Randomization, targeting equal 1:1 allocation

The following settings are assumed:

• There are two treatment arms investigated in a trial:
  • $E$ is an experimental treatment arm.
  • $C$ is a control treatment arm.
• $n$ is a total sample size.
• $N_1(j)$ and $N_2(j)$ are treatment numbers, i.e., sample sizes on treatments after the $j^\text{th}$ allocation step ($N_1(j)+N_2(j) = j, \: j = 1, \ldots, n$).
• $\delta_j$ is a treatment indicator:

\[\delta_j = \left\{\begin{array}{rl} 1, &\text{if treatment }E \\ 0, &\text{if treatment }C \end{array}\right. .\]

Under these assumptions, a restricted randomization procedure is defined as

\[\begin{aligned} \phi_1 &= \Pr(\delta_1 = 1) = 0.5; \\ \phi_j &= \Pr(\delta_j = 1|\delta_1, \ldots, \delta_{j-1}), \: j = 2, \ldots, n. \end{aligned}\]