Calculate updated allocation probability for each arm based on the accumulative data with binary outcomes
next_allocation_rate_BAR(n, success_count, tot_num,
power_c = "n/2N",
lower_bound = .05,
control_arm = "",
seed = 100)next_allocation_rate_BAR() returns the updated allocation probability for each arm
the number of patients enrolled for each arm
the number of responders for each arm
the total number of patients enrolled for the trial. If this number cannot be pre-planned, the user can choose argument "power_c" to be numeric instead of "n/2N". In this case, even if the "tot_num" is given a number, this number will not be used
the power correction of allocation probability. The default value is power_c = "n/2N" and can also be numeric, e.g., power_c = .5
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)
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)
the seed. The default value is seed = 100
Chia-Wei Hsu, Haitao Pan
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.
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.
## power_c = "n/2N"
next_allocation_rate_BAR(n = c(30, 30, 30),
success_count = c(5, 6, 12),
tot_num = 150)
## power_c = .5
next_allocation_rate_BAR(n = c(30, 30, 30),
success_count = c(5, 6, 12),
tot_num = 150, power_c = .5)
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