Is it possible to to have more than 2 groups for biased coin randomization? If no, is there any modifications supporting multiple groups?

Question

1. Is it possible to have more than 2 groups for biased coin randomization?

2. If no, is there any modifications of biased coin randomization supporting multiple groups?

This is my R code for generating randomization sequence for 2 groups (this code works fine.):

library(randomizeR)

N <- 100
p <- 0.5
groups <- LETTERS[1:2]

main_path="c:/del/4/"


obj_my_par<-ebcPar(N, p, groups)



(ebc_seq<-genSeq(obj_my_par, r=1))


(mylist<-getRandList(ebc_seq))
table(mylist)

But if I change LETTERS[1:2] to LETTERS[1:4] (increase number of groups) I am getting error:

Error in validObject(.Object) : 
  invalid class “ebcPar” object: Length of groups is 4. Should have length 2.

Answer 1

Efron's biased coin design is predicated on two groups. From the error message you're getting, it seems like the R function you're using has not been extended to more than 2 groups. Proposing an extension would be pretty straightforward, although establishing its mathematical properties would probably be non-trivial, given the amount of maths in Efron's paper with just two groups.

Suppose there are $J$ possible treatment assignments and an available sample size of $N$. Let $T_i\in\{1,\ldots,J\}$ be the $i$th treatment assignment, $i=1,\ldots, N$. Let $n_{ij}=\sum_{i'=0}^{i-1} 1[T_{i'} = j]$ denote the number of assignments to arm $j$ just prior to assignment $i$. Under simple randomization, we would use $\Pr(T_i = j)\propto 1$. Under a biased randomization analogous to Efron's proposal, given some weight $\theta > 1$, you could consider \begin{align} \Pr(T_i=j)\propto \begin{cases} 1, & n_{ij} = \max_{j'} n_{ij'}\\ \theta, & n_{ij} < \max_{j'} n_{ij'} \end{cases} \end{align} When $J=2$, then setting $\theta = p/(1-p)$, where $p$ is as defined at the top of page 405 of Efron's paper, reduces to his original design.