Randomization, targeting unequal allocation in two-arm trial and equal/unequal allocation in multi-arm trial

The following settings are assumed:

• $K$ is a number of treatment arms in a trial ($K \geq 2$).
• $\mathbf{w} = \left(w_1,\ldots, w_K\right)'$ is the fixed allocation ratio, $w_1:\ldots :w_K$, where $w_k$'s are positive, not necessarily equal numbers (usually, integers) with the greatest common divisor of 1.
• $\boldsymbol{\rho} = \left(\rho_1, \ldots, \rho_K\right)'$ is a vector of target allocation proportions, where

\[\rho_k = \frac{w_k}{\sum_{k=1}^{K}{w_k}}, 0 \leq \rho_k \leq 1, \text{ and }\sum_{k=1}^K{\rho_k} = 1.\]

• $n$ is a total sample size.
• $\mathbf{N}(j) = \left(N_1(j), \ldots, N_K(j)\right)'$ is a vector of treatment numbers, i.e., numbers of subjects allocated to $K$ treatments after $j$ allocations ($1 \leq j \leq n$).
  • Note that, in general, $N_k(j)$'s are random variables with $\sum_{k = 1}^K{N_k(j)} = j$.
• $\mathbf{P}(j) = \left(P_1(j), \ldots, P_K(j)\right)'$ is a vector of randomization (allocation) probabilities for the $j^\text{th}$subject.
  • Note that $0 \leq P_k(j) \leq 1$, and $\sum_{k = 1}^K{P_k(j)} = 1$ for each $j = 1, 2, \ldots, n$.
  • Also, note that in general, $\mathbf{P}(j)$ depends on $\mathbf{N}(j-1)$ (in generalization of Efron's BCD) or on $\frac{\mathbf{N}(j-1)}{j-1}$ (in generalization of Wei's urn design).

Under these assumptions, a restricted randomization procedure is defined as

\[\begin{aligned} P_k(1) &= \Pr(\delta_1 = k) = \rho_k, \: k = 1, \ldots, K; \\ P_k(j) &= \Pr(\delta_j = k|\delta_1, \ldots, \delta_{j-1}), \: k = 1, \ldots, K; \: j = 2, \ldots, n. \end{aligned}\]