Restricted randomization for two-arm clinical trials with equal \(1{:}1\) allocation

Background

This repository collects a set of Julia functions to simulate several restricted randomization procedures, targeting equal \(1{:}1\) allocation in two-arm clinical trials.

There are case studies included in the repository.

Restricted Randomization

Let us assume that there are two treatment arms considered in a clinical trial:

• Experimental (\(E\));
• Control (\(C\)).

Sample sizes on treatments are \(N_1\) (\(n_E\)) and \(N_2\) (\(n_c\)) respectively, and \(n = n_E + n_C\) is a total sample size.

A restricted randomization procedure is defined as

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

where \(j\) is a number of a subject currently randomized in a trial (or an allocation step), and \(\phi_j\)is a probability of tretament assignment for the \(j^\text{th}\) subject, \(j = 1, \ldots, n\).

\[ \delta_j = \left\{ \begin{array}{rl} 1, \text{ if tretament }E \\ 0, \text{ if tretament }C \end{array} \right. \tag{2}\]

There are several restricted randomziation procedures implemented in the current repository.

Completely Randomized Design, \(\text{CRD}\)

\[ \phi_j = 0.5, \: j = 1, \ldots, n. \tag{3}\]

Truncated Binomial Design, \(\text{TBD}\)

Treatment assignments are made with probability 0.5 until one of the treatments receives its quota of \(\frac{n}{2}\) subjects; thereafter all remaining assignments are made deterministically to the opposite treatment.

For the current allocation step \(j\), let \(N_1\) and \(N_2\) be the numbers of subjects allocated to treatments s.t. \(N_1 + N_2 = j-1\). Then,

\[ \phi_j = \left\{ \begin{array}{rl} 0.5, & \max(N_1, N_2) < \frac{n}{2} \\ 1, & N_1 < N_2 \\ 0, & N_1 > N_2 \end{array} \right.,\: j = 1, \ldots, n. \tag{4}\]

Permuted Block Design, \(\text{PBD}(b)\)

Treatment assignments are made in blocks of size \(b\), where \(b\) is an even number. The probabilities of treatment assignments within each block are changed according the current imbalance in a block.

At the \(j^\text{th}\) allocation step, let \(k\) be a number of the \(j^\text{th}\) subject in a current block and \(n_1\) be a number of subjects in a block allocated to treatment 1 (\(E\)). Then,

\[ \phi_j = \frac{0.5b-n_1}{b-k+1}, \: j = 1, \ldots, n. \tag{5}\]

Random Allocation Rule, \(\text{Rand}\)

A version of PBD, when the block size \(b\) equals to the total sample size \(n\). At the \(j^\text{th}\) allocation step,

\[ \phi_j = \frac{0.5n-N_1}{n-j+1}, \: j = 1, \ldots, n, \tag{6}\]

and \(N_1\) is a number of subject already allocated to the treatment 1 (\(E\)).

Efron’s Biased Coin Design, \(\text{BCD}(p)\)

At any allocation step, if treatment numbers \(N_1\) and \(N_2\) are balanced, the next assignment is made with probability 0.5; otherwise, the underrepresented treatment is assigned with probability \(p\), where \(0.5 < p \leq 1\) is a fixed and pre-specified parameter that determines the tradeoff between balance and randomness.

\[ \phi_j = \left\{ \begin{array}{rl} 0.5, & N_1 = N_2 \\ p, & N_1 < N_2 \\ 1-p, & N_1 > N_2 \end{array} \right.,\: j = 1, \ldots, n. \tag{7}\]

Note that \(p=1\) corresponds to PBD with block size \(b=2\).

Adjustable Biased Coin Design, \(\text{ABCD}(a)\)

An extension of Efron’s BCD. At the \(j^\text{th}\) allocation step, given tretament numbers \(N_1\) and \(N_2\), \(N_1 + N_2\), s.t. \(N_1 + N_2 = j-1\), and imbalance \(d = N_1-N_2\),

\[ \phi_j = \left\{ \begin{array}{rl} 0.5, & |d| \leq 1 \\ \frac{|x|^a}{|x|^a + 1}, & d < -1 \\ \frac{1}{|x|^a + 1}, & d > 1 \end{array} \right.,\: j = 1, \ldots, n, \tag{8}\]

where \(a>0\) is a parameter of the randomization procedure.

Generalized Biased Coin Design, \(\text{GBCD}(\gamma)\)

A generalization of Efron’s BCD. At the \(j^\text{th}\) allocation step, given tretament numbers \(N_1\) and \(N_2\), \(N_1 + N_2\), s.t. \(N_1 + N_2 = j-1\),

\[ \phi_j = \left\{ \begin{array}{rl} 0.5, & j = 1 \\ \frac{N_2^\gamma}{N_1^\gamma + N_2^\gamma}, & j = 2, \ldots, n, \end{array} \right. \tag{9}\]

where \(\gamma > 0\) is a parameter of the randomization procedure.

Big Stick Design, \(\text{BSD}(mti)\)

An example of maximum tolerated imbalance (MTI) procedures. It makes prediction of the future treatment allocations more difficult (even knowing the current sizes of the treatment groups) and controls treatment imbalance at a predefined threshold throughout the experiment. A general MTI procedure specifies a certain boundary for treatment imbalance, say \(mti > 0\), that cannot be exceeded.

At the \(j^\text{th}\) allocation step, given tretament numbers \(N_1\) and \(N_2\), \(N_1 + N_2\), s.t. \(N_1 + N_2 = j-1\), and imbalance \(d = N_1-N_2\),

\[ \phi_j = \left\{ \begin{array}{rl} 0.5, & |d| < mti \\ 0, & d = mti \\ 1, & d = -mti \end{array} \right.,\: j = 1, \ldots, n. \tag{10}\]

Biased Coin Design With Imbalance Tolerance, \(\text{BCDWIT}(p, mti)\)

A combination of Efron’s BCD and BSD. At the \(j^\text{th}\) allocation step, given tretament numbers \(N_1\) and \(N_2\), \(N_1 + N_2\), s.t. \(N_1 + N_2 = j-1\), and imbalance \(d = N_1-N_2\),

\[ \phi_j = \left\{ \begin{array}{rl} 0.5, & |d| < mti~\&~d = 0\\ p, & |d| < mti~\&~d < 0 \\ 1-p, & |d| < mti~\&~d > 0 \\ 0, & d = mti \\ 1, & d = -mti \end{array} \right.,\: j = 1, \ldots, n. \tag{11}\]

Operational Characteristics

Two important characteristics of a randomization procedure are imbalance and randomness.

Imbalance

After the \(j^\text{th}\) allocations (\(1 \leq j \leq b\)),

\[ D(j) = N_1 - N_2 \tag{12}\]

is a tretament imbalance, where \(N_1\) and \(N_2\) are the numbers of treatment assigments to treatment 1 (\(E\)) and 2 (\(C\)) respectively.

For any \(j \geq 1\), \(D(j)\) is a random variable with a probability distribution determined by the chosen randomization procedure.

Several measures of imbalance can be considered, for example, the absolute value of imbalance,

\[ |D(j)|, \: j = 1, \ldots, n. \tag{13}\]

\(|D(i)| = 0\) indicates that the trial is perfectly balanced.

One can also consider \(\Pr(|D(j)| = 0)\), the probability of achieving exact balance after \(j\) allocation steps. In particular, \(\Pr(|D(n)| = 0)\) is the probability that the final treatment numbers are balanced.

Two other useful summary measures are the expected imbalance at the \(j^\text{th}\) step, \(E|D(j)|\) and the expected value of the maximum imbalance of the entire randomization sequence, \(E\left(\max\limits_{1\leq j\leq n}|D(j)|\right)\).

Randomness

Greater forcing of balance implies lack of randomness. A procedure that lacks randomness may be susceptible to selection bias.

A classic approach to quantify the degree of susceptibility of a procedure to selection bias is the Blackwell-Hodges model.

Let \(G_j = 1\) (or \(0\)), if at the \(j^\text{th}\) allocation step an investigator makes a correct (or incorrect) guess on treatment assignment \(\delta_j\), given past allocations \(\boldsymbol{\delta}_{i−1}\).

Then. the predictability of the design at the \(j^\text{th}\) step is the expected value of \(G_j\), i.e. \(E(G_j) = \Pr(G_j = 1)\).

Blackwell and Hodges considered the expected bias factor, the difference between expected total number of correct guesses of a given sequence of random assignments and the similar quantity obtained from CRD for which treatment assignments are made independently with equal probability:

\[ E(F) = E\left(\sum_{j=1}^nG_j\right) - \frac{n}{2}. \]

This quantity equal to zero for CRD, and it is positive for restricted randomization procedures (greater values indicate higher expected bias).

One can also consider an expected proportion of deterministic assignments in a sequence as another measure of lack of randomness.