Steady-state statistical properties and implementation of randomization designs with maximum tolerated imbalance restriction for two-arm equal allocation clinical trials

Abstract

In recent decades, several randomization designs have been proposed in the literature as better alternatives to the traditional permuted block design (PBD), providing higher allocation randomness under the same restriction of the maximum tolerated imbalance (MTI) . However, PBD remains the most frequently used method for randomizing subjects in clinical trials. This status quo may reflect an inadequate awareness and appreciation of the statistical properties of these randomization designs, and a lack of simple methods for their implementation. This manuscript presents the analytic results of statistical properties for five randomization designs with MTI restriction based on their steady-state probabilities of the treatment imbalance Markov chain and compares them to those of the PBD. A unified framework for randomization sequence generation and real-time on-demand treatment assignment is proposed for the straightforward implementation of randomization algorithms with explicit formulas of conditional allocation probabilities. Topics associated with the evaluation, selection, and implementation of randomization designs are discussed. It is concluded that for two-arm equal allocation trials, several randomization designs offer stronger protection against selection bias than the PBD does, and their implementation is not necessarily more difficult than the implementation of the PBD.

1. INTRODUCTION

The primary goal of subject randomization in clinical trials is to prevent selection bias,¹ which occurs when an investigator can predict future treatment allocation with a success probability higher than a purely random guess, and subsequently is influenced by such knowledge on his/her decision about when and what type of subjects to enroll. With perfect concealment or real-time randomization, completely random assignment eliminates all selection bias but may result in unacceptable treatment imbalance. For example, using completely random assignment in a two-arm equal allocation trial with a sample size of n, the size of either arm is a binomial random variable with a mean of $\frac{n}{2}$ and a variance of $\frac{n}{4}$. The treatment imbalance, measured by the difference between the sizes of the two arms, has a standard deviation of $\sqrt{n}$. In other words, when $n=100$, there are 32% chance of seeing a treatment imbalance with an absolute value greater than or equal to 10, which could decrease the trial efficiency.

Many restricted randomization designs have been proposed to control treatment imbalances while maintaining a certain level of allocation randomness. Due to the tradeoff between imbalance control and allocation randomness within each randomization algorithm, valid comparisons between different randomization designs must evaluate both treatment imbalance and allocation randomness quantitatively.² Commonly used measures for treatment imbalances include the maximum and the standard deviation of treatment imbalances during the trial. Allocation randomness can be assessed by the probability of deterministic assignments and the correct guess probability based on the convergence guessing strategy described by Blackwell and Hodges.³ Comprehensive research comparing treatment imbalance control and allocation randomness between different randomization designs has been reported in recent years, using computer simulations.^4,5,6 Results of simulation studies are affected by factors other than the randomization algorithm, such as the length of the allocation sequence and the implementation method (stratified or not).

This manuscript focuses on the steady-state properties of randomization designs with MTI restriction for two-arm equal allocation trials. The background section provides a review of the mechanisms, implementation methods, and applicable trial setting types of sixteen currently available randomization designs and provides the rational for selecting randomization designs to be compared with PBD. The method section derives the allocation probability conditioned on current treatment imbalances for these randomization designs and proposes a unified framework for real-time on-demand randomization. The steady-state properties of treatment imbalances are obtained based on the conditional allocation probabilities, and explicit formulas for the statistical properties of these randomization designs are provided. The Results section presents a comparison of the treatment imbalance and allocation randomness for these randomization designs quantitatively and graphically under different values of the MTI. In Section 5, topics associated with comparison measures, research approaches, implementation methods and extensions to multi-arm and unequal allocation trial settings are discussed, followed by conclusions in Section 6.

2. BACKGROUND

Randomized controlled clinical trial has been recognized as a valid approach for comparing safety and efficacy among different treatments. Many restricted randomization designs have been proposed, including the sixteen designs listed in Table 1. Some of them are commonly used and well-studied, and some others are newly proposed and are relatively unknown to many working in clinical trial field. Initially introduced by Hill in 1951,⁷ the permuted block design (PBD) is the most used randomization method in clinical trial practice. It provides a consistent treatment imbalance control, applicable to two or multi-arm trials with equal or unequal allocations. Its perceived easy implementation is an important reason for its wide acceptance, especially in interactive response technology (IRT) systems. In recent decades, the PBD has been criticized for its low allocation randomness caused by the enforced block-end balance.^1,4,5,6,8 Matts and Lachin approved that using PBD with a block size of $b=2\delta$ in a two-arm equal allocation trial, the probability of deterministic assignment is $\frac{1}{\delta +1}$.⁹ When the block size is 4, 6, or 8, the probability of deterministic assignment is 33%, 25%, or 20% respectively. Zhao and Weng obtained the formula of the probability of deterministic assignment of PBD for two or multi-arm equal or unequal allocation trials as $\frac{1}{b}{{\displaystyle \sum }}_{j=1}^m\frac{b{r}_j}{b-b{r}_j+1}$.¹⁰ Here b is the block size, m is the number of arms, $r_j>0$ is the target allocation ratio for arm j with ${{\displaystyle \sum }}_{j=1}^m{r}_j=1$. For example, in a three-arm trial with allocation 1:2:3 and a block size of 6 or 12, the probability of deterministic assignment is 22% and 12% respectively. The high probability of deterministic assignment results from the enforced block-end balancing. The primary motivation of several recently proposed randomization designs is to eliminate the block-end balance while maintaining the MTI restriction.

Table 1.

Commonly studied and newly proposed randomization designs and their attributes

Randomization design	Author(s)	Maximum Imbalance Control	Explicit conditional allocation probability	Without fixed sequence length	Unequal allocation	Multi-arm
Permuted block design	Hill (1951)7	✔	✔	✔	✔	✔
Biased coin design	Efron (1971)11		✔	✔
Urn design	Wei (1977)12		✔	✔
Big Stick Design	Sores and Wu (1983)13	✔	✔	✔
BCD with Imbalance Tolerance	Chen (1999)14	✔	✔	✔
Ehrenfest Urn Design	Chen (2000)15	✔	✔	✔
Random Allocation Rule	Rosenberger and Lachin (2002)16		✔
Truncated Binomial Design	Rosenberger and Lachin (2002)16		✔
Maximal Procedure	Berger et al. (2003)17	✔			✔
Brick Tunnel Design	Kuznetsova and Tymofyeyev (2011)18	✔		✔	✔	✔
Block Urn Randomization	Zhao and Weng (2011)19	✔	✔	✔	✔	✔
Variable Block Design	Efird (2011)20	✔		✔	✔	✔
Wide Brick Tunnel Randomization	Kuznetsova and Tymofyeyev (2014)21	✔		✔	✔	✔
Mass-weighted urn design	Zhao (2015)22	✔	✔	✔	✔	✔
Asymptotic Maximal Procedure	Zhao et al. (2018)23	✔	✔	✔	✔
Meagered Block Design	van der Pas (2019)24	✔		✔	✔	✔

Efron’s biased coin design (BCD) uses a pre-specified biased coin probability to reduce treatment imbalance in two-arm equal allocation trials.¹¹ Zhao et al. revealed that the BCD has the worst performance between the cost in allocation randomness and benefit in treatment imbalance control among fourteen randomization designs evaluated in a simulation study.⁴ Wei’s urn design (UD) was defined for two-arm equal allocation trials.¹² Its treatment imbalance control capacity decreases as the length of the allocation sequence increases, and therefore is not a good choice for trials with medium or large sample sizes.⁴

The big stick design (BSD) was originally defined for two-arm equal allocation trials. It uses deterministic assignments to enforce the MTI and completely random assignments by default.¹³ Previous simulation studies indicated that BSD has the lowest correct guess probability under the same restriction of MTI.⁴ The biased coin design with imbalance tolerance (BCDWIT) combines the features of BCD and BSD so that when the treatment imbalance reaches the MTI, a deterministic assignment is used to reduce the imbalance. Otherwise, a pre-specified biased coin probability is used when imbalance occurs but not reaches the MTI yet.¹⁴ The Ehrenfest urn design (EUD) is a modification of the BCDWIT by replacing the fixed biased coin probability with an adjustment proportional to the imbalance magnitude and in the opposite direction.¹⁵

Random allocation rule (RAR) was defined for two-arm equal allocation trials with a fixed allocation sequence length. It can be viewed as a permuted block with the entire allocation sequence.¹⁶ The truncated binomial design (TBD) uses completely random assignment until one arm reaches half of the entire allocation sequence length. After that deterministic assignments are used.¹⁶ Both RAR and TBD have a maximum possible treatment imbalance equal to half the allocation sequence length, and therefore, have rarely been considered in clinical trial practice.

The maximal procedure (MP) was created by Berger et al. in 2003 for two-arm trials with equal or unequal allocations.¹⁷ It maximizes the number of allocation sequences under the restriction of the target allocation, the MTI and the terminal balance. The MP calculates the conditional allocation probability based on the requirement that all feasible allocation sequences have the same probability. Therefore, the conditional allocation probability depends on the current imbalance as well as the order of the assignment in the allocation sequence. For long allocation sequences, the calculation could be complex. The asymptotic maximal procedure (AMP) is a modification of the MP by releasing the restrictions of the allocation sequence length and therefore the sequence-end balance.²³

The brick tunnel randomization (BTR) was proposed by Kuznetsova and Tymofyeyev in 2011 for two or multi-arm trials with unequal allocations.¹⁸ It preserves the unconditional allocation probabilities for each treatment assignment while minimizing treatment imbalance at the cost of allocation randomness. The wide brick tunnel randomization (WBTR) is a modification of the BTR with the MTI being selected.²¹ It also preserves the allocation ratio for each assignment in the allocation sequence and is applicable to two or multi-arm trials with equal or unequal allocations. Both BTR and WBTR do not have an explicit formula for the conditional allocation probability, creating challenges in implementation.

The block urn design (BUD) proposed by Zhao and Weng in 2011 is a modification of the PBD that eliminates the block-end balance.¹⁹ It is applicable to all trial settings where the PBD applies. When used in trials with unequal allocations, it also preserves the unconditional allocation probability for each assignment. Compared to MP and WBTR, it may have a fewer number of feasible allocation sequences. Its conditional allocation probability can be expressed as a simple function of the current imbalance and, therefore, is easy to implement. The mass-weighted urn design (MWUD) proposed by Zhao in 2015 is aimed to target two or multi-arm unequal allocations with irrational numbers.²² MWUD does not preserve the allocation ratios. When used in two-arm equal allocation trials, it is reduced to the EUD.

The variable block design (VBD) uses a randomly selected block size intended to reduce the treatment predictability associated with a fixed block size.²⁰ However, with the same MTI defined by the largest block size, the use of any smaller blocks in the randomization sequence will reduce the allocation randomness. The VBD will have a higher probability of deterministic assignments and a higher correct guess probability than PBD under the same restriction of the MTI. Therefore, it is not suggested to be used. The merged block design (MBD) uses two PBD sequences and picks the next available treatment assignment in a PBD sequence selected at random to compose the final allocation sequence.²⁴ It is anticipated that MBD will have a higher allocation randomness than PBD. Both VBD and MBD do not have an explicit formula for the conditional allocation probability.

The primary goal of this manuscript is to determine the statistical properties of randomization designs with the potential to challenge the dominance of the PBD in two-arm equal allocation trials. Five randomization designs, including BDS, BCDWIT, EUD, BUD, and AMP, are selected because they all share the good features that PBD has, use MTI, and have an explicit conditional allocation probability for easy implementation, but provide higher allocation randomness.

3. METHODS

3.1. Conditional Allocation Probability of Randomization Designs

Randomization designs are often named after conceptual operation models, such as block, urn, and biased coin. While an operation model helps to illustrate the randomization procedure, the conditional allocation probability provides a unified framework for the description, comparison, and implementation of different randomization designs.

All randomization designs with MTI restriction use a completely random assignment when the current treatment imbalance is zero and a deterministic assignment when the treatment imbalance reaches the MTI. Differences are exclusively based on their responses to imbalances below the MTI. The conditional allocation probability defines the probability of assigning the current subject to a specific arm based on the prior treatment assignment history. It determines the statistical and operational properties of the randomization design, and thus serves as a basis for comparing different randomization designs.

To derive the conditional allocation probabilities for randomization designs with MTI restriction for two-arm equal allocation (1:1) trials, the following parameters are defined.

$n_A$: number of subjects previously randomized to arm A.

$n_B$: number of subjects previously randomized to arm B.

$n=n_A+n_B$: number of subjects previously randomized.

$d=n_A-n_B$: treatment imbalance among the n subjects previously randomized.

$\delta$: pre-specified MTI, the maximum tolerated treatment imbalance ($\delta >0$).

$p_A$: conditional allocation probability assigning the current subject to arm A.

$p_B=1-p_A$: conditional allocation probability assigning the current subject to arm B.

Additionally, two mathematical functions are used for the expression of conditional allocation probabilities throughout the manuscript:

Function $int\left(x\right)$ returns an integer less than or equal to x.

Function $sign\left(x\right)$ returns value $-1, 0, 1$ if x is negative, zero, or positive, respectively.

3.1.1. Permuted Block Design (PBD)

PBD can be described as a procedure of repeated random draws of permuted blocks with replacement. Rosenberger and Lachin indicated that the PBD can be considered as repeated random allocation rules, which follows a hypergeometric distribution and can be illustrated with an urn model.¹⁶ The PBD starts from an active urn with δ balls for arm A and δ balls for arm B and an empty inactive urn. When a subject is ready for randomization, randomly draw a ball from the active urn, assign the subject to the arm associated with the ball, and place the ball in the inactive urn. When the active urn is empty, return all balls from the inactive urn to the active urn. Repeat the procedure until the end of the study. The conditional allocation probability for assigning the current subject to arm A equals the proportion of balls associated with arm A in the active urn.

$$p_A\left(PBD\right)= \frac{\delta +\delta \times int\left(\frac{n}{2\delta }\right)-n_A}{2\delta +2\delta \times int\left(\frac{n}{2\delta }\right)-n}=0.5-0.5\times \frac{d}{2\delta +2\delta \times int\left(\frac{n}{2\delta }\right)-n}$$ (1)

Here the value of $int\left(\frac{n}{2\delta }\right)$ is the number of completed blocks in the n assignments prior to the current subject.

3.1.2. Big Stick Design (BSD)

The BSD uses completely random assignments by default and deterministic assignments when the MTI is reached.¹³ The conditional allocation probability for assigning the current subject to arm A can be written as the sum of two components, the default probability of 0.5 and the adjustment of 0.5 in the opposite direction of the treatment imbalance when it reaches the MTI:

$$p_A\left(BSD\right)=0.5-0.5\times sign\left(d\right)\times int\left(\frac{\left|d\right|}{\delta }\right)$$ (2)

3.1.3. Biased Coin Design with Imbalance Tolerance (BCDWIT)

The BCDWIT uses a biased coin probability $p_{bc}\in \left[0.5, 1\right]$ favoring of reducing the treatment imbalance when it occurs but has not yet reached the MTI, and a deterministic assignment when it reaches the MTI.¹⁴ Its conditional allocation probability can be obtained by adding $\left(p_{bc}-0.5\right)$ to the conditional allocation probability of the BSD in the opposite direction of the treatment imbalance when $\left|d\right|<\delta$.

$$p_A\left(BCDWIT\right)=0.5-0.5\times sign\left(d\right)\times int\left(\frac{\left|d\right|}{\delta }\right)- \left(p_{bc}-0.5\right)\times sign\left(d\right) \times \left[1-int\left(\frac{\left|d\right|}{\delta }\right)\right]$$ (3)

The performance of the BCDWIT depends on the value of $p_{bc}$. When $p_{bc}=0.5$, the BCDWIT behaves the same as the BSD. When $p_{bc}=1$, the BCDWIT becomes the PBD with a block size of 2, which has 50% completely random assignments and 50% deterministic assignments.

3.1.4. Ehrenfest Urn Design (EUD)

The EUD can be illustrated with the urn model used by the PBD. Both the active urn and the inactive urn start with δ balls for arm A and δ balls for arm B. When a ball is randomly drawn from the active urn and the subject is assigned to the arm accordingly, the ball is placed in the inactive urn, and a ball of the opposite arm in the inactive urn is moved to the active urn. In this way, the number of balls for arm A in the inactive urn minus the number of balls for arm B in the inactive urn equals two times of the treatment imbalance. The conditional allocation probability for assigning the current subject to arm A is the proportion of balls for arm A in the active urn:

$$p_A\left(EUD\right)=0.5-\frac{n_A-n_B}{2\delta }=0.5-0.5\times \frac{d}{\delta }$$ (4)

3.1.5. Block Urn Design (BUD)

The BUD is a modification of the PBD.¹⁹ It can be illustrated with the same urn model as the PBD. Instead of waiting until the active urn is empty to return all balls from the inactive urn to the active urn, the BUD returns a balanced set of balls from the inactive urn to the active urn as soon as the set becomes available. For a two-arm equal allocation trial, a balanced set of balls contains one ball for arm A and one ball for arm B. With this ball return policy, balls left in the inactive urn will be for the same arm. The number of balls in the inactive urn represents the absolute value of treatment imbalance. The conditional allocation probability for assigning the current subject to arm A is:

$$p_A\left(BUD\right)=0.5 - \frac{n_A-n_B}{4\delta -2\times \left|n_A-n_B\right|}=0.5-0.5\times \frac{d}{2\delta -\left|d\right|}$$ (5)

3.1.6. Asymptotic Maximal Procedure (AMP)

The AMP is a modification of the MP by releasing the restrictions of the allocation sequence length and the sequence-end balance.²³ This modification allows for stable conditional allocation probabilities to be independent of the assignment order in the allocation sequence. For two-arm equal-allocation trials with MTI from 2 to 5, the conditional allocation probabilities have been obtained as below:

$$p_A\left(AMP\right)=0.5-\left\{\begin{array}{ll}0.5\times sign\left(d\right)\hfill & if \left|d\right|=\delta or d=0\hfill \\ 0.16667\times sign\left(d\right)\hfill & if \delta =2 and \left|d\right|=1\hfill \\ 0.08578\times sign\left(d\right)\hfill & if \delta =3 and \left|d\right|=1\hfill \\ 0.20711\times sign\left(d\right)\hfill & if \delta =3 and \left|d\right|=2\hfill \\ 0.05279\times sign\left(d\right)\hfill & if \delta =4 and \left|d\right|=1\hfill \\ 0.11803\times sign\left(d\right)\hfill & if \delta =4 and \left|d\right|=2\hfill \\ 0.22361\times sign\left(d\right)\hfill & if \delta =4 and \left|d\right|=3\hfill \\ 0.03590\times sign\left(d\right)\hfill & if \delta =5 and \left|d\right|=1\hfill \\ 0.07735\times sign\left(d\right)\hfill & f \delta =5 and \left|d\right|=2\hfill \\ 0.13397\times sign\left(d\right)\hfill & f \delta =5 and \left|d\right|=3\hfill \\ 0.23205\times sign\left(d\right)\hfill & f \delta =5 and \left|d\right|=4\hfill \end{array}\right.$$ (6)

Quantitative comparisons show that a noticeable difference in the conditional allocation probability between MP and AMP exists only for the first few assignments in the allocation sequence.⁹ For example, with $\delta =3$, the difference between AMP and MP in the conditional allocation probability is less than 0.0025 after the 3^rd assignment in the allocation sequence. When used for unequal allocation trials, both MP and AMP do not preserve the unconditional allocation probability.

3.2. Unified Framework for MTI Randomization Sequence Generation

With the availability of explicit formulas for conditional allocation probabilities, a unified framework for the generating randomization sequences can be created for different randomization designs. For two-arm equal allocation trials, the conditional allocation probability contains the target allocation probability of 0.5 and an imbalance-adaptive adjustment. The direction of the adjustment is opposite of the direction of the imbalance. For BSD, EUD, BCDWIT, BUD, and AMP, the magnitude of the adjustment depends on the absolute value of the current imbalance $d$ only.

$$p_A\left(d\right)=0.5-0.5\times sign\left(d\right)\times g\left(d\right)$$ (7)

Here $g\left(d\right)$ represents a function of imbalance d. For example, BSD has $g\left(d\right)=int\left(\frac{\left|d\right|}{\delta }\right)$, EUD has $g\left(d\right)=\frac{\left|d\right|}{\delta }$, and BUD has $g\left(d\right)=\frac{\left|d\right|}{2\delta -\left|d\right|}$. With the conditional allocation probability function (7) and pre-specified MTI threshold δ, randomization sequences can be obtained with a list of random numbers with a uniform distribution on (0,1). When the random number R is less than or equal to the conditional allocation probability$p_A$, the subject is assigned to arm A, otherwise the subject is assigned to arm B.

$$Tx=\left\{\begin{array}{cc}A& if R\le p_A\\ B& if R>p_B\end{array}\right.$$ (8)

Furthermore, if the value of the random number R is generated when a subject randomization is requested, a real-time on-demand treatment assignment can be obtained based on (7) and (8).

3.3. Steady-State Probabilities of Randomization Procedures

Formulas (2–6) indicate that these five randomization procedures satisfy the conditions of Markov property in a specific way such that their conditional allocation probability for the next treatment assignment depend only on the current imbalance status and do not depend on past treatment assignment history. Therefore, the statistical properties of these randomization designs can be obtained directly from their steady-state probabilities of the treatment imbalance. For two-arm equal allocation trials, the treatment imbalance can change by one unit after each treatment assignment. For randomization designs with MTI restriction, the absolute value of treatment imbalance $\left|d\right|$ can only be one of the integers from 0 to the MTI threshold δ. Let $P_{i,j}$ be the probability for $\left|d\right|$ transferring from i to j. Based on the conditional allocation probability defined in equations (2–6), there are:

$$P_{0,1}=1$$ (9)

$$P_{\delta ,\delta -1}=1$$ (10)

$$P_{i,j}=0 \left(\left|i-j\right|\ne 1\right)$$ (11)

$$0.5\ge P_{i,i+1}>0 (0<i<\delta )$$ (12)

$$1>P_{i,i-1}\ge 0.5 (0<i<\delta )$$ (13)

Here (9) indicates that if currently balanced, the next treatment assignment will certainly increase $\left|d\right|$ to 1. Equation (10) reflects the fact that when the current imbalance reaches the MTI, the next assignment will be deterministic and $\left|d\right|$ will be reduced to $\delta -1$. Equation (11) indicates that each treatment assignment can only and will certainly change the imbalance by 1 unit. Equations (12) and (13) indicate that when $0<\left|d\right|<\delta$, the probability for $\left|d\right|$ being increased is no greater than 0.5, and the probability for $\left|d\right|$ being decreased is no less than 0.5. The value of this probability depends on the randomization design. Equations (14) and (15) show the transition probability matrix of BSD and BUD with $\delta =3$.

$$\mathit{P}\left(BSD, \delta =3\right)=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0.5& 0& 0.5& 0\\ 0& 0.5& 0& 0.5\\ 0& 0& 1& 0\end{array}\right]$$ (14)

$$\mathit{P}\left(BUD, \delta =3\right)=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0.6& 0& 0.4& 0\\ 0& 0.75& 0& 0.25\\ 0& 0& 1& 0\end{array}\right]$$ (15)

In general, for all randomization designs satisfying the conditions of imbalance-status-dependent Markov property, the treatment imbalance transition probability matrix can be written as:

$$\mathit{P}=\left[\begin{array}{cccccc}0& 1& 0& \cdots & 0& 0\\ P_{1,0}& 0& 1-P_{1,0}& \cdots & 0& 0\\ 0& P_{2,1}& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& 0& 0& 0\\ 0& 0& 0& P_{\delta -1,1}& 0& 1-P_{\delta -1,1}\\ 0& 0& 0& 0& 1& 0\end{array}\right]$$ (16)

Let $\mathit{\pi }=\left[{\pi }_0,{\pi }_1,\cdots ,{\pi }_{\delta }\right]$ be the steady-state probability vector, with ${{\displaystyle \sum }}_{i=0}^{\delta }{\pi }_i=1$, there is:

$$\mathit{P}\mathit{\pi }=\mathit{\pi }$$ (17)

Insert (16) into (17), the steady-state probabilities can be obtained for specific values of δ:

$$\mathit{\pi }\left( \delta =2\right)=\frac{1}{2}{\left[\begin{array}{l}{P}_{1,0}\hfill \\ 1\hfill \\ 1-P_{1,0}\hfill \end{array}\right]}^T$$ (18)

$$\mathit{\pi }\left(\delta =3\right)=\frac{1}{2\left(1-P_{1,0}+P_{1,0}{P}_{2,1}\right)}{\left[\begin{array}{l}{P}_{1,0}{P}_{2,1}\hfill \\ P_{2,1}\hfill \\ 1-P_{1,0}\hfill \\ \left(1-P_{1,0}\right)\left(1-P_{2,1}\right)\hfill \end{array}\right]}^T$$ (19)

$$\mathit{\pi }\left( \delta =4\right)=\frac{1}{2\left[\left(1-P_{1,0}\right)\left(1-P_{2,1}\right)+P_{2,1}{P}_{3,2}\right]}{\left[\begin{array}{l}{P}_{1,0}{P}_{2,1}{P}_{3,2}\hfill \\ P_{2,1}{P}_{3,2}\hfill \\ \left(1-P_{1,0}\right)P_{3,2}\hfill \\ \left(1-P_{1,0}\right)\left(1-P_{2,1}\right)\hfill \\ \left(1-P_{1,0}\right)\left(1-P_{2,1}\right)\left(1-P_{3,2}\right)\hfill \end{array}\right]}^T$$ (20)

$$\mathit{\pi }\left( \delta =5\right)=\frac{1}{2\left\{\left(1-P_{2,1}\right)\left[\left(1-P_{1,0}\right)\left(1-P_{3,2}\right)-P_{1,0}{P}_{3,2}{P}_{4,3}\right]+P_{3,2}{P}_{4,3}\right\}}{\left[\begin{array}{l}{P}_{1,0}{P}_{2,1}{P}_{3,2}{P}_{4,3}\hfill \\ P_{2,1}{P}_{3,2}{P}_{4,3}\hfill \\ \left(1-P_{1,0}\right)P_{3,2}{P}_{4,3}\hfill \\ \left(1-P_{1,0}\right)\left(1-P_{2,1}\right)P_{4,3}\hfill \\ \left(1-P_{1,0}\right)\left(1-P_{2,1}\right)\left(1-P_{3,2}\right)\hfill \\ \left(1-P_{1,0}\right)\left(1-P_{2,1}\right)\left(1-P_{3,2}\right)\left(1-P_{4,3}\right)\hfill \end{array}\right]}^T$$ (21)

Here ${\pi }_0$ is the steady-state probability of completely random assignment when treatment imbalance $d=0$ and ${\pi }_{\delta }$ is the probability of deterministic assignment when $\left|d\right|=\delta$. Equations (18–21) apply generically to all randomization designs with MTI restriction, satisfying the condition of imbalance-status-dependent Markov property for two-arm equal allocation trials. Analytical results for $\delta$ larger than 5, corresponding to block sizes larger than 10, can be obtained through lengthy mathematic derivation. It may be easier to find the numerical steady-state probabilities of treatment imbalance by solving equation (17) with elements of the transition matrix for the specific randomization design.

3.4. Statistical Properties of Randomization Designs with MTI Restriction

Statistical properties of randomization designs can be measured by the probability of deterministic assignment (DA), the probability of completely random assignment (CR), the correct guess probability (CG), and the standard deviation of treatment imbalance (SD). CG is defined based on the Blackwell and Hodges’ convergence guessing strategy; predicting the treatment which has hitherto occurred less often be the next assignment.³ SD is defined as the standard deviation of treatment imbalance across the randomization sequence. Let $P_A$ be a random variable of the probability of assigning a subject to arm $A$ and $D$ be a random variable of the treatment imbalance after that assignment, the four statistical property measures can be defined as:

$$DA=E\left(P_A=0\right)+E\left(P_A=1\right)$$ (22)

$$CR=E\left(P_A=0.5\right)$$ (23)

$$CG=0.5+E\left(\left|P_A-0.5\right|\right)$$ (24)

$$SD=\sqrt{E\left(D^2\right)}$$ (25)

These measures can be directly obtained from their steady-state probabilities for all randomization designs satisfying the condition of the imbalance-status-dependent Markov property.

$$DA={\pi }_{\delta }$$ (26)

$$CR={\pi }_0+{{\displaystyle \sum }}_{j=1}^{\delta -1}int\left({\pi }_j{P}_{j,j+1}\right)$$ (27)

$$CG=0.5\times {\pi }_0+{{\displaystyle \sum }}_{j=1}^{\delta }{\pi }_j{P}_{j,j-1}$$ (28)

$$SD=\sqrt{{{\displaystyle \sum }}_{j=1}^{\delta }{\pi }_j{j}^2}$$ (29)

It is important to note that all these statistical properties are defined based on infinite randomization sequence lengths. If the length of the allocation sequence is extremely small, such as on the same scale of the maximum tolerated imbalance, the statistical properties of these randomization designs will shift towards those of the completely random assignment. For example, if the allocation sequence length is 1, the treatment assignment is always completely random. If the allocation sequence length is 2 and the MTI is 3, the probability of deterministic assignment is zero.

3.5. Statistical Properties of the Permuted Block Design

PBD has its conditional allocation probability depending not only on the current treatment imbalance, but also on the sequence order of the assignment within the block. Some statistical properties of the PBD have been studied via both analytical and simulation approaches.^4,10,16 Based on definitions (22–25), analytical results for the four statistical property measures can be obtained by listing all block permutations. Let $p_{ij}$ be the conditional allocation probability for arm A and $d_{ij}$ be the treatment imbalance for assignment $i \left(i=1,2,\cdots ,b\right)$ within block $j \left(j=1,2,\cdots ,h\right)$. Both $p_{ij}$ and $d_{ij}$ can be specified for each given block sequence permutation. For example, for block j of AABB, there are $p_{1j}=1/2$, $p_{2j}=1/3$, $p_{3j}=0$, $p_{4j}=0$, $d_{1j}=1$, $d_{2j}=2$, $d_{3j}=1$, and $d_{4j}=0$. The analytical results of the four statistical property measures for PBD can be written as functions of $p_{ij}\left(i=1,2,\cdots ,b;j=1,2,\cdots ,h\right)$ and $d_{ij}\left(i=1,2,\cdots ,b;j=1,2,\cdots ,h\right)$:

$$DA=\frac{1}{bh}\left[{{\displaystyle \sum }}_{i=1}^b{{\displaystyle \sum }}_{j=1}^{h}int\left(\frac{\left|p_{ij}-0.5\right|}{0.5}\right)\right]$$ (30)

$$CR=1-\frac{1}{bh}\left[{{\displaystyle \sum }}_{i=1}^b{{\displaystyle \sum }}_{j=1}^{h}sign\left(\left|p_{ij}-0.5\right|\right)\right]$$ (31)

$$CG=0.5+\frac{1}{bh}\left[{{\displaystyle \sum }}_{i=1}^b{{\displaystyle \sum }}_{j=1}^h\left|p_{ij}-0.5\right|\right]$$ (32)

$$SD=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^b{{\displaystyle \sum }}_{j=1}^h{d}_{ij}^2}{bh}}$$ (33)

4. RESULTS

Based on formulas (1–6), conditional allocation probabilities can be obtained for the six randomization designs with MTI restriction for specific values of the MTI. Table 2 lists the results with $MTI=3$.

Table 2.

Conditional allocation probability $p_A$ for randomization designs with MTI restriction (For two-arm equal-allocation trials with MTI = 3)

Randomization Design		Current Imbalance d = nA-nB
		−3	−2	−1	0	1	2	3
Big Stick Design		1	0.5	0.5	0.5	0.5	0.5	0
Biased Coin Design with Imbalance Tolerance (pbc = 0.75)		1	0.75	0.75	0.5	0.25	0.25	0
Ehrenfest Urn Design		1	0.8333	0.6667	0.5	0.3333	0.1667	0
Block Urn Design		1	0.75	0.6	0.5	0.4	0.25	0
Asymptotic Maximal Procedure		1	0.7071	0.5858	0.5	0.4142	0.2929	0
Permuted Block Design (block size b = 6)	1st assignment in the block				0.5
	2nd assignment in the block			0.6		0.4
	3rd assignment in the block		0.75		0.5		0.25
	4th assignment in the block	1		0.6667		0.3333		0
	5th assignment in the block		1		0.5		0
	6th assignment in the block			1		0

p_A: conditional allocation probability assigning the subject to arm A. An assignment is deterministic if p_A = 0 or p_A = 1.

Table 2 reveals two important differences between PBD and the other five randomized designs with MTI constraints. First, the conditional allocation probabilities of PBD depend on the current imbalance and the sequence order within the block, whereas the conditional allocation probabilities of the other five randomized designs depend entirely on the current imbalance. Second, forcing block-end balancing requires PBD to use deterministic allocation not only when the MTI is reached but also when the absolute value of the imbalance is equal to the number of remaining allocations in the block, whereas the other five randomized designs only use deterministic allocation when the MTI is reached. This explains why PBD has a higher deterministic assignment probability than the other five randomized designs with the same MTI.

Using the conditional allocation probability formulas (1–6) and the treatment assignment formula (8), allocation sequences for these randomization designs can be easily created in a generic approach. This will eliminate the implementation burdens of replacing PBD with better alternatives when pre-generated allocation sequences are required. When real-time randomization is used, the conditional allocation probability formulas (1–6) can be directly used for treatment assignment for the six randomization designs listed in Table 3 without critical differences. Table 3 shows sample randomization sequences for two-arm equal allocation trials with an MTI of 3.

Table 3.

Example of randomization sequence generation based on conditional allocation probability (For two-arm equal allocation trials with MTI = 3)

n	R	Permuted Block Randomization			Block Urn Design			Ehrenfest Urn Design			Big Stick Design			Biased Coin Design with Imbalance Tolerance (pbc =0.75)			Asymptotic Maximal Procedure
		d	PA	Tx	d	PA	Tx	d	PA	Tx	d	PA	Tx	d	PA	Tx	d	PA	Tx
0	0.2199	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A
1	0.6358	1	0.4	B	1	0.4	B	1	0.3333	B	1	0.5	B	1	0.25	B	1	0.4142	B
2	0.0891	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A
3	0.1204	1	0.3333	A	1	0.4	A	1	0.3333	A	1	0.5	A	1	0.25	A	1	0.4142	A
4	0.0240	2	0	B	2	0.25	A	2	0.1667	A	2	0.5	A	2	0.25	A	2	0.2929	A
5	0.9961	1	0	B	3	0	B	3	0	B	3	0	B	3	0	B	3	0	B
6	0.9307	0	0.5	B	2	0.25	B	2	0.1667	B	2	0.5	B	2	0.23	B	2	0.2929	B
7	0.4480	−1	0.6	A	1	0.4	B	1	0.3333	B	1	0.5	A	1	0.25	B	1	0.4142	B
8	0.7067	0	0.5	B	0	0.5	B	0	0.5	B	2	0.5	B	0	0.5	B	0	0.5	B
9	0.4948	−1	0.6667	A	−1	0.6	A	−1	0.6667	A	1	0.5	A	−1	0.75	A	−1	0.5858	A
10	0.6170	0	0.5	B	0	0.5	B	0	0.5	B	2	0.5	B	0	0.5	B	0	0.5	B
11	0.4433	−1	1	A	−1	0.6	A	−1	0.6667	A	1	0.5	A	−1	0.75	A	−1	0.5858	A
12	0.2353	0	0.5	A	0	0.5	A	0	0.5	A	2	0.5	A	0	0.5	A	0	0.5	A
13	0.3359	1	0.4	A	1	0.4	A	1	0.3333	B	3	0	B	1	0.25	B	1	0.4142	A
14	0.2381	2	0.25	A	2	0.25	A	0	0.5	A	2	0.5	A	0	0.5	A	2	0.2929	A
15	0.2577	3	0	B	3	0	B	1	0.3333	A	3	0	B	1	0.25	B	3	0	B
16	0.4998	2	0	B	2	0.25	B	2	0.1667	B	2	0.5	A	0	0.5	A	2	0.2929	B
17	0.2268	1	0	B	1	0.4	1	1	0.3333	A	3	0	B	1	0.25	A	1	0.4142	A
18	0.6486	0	0.5	B	2	0.25	B	2	0.1667	B	2	0.5	B	2	0.25	B	2	0.2929	B
19	0.5979	−1	0.6	A	1	0.4	B	1	0.3333	B	1	0.5	B	1	0.25	B	1	0.4142	B
20	0.0380	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A	0	0.5	A

n: number of previous assigned subjects

R: random number with uniform distribution on (0,1)

Tx: treatment assignment

d = n_A – n_B, treatment imbalance

P_A = probability of assigning current subject to arm A

Figure 1 shows responses in the conditional allocation probability of different randomization designs to treatment imbalance. The order of response sensitivity from high to low is: EUD, BUD, AMP, and BSD. The performance of BCDWIT varies depending on the biased coin probability. If $p_{bc}=0.5$, BCDWTI becomes BSD. BCDWIT can be the most sensitive procedure if a higher biased coin probability is used. The response of PBD to treatment imbalance varies based on the allocation sequence order within the block. Its least sensitive response occurs at the beginning of the block and overlaps with the response of BUD.

Figure 1.

Responses in conditional allocation probability to treatment imbalance (For two-arm equal allocation trials with MTI = 5)

Sensitive responses to imbalances smaller than the MTI reduce the chance of imbalance reaching MTI, thereby reducing the probability of deterministic assignments. However, using more biased coin probabilities far away from 0.5 may increase the correct guess probability.

Table 4 lists the statistical properties of the six randomization designs with MTI restriction. Results for the five designs satisfying the imbalance-status-dependent Markov property are obtained from the steady-state probabilities (18–21) and formulas for the statistical property measures (26–29). The results for PBD are obtained based on formulas (30–33) with details shown in Table 5.

Table 4.

Statistical properties of six randomization designs with MTI restriction (For two-arm equal allocation trials with MTI from 2 to 5)

Maximum Tolerated Imbalance δ	Statistical Property	Block Urn Design	Ehrenfest Urn Design	Big Stick Design	Asymptotic Maximal Procedure	Biased Coin Design with Imbalance Tolerance			Permuted Block Design
						Pbc=0.65	Pbc=0.75	Pbc=0.85
2	CR	0.3333	0.375	0.75	0.3333	0.325	0.375	0.425	0.4167
	${\pi }_1$	0.5	0.5	0.5	0.5	0.5	0.5	0.5
	DA	0.1667	0.125	0.25	0.1667	0.175	0.125	0.075	0.3333
	CG	0.6667	0.6875	0.625	0.6667	0.6625	0.6875	0.7125	0.7083
	SD	1.0801	1	1.2247	1.0801	1.0954	1	0.8944	0.9325
3	CR	0.2647	0.3125	0.8333	0.25	0.2735	0.3462	0.4140	0.3667
	${\pi }_1$	0.4412	0.4686	0.3333	0.4268	0.4207	0.4615	0.4871
	${\pi }_2$	0.2353	0.1875	0.3333	0.25	0.2265	0.1539	0.0860
	DA	0.0588	0.0313	0.1667	0.0732	0.0793	0.0385	0.0129	0.25
	CG	0.6235	0.6563	0.5833	0.625	0.6367	0.6731	0.7070	0.6833
	SD	1.3827	1.2247	1.7795	1.4442	1.4284	1.1929	0.9731	1.0847
4	CR	0.2253	0.2734	0.875	0.2	0.2520	0.3375	0.4122	0.3321
	${\pi }_1$	0.3944	0.4375	0.25	0.3618	0.3876	0.45	0.4849
	${\pi }_2$	0.2535	0.2188	0.25	0.2618	0.2087	0.15	0.0856
	${\pi }_3$	0.1056	0.0625	0.25	0.1382	0.1124	0.05	0.0151
	DA	0.0211	0.0078	0.125	0.0382	0.0393	0.0125	0.0023	0.2
	CG	0.6127	0.6367	0.5625	0.6	0.6260	0.6688	0.7061	0.6661
	SD	1.6423	1.4142	2.3452	1.8067	1.6921	1.3038	0.9997	1.2258
5	CR	0.1992	0.2461	0.9	0.1647	0.2417	0.3347	0.4118	0.3063
	${\pi }_1$	0.3585	0.4102	0.2	0.3074	0.3719	0.4463	0.4845
	${\pi }_2$	0.2549	0.2344	0.2	0.2471	0.2002	0.1488	0.0855
	${\pi }_3$	0.1338	0.0879	0.2	0.1690	0.1078	0.0496	0.0151
	${\pi }_4$	0.0459	0.0198	0.2	0.0882	0.0581	0.0165	0.0027
	DA	0.0077	0.0020	0.1	0.0236	0.0203	0.0041	0.0004	0.1667
	CG	0.5996	0.6231	0.55	0.5824	0.6209	0.6674	0.7059	0.6532
	SD	1.8730	1.5811	2.9155	2.1949	1.8921	1.3621	1.0074	1.3543

CR: Probability of completely random assignment. For BUD, EUD, AMP and BCDWIT, CR=π₀. For BSD, CR=1-π_δ.

DA: Probability of deterministic assignment. For BUD, EUD, BSD, AMP and BCDWIT, DA= π_δ.

CG: Probability of making a correct guess based on Blackwell and Hodges’ convergence guessing strategy.

SD: Standard deviation of treatment imbalance.

Data for PBD is obtained from formulas (30–33).

Table 5.

Statistical properties of permuted block design (For two-arm equal allocation trials)

Block Size b = 2δ	Number of Block Permutations h	Conditional Allocation Probability pA	Number of pA Occurrence in all Block Permutations	Probability of Completely Random Assignment	Probability of Deterministic Assignment	Correct Guess Probability	Standard Deviation of Treatment Imbalance
4	6	0 or 1	8	0.4167	0.3333	0.7083	0.9325
		1/3 or 2/3	6
		1/2	10
6	20	0 or 1	30	0.3667	0.25	0.6833	1.0847
		1/4 or 3/4	8
		1/3 or 2/3	18
		2/5 or 3/5	20
		1/2	44
8	70	0 or 1	112	0.3321	0.2	0.6661	1.2258
		1/5 or 4/5	10
		1/4 or 3/4	32
		1/3 or 2/3	90
		2/5 or 3/5	60
		3/7 or 4/7	70
		1/2	186
10	252	0 or 1	420	0.3063	0.1667	0.6532	1.3543
		1/6 or 5/6	12
		1/5 or 4/5	50
		1/4 or 3/4	120
		2/7 or 5/7	42
		1/3 or 2/3	330
		3/8 or 5/8	112
		2/5 or 3/5	200
		3/7 or 4/7	210
		4/9 or 5/9	252
		1/2	772

A tradeoff between allocation randomness and treatment imbalance control exists in all restricted randomization designs. Any gain in treatment imbalance control comes with a cost in allocation randomness. However, for the same amount of gain in treatment imbalance control, different randomization designs may pay different cost in allocation randomness. Figure 2 shows the benefit measured by the standard deviation of treatment imbalance and associated cost measured by the correct guess probability for the six randomization designs with MTI restriction.

Figure 2.

Comparsion of statistical properties - Correct guess probability vs. standard deviation of treatment imbalance For two-arm equal allocation trials with MTI varying from 2 (low right) to 5 (up left)

Both the probability of deterministic assignment (DA) and the correct guess probability (CG) are the measures for treatment allocation randomness and are positively correlated with each other. Figure 3 shows DA and CG for the six randomization designs with MTI varying from 2 to 5.

Figure 3.

Comparsion of statistical properties - Correct guess probability of deterministic assignment For two-arm equal allocation trials with MTI varying from 2 (up right) to 5 (low left)

5. DISCUSSION

5.1. Pre-generated Allocation Sequence vs Real-time Treatment Assignment

Selection bias in clinical trials comes from two sources: concealment failure and excess correct guess probability. Concealment failure is related to the randomization implementation method and is not affected by the randomization algorithm.²⁵ It can occur only if the allocation sequence is pre-generated. Therefore, the best way to eliminate concealment failure is to use real-time treatment assignments.

Excess correct guess probability is the consequence of low allocation randomness. It comes from the unblinding of previous treatment assignments and knowledge of the randomization design. A deterministic assignment can be predicted with certainty even if the concealment of the allocation sequence is perfect, or the treatment assignment is made in real-time. For example, knowing the first two subjects were assigned to arm A and that a PBD with a block size of 4 is used in the two-arm equal allocation trial, the investigator can predict that the next two subjects will be allocated to arm B with certainty. To reduce the risk of selection bias, access to information about the randomization design and its implementation parameters should be limited to people responsible for designing, approving, implementing, and validating the randomization algorithm. The study protocol and other trial operation documents accessible by site investigators should avoid releasing detailed information about the randomization design.

Interactive response technology (IRT) systems are widely used in clinical trials for subject randomization. Most IRT systems use pre-generated PBD allocation sequences. Statisticians responsible for selecting randomization designs often face an implementation burden from IRT systems when considering using advanced randomization designs. The current technical methods for randomization algorithm implementation lag behind the research achievements on the statistical properties of randomization designs. It has been reported that real-time on-demand randomization function can be programmed in a web-based clinical trial management system integrated with the electronic data capture system, instead of a silo IRT system.²⁶ The conditional allocation probability formulas (1–6) prove that for both allocation sequence generation and real-time treatment assignment there is no critical difference between the PBD and other randomization designs with MTI restriction. As such, it is expected that real-time randomization will be used more often in clinical trials and statisticians can select the best randomization design for their trials based on the statistical properties, rather than limitations in implementation technology.

5.2. Steady-state Analytical Results vs Simulated Results for Statistical Properties

Statistical properties of randomization designs have been studied with computer simulations under specific trial settings and implementation conditions, such as the allocation sequence lengths and the stratification settings.^4,5,6 For this reason, computer simulation results for the same randomization algorithm may differ from each other. Statistical properties presented in Section 4 are steady-state results, assuming the length of the randomization sequence is infinite. For a given randomization design with a specific MTI, there is only one steady-state result for each of the four statistical property measures, including the probability of deterministic assignments, the probability of completely random assignments, the correct guess probability, and the standard deviation of treatment imbalance. These results represent the characteristics of the randomization design itself, without the influence of the implementation conditions. The difference between steady-state performances and those under a limited allocation sequence length is worthy of further research. In general, the steady-state allocation randomness reflects the cap of allocation randomness under different allocation sequence lengths. For example, with MTI=3 the probability of deterministic assignment for PBD, BSD, BCDWIT(P_bc=0.65), AMP, BUD, and EUD is 25%, 16.67%, 7.93%, 7.32%, 5.88% and 3.13% respectively. For shorter allocation sequences, these numbers will be a little smaller, but the relative ranking order will be the same, the PBD will have the highest and the EUD will have the lowest. Simulation study is a useful tool to evaluate performances of randomization designs under specific trial settings. In this case, analytical results can be used to validate the simulation results.

5.3. Probability of Deterministic Assignment vs Correct Guess Probability

The probability of deterministic assignment (DA) and the correct guess probability (CG) are commonly used measured for allocation randomness and are positively correlated with each other. As shown in Figure 3, a design could have a lower DA and a higher CG when compared to another design. For example, with MTI=3, BSD has a higher DA of 16.67% and a lower CG of 0.5833 when compared to BUD, which has a DA of 5.88% and a CG of 0.6235. If the risk of selection bias associated with treatment prediction with certainty is the major concern, BUD will be a better choice than BSD. However, if the risk of selection bias is defined based on Blackwell and Hodges’ convergence guessing strategy, BSD will be a better option. Compared to DA, CG provides a more comprehensive assessment for allocation randomness because it covers all treatment assignments. The CG is also less sensitive to the distribution of the random number used for treatment assignment as defined in formulas (8) and (24) and therefore will be more stable than DA for the measurement of allocation randomness of the entire allocation sequence. When evaluating the performance of a randomization design, it is suggested to use CG as the primary measure for allocation randomness.

5.4. Tradeoff between Allocation Randomness and Treatment Imbalance Control

Allocation randomness and treatment imbalance control are competing demands within restricted randomization designs. When comparing the tradeoff performance of different randomization designs, SD and CG are used as primary measures for treatment imbalance and allocation randomness respectively. An overall performance score, in the format of Euclidean distance, combining CG and SD with weight and unified scaling has been used in literature.^4,5 However, in clinical trial practice, tolerances to treatment imbalance and the risk of selection bias are often specifically connected to the trial operation characteristics, such as the treatment blinding type, the efficacy outcome assessment type, and the sample size. Complete information on the tradeoff performance shown in Figure 2 is more valuable than a composed score.

Figure 2 shows that, under the same MTI, the PBD has a smaller SD and higher CG than the other five randomization designs. If treatment imbalance control is important and the risk of selection bias associated with correct guess probability is low, PBD(δ =2) with a block size of 4 can be considered. The BSD is a good choice if selection bias associated with correct guess probability is the major concern. In case the probability of deterministic assignment is a sensitive issue, EUD and BUD will be better alternatives to PBD and BSD. If the SD is considered more important than the MTI, randomization designs with similar SD but different MTI can be compared. For example, PBD(δ=3) has $SD=1.0847$, $CG=0.6833$ and $DA=25\%$, whereas BUD(δ=2) has $SD=1.0801$, $CG=0.6667$ and $DA=16.67\%$. Therefore, BUD(δ=2) performances better than PBD(δ=3) in terms of both allocation randomness and imbalance control.

5.5. Statistical Properties of Stratified Randomization

Stratified randomization is widely used in multicenter trials, or trials with important confounding factors to be balanced. With stratified randomization, the selected randomization algorithm runs independently within each stratum. The allocation randomness measured by DA and CG for the entire study will be the same as the DA and CG of the randomization algorithm, or slightly lower because of the small average stratum size. The treatment imbalance in a covariate margin will be the sum of imbalances in strata associated with the covariate margin, and the overall treatment imbalance will be the sum of imbalances in all strata. At any given timepoint for a two-arm equal allocation trial, treatment imbalance in each stratum is an independent random variable with the expected value of zero and the standard deviation specified by equation (29) or (33). Therefore, if a covariate margin includes m strata, and the entire study has s strata, the standard deviation of treatment imbalance within that margin and across the entire study will be $\sqrt{m}$ and $\sqrt{s}$ times the standard deviation of the treatment imbalance for the randomization algorithm respectively. For example, for a two-arm equal allocation trial stratified by clinical sites (10 categories) and sex (2 categories), PBD(δ=2) has $S{D}_{total}=4.1703$, $CG=0.7803$ and $DA=33.33\%$. Under the same trial setting, EUD(δ=2) has $S{D}_{total}=4.4721$, $CG=0.6875$ and $DA=12.50\%$ BUD(δ=3) has $S{D}_{total}=6.1836$, $CG=0.6235$ and $DA=5.88\%$. In general, the overall treatment imbalance in stratified randomization is amplified by the square root of the total number of strata. Tightened imbalance control within each stratum will be required if overall imbalance control is important. PBD(δ=2) can be the risk of selection bias associated with allocation randomness is low. When the number of stratifying factors is not small and the total number of strata is large, the minimal sufficient balance is a better alternative to the stratified randomization.²⁷ It uses completely random assignment by default and biased coin assignments to controls imbalances in covariate margins as well as the entire study when these imbalances are exceed their pre-specified thresholds.

5.6. Formal Definition of MTI Procedures

The team “maximum tolerated imbalance” (MTI) was introduced by Berger et al. in 2003 when proposing the maximal procedure for two-arm unequal allocation trilas.¹⁷ The team “MTI procedure” was initially used by Berger et al. when comparing several randomization designs with MTI restriction to the PBD in 2016.⁸ It is arguable whether or not to include PBD in the class of MTI procedure. Apparently, the concept of MTI procedure class was originally created to distinguish it from PBD. Therefore, it is assumed that the PBD is excluded from the class of MTI procedure. Nevertheless, the PBD does obey the restriction of MTI, like those randomization designs included in the class of MTI procedure. An additional condition is needed to define the MTI procedure to exclude PBD. For example, a randomization design can be considered as an MTI procedure if it uses deterministic assignment when and only when the treatment imbalance reaches the MTI threshold. This definition works for two-arm equal allocation trials only. It will not be generalizable to multi-arm or unequal allocation trials. For example, a multi-arm equal allocation trial could have more than one arm with the same smallest size. When treatment imbalance is defined as the difference between the largest and the smallest arms, the next subject will be randomly allocated to one of the smallest arms, not necessarily deterministically. For unequal allocation trials, not all randomization designs can achieve all treatment allocation nodes within the spaces defined by the target allocation and the MTI threshold. Therefore, deterministic assignments could be used when the MTI has not been reached yet. To avoid potential confusion, this manuscript replaces the term “MTI procedure” with “randomization designs with MTI restriction” and includes the PBD in this group.

6. CONCLUSION

For two-arm equal allocation trials, under the same MTI restriction, BSD, BUD, EUD, AMP, and BCDWIT provide a higher allocation randomness than the PBD. These five randomization designs are belter alternatives to PBD for unstratified trials. Among the five designs, BSD has the lowest correct guess probability and EUD has the lowest probability of deterministic assignment. BUD and AMP offer a good combination between the two measures of allocation randomness. BCDWIT with high biased coin probability is not encouraged due to its high correct guess probability. PBD has the tightest imbalance control measured by the standard deviation of treatment imbalance. When the risk of selection bias associated with allocation randomness is low, PBD with a block size of 4 can be considered, especially in stratified randomization. BUD and EUD are better alternatives to PBD with block size greater than 4. The availability of explicit formulas for conditional allocation probabilities of the six randomization designs discussed in this manuscript paves the road for a unified framework for randomization sequence generation as well as real-time on-demand randomization. The authors believe that the quality and credibility of clinical trials will be better protected by using real-time on-demand treatment assignment to eliminate concealment failure and by choosing optimal randomization designs with high allocation randomness to reduce the risk of selection bias.