sim_brar_unknown_var {RARtrials}

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

Pats

the number of patients accrued within a certain time frame indicates the count of individuals who have been affected by the disease during that specific period, for example, a month or a day. If this number is 10, it represents that 10 people have got the disease within the specified time frame.

nMax

the assumed maximum accrued number of patients with the disease in the population, this number should be chosen carefully to ensure a sufficient number of patients are simulated, especially when considering the delay mechanism.

TimeToOutcome

the distribution of delayed response times or a fixed delay time for responses. The delayed time could be a month, a week or any other time frame. When the unit changes, the number of TimeToOutcome should also change. It can be in the format of expression(rnorm( length( vStartTime ),30, 3)), representing delayed responses with a normal distribution, where the mean is 30 days and the standard deviation is 3 days.

enrollrate

probability that patients in the population can enroll in the trial. This parameter is related to the number of people who have been affected by the disease in the population, following an exponential distribution.

N1

number of participants with equal randomization in the 'initialization' period. Recommend using 10 percent of the total sample size.

armn

number of total arms in the trial.

au

a vector of cut-off values in the final selection at the end of the trial, with a length equal to the number of arms minus 1.

N2

maximal sample size for the trial.

tp

tuning parameter. Some commonly used numbers are 0.5, 1 and n/2N.

armlabel

a vector of treatment labels with an example of c(1, 2), where 1 and 2 describe how each arm is labeled in a two-armed trial.

blocksize

size of block used for equal randomization regarding participants in the 'initialization' period. Recommend to be an even multiple of the number of total arms.

mean

a vector of means in hypotheses, for example, as c(10,10) where 10 stands for the mean in both groups. Another example is c(10,12) where 10 and 12 stand for the mean for the control and the other treatment group, respectively.

sd

a vector of standard deviations in hypotheses, for example, as c(2,2) where 2 stands for the standard deviation in both groups. Another example is c(1,2) where 1 and 2 stand for the standard deviation for the control and the other treatment group, respectively.

minstart

a specified number of participants when one starts to check decision rules.

deltaa

a vector of minimal effect expected to be observed for early futility stopping in each arm is approximately 1\%. The length of this parameter is armn-1.

tpp

indicator of tp equals to n/2N. When tp is n/2N, tpp should be assigned 1. Default value is set to 0.

deltaa1

a vector of pre-specified minimal effect size expected to be observed at the final stage for each arm. The length of this parameter is armn-1.

V01, a01, b01, m01

prior parameters m, V, a, b in NIG(V,m,a,b) of arm 1 in the trial, which stands for the control.

V02, a02, b02, m02

prior parameters m, V, a, b in NIG(V,m,a,b) of arm 2 in the trial. Default value is set to V01, a01, b01 and m01.

V03, a03, b03, m03

prior parameters m, V, a, b in NIG(V,m,a,b) of arm 3 in the trial. Default value is set to V01, a01, b01 and m01.

V04, a04, b04, m04

prior parameters m, V, a, b in NIG(V,m,a,b) of arm 4 in the trial. Default value is set to V01, a01, b01 and m01.

V05, a05, b05, m05

prior parameters m, V, a, b in NIG(V,m,a,b) of arm 5 in the trial. Default value is set to V01, a01, b01 and m01.

side

direction of a one-sided test, with values 'upper' or 'lower'.

...

additional arguments to be passed to integrate (such as rel.tol) from this function.

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')