get_oc_BAR {BAR}

Generate operating characteristics for Bayesian adaptive randomization

Description

Randomization is the established method for obtaining scientifically valid comparisons of competing treatments in clinical trials and other experiments. Bayesian adaptive randomization (BAR) allows changes to be made to the randomization probabilities to treatments during the trial. The aim of the procedure is to allocate a greater proportion of patients to treatments that have so far demonstrated evidence of a better performance than other arms. Binary outcomes are considered in this package

Usage

get_oc_BAR(success_prob, n_burn_in, tot_num, block_size,
           power_c = "n/2N", lower_bound = .05, reptime,
           control_arm = "", output = "", seed = 100)

Arguments

success_prob

the successful probability for each arm (the first slot refers to the control arm)

n_burn_in

the number of burn-in for each arm

tot_num

the total number of patients enrolled for the trial

block_size

the block size

power_c

the power correction of allocation probability. The default value is power_c = "n/2N" and can also be numeric, e.g., power_c = .5

lower_bound

the lower bound of the allocation probability. It must between 0 and \frac{1}{K}. The default value is lower_bound = .05; K indicates total number of arms (including control arm)

reptime

the number of simulated trials

control_arm

if this argument is "fixed", then allocation probability of control arm (the first slot) will be fixed to \frac{1}{K}. The default of this argument will return unfixed results; K indicates total number of arms (including control arm)

output

if this argument is "raw", then the function will return updated allocation probability path after burn-in for each arm for each simulated trial. The default of this argument will return the average allocation probability and the average number of patients assigned to each arm

seed

the seed. The default value is seed = 100

Details

We show how the updated allocation probabilities for each arm are calculated.

Treatments are denoted by k = 1,\ldots,K.\;N is the total sample size. If no burn-in(s), the BAR will be initiated start of a study, that is, for each enrolled patient, n = 1,\ldots,N, the BAR will be used to assign each patient. Denoting the true unknown response rates of K treatments by \pi_{1},\ldots,\pi_{K},\;we can compute K posterior probabilities: r_{k,n} = Pr(\pi_{k} = max\{\pi_{1},\ldots,\pi_{K}\}\;|\;Data_{n}), here, n refers to the n-th patient and k refers to the k-th arm. We calculate the updated probabilities of the BAR algorithm according to the following steps.
\;
Step 1: (Normalization) Normalize r_{k,n} as r_{k,n}^{(c)} = \frac{(r_{k,n})^{c}}{\sum_{j=1}^{K}(r_{j,n})^{c}}, here \;c = \frac{n}{2N}.

Step 2: (Restriction) To avoid the BAR sticking to very low/high probabilities, a restriction rule to the posterior probability r_{k,n}^{(c)} will be applied:

Lower\;Bound \le r_{k,n}^{(c)} \le 1 - (K - 1) \times Lower\;Bound,

0 \le Lower\;Bound \le \frac{1}{K}

After restriction, the posterior probability is denoted as r_{k,n}^{(c,re)}.

Step 3: (Re-normalization) Then, we can have the updated allocation probabilities by the BAR denoted as:

r_{k,n}^{(f)} = \frac{r_{k,n}^{(c,re)}\times(\frac{r_{k,n}^{(c,re)}}{\frac{n_{k}}{n}})^{2}}{\sum_{j=1}^{K}\{r_{j,n}^{(c,re)}\times(\frac{r_{j,n}^{(c,re)}}{\frac{n_{j}}{n}})^{2}\}}

where n_{k} is the number of patients enrolled on arm k up-to-now.

Step 4: (Re-restriction) Finally, restricts again by using

Lower\;Bound \le r_{k,n}^{(f)} \le 1 - (K-1) \times Lower\;Bound,

0 \le Lower\;Bound \le \frac{1}{K}

and denote r_{k,n}^{(ff)} as the allocation probability used in the BAR package.

Value

get_oc_BAR() depending on the argument "output", it returns:

default: (1) the average allocation probability (2) the average number of patients assigned to each arm

raw: (1) updated allocation probability path after burn-in for each arm for each simulated trial

Author(s)

Chia-Wei Hsu, Haitao Pan

References

Wathen JK, Thall PF. A simulation study of outcome adaptive randomization in multi-arm clinical trials. Clin Trials. 2017 Oct; 14(5): 432-440. doi: 10.1177/1740774517692302.

Xiao, Y., Liu, Z. & Hu, F. Bayesian doubly adaptive randomization in clinical trials. Sci. China Math. 60, 2503-2514 (2017). doi: 10.1007/s11425-016-0056-1.

Hu F, Zhang L X. Asymptotic properties of doubly adaptive biased coin designs for multi-treatment clinical trials. Ann Statist, 2004, 30: 268–301.

Examples

## power_c = "n/2N"
get_oc_BAR(success_prob = c(.1, .5, .8), n_burn_in = 10,
           tot_num = 150, block_size = 1, reptime = 5)

## power_c = .5
get_oc_BAR(success_prob = c(.1, .5, .8), n_burn_in = 10,
           tot_num = 150, block_size = 1, power_c = .5,
           reptime = 5)