flgi_cut_off_binary {RARtrials}

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

Gittinstype

type of Gittins indices, should be set to 'binary' in this function.

df

discount factor which is the multiplier for loss at each additional patient in the future. Available values are 0, 0.5, 0.7, 0.99 and 0.995. The maximal sample size can be up to 2000.

gittins

user specified Gittins indices for calculation in this function. Recommend using the bmab_gi_multiple_ab function from gittins package. If gittins is provided, Gittinstype and df should be NULL.

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.

I0

a matrix with K rows and 2 columns, where the numbers inside are equal to the prior parameters, and K is equal to the total number of arms. For example, matrix(1,nrow=2,ncol=2) means that the prior distributions for two-armed trials are beta(1,1); matrix(c(2,3),nrow=2,ncol=2,byrow = TRUE) means that the prior distributions for two-armed trials are beta(2,3). The first column represents the prior of the number of successes, and the second column represents the prior of the number of failures.

K

number of total arms in the trial.

noRuns2

number of simulations for simulated allocation probabilities within each block. Default value is set to 100, which is recommended in (Villar et al. 2015).

Tsize

maximal sample size for the trial.

ptrue

a vector of hypotheses, for example, as c(0.1,0.1) where 0.1 stands for the success probability for both groups. Another example is c(0.1,0.3) where 0.1 and 0.3 stand for the success probability for the control and the treatment group, respectively.

block

block size.

rule

rules can be used in this function, with values 'FLGI PM', 'FLGI PD' or 'CFLGI'. 'FLGI PM' stands for making decision based on posterior mean; 'FLGI PD' stands for making decision based on posterior distribution; 'CFLGI' stands for controlled forward-looking Gittins Index.

ztype

Z test statistics, with choice of values from 'pooled' and 'unpooled'.

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.