mu,\sigma^2) \sim NIG({\sf mean}=m,{\sf variance}=V \times \sigma^2,{\sf shape}=a,{\sf rate}=b)) distribution should be converted to parameters equivalent in a Normal-Inverse-Chi-Squared ((\mu,\sigma^2) \sim NIX({\sf mean}=\mu,{\sf effective sample size}=\kappa,{\sf degrees of freedom}=\nu,{\sf variance}=\sigma^2/\kappa)) distribution using convert_gamma_to_chisq before applying this function.

Value

a posterior probability of Pr(\mu_k>\mu_{control}+\delta|data) with side equals to 'upper'; a posterior probability of Pr(\mu_{control}>\mu_k+\delta|data) with side equals to 'lower'.

References

Wathen J, Thall P (2017). “A simulation study of outcome adaptive randomization in multi-arm clinical trials.” Clinical Trials, 14, 174077451769230. doi:10.1177/1740774517692302.

Murphy K (2007). “Conjugate Bayesian analysis of the Gaussian distribution.” University of British Columbia. https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf.

Examples

para<-list(V=1/2,a=0.5,m=9.1/100,b=0.00002)
par<-convert_gamma_to_chisq(para)
set.seed(123451)
y1<-rnorm(100,0.091,0.009)
par1<-update_par_nichisq(y1, par)
set.seed(123452)
y2<-rnorm(90,0.09,0.009)
par2<-update_par_nichisq(y2, par)
pgreater_NIX(par1=par1,par2=par2, side='upper')
pgreater_NIX(par1=par1,par2=par2, side='lower')

Calculate the Futility Stopping Probability for Binary Endpoint with Beta Distribution

Description

Calculate the futility stopping probability in Bayesian response-adaptive randomization with a control group using the Thall \& Wathen method for binary outcomes. The conjugate prior distributions follow Beta (Beta(\alpha,\beta)) distributions and can be specified individually for each treatment group.

Usage

pgreater_beta(a1, b1, a2, b2, delta, side, ...)

Arguments

Details

This function calculates the results of Pr(p_k>p_{{\sf control}}+\delta|{\sf data}) for side equals to 'upper' and the results of Pr(p_{{\sf control}}>p_k+\delta|{\sf data}) for side equals to 'lower'. The result indicates the posterior probability of stopping a treatment group due to futility around 1\% in Bayesian response-adaptive randomization with a control arm using Thall \& Wathen method, with accumulated results during the conduct of trials.

Value

a posterior probability of Pr(p_k>p_{{\sf control}}+\delta|{\sf data}) with side equals to 'upper'; a posterior probability of Pr(p_{{\sf control}}>p_k+\delta|{\sf data}) with side equals to 'lower'.

References

Wathen J, Thall P (2017). “A simulation study of outcome adaptive randomization in multi-arm clinical trials.” Clinical Trials, 14, 174077451769230. doi:10.1177/1740774517692302.

Examples

pgreater_beta(a1=8, b1=10,a2=5, b2=19, delta=0.1, side='upper')
pgreater_beta(a1=65, b1=79,a2=58, b2=68, delta=0, side='lower')

Calculate the Futility Stopping Probability for Continuous Endpoint with Known Variances Using Normal Distribution

Description

Calculate the futility stopping probability in Bayesian response-adaptive randomization with a control group using the Thall \& Wathen method for continuous outcomes with known variances. The conjugate prior distributions follow Normal (N(mean,sd)) distributions and can be specified individually for each treatment group.

Usage

pgreater_normal(
  mean1 = NULL,
  sd1 = NULL,
  mean2 = NULL,
  sd2 = NULL,
  delta = 0,
  side,
  ...
)

Arguments

Details

This function calculates the results of Pr(\mu_k>\mu_{{\sf control}}+\delta|{\sf data}) for side equals to 'upper' and the results of Pr(\mu_{{\sf control}}>\mu_k+\delta|{\sf data}) for side equals to 'lower'. The result indicates the posterior probability of stopping a treatment group due to futility around 1\% in Bayesian response-adaptive randomization with a control arm using Thall \& Wathen method, with accumulated results during the conduct of trials.

Value

a posterior probability of Pr(\mu_k>\mu_{{\sf control}}+\delta|{\sf data}) with side equals to 'upper'; a posterior probability of Pr(\mu_{{\sf control}}>\mu_k+\delta|{\sf data}) with side equals to 'lower'.

References

Wathen J, Thall P (2017). “A simulation study of outcome adaptive randomization in multi-arm clinical trials.” Clinical Trials, 14, 174077451769230. doi:10.1177/1740774517692302. Murphy K (2007). “Conjugate Bayesian analysis of the Gaussian distribution.” University of British Columbia. https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf.

Examples

pgreater_normal(mean1=0.091,sd1=0.09,mean2=0.097,sd2=0.08,delta=0,side='upper')
pgreater_normal(mean1=0.091,sd1=0.09,mean2=0.087,sd2=0.1,delta=0,side='lower')

Posterior Probability that a Particular Arm is the Best for Continuous Endpoint with Unknown Variances

Description

Calculate posterior probability that a particular arm is the best in a trial using Bayesian response-adaptive randomization with a control group (the Thall \& Wathen method). The conjugate prior distributions follow Normal-Inverse-Chi-Squared (NIX) distributions for continuous outcomes with unknown variance in each arm and can be specified individually.

Usage

pmax_NIX(
  armn,
  par1 = NULL,
  par2 = NULL,
  par3 = NULL,
  par4 = NULL,
  par5 = NULL,
  side,
  ...
)

Arguments

Details

This function calculates the results of formula Pr(\mu_k={\sf max}\{\mu_1,...,\mu_k\}) for side equals to 'upper' and the results of formula Pr(\mu_k={\sf min}\{\mu_1,...,\mu_k\}) for side equals to 'lower'. This function returns the probability that the posterior probability of arm k is maximal or minimal in trials with up to five arms. Parameters used in a Normal-Inverse-Gamma ((\mu,\sigma^2) \sim NIG({\sf mean}=m,{\sf variance}=V \times \sigma^2,{\sf shape}=a,{\sf rate}=b)) distribution should be converted to parameters equivalent in a Normal-Inverse-Chi-Squared ((\mu,\sigma^2) \sim NIX({\sf mean}=\mu,{\sf effective sample size}=\kappa,{\sf degrees of freedom}=\nu,{\sf variance}=\sigma^2/\kappa)) distribution using convert_gamma_to_chisq before applying this function.

Value

a probability that a particular arm is the best in trials up to five arms.

Examples

para<-list(V=1/2,a=0.8,m=9.1,b=1/2)
par<-convert_gamma_to_chisq(para)
set.seed(123451)
y1<-rnorm(100,9.1,1)
par11<-update_par_nichisq(y1, par)
set.seed(123452)
y2<-rnorm(90,9,1)
par22<-update_par_nichisq(y2, par)
set.seed(123453)
y3<-rnorm(110,8.92,1)
par33<-update_par_nichisq(y3, par)
y4<-rnorm(120,8.82,1)
par44<-update_par_nichisq(4, par)
pmax_NIX(armn=4,par1=par11,par2=par22,par3=par33,par4=par44,side='upper')
pmax_NIX(armn=4,par1=par11,par2=par22,par3=par33,par4=par44,side='lower')

para<-list(V=1/2,a=0.5,m=9.1/100,b=0.00002)
par<-convert_gamma_to_chisq(para)
set.seed(123451)
y1<-rnorm(100,0.091,0.009)
par11<-update_par_nichisq(y1, par)
set.seed(123452)
y2<-rnorm(90,0.09,0.009)
par22<-update_par_nichisq(y2, par)
set.seed(123453)
y3<-rnorm(110,0.0892,0.009)
par33<-update_par_nichisq(y3, par)
pmax_NIX(armn=3,par1=par11,par2=par22,par3=par33,side='upper')
pmax_NIX(armn=3,par1=par11,par2=par22,par3=par33,side='lower')

Posterior Probability that a Particular Arm is the Best for Binary Endpoint

Description

Calculate posterior probability that a particular arm is the best in a trial using Bayesian response-adaptive randomization with a control group (the Thall \& Wathen method). The conjugate prior distributions follow Beta (Beta(\alpha,\beta)) distributions for binary outcomes in each arm and can be specified individually.

Usage

pmax_beta(
  armn,
  a1 = NULL,
  b1 = NULL,
  a2 = NULL,
  b2 = NULL,
  a3 = NULL,
  b3 = NULL,
  a4 = NULL,
  b4 = NULL,
  a5 = NULL,
  b5 = NULL,
  side,
  ...
)

Arguments

Details

This function calculates the results of formula Pr(p_k={\sf max}\{p_1,...,p_K\}) for side equals to 'upper' and the results of formula Pr(p_k={\sf min}\{p_1,...,p_K\}) for side equals to 'lower'. This function returns the probability that the posterior probability of arm k is maximal or minimal in trials with up to five arms.

Value

a probability that a particular arm is the best in trials up to five arms.

Examples

pmax_beta(armn=5,a1=8,b1=10,a2=5,b2=19,a3=8,b3=21,
a4=6, b4=35, a5=15, b5=4, side='upper')
pmax_beta(armn=4,a1=56,b1=98,a2=25,b2=70,a3=87,b3=107,
a4=106, b4=202, side='lower')
pmax_beta(armn=3,a1=60,b1=46,a2=55,b2=46,a3=35,b3=36,side='upper')

Posterior Probability that a Particular Arm is the Best for Continuous Endpoint with Known Variances

Description

Calculate posterior probability that a particular arm is the best in a trial using Bayesian response-adaptive randomization with a control group (the Thall \& Wathen method). The conjugate prior distributions follow Normal (N({\sf mean},{\sf sd})) distributions for continuous outcomes with known variance in each arm and can be specified individually.

Usage

pmax_normal(
  armn,
  mean1 = NULL,
  sd1 = NULL,
  mean2 = NULL,
  sd2 = NULL,
  mean3 = NULL,
  sd3 = NULL,
  mean4 = NULL,
  sd4 = NULL,
  mean5 = NULL,
  sd5 = NULL,
  side,
  ...
)

Arguments

Details

This function calculates the results of formula Pr(\mu_k={\sf max}\{\mu_1,...,\mu_K\}) for side equals to 'upper' and the results of formula Pr(\mu_k={\sf min}\{\mu_1,...,\mu_K\}) for side equals to 'lower'. This function returns the probability that the posterior probability of arm k is maximal or minimal in trials with up to five arms.

Value

a probability that a particular arm is the best in trials up to five arms.

Examples

pmax_normal(armn=5,mean1=0.8,sd1=0.2,mean2=0.5,sd2=0.1,mean3=0.8,
sd3=0.5,mean4=0.6,sd4=0.2,mean5=0.6,sd5=0.2,side='upper')
pmax_normal(armn=4,mean1=8,sd1=2,mean2=8.5,sd2=2,mean3=8.3,
sd3=1.8,mean4=8.7,sd4=2,side='lower')
pmax_normal(armn=3,mean1=80,sd1=20,mean2=50,sd2=10,mean3=80,
sd3=15,side='upper')

Simulate a Trial Using A-Optimal Allocation for Continuous Endpoint with Known Variances

Description

sim_A_optimal_known_var simulates a trial for continuous endpoints with known variances, and allocation ratios are fixed.

Usage

sim_A_optimal_known_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N2,
  armn,
  mean,
  sd,
  alphaa = 0.025,
  armlabel,
  side
)

Arguments

Details

This function aims to minimize the criteria tr[M^{-1}(\mathbf{\rho})] and minimize the overall variance of pairwise comparisons. It is generalized Neyman allocation, specifically designed for continuous endpoints with known variances. With more than two arms the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_A_optimal_known_var returns an object of class "aoptimal". An object of class "aoptimal" is a list containing final decision based on the Z test statistics with 1 stands for selected and 0 stands for not selected, Z test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

Author(s)

Chuyao Xu, Thomas Lumley, Alain Vandal

References

Sverdlov O, Rosenberger W (2013). “On Recent Advances in Optimal Allocation Designs in Clinical Trials.” Journal of Statistical Theory and Practice, 7, 753-773. doi:10.1080/15598608.2013.783726.

Examples

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under null hypothesis
#in a two-armed trial
sim_A_optimal_known_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm(length( vStartTime ),30, 3)),enrollrate=0.9,N2=88,armn=2,
mean=c(9.1/100,9.1/100),sd=c(0.009,0.009),alphaa=0.025,armlabel = c(1,2),side='lower')

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under alternative hypothesis
#in a two-armed trial
sim_A_optimal_known_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm(length( vStartTime ),30, 3)),enrollrate=0.9,N2=88,armn=2,
mean=c(9.1/100,8.47/100),sd=c(0.009,0.009),alphaa=0.025,armlabel = c(1,2),side='lower')

Simulate a Trial Using A-Optimal Allocation for Continuous Endpoint with Unknown Variances

Description

sim_A_optimal_unknown_var simulates a trial for continuous endpoints with unknown variances, and the allocation probabilities change based on results of accumulated participants in the trial.

Usage

sim_A_optimal_unknown_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  N2,
  armn,
  mean,
  sd,
  alphaa = 0.025,
  armlabel,
  side
)

Arguments

Details

This function aims to minimize the criteria tr[M^{-1}(\mathbf{\rho})] and to minimize the overall variance of pairwise comparisons. It is generalized Neyman allocation, specifically designed for continuous endpoints with known variances. With more than two arms the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_A_optimal_unknown_var returns an object of class "aoptimal". An object of class "aoptimal" is a list containing final decision based on the T test statistics with 1 stands for selected and 0 stands for not selected, T test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

Author(s)

Chuyao Xu, Thomas Lumley, Alain Vandal

References

Sverdlov O, Rosenberger W (2013). “On Recent Advances in Optimal Allocation Designs in Clinical Trials.” Journal of Statistical Theory and Practice, 7, 753-773. doi:10.1080/15598608.2013.783726.

Examples

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under null hypothesis
#in a three-armed trial
sim_A_optimal_unknown_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm( length( vStartTime ),30, 3)),enrollrate=0.1,N1=12,N2=132,armn=3,
mean=c(9.1/100,9.1/100,9.1/100),sd=c(0.009,0.009,0.009),alphaa=0.025,
armlabel = c(1,2,3),side='upper')

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under alternative hypothesis
#in a three-armed trial
sim_A_optimal_unknown_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm( length( vStartTime ),30, 3)),enrollrate=0.1,N1=12,N2=132,armn=3,
mean=c(9.1/100,9.28/100,9.28/100),sd=c(0.009,0.009,0.009),alphaa=0.025,
armlabel = c(1,2,3),side='upper')

Simulate a Trial Using Aa-Optimal Allocation for Continuous Endpoint with Known Variances

Description

sim_Aa_optimal_known_var simulates a trial for continuous endpoints with known variances, and the allocation probabilities are fixed.

Usage

sim_Aa_optimal_known_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N2,
  armn,
  mean,
  sd,
  alphaa = 0.025,
  armlabel,
  side
)

Arguments

Details

This function aims to minimize the criteria tr[A^TM^{-1}(\rho)A] and minimize the overall variance of pairwise comparisons. It is analogous to Neyman allocation, favoring a higher allocation ratio to the control group. With more than two treatment groups the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_Aa_optimal_known_var returns an object of class "Aaoptimal". An object of class "Aaoptimal" is a list containing final decision based on the Z test statistics with 1 stands for selected and 0 stands for not selected, Z test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

References

Sverdlov O, Rosenberger W (2013). “On Recent Advances in Optimal Allocation Designs in Clinical Trials.” Journal of Statistical Theory and Practice, 7, 753-773. doi:10.1080/15598608.2013.783726.

Examples

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under null hypothesis
#in a two-armed trial
sim_Aa_optimal_known_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm(length( vStartTime ),30, 3)),enrollrate=0.9,N2=88,armn=2,
mean=c(9.1/100,9.1/100),sd=c(0.009,0.009),alphaa=0.025,armlabel = c(1,2),side='lower')

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under alternative hypothesis
#in a two-armed trial
sim_Aa_optimal_known_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm(length( vStartTime ),30, 3)),enrollrate=0.9,N2=88,armn=2,
mean=c(9.1/100,8.47/100),sd=c(0.009,0.009),alphaa=0.025,armlabel = c(1,2),side='lower')

Simulate a Trial Using Aa-Optimal Allocation for Continuous Endpoint with Unknown Variances

Description

sim_Aa_optimal_unknown_var simulates a trial for continuous endpoints with unknown variances, and the allocation probabilities change based on results of accumulated participants in the trial.

Usage

sim_Aa_optimal_unknown_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  N2,
  armn,
  mean,
  sd,
  alphaa = 0.025,
  armlabel,
  side
)

Arguments

Details

This function aims to minimize the criteria tr[A^TM^{-1}(\rho)A] and minimize the overall variance of pairwise comparisons. It is analogous to Neyman allocation, favoring a higher allocation ratio to the control group. With more than two treatment groups the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_Aa_optimal_known_var returns an object of class "Aaoptimal". An object of class "Aaoptimal" is a list containing final decision based on the T test statistics with 1 stands for selected and 0 stands for not selected, T test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

Author(s)

Chuyao Xu, Thomas Lumley, Alain Vandal

References

Sverdlov O, Rosenberger W (2013). “On Recent Advances in Optimal Allocation Designs in Clinical Trials.” Journal of Statistical Theory and Practice, 7, 753-773. doi:10.1080/15598608.2013.783726.

Examples

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under null hypothesis
#in a three-armed trial
sim_Aa_optimal_unknown_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm( length( vStartTime ),30, 3)),enrollrate=0.1,N1=12,N2=132,armn=3,
mean=c(9.1/100,9.1/100,9.1/100),sd=c(0.009,0.009,0.009),alphaa=0.025,
armlabel = c(1,2,3),side='upper')

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under alternative hypothesis
#in a three-armed trial
sim_Aa_optimal_unknown_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm( length( vStartTime ),30, 3)),enrollrate=0.1,N1=12,N2=132,armn=3,
mean=c(9.1/100,9.28/100,9.28/100),sd=c(0.009,0.009,0.009),alphaa=0.025,
armlabel = c(1,2,3),side='upper')

Simulate a Trial Using Randomized Play-the-Winner Rule for Binary Endpoint

Description

Simulate randomized play-the-winner rule in a two-armed trial with binary endpoint.

Usage

sim_RPTW(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  na0,
  nb0,
  na1,
  nb1,
  h,
  alphaa = 0.025,
  N2,
  side,
  Z = NULL
)

Arguments

Details

This function simulates trials using the randomized play-the-winner rule under both no delay and delayed scenarios. This method is a type of urn design with the motivation to allocate more participants to the better treatment group. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_RPTW returns an object of class "rptw". An object of class "rptw" is a list containing final decision based on the Z test statistics with 1 stands for selected and 0 stands for not selected, Z test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result with 1 stand for selected and 0 stand for not selected.

References

Wei LJ, Durham S (1978). “The Randomized Play-the-Winner Rule in Medical Trials.” Journal of the American Statistical Association, 73(364), 840–843.

Examples

#sim_RPTW with no delay responses
sim_RPTW(Pats=10,nMax=50000,TimeToOutcome=0,enrollrate=0.9,na0=1,nb0=1,na1=1,nb1=1,
h=c(0.1,0.3),alphaa=0.025,N2=168,side='upper')
#sim_RPTW with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days
sim_RPTW(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm( length( vStartTime ),30, 3)),
enrollrate=0.9,na0=1,nb0=1,na1=1,nb1=1,h=c(0.1,0.3),alphaa=0.025,N2=168,side='upper')

Simulate a Trial Using Generalized RSIHR Allocation for Continuous Endpoint with Known Variances

Description

sim_RSIHR_optimal_known_var simulates a trial for continuous endpoints with known variances, and the allocation probabilities are fixed.

Usage

sim_RSIHR_optimal_known_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  N2,
  armn,
  mean,
  sd,
  alphaa = 0.025,
  armlabel,
  cc,
  side
)

Arguments

Details

This function aims to minimize the criteria \sum_{i=1}^{K}n_i\Psi_i with constraints \frac{\sigma_1^2}{n_1}+\frac{\sigma_k^2}{n_k}\leq C, where k=2,...,K for some fixed C. It is equivalent to generalized RSIHR allocation for continuous endpoints with known variances. With more than two arms the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_RSIHR_optimal_known_var returns an object of class "RSIHRoptimal". An object of class "RSIHRoptimal" is a list containing final decision based on the Z test statistics with 1 stands for selected and 0 stands for not selected, Z test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

References

Biswas A, Mandal S, Bhattacharya R (2011). “Multi-treatment optimal response-adaptive designs for phase III clinical trials.” Journal of the Korean Statistical Society, 40(1), 33-44. ISSN 1226-3192, doi:10.1016/j.jkss.2010.04.004.

Examples

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under null hypothesis
#in a three-armed trial
sim_RSIHR_optimal_known_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm(length( vStartTime ),30, 3)),enrollrate=0.9,N1=9,N2=132,armn=3,
mean=c(9.1/100,9.1/100,9.1/100),sd=c(0.009,0.009,0.009),alphaa=0.025,
armlabel=c(1,2,3),cc=mean(c(9.1/100,9.1/100,9.1/100)),side='lower')

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under alternative hypothesis
#in a three-armed trial
sim_RSIHR_optimal_known_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm(length( vStartTime ),30, 3)),enrollrate=0.9,N1=9,N2=132,armn=3,
mean=c(9.1/100,8.47/100,8.47/100),sd=c(0.009,0.009,0.009),alphaa=0.025,
armlabel=c(1,2,3),cc=mean(c(9.1/100,8.47/100,8.47/100)),side='lower')

Simulate a Trial Using Generalized RSIHR Allocation for Continuous Endpoint with Unknown Variances

Description

sim_RSIHR_optimal_unknown_var simulates a trial for continuous endpoints with unknown variances, and the allocation probabilities change based on results of accumulated participants in the trial.

Usage

sim_RSIHR_optimal_unknown_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  N2,
  armn,
  mean,
  sd,
  alphaa = 0.025,
  armlabel,
  cc,
  side
)

Arguments

Details

This function aims to minimize the criteria \sum_{i=1}^{K}n_i\Psi_i with constraints \frac{\sigma_1^2}{n_1}+\frac{\sigma_k^2}{n_k}\leq C, where k=2,...,K for some fixed C. It is equivalent to generalized RSIHR allocation for continuous endpoints with unknown variances. With more than two arms the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_RSIHR_optimal_unknown_var returns an object of class "RSIHRoptimal". An object of class "RSIHRoptimal" is a list containing final decision based on the T test statistics with 1 stands for selected and 0 stands for not selected, T test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

References

Biswas A, Mandal S, Bhattacharya R (2011). “Multi-treatment optimal response-adaptive designs for phase III clinical trials.” Journal of the Korean Statistical Society, 40(1), 33-44. ISSN 1226-3192, doi:10.1016/j.jkss.2010.04.004.

Examples

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under null hypothesis
#in a three-armed trial
sim_RSIHR_optimal_unknown_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm( length( vStartTime ),30, 3)),enrollrate=0.9,N1=8,N2=88,armn=2,
mean=c(9.1/100,9.1/100),sd=c(0.009,0.009),alphaa=0.025,armlabel=c(1,2),
cc=mean(c(9.1/100,9.1/100)),side='upper')

#Run the function with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under alternative hypothesis
#in a three-armed trial
sim_RSIHR_optimal_unknown_var(Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm( length( vStartTime ),30, 3)),enrollrate=0.9,N1=8,N2=88,armn=2,
mean=c(9.1/100,8.47/100),sd=c(0.009,0.009),alphaa=0.025,armlabel=c(1,2),
cc=mean(c(9.1/100,8.47/100)),side='upper')

Simulate a Trial Using Bayesian Response-Adaptive Randomization with a Control Group for Binary Outcomes

Description

sim_brar_binary simulate a trial with two to five arms using Bayesian Response-Adaptive Randomization with a control group for binary outcomes. The conjugate prior distributions follow Beta (Beta(\alpha,\beta)) distributions and can be specified individually for each arm.

Usage

sim_brar_binary(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  armn,
  h,
  au,
  N2,
  tp,
  armlabel,
  blocksize,
  alpha1 = 1,
  beta1 = 1,
  alpha2 = alpha1,
  beta2 = beta1,
  alpha3 = alpha1,
  beta3 = beta1,
  alpha4 = alpha1,
  beta4 = beta1,
  alpha5 = alpha1,
  beta5 = beta1,
  minstart,
  deltaa,
  tpp = 0,
  deltaa1,
  side,
  ...
)

Arguments

Details

This function generates a designed trial using Bayesian response-adaptive randomization with a control group under no delay and delayed scenarios for binary outcomes. The function can handle trials with up to 5 arms. This function uses the formula \frac{Pr(p_k={\sf max}\{p_1,...,p_K\})^{tp}} {\sum_{k=1}^{K}{Pr(p_k={\sf max}\{p_1,...,p_K\})^{tp}}} with side equals to 'upper', and \frac{Pr(p_k={\sf min}\{p_1,...,p_K\})^{tp}} {\sum_{k=1}^{K}{Pr(p_k={\sf min}\{p_1,...,p_K\}){tp}}} with side equals to 'lower', utilizing available data at each step. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_brar_binary returns an object of class "brar". An object of class "brar" is a list containing final decision, test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' results. In the final decision, 'Superiorityfinal' refers to the selected arm, while 'Not Selected' indicates the arm stopped due to futility, and 'Control Selected' denotes the control arm chosen because other arms did not meet futility criteria before the final stage or were not deemed effective at the final stage. Note that before final stage of the trial, test statistics is calculated from deltaa, and test statistics is calculated from deltaa1 at the final stage.

References

Wathen J, Thall P (2017). “A simulation study of outcome adaptive randomization in multi-arm clinical trials.” Clinical Trials, 14, 174077451769230. doi:10.1177/1740774517692302.

Examples

#sim_brar_binary with delayed responses follow a normal distribution with a mean
#of 30 days and a standard deviation of 3 days, where h1=c(0.2,0.4) and tp=0.5.
sim_brar_binary(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm( length( vStartTime ),30, 3)),
enrollrate=0.1,N1=24,armn=2,h=c(0.2,0.4),au=0.36,N2=224,tp=0.5,armlabel=c(1,2),blocksize=4,
alpha1=1,beta1=1,alpha2=1,beta2=1,minstart=24,deltaa=-0.01,tpp=0,deltaa1=0.1,side='upper')

Simulate a Trial Using Bayesian Response-Adaptive Randomization with a Control Group for Continuous Endpoint with Known Variances

Description

sim_brar_known_var simulate a trial with two to five arms using Bayesian Response-Adaptive Randomization with a control group for continuous outcomes with known variances. The conjugate prior distributions follow Normal (N(mean,sd)) distributions and can be specified individually for each arm.

Usage

sim_brar_known_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  armn,
  au,
  N2,
  tp,
  armlabel,
  blocksize,
  mean,
  sd,
  minstart,
  deltaa,
  tpp,
  deltaa1,
  mean10 = 0,
  mean20 = mean10,
  mean30 = mean10,
  mean40 = mean10,
  mean50 = mean10,
  sd10 = 1,
  sd20 = sd10,
  sd30 = sd10,
  sd40 = sd10,
  sd50 = sd10,
  n10 = 1,
  n20 = n10,
  n30 = n10,
  n40 = n10,
  n50 = n10,
  side,
  ...
)

Arguments

Details

This function generates a designed trial using Bayesian response-adaptive randomization with a control group under no delay and delayed scenarios for continuous outcomes with known variances. The function can handle trials with up to 5 arms. This function uses the formula \frac{Pr(\mu_k={\sf max}\{\mu_1,...,\mu_K\})^{tp}} {\sum_{k=1}^{K}{Pr(\mu_k={\sf max}\{\mu_1,...,\mu_K\})^{tp}}} with side equals to 'upper', and \frac{Pr(\mu_k={\sf min}\{\mu_1,...,\mu_K\})^{tp}} {\sum_{k=1}^{K}{Pr(\mu_k={\sf min}\{\mu_1,...,\mu_K\}){tp}}} with side equals to 'lower', utilizing available data at each step. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_brar_known_var returns an object of class "brar". An object of class "brar" is a list containing final decision, test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' results. In the final decision, 'Superiorityfinal' refers to the selected arm, while 'Not Selected' indicates the arm stopped due to futility, and 'Control Selected' denotes the control arm chosen because other arms did not meet futility criteria before the final stage or were not deemed effective at the final stage. Note that before final stage of the trial, test statistics is calculated from deltaa, and test statistics is calculated from deltaa1 at the final stage.

References

Wathen J, Thall P (2017). “A simulation study of outcome adaptive randomization in multi-arm clinical trials.” Clinical Trials, 14, 174077451769230. doi:10.1177/1740774517692302.

Examples

#sim_brar_known_var with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days, where mean=c(8.9/100,8.74/100,8.74/100),
#sd=c(0.009,0.009,0.009), tp=0.5 and the minimal effect size is 0.
sim_brar_known_var(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm(
length(vStartTime),30, 3)),enrollrate=0.1, N1=21,armn=3,au=c(0.973,0.973),
N2=189,tp=0.5,armlabel=c(1,2,3),blocksize=6,mean=c(8.9/100,8.74/100,8.74/100),
sd=c(0.009,0.009,0.009),minstart=21,deltaa=c(0.0005,0.0005),tpp=0,deltaa1=c(0,0),
mean10=0.09,mean20=0.09,mean30=0.09,sd10=0.01,sd20=0.01,sd30=0.01,n10=1,n20=1,n30=1,side='lower')

Simulate a Trial Using Bayesian Response-Adaptive Randomization with a Control Group for Continuous Endpoint with Unknown Variances

Description

sim_brar_unknown_var simulate a trial with two to five arms using Bayesian Response-Adaptive Randomization with a control group for continuous outcomes with unknown variances. The conjugate prior distributions follow Normal-Inverse-Gamma (NIG) ((\mu,\sigma^2) \sim NIG(mean=m,variance=V \times \sigma^2,shape=a,rate=b)) distributions and can be specified individually for each arm.

Usage

sim_brar_unknown_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  armn,
  au,
  N2,
  tp,
  armlabel,
  blocksize,
  mean = 0,
  sd = 1,
  minstart,
  deltaa,
  tpp,
  deltaa1,
  V01,
  a01,
  b01,
  m01,
  V02 = V01,
  V03 = V01,
  V04 = V01,
  V05 = V01,
  a02 = a01,
  a03 = a01,
  a04 = a01,
  a05 = a01,
  b02 = b01,
  b03 = b01,
  b04 = b01,
  b05 = b01,
  m02 = m01,
  m03 = m01,
  m04 = m01,
  m05 = m01,
  side,
  ...
)

Arguments

Details

This function generates a designed trial using Bayesian response-adaptive randomization with a control group under no delay and delayed scenarios for continuous outcomes with unknown variances. The function can handle trials with up to 5 arms. This function uses the formula \frac{Pr(\mu_k=max\{\mu_1,...,\mu_K\})^{tp}} {\sum_{k=1}^{K}{Pr(\mu_k=max\{\mu_1,...,\mu_K\})^{tp}}} with side equals to 'upper', and \frac{Pr(\mu_k=min\{\mu_1,...,\mu_K\})^{tp}} {\sum_{k=1}^{K}{Pr(\mu_k=min\{\mu_1,...,\mu_K\}){tp}}} with side equals to 'lower', utilizing available data at each step. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_brar_unknown_var returns an object of class "brar". An object of class "brar" is a list containing final decision, test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' results. In the final decision, 'Superiorityfinal' refers to the selected arm, while 'Not Selected' indicates the arm stopped due to futility, and 'Control Selected' denotes the control arm chosen because other arms did not meet futility criteria before the final stage or were not deemed effective at the final stage. Note that before final stage of the trial, test statistics is calculated from deltaa, and test statistics is calculated from deltaa1 at the final stage.

References

Murphy K (2007). “Conjugate Bayesian analysis of the Gaussian distribution.” University of British Columbia. https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf.

Examples

#sim_brar_unknown_var with delayed responses follow a normal distribution with
#a mean of 30 days and a standard deviation of 3 days under the null hypothesis,
#where mean=c(9.19/100,8.74/100,8.74/100), sd=c(0.009,0.009,0.009), tp=1 and 
#the minimal effect size is 0.
sim_brar_unknown_var(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm(
length(vStartTime ),30,3)),enrollrate=0.1, N1=48,armn=3,au=c(0.85,0.85),
N2=480,tp=1,armlabel=c(1, 2,3),blocksize=6,mean=c(9.19/100,8.74/100,8.74/100),
sd=c(0.009,0.009,0.009), minstart=48,deltaa=c(-0.000325,-0.000325),
tpp=0,deltaa1=c(0,0),V01=1/2,a01=0.3,m01=9/100,b01=0.00001,side='lower')

Simulate a Trial Using Doubly Adaptive Biased Coin Design with Maximal Power Strategy for Binary Endpoint

Description

sim_dabcd_max_power can be used for doubly adaptive biased coin design with maximal power strategy for binary outcomes, targeting generalized Neyman allocation and generalized RSIHR allocation.

Usage

sim_dabcd_max_power(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  N2,
  armn,
  armlabel,
  h,
  BB,
  type,
  gamma = 2,
  alphaa = 0.025,
  side
)

Arguments

Details

The function simulates a trial for doubly adaptive biased coin design with maximal power strategy targeting generalized Neyman allocation with 2-5 arms which is provided in (Tymofyeyev et al. 2007) and generalized RSIHR allocation with 2-3 arms which is provided in (Jeon and Feifang 2010), with modifications for typos in (Sabo and Bello 2016). All of those methods are not smoothed. The output of this function is based on Hu \& Zhang's formula (Hu and Zhang 2004). With more than two armd the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_dabcd_max_power returns an object of class "dabcd". An object of class "dabcd" is a list containing final decision based on the Z test statistics with 1 stands for selected and 0 stands for not selected, Z test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

References

Hu F, Zhang L (2004). “Asymptotic Properties of Doubly Adaptive Biased Coin Designs for Multitreatment Clinical Trials.” The Annals of Statistics, 32(1), 268–301.

Tymofyeyev Y, Rosenberger WF, Hu F (2007). “Implementing Optimal Allocation in Sequential Binary Response Experiments.” Journal of the American Statistical Association, 102(477), 224-234. doi:10.1198/016214506000000906.

Jeon Y, Feifang H (2010). “Optimal Adaptive Designs for Binary Response Trials With Three Treatments.” Statistics in Biopharmaceutical Research, 2, 310-318. doi:10.1198/sbr.2009.0056.

Sabo R, Bello G (2016). “Optimal and lead-in adaptive allocation for binary outcomes: a comparison of Bayesian methodologies.” Communications in Statistics - Theory and Methods, 46.

Examples

sim_dabcd_max_power(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm(
length( vStartTime ),30, 3)),enrollrate=0.9,N1=30,N2=300,armn=3,
armlabel=c(1,2,3),h=c(0.2,0.3,0.2),BB=0.1,type='Neyman',
side='upper')
sim_dabcd_max_power(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm(
length( vStartTime ),60, 3)),enrollrate=0.1,N1=50,N2=500,armn=3,
armlabel=c(1,2,3),h=c(0.2,0.3,0.3),BB=0.15,type='RSIHR',
side='upper')

Simulate a Trial Using Doubly Adaptive Biased Coin Design with Minmial Variance Strategy for Binary Endpoint

Description

sim_dabcd_min_var can be used for doubly adaptive biased coin design with minimal variance strategy for binary outcomes, targeting generalized Neyman allocation and generalized RSIHR allocation with 2-5 arms.

Usage

sim_dabcd_min_var(
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  N1,
  N2,
  armn,
  armlabel,
  h,
  type,
  gamma = 2,
  alphaa = 0.025,
  side
)

Arguments

Details

The output of this function is based on Hu \& Zhang's formula (Hu and Zhang 2004). With more than two arms the one-sided nominal level of each test is alphaa divided by arm*(arm-1)/2; a Bonferroni correction. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_dabcd_min_var returns an object of class "dabcd". An object of class "dabcd" is a list containing final decision based on the Z test statistics with 1 stands for selected and 0 stands for not selected, Z test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

References

Hu F, Zhang L (2004). “Asymptotic Properties of Doubly Adaptive Biased Coin Designs for Multitreatment Clinical Trials.” The Annals of Statistics, 32(1), 268–301.

Examples

sim_dabcd_min_var(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm(
length( vStartTime ),30, 3)),enrollrate=0.9,N1=30,N2=300,armn=3,
armlabel=c(1,2,3),h=c(0.2,0.3,0.2),type='Neyman',
side='upper')
sim_dabcd_min_var(Pats=10,nMax=50000,TimeToOutcome=expression(rnorm(
length( vStartTime ),60, 3)),enrollrate=0.1,N1=50,N2=500,armn=3,
armlabel=c(1,2,3),h=c(0.2,0.3,0.3),type='RSIHR',
side='lower')

Simulate a Trial Using Forward-Looking Gittins Index for Binary Endpoint

Description

Function for simulating a trial using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule for binary outcomes in trials with 2-5 arms. The conjugate prior distributions follow Beta (Beta(\alpha,\beta)) distributions and should be the same for each arm.

Usage

sim_flgi_binary(
  Gittinstype,
  df,
  gittins = NULL,
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  I0,
  K,
  noRuns2 = 100,
  Tsize,
  ptrue,
  block,
  rule,
  ztype,
  stopbound,
  side
)

Arguments

Details

This function simulates a trial using the forward-looking Gittins Index rule or the controlled forward-looking Gittins Index rule under both no delay and delayed scenarios. The cut-off value used for stopbound is obtained by simulations using flgi_stop_bound_binary. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_flgi_binary returns an object of class "flgi". An object of class "flgi" is a list containing final decision based on the Z test statistics with 1 stands for selected and 0 stands for not selected, final decision based on the maximal Gittins Index value at the final stage, Z test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

References

Villar S, Wason J, Bowden J (2015). “Response-Adaptive Randomization for Multi-arm Clinical Trials Using the Forward Looking Gittins Index Rule.” Biometrics, 71. doi:10.1111/biom.12337.

Examples

#The forward-looking Gittins Index rule with delayed responses follow a normal distribution
#with a mean of 60 days and a standard deviation of 3 days

sim_flgi_binary(Gittinstype='Binary',df=0.5,Pats=10,nMax=50000,TimeToOutcome=expression(
rnorm( length( vStartTime ),60, 3)),enrollrate=0.9,I0= matrix(1,nrow=2,2),
K=2,Tsize=992,ptrue=c(0.6,0.7),block=20,rule='FLGI PM',ztype='unpooled',
stopbound=1.9991,side='upper')

Simulate a Trial Using Forward-Looking Gittins Index for Continuous Endpoint with Known Variances

Description

Function for simulating a trial using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule for continuous outcomes with known variances in trials with 2-5 arms. The conjugate prior distributions follow Normal (N({\sf mean},{\sf sd})) distributions and should be the same for each arm.

Usage

sim_flgi_known_var(
  Gittinstype,
  df,
  gittins = NULL,
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  K,
  noRuns2,
  Tsize,
  block,
  rule,
  prior_n,
  prior_mean,
  stopbound,
  mean,
  sd,
  side
)

Arguments

Details

This function simulates a trial using the forward-looking Gittins Index rule or the controlled forward-looking Gittins Index rule under both no delay and delayed scenarios. The cut-off value used for stopbound is obtained by simulations using flgi_stop_bound_flgi_known_var. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_flgi_known_var returns an object of class "flgi". An object of class "flgi" is a list containing final decision based on the Z test statistics with 1 stands for selected and 0 stands for not selected, final decision based on the maximal Gittins Index value at the final stage, Z test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

References

Williamson SF, Villar S (2019). “A Response‐Adaptive Randomization Procedure for Multi‐Armed Clinical Trials with Normally Distributed Outcomes.” Biometrics, 76. doi:10.1111/biom.13119.

Examples

#The forward-looking Gittins Index rule with delayed responses follow a normal distribution
#with a mean of 30 days and a standard deviation of 3 days

sim_flgi_known_var(Gittinstype='KV',df=0.995,Pats=10,nMax=50000,
TimeToOutcome=expression(rnorm( length( vStartTime ),30, 3)),enrollrate=0.5,
K=3,noRuns2=100,Tsize=852,block=20,rule='FLGI PM',prior_n=rep(1,3),
prior_mean=rep(9/100,3),stopbound=(-2.1725),mean=c(9.1/100,8.83/100,8.83/100),
sd=c(0.009,0.009,0.009),side='lower')

#The controlled forward-looking Gittins Index rule with delayed responses follow a 
#normal distribution with a mean of 30 days and a standard deviation of 3 days

sim_flgi_known_var(Gittinstype='KV',df=0.995,Pats=10,nMax=50000,
TimeToOutcome=expression(rnorm( length( vStartTime ),30, 3)),enrollrate=0.1,
K=3,noRuns2=100,Tsize=852,block=20,rule='CFLGI',prior_n=rep(1,3),
prior_mean=rep(9/100,3),stopbound=(-2.075),mean=c(9.1/100,8.83/100,8.83/100),
sd=c(0.009,0.009,0.009),side='lower')

Simulate a Trial Using Forward-Looking Gittins Index for Continuous Endpoint with Unknown Variances

Description

Function for simulating a trial using the forward-looking Gittins Index rule and the controlled forward-looking Gittins Index rule for continuous outcomes with unknown variances in trials with 2-5 arms. The prior distributions follow Normal-Inverse-Gamma (NIG) ((\mu,\sigma^2) \sim NIG({\sf mean}=m,{\sf variance}=V \times \sigma^2,{\sf shape}=a,{\sf rate}=b)) distributions and should be the same for each arm.

Usage

sim_flgi_unknown_var(
  Gittinstype,
  df,
  gittins = NULL,
  Pats,
  nMax,
  TimeToOutcome,
  enrollrate,
  K,
  noRuns2,
  Tsize,
  block,
  rule,
  prior_n,
  prior_mean1,
  prior_sd1,
  stopbound,
  mean,
  sd,
  side
)

Arguments

Details

This function simulates a trial using the forward-looking Gittins Index rule or the controlled forward-looking Gittins Index rule under both no delay and delayed scenarios. The cut-off value used for stopbound is obtained by simulations using flgi_stop_bound_flgi_unk_var. Considering the delay mechanism, Pats (the number of patients accrued within a certain time frame), nMax (the assumed maximum accrued number of patients with the disease in the population) and TimeToOutcome (the distribution of delayed response times or a fixed delay time for responses) are parameters in the functions adapted from https://github.com/kwathen/IntroBayesianSimulation. Refer to the website for more details.

Value

sim_flgi_unknown_var returns an object of class "flgi". An object of class "flgi" is a list containing final decision based on the T test statistics with 1 stands for selected and 0 stands for not selected, final decision based on the maximal Gittins Index value at the final stage, T test statistics, the simulated data set and participants accrued for each arm at the time of termination of that group in one trial. The simulated data set includes 5 columns: participant ID number, enrollment time, observed time of results, allocated arm, and participants' result.

References

Williamson SF, Villar S (2019). “A Response‐Adaptive Randomization Procedure for Multi‐Armed Clinical Trials with Normally Distributed Outcomes.” Biometrics, 76. doi:10.1111/biom.13119.

Examples

#The forward-looking Gittins Index rule with delayed responses follow a normal 
#with a mean of 60 days and a standard deviation of 3 days

sim_flgi_unknown_var(Gittinstype='UNKV',df=0.5,Pats=10,nMax=50000,
TimeToOutcome=expression(rnorm( length( vStartTime ),60, 3)),enrollrate=0.9,
K=3,noRuns2=100,Tsize=852,block=20,rule='FLGI PM',prior_n=rep(2,3),
prior_mean1=rep(9/100,3),stopbound=(-1.9298),prior_sd1=rep(0.006324555,3),
mean=c(9.1/100,8.83/100,8.83/100),sd=c(0.009,0.009,0.009),side='lower')

Update Parameters of a Normal-Inverse-Chi-Squared Distribution with Available Data

Description

Update parameters of a Normal-Inverse-Chi-Squared distribution

Usage

update_par_nichisq(y, par)

Arguments

Details

This function updates parameters of a Normal-Inverse-Chi-Squared ((\mu,\sigma^2) \sim NIX( {\sf mean}=\mu, {\sf effective sample size}=\kappa, {\sf degrees of freedom}=\nu, {\sf variance}=\sigma^2/\kappa)) distribution with available data to parameters of a posterior Normal-Inverse-Gamma ((\mu,\sigma^2) \sim NIG({\sf mean}=m,{\sf variance}=V \times \sigma^2,{\sf shape}=a,{\sf rate}=b))distribution. Those updated parameters can be converted to parameters in a Normal-Inverse-Gamma distribution for continuous outcomes with unknown variances using convert_chisq_to_gamma.

Value

A list of parameters including mu, kappa, nu, sigsq for a posterior Normal-Inverse-Chi-Squared distribution incorporating available data.

References

Murphy K (2007). “Conjugate Bayesian analysis of the Gaussian distribution.” University of British Columbia. https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf.

Examples

para<-list(V=1/2,a=0.5,m=9.1/100,b=0.00002)
par<-convert_gamma_to_chisq(para)
set.seed(123451)
y1<-rnorm(100,0.091,0.009)
update_par_nichisq(y1, par)
set.seed(123452)
y2<-rnorm(90,0.09,0.009)
update_par_nichisq(y2, par)