The asymptotic maximal procedure for subject randomization in clinical trials

Abstract

The maximal procedure is a restricted randomization method that maximizes the number of feasible allocation sequences under the constraints of the maximum tolerated imbalance and the allocation sequence length. It assigns an equal probability to all feasible sequences. However, its implementation is not easy due to the lack of the Markovian property of the conditional allocation probabilities. In this paper, we propose the asymptotic maximal procedure, which replaces the sequence-length-dependent conditional allocation probabilities with their asymptotic values. The new randomization procedure is compared with the original maximal procedure and few other randomization procedures with the maximum tolerated imbalance via simulations and is found to be a practical choice for future clinical trials.

Keywords

Allocation predictability treatment imbalance asymptotic maximal procedure randomized controlled trial subject randomization

1 Introduction

The maximal procedure is a randomization method proposed by Berger et al. in 2003, aiming to minimize the treatment allocation predictability while retaining the maximum tolerated imbalance (MTI).¹ Like the big stick design,² Chen’s biased coin design with imbalance tolerance³ and the block urn design,⁴ the maximal procedure has been considered as a better alternative to the commonly used permuted block randomization due to the improvement in the allocation randomness measured by the proportion of deterministic assignments and the correct guess probability.^1,4–7 A treatment assignment is deterministic when the conditional allocation probability equals to 1.0 for one treatment arm and zero for the other arm(s). The correct guess probability is defined based on the Blackwell–Hodges optimal guessing strategy,⁸ which equals to the conditional allocation probability for the least assigned treatment arm with regard to the target allocation. Among the aforementioned restricted randomization designs, the permuted block randomization has the highest proportion of deterministic assignments and the highest correct guess probability and has been considered as the most vulnerable design for selection bias.^5–7,9,10 For a two-arm balanced trial using a block size of 4, 6, or 8, the proportion of deterministic assignment is 33, 25, or 20%, respectively.¹¹ Some investigators advocate varying block sizes for the putative benefit in allocation randomness.¹² Quantitative analyses indicate that such benefit does not exist.^7,13 In fact, with the maximal imbalance determined by the largest block size, mixing with smaller blocks only increases the proportion of deterministic assignments and the correct guess probability. All MTI procedures use deterministic assignments when the MTI is reached. The difference is in how they respond to imbalances smaller than the MTI. The big stick design takes no action. Chen’s procedure uses a prespecified biased coin probability, such as 0.7, favoring reducing the imbalance. The block urn design and the maximal procedure use an imbalance-adaptive biased coin probability favoring reducing the imbalance. Both the big stick design and Chen’s procedure were defined for two-arm balanced trials only.^2,3 The maximal procedure and the block urn design are generally applicable to two or multi-arm trials with balanced or unbalanced allocations.^1,4 When both the proportion of deterministic assignments and the correct guess probability are taken into account, the maximal procedure and the block urn design offer better properties than the big stick design and Chen’s procedure.^4,5

Unlike the block urn design, which uses an urn model to generate the conditional allocation probability, the maximal procedure obtains the conditional allocation probability by placing a uniform distribution on all feasible allocation sequences. This property is desirable when an assumption-free randomization test for the equality of the treatments is used. In this case, the test statistic is computed for all feasible sequences and compared to the observed one to obtain the p-value.¹³ Theoretically speaking, for equal allocation trials, a precise randomization test requires each pair of mirroring sequences, such as ABAABAB and BABBABA, which have the same probability being selected. While this condition could be difficult to verify for other randomization sequences, it has been ensured under the maximal procedure, because all feasible allocation sequences will have the same chance being selected. In addition, for unequal allocation trials, the maximal procedure has more feasible allocation sequences than the block urn design does in most cases.⁴ More feasible allocation sequences result in lower allocation predictability and makes the randomization test to be more powerful.¹ This uniform distribution on all feasible sequences comes with a price, however. It eliminates the Markovian property from the maximal procedure,¹ meaning that the conditional allocation probabilities depend not only on the current imbalance level but also on the sequence length.^14,15 In 2008, Salama et al.¹⁶ proposed a two-step process for the generation of the maximal procedure allocation sequences. However, due to the lack of the Markovian property, an explicit formula for the calculation of the conditional allocation probability remains missing, thereby causing difficulties in the implementation of the maximal procedure. Recently, Berger et al.¹⁴ proposed a modification on the maximal procedure by dropping the terminal balance requirement. However, the details of the algorithm for the calculation of the conditional allocation probability have not been revealed and a computer application for the generation of the allocation sequence for this modified maximal procedure is under development.¹⁴ In this paper, the asymptotic maximal procedure is proposed by replacing the sequence-length-dependent conditional allocation probabilities with their asymptotic values. The new randomization procedure is then compared to both the original maximal procedure and other MTI procedures based on the proportion of deterministic assignments and the correct guess probability. In Section 2, the performance of the maximal procedure is explored with the focus on the impact of the sequence length on the conditional allocation probability and the overall allocation randomness of the maximal procedure. In Section 3, the asymptotic maximal procedure is defined and its conditional allocation probabilities are provided. Section 4 presents the simulation results comparing the asymptotic maximal procedure with other MTI procedures, followed by discussions in Section 5.

2 Background

The maximal procedure assigns an equal probability to all feasible allocation sequences under the constraints of the target allocation ratio, the sequence length, the MTI, and perfect terminal balance. While the maximal procedure’s statistical properties are desirable, they do not directly lead to a generalizable implementation method. Salama et al.¹⁶ provided an algorithm for the construction of maximal procedure allocation sequences using a graphical approach. Shown in Figure 1 is an example for a two-arm trial with a balanced allocation, a final sequence length of 10, and $MTI = 3$ . A circle represents a feasible allocation node. The two numbers in the upper portion of the circle represent the numbers of subjects randomized to treatment arms A and B, respectively, prior to that node. The number in the lower portion of the circle represents the number of unique feasible allocation sequences from that node to the terminal point. It equals the sum of sequence numbers in the right-connected nodes. The edge between two nodes represents a feasible treatment assignment. Upper-right edges and lower-right edges represent assignments to treatment A and B, respectively. According to maximal procedure’s equal probability principle, the conditional allocation probability on each edge connecting the current node and the next node is proportional to the number of allocation sequences from the current node to the terminal node passing through the next node.

Figure 1 indicates that both the number of sequences from a node to the terminal point and the conditional allocation probability from the current node to the next node are affected not only by the imbalance level of the current node but also the distance between the current node and the terminal point. For example, nodes (2,1), (3,2), and (4,3) all have an imbalance level of 1; the conditional allocation probabilities for assigning the current subject to arm A at the three nodes are 7/17, 2/5, and 1/3, respectively. In practice, the sequence length must be prespecified in order to calculate the conditional allocation probability and to generate the maximal procedure randomization sequence. The sequence length represents the number of subjects to be randomized in the trial or the stratum. Restricted randomization is often stratified by some important baseline covariates, such as site, baseline disease severity category, and age group. It is well known that in a sequential clinical trial, investigators cannot predict the baseline stratification information for the next patient. Therefore, the actual size of a stratum remains unknown until the end of the trial. Possible solutions for this problem are combining several short allocation sequences together or using a sufficiently long sequence for each stratum. Both ways cannot guarantee that the stratum size at the end of the study matches the planned sequence length. In other words, the perfectly balanced terminal point is unlikely to achieve in each stratum.

The sequence length not only affects the calculation of the conditional allocation probability but also affects the performance of the maximal procedure. Table 1 demonstrates the generation of the maximal procedure allocation sequence with

MTI = 3

and a sequence length of 10. For this specific sample sequence, the proportion of deterministic assignments is 30%, and the correct guess probability is 60%. Among the three deterministic assignments, one is required to enforce the MTI, and the other two are needed for the terminal balance.

Table 1.

Maximal procedure allocation sequence example (two-arm trial with a balanced allocation, a sequence length of 10, and MTI of 3).

Sequence number	Current node	Next node option 1	Next node option 2	Sequences through next node opt. 1	Sequences through next node opt. 2	Random number	Conditional allocation probability	Treatment assignment
1	(0,0)	(1,0)	(0,1)	116	116	0.710929	0.5	B
2	(0,1)	(1,1)	(0,2)	68	48	0.901431	0.586207	B
3	(0,2)	(1,2)	(0,3)	34	14	0.871400	0.708333	B
4	(0,3)	(1,3)	NA	14	0	0.154093	1	A ^a
5	(1,3)	(2,3)	(1,4)	10	6	0.339842	0.714286	A
6	(2,3)	(3,3)	(2,4)	6	4	0.361661	0.6	A
7	(3,3)	(4,3)	(3,4)	3	3	0.429304	0.5	A
8	(4,3)	(5,3)	(4,4)	1	2	0.277114	0.333333	A
9	(5,3)	NA	(5,4)	0	1	0.217608	0	B ^a
10	(5,4)	NA	(5,5)	0	1	0.185001	0	B ^a

MTI: maximum tolerated imbalance.

(n_A,n_B): allocation node with n_A and n_B subjects previously enrolled in arms A and B, respectively.

Deterministic assignment.

This example does not reflect the superior allocation randomness of the maximal procedure. It is clear that the values of the 10 random numbers played a role. It is possible that with another set of random numbers, one can obtain an allocation sequence with only one deterministic assignment, which occurs at the end of the sequence. It is also clear that the expected number of deterministic assignments for adhering to the MTI condition is proportional to the sequence length, while those needed for the terminal balance is independent of the sequence length. Therefore, as the sequence length increases, the expected proportion of deterministic assignments should decrease. To quantitatively evaluate the impact of the sequence length on the allocation randomness of the maximal procedure, a simulation study is conducted with the results shown in Figure 2.

Figure 2 indicates that both the proportion of deterministic assignments and the correct guess probability decrease as the allocation sequence length increases. The superior allocation randomness of the maximal procedure is available only when the allocation sequence length is sufficiently large. When $MTI = 3$ is used to a sequence of six assignments, the maximal procedure behaves almost the same as the permuted block randomization. The proportion of deterministic assignments for the maximal procedure drops from 25 to 8% when the sequence length increases from 6 to 100.

Knowing that the constraint of the terminal balance is unnecessary due to the uncertainty of the actual stratum size in sequence trials, and that the allocation randomness improves as the sequence length increases, it is reasonable to use maximal procedure with an infinity sequences length for all strata, disregards the actual sample/stratum size at the end of the study. This leads to the concept of the asymptotic maximal procedure.

3 Method

3.1 Notations

As depicted in Figure 1, let b denote the sequence length, $n_{A} and n_{B}$ represent the number of subjects assigned to arms A and B, respectively, prior to node $(n_{A}, n_{B})$ . Let $k = n_{A} - n_{B}$ be the treatment imbalance level. Define $j \geq 0$ as the allocation wave number; which equals the total number of nodes with the same imbalance level as the current node prior to the terminal point. For example in Figure 1, node (3,2) has imbalance level $k = 1$ . Between node (3,2) and terminal point (5,5), there are two nodes, i.e. (4,3) and (5,4), with the same imbalance level of 1. Therefore, node (3,2) has $j = 2$ . The group of all allocation nodes with the same j value is considered as an allocation wave. A complete allocation wave always consists of $(2 MTI + 1)$ nodes, one for each tolerated imbalance level k. For example, wave 1 consists of nodes (1,4), (2,4), (3,4), (4,4), (4,3), (4,2), and (4,1). For two-arm balanced allocation trials, the allocation wave number j equals $min (b / 2 - n_{A}, b / 2 - n_{B})$ . The combination of the allocation wave number j and the imbalance level k uniquely identifies an allocation node. For this reason, an allocation node can be interchangeably represented by $(n_{A}, n_{B})$ or ${node}_{jk}$ . For example, node (1,2) is equivalent to ${node}_{3, - 1}$ . The ${node}_{0, 0}$ represents the balanced terminal node.

3.2 The feasible sequence number and the conditional allocation probability

Denote $s_{jk}$ as the number of feasible sequences from ${node}_{jk}$ to the terminal ${node}_{0, 0}$ . Let $p_{jkA}$ and $p_{jkB}$ be the conditional allocation probabilities at ${node}_{jk}$ for assigning the current subject to treatment A and B, respectively. For two-arm balanced trials, the number of feasible allocation sequences from ${node}_{jk}$ to the terminal ${node}_{0, 0}$ can be obtained from a recursive function of parameters j and k

\begin{matrix} s_{jk} = {\begin{matrix} 1 & j = 0; k \neq 0 \\ s_{j, 1 - MTI} & j > 0; k = - MTI \\ s_{j, k + 1} + s_{j - 1, k - 1} & j > 0; - MTI < k < 0 \\ s_{j - 1, - 1} + s_{j - 1, 1} & j > 0; k = 0 \\ s_{j, k - 1} + s_{j - 1, k + 1} & j > 0; 0 < k < MTI \\ s_{j, MTI - 1} & j > 0; k = MTI \end{matrix} \end{matrix}

(1)

Examples for the above six scenarios there are $s_{0, 2} = 1$ , $s_{1, - 3} = s_{1, - 2} = 4$ , $s_{2, - 1} = s_{2, 0} + s_{1, - 1} = 10$ , $s_{3, 0} = s_{2, 1} + s_{2, - 1} = 20$ , $s_{3, 1} = s_{2, 2} + s_{2, 0} = 34$ , and $s_{3, 3} = s_{3, 2} = 14$ . Based on the maximal procedure criteria, all feasible sequences are assigned with the same probability. Therefore, $p_{jkA}$ and $p_{jkB}$ are proportional to the number of sequences passing through the current node $(n_{A}, n_{B})$ to the next node $(n_{A} + 1, n_{B})$ or $(n_{A}, n_{B} + 1)$ , respectively. For two-arm balanced allocation trials, we obtain these conditional allocation probabilities

\begin{matrix} p_{jkA} = {\begin{matrix} 1 & (j = 0 and k < 0) or (k = - MTI) \\ \frac{s_{j, k + 1}}{s_{j, k + 1} + s_{j - 1, k - 1}} & (j > 0) and (- MTI < k < 0) \\ \frac{s_{j - 1, 1}}{s_{j - 1, - 1} + s_{j - 1, 1}} = \frac{1}{2} & k = 0 \\ \frac{s_{j - 1, k + 1}}{s_{j, k - 1} + s_{j - 1, k + 1}} & (j > 0) and (0 < k < MTI) \\ 0 & (j = 0 and k > 0) or (k = MTI) \end{matrix} \end{matrix}

(2)

Table 2 shows the results for two-arm trials with balanced allocations and MTI of 3 and 4, respectively. Due to the symmetric nature for balanced allocations, there are

s_{jk} = s_{j, - k}

and

p_{jkA} = p_{j, - k, B} = 1 - p_{j, - k, A}

for

0 < k \leq MTI

. Therefore, only results for nonnegative k values are listed in Table 2. It is clear that the conditional allocation probability

p_{jkA}

changes as the allocation wave changes. It is important to indicate that

s_{jk}

and

p_{jkA}

data listed in Table 2 are not affected by the allocation sequence length, because they are fully determined backwards from the terminal nodes. As long as the sequence includes

{node}_{jk}

, corresponding

s_{jk}

and

p_{jkA}

remain the same.

Table 2.

Number of feasible sequences and conditional allocation probabilities for maximal procedure in two-arm equal allocation trials.

		$s_{jk}$					$p_{jkA}$
MTI	j	k = 0	k = 1	k = 2	k = 3		k = 0	k = 1	k = 2	k = 3
3	0		1	1	1			0	0	0
	1	2	3	4	4		0.5	0.3333	0.2500	0
	2	6	10	14	14		0.5	0.4000	0.2857	0
	3	20	34	48	48		0.5	0.4118	0.2917	0
	4	68	116	164	164		0.5	0.4138	0.2927	0
	5	232	396	560	560		0.5	0.4141	0.2929	0
	6	792	1,352	1,912	1,912		0.5	0.4142	0.2929	0
	7	2,704	4,616	6,528	6,528		0.5	0.4142	0.2929	0
	8	9,232	15,760	22,288	22,288		0.5	0.4142	0.2929	0
MTI	j	k = 0	k = 1	k = 2	k = 3	k = 4	k = 0	k = 1	k = 2	k = 3	k = 4
4	0		1	1	1	1		0	0	0	0
	1	2	3	4	5	5	0.5	0.3333	0.2500	0.2000	0
	2	6	10	15	20	20	0.5	0.4000	0.3333	0.2500	0
	3	20	35	55	75	75	0.5	0.4286	0.3636	0.2667	0
	4	70	125	200	275	275	0.5	0.4400	0.3750	0.2727	0
	5	250	450	725	1000	1000	0.5	0.4444	0.3793	0.2750	0
	6	900	1,625	2,625	3,625	3,625	0.5	0.4462	0.3810	0.2759	0
	7	3,250	5,875	9,500	13,125	13,125	0.5	0.4468	0.3816	0.2762	0
	8	11,750	21,250	34,375	47,500	47,500	0.5	0.4471	0.3818	0.2763	0
	9	42,500	76,875	123,475	171,875	171,875	0.5	0.4472	0.3819	0.2764	0
	10	153,750	278,125	450,000	621,875	621,875	0.5	0.4472	0.3819	0.2764	0

MTI: maximum tolerated imbalance.

j = allocation wave number.

k = treatment imbalance level.

p_j,k = conditional allocation probability for assigning the next subject to treatment A.

s_j,k = number of feasible sequences to the balanced terminal node.

With minor modification, the strategy of using the allocation wave j and treatment imbalance k to calculate the conditional allocation probability for the maximal procedure applies to unequal allocations. Consider a two-arm trial with allocation ratio $r_{A} : r_{B}$ , with both r_A and r_B being positive integers. The allocation wave number j and imbalance level k are defined as follows

\begin{matrix} j = min {int [b / (r_{A} + r_{B}) - n_{A} / r_{A}], int [b / (r_{A} + r_{B}) - n_{B} / r_{B}]} \\ k = r_{B} n_{A} - r_{A} n_{B} \end{matrix}

Here function $int (x)$ returns the largest integer less than or equal to x. Figure 3 depicts the allocation waves for a trial with allocation $r_{A} : r_{B} = 3 : 2$ , sequence length $b = 20$ , and $MTI = 6$ . For example, there are 444 allocation sequences from ${node}_{2, 1}$ to terminal ${node}_{0.0}$ , 238 pass through ${node}_{2, 3}$ and 206 pass through ${node}_{2, - 2}$ when the current subject is assigned to arm A and B, respectively.

Based on the definition of feasible allocation sequence and Maximal procedure’s equal probability sequence feature, the conditional allocation probability assigning the current subject to arm A is proportional to the number of feasible allocation sequences from the current node to the immediate upper-right connected nodes. For example,

p_{2, 1, A} = 238 / 444 = 0.5360

. Table 3 lists the number of feasible allocation sequences and conditional allocation probabilities for five allocation waves.

Table 3.

Number of feasible sequences and conditional allocation probabilities for maximal procedure in two-arm trials with 3:2 allocation and MTI = 6.

	Wave j	Treatment imbalance level k = 2n_A–3n_B
	Wave j	–6	–5	–4	–3	–2	–1	0	1	2	3	4	5	6
$s_{jk}$	0	1	4	1	4	1	3	1	2	6	1	3	6	1
	1	14	58	14	58	14	44	10	30	90	16	46	90	16
	2	206	856	206	856	206	650	148	444	1,332	238	682	1,332	238
	3	3,044	12,652	3,044	12,652	3,044	9,608	2,188	6,564	19,692	3,520	10,084	19,692	3,520
	4	44,996	187,024	44,996	187,024	44,996	142,028	32,344	97,032	291,096	52,036	149,068	291,096	52,036
$p_{jkA}$	0	1	1	1	0.75	1	0.6667		0.5	0.5	0	0.3333	0	0
	1	1	1	1	0.7586	0.7143	0.6818	0.6	0.5333	0.5111	0.3750	0.3478	0	0
	2	1	1	1	0.7593	0.7184	0.6831	0.6081	0.5360	0.5120	0.3782	0.3490	0	0
	3	1	1	1	0.7594	0.7188	0.6832	0.6088	0.5363	0.5121	0.3784	0.3491	0	0
	4	1	1	1	0.7594	0.7188	0.6832	0.6088	0.5363	0.5121	0.3784	0.3491	0	0

MTI: maximum tolerated imbalance.

As shown in Table 3, when MTI is enforced in trials with unequal allocation, a deterministic assignment for a treatment arm is used when otherwise the imbalance will go beyond the MTI. For example, at ${node}_{1, 5}$ the imbalance has not reached the MTI yet. However, a treatment assignment to arm A will yield an imbalance of 7, which exceeds the MTI of 6. Therefore, the subject will be assigned to arm B deterministically. Similar scenarios occur at ${node}_{jk}$ when $j > 0 and k = - 4, - 5, or 5 .$

Figures 1 and 3 also indicate that the number of allocation nodes between two consecutive waves with the same imbalance level equals the minimal number of treatment assignments in a balanced set. For example, in Figure 1 with 1:1 allocation, two treatment assignments are made from node_j+1,k to node_jk. In Figure 3 with 2:3 allocation, it takes five treatment assignments.

3.3 Asymptotic conditional allocation probability

The entries listed in Tables 2 and 3 show that with a given allocation wave j, the conditional allocation probability $p_{jkA}$ varies as treatment imbalance level k changes. However, for a given imbalance level k, the value of $p_{jkA}$ stabilizes after a few allocation waves. When the sequence length is large enough, it is expected that $p_{jkA}$ is expected to converge to its asymptotic value, denoted as $p_{kA}$ , which is no longer dependent on the allocation wave number j. In fact, it is solely determined by the treatment imbalance level k and the MTI. This leads to the asymptotic maximal procedure, a modification of the original maximal procedure. Using these asymptotic $p_{kA}$ as the conditional allocation probability for all treatment assignments, we ensure that the resulting asymptotic maximal procedure will have the Markovian property, and that its implementation will become much more straightforward.

For two-arm balanced trials, it is always the case that $p_{j, 0, A} = 1 / 2$ , $p_{j, MTI, A} = 0$ , and $p_{j, - k, A} = 1 - p_{j, k, A}$ for $j > 0$ . Therefore, the focus is to find the asymptotic value of $p_{jkA}$ for $(0 < k < MTI)$ . As the conditional allocation probability converges, there is ${lim}_{j \to \infty} (p_{jkA} - p_{j + 1, kA}) = 0$ , i.e.

{lim}_{j \to \infty} (\frac{s_{j - 1, k + 1}}{s_{j, k - 1} + s_{j - 1, k + 1}} - \frac{s_{j, k + 1}}{s_{j + 1, k - 1} + s_{j, k + 1}}) = {lim}_{j \to \infty} (\frac{1}{\frac{s_{j, k - 1}}{s_{j - 1, k + 1}} + 1} - \frac{1}{\frac{s_{j + 1, k - 1}}{s_{j, k + 1}} + 1}) = 0

(3)

This leads to ${lim}_{j \to \infty} \frac{s_{j, k - 1}}{s_{j - 1, k + 1}} = {lim}_{j \to \infty} \frac{s_{j + 1, k - 1}}{s_{j, k + 1}}$ . When two fractions are equal, their numerators and denominators must be proportional to each other. Therefore, there is

{lim}_{j \to \infty} \frac{s_{j + 1, k - 1}}{s_{j, k - 1}} = {lim}_{j \to \infty} \frac{s_{j, k + 1}}{s_{j - 1, k + 1}} = ρ_{MTI} (0 < k < MTI)

(4)

In other words, the ratio between the feasible allocation sequence numbers for the two nodes in consecutive allocation waves with the same imbalance level converges to a constant $ρ_{MTI}$ , which is dependent on the MTI only. The value of $ρ_{MTI}$ can be obtained by implementing equation (4) into equations (1) and (2). For $MTI = 2$ , there are

s_{j, 1} = s_{j, 0} + s_{j - 1, 2} = s_{j - 1, - 1} + s_{j - 1, 1} + s_{j - 1, 1} = 3 s_{j - 1, 1} \to ρ_{2} = \frac{s_{j, 1}}{s_{j - 1, 1}} = 3

(5)

Similarly, for $MTI = 3$ , there are

\begin{matrix} {\begin{matrix} s_{j, 1} = 2 s_{j - 1, 1} + s_{j - 1, 2} \\ s_{j, 2} = s_{j, 1} + s_{j - 1, 2} \end{matrix} \to 4 s_{j, 1} = s_{j + 1, 1} + 2 s_{j - 1, 1} \end{matrix}

(6)

With j large enough, based on equation (4), there are $s_{j + 1, k} / s_{j, k} = ρ_{3}$ and

ρ_{3}^{2} - 4 ρ_{3} + 2 = 0 \to ρ_{3} = 2 + \sqrt{2} = 3.41421

(7)

Likewise, for $MTI = 4$ , there are

\begin{matrix} {\begin{matrix} s_{j, 1} = 2 s_{j - 1, 1} + s_{j - 1, 2} \\ s_{j, 2} = s_{j, 1} + s_{j - 1, 3} \\ s_{j, 3} = s_{j, 2} + s_{j - 1, 3} \end{matrix} \to {\begin{matrix} s_{j - 1, 2} = s_{j, 1} - 2 s_{j - 1, 1} \\ s_{j, 3} = 2 s_{j + 1, 1} - 5 s_{j, 1} \end{matrix} \to s_{j + 1, 1} - 5 s_{j, 1} + 5 s_{j - 1, 1} = 0 \end{matrix}

(8)

With j large enough, based on equation (4), there are $s_{j + 1, k} / s_{j, k} = ρ_{4}$ and

ρ_{4}^{2} - 5 ρ_{4} + 5 = 0 \to ρ_{4} = \frac{5 + \sqrt{5}}{2} = 3.61803

(9)

Asymptotic values of the conditional allocation probabilities can be calculated from equations (2) and equations (5) to (9) and $ρ_{MTI}$

\begin{matrix} p_{1 A} (MTI = 2) = {lim}_{j \to \infty} \frac{s_{j - 1, 2}}{s_{j, 0} + s_{j - 1, 2}} = {lim}_{j \to \infty} \frac{s_{j - 1, 1}}{2 s_{j - 1, 1} + s_{j - 1, 1}} = \frac{1}{3} \\ p_{1 A} (MTI = 3) = {lim}_{j \to \infty} \frac{s_{j - 1, 2}}{s_{j, 0} + s_{j - 1, 2}} = {lim}_{j \to \infty} \frac{s_{j - 1, 2}}{3 s_{j - 1, 2} - 2 s_{j - 2, 2}} = \frac{1}{3 - 2 / ρ_{3}} = 0.41421 \\ p_{2 A} (MTI = 3) = {lim}_{j \to \infty} \frac{s_{j - 1, 3}}{s_{j, 1} + s_{j - 1, 3}} = {lim}_{j \to \infty} \frac{s_{j - 1, 2}}{s_{j, 2}} = \frac{1}{ρ_{3}} = 0.29289 \\ p_{1 A} (MTI = 4) = {lim}_{j \to \infty} \frac{s_{j - 1, 2}}{s_{j, 0} + s_{j - 1, 2}} = {lim}_{j \to \infty} \frac{s_{j, 1} - 2 s_{j - 1, 1}}{s_{j, 1}} = \frac{ρ_{4} - 2}{ρ_{4}} = 0.44721 \\ p_{2 A} (MTI = 4) = {lim}_{j \to \infty} \frac{s_{j - 1, 3}}{s_{j, 1} + s_{j - 1, 3}} = {lim}_{j \to \infty} \frac{2 s_{j, 1} - 5 s_{j - 1, 1}}{3 s_{j, 1} - 5 s_{j - 1, 1}} = \frac{2 ρ_{4} - 5}{3 ρ_{4} - 5} = 0.38197 \\ p_{3 A} (MTI = 4) = {lim}_{j \to \infty} \frac{s_{j - 1, 4}}{s_{j, 2} + s_{j - 1, 4}} = {lim}_{j \to \infty} \frac{s_{j - 1, 3}}{s_{j, 3}} = \frac{1}{ρ_{4}} = 0.27639 \end{matrix}

For unbalanced allocations, the calculation of the analytical results of conditional allocation probabilities will be more complex. However, as shown in both Tables 2 and 3,

p_{jkA}

converges quickly after few allocation waves. Therefore, it is sufficient to use the value of

p_{jkA}

with large allocation wave number j as its asymptotic value. Table 4 provides the conditional allocation probabilities of the asymptotic maximal procedure for two-arm trials with commonly used unequal allocation ratios and MTI values. With the data in Table 4, the asymptotic maximal procedure can be easily implemented. Treatment allocation randomness for various allocation ratios and MTI values is evaluated with simulations. Based on the results of these simulations, for two-arm equal trials, we suggest that minimal MTI values of 3, 5, and 7 be used for allocations 1:1, 2:1, and 3:2, respectively.

Table 4.

Conditional allocation probability and allocation randomness for asymptotic maximal procedure.

Allocation		1:1 (k = n_A – n_B)				2:1 (k = n_A – 2n_B)				3:2 (k = 2n_A – 3n_B)
MTI		2	3	4	5	3	4	5	6	5	6	7	8
Treatment imbalance level k	−8												1
	−7											1	1
	−6								1		1	1	1
	−5				1			1	1	1	1	1	0.78878
	−4			1	0.73205		1	1	0.82843	1	1	0.77740	0.75252
	−3		1	0.72361	0.63397	1	1	0.81913	0.79289	1	0.75941	0.73905	0.73221
	−2	1	0.70711	0.61803	0.57735	1	0.80254	0.77920	0.73880	0.72871	0.71882	0.71366	0.67113
	−1	0.66667	0.58578	0.55279	0.53590	0.76759	0.75396	0.71663	0.70711	0.68614	0.68319	0.64690	0.63427
	0	0.5	0.5	0.5	0.5	0.69722	0.67368	0.67609	0.67157	0.62772	0.60884	0.59877	0.61348
	1	0.33333	0.41421	0.44721	0.46410	0.62925	0.61125	0.62670	0.63870	0.54257	0.53628	0.57568	0.56609
	2	0	0.29289	0.38197	0.42265	0.41080	0.52049	0.57313	0.6	0.40693	0.51209	0.50478	0.53868
	3		0	0.27639	0.36603	0	0.37936	0.49644	0.55228	0.38564	0.37843	0.47397	0.48558
	4			0	0.26795		0	0.36433	0.48223	0	0.34908	0.36535	0.45681
	5				0			0	0.35579	0	0	0.33545	0.35009
	6								0		0	0	0.32686
	7											0	0
	8												0
Deterministic assignments^a		16.34%	7.16%	3.61%	2.08%	21.50%	11.94%	7.11%	4.57%	23.85%	16.10%	11.33%	8.30%
Correct guess probability^b		66.49%	62.29%	59.75%	58.04%	68.85%	66.69%	63.72%	61.77%	69.80%	67.32%	65.04%	63.38%

MTI: maximum tolerated imbalance.

Allocation randomness are evaluated with sample size = 100, simulation = 50,000 per scenario.

Conditional allocation probability p_A equals 0 or 1.

Guess the next assignment being the least represented arm based on the target allocation.

4 Results

With the treatment imbalance contained by the MTI, the comparison of different randomization designs focuses on the allocation randomness. The proportion of deterministic assignments and the correct guess probability are two commonly used measures for allocation randomness. Figure 4(a) and (b) compares six randomization designs with 1:1 allocation, $MTI = 3$ (equivalent to block size 6 in permuted block randomization), and sample size varying from 10 to 100. The permuted block randomization has both the highest proportion of deterministic assignments and the highest correct guess probability; when the sample size is a multiple of the block size, the allocation randomness is worsened due to the enforced perfect balance at the end of each block. Chen’s procedure uses a fixed biased coin probability to manage two competing demands. A higher biased coin probability will decrease the proportion of deterministic assignments at the cost of a higher correct guess probability, and a lower biased coin probability can reduce the correct guess probability at the cost of a higher proportion of deterministic assignments. When the biased coin probability is reduced to 0.5, Chen’s procedure is equivalent to the big stick design, which has the lowest correct guess probability and a medium proportion of deterministic assignments. The maximal procedure performs better when the sample size is larger. Compared to the block urn design, the asymptotic maximal procedure has a slightly lower correct guess probability and a slightly higher proportion of deterministic assignments. Both perform better overall than the big stick design and Chen’s procedure.

Figures 4(c) and (d) show the simulation results for unequal allocation 2:1 with MTI of 6 (equivalent to block size of 9 in permuted block randomization) and sample size varying from 9 to 99. Once again, the permuted block randomization has the worst performance. The maximal procedure has good performance when the sample size is larger than 40. The block urn design and the asymptotic maximal procedure have the same low proportion of deterministic assignments. The asymptotic maximal procedure has a lower correct guess probability. The big stick design and Chen’s procedure are not included in this comparison as both were not defined for unequal allocations.

5 Discussion

Restricted randomization designs are applied in clinical trials to consistently control the treatment imbalances for the purpose of preventing potential chronological biases. Behind any gain in treatment balance there is a cost in allocation randomness. However, for the same amount of gain in treatment balance defined by the MTI, different randomization designs pay different costs in the allocation randomness.

The obvious inferiority of the permuted block randomization is the consequence of the enforced balance at the end of each block. Under the control of the MTI throughout the study, the periodical perfect block end balance does not add gains in treatment balance, but does add deterministic assignments, making the permuted block randomization the most vulnerable design to selection bias. We noticed that easy-to-use is often the only reason for using the permuted block randomization in clinical trial practice. In reality, the big stick design and Chen’s procedure are even easier than the permuted block randomization. With several better and easy-to-use alternatives available, such as the asymptotic maximal procedure and the block urn design shown in Figure 4, investigators should stop using the permuted block randomization in order to better prevent selection bias and protect the integrity of the trial.^10,14,15,17 The fact that the permuted block randomization remains the most commonly used method in clinical trial practice calls for more efforts to disclose its inferiority and its potential damages, and to introduce better alternatives to investigators. The asymptotic maximal procedure significantly improves the performance of the original maximal procedure. It reduces the proportion of deterministic assignments by releasing the enforced terminal balance, and more importantly, provides the Markovian property for easy implementation.

The proportion of deterministic assignments and the correct guess probability are two related and competing measures of allocation randomness for all MTI procedures and are affected by the conditional allocation probability used when the imbalance is within the MTI boundaries. In general, a high biased coin probability (i.e. close to 1.0) increases the correct guess probability and meanwhile makes the imbalance more likely being reduced before achieving the MTI, and therefore, is less likely to trigger deterministic assignments. On the other hand, a small biased coin probability (i.e. close to 0.5) can reduce the correct guess probability, and at the same time, increase the chance for calling deterministic assignments to contain the imbalance. The big stick design uses complete randomization when imbalance is within the MTI. Therefore, it has the lowest correct guess probability and relatively higher proportion of deterministic assignments. Chen’s procedure uses a fixed biased coin probability for all treatment imbalances within the MTI. When MTI is small, there are few discrete levels of imbalance between zero and MTI, so a properly selected biased coin probability yields good performances. Simulation study suggests that $p_{bc} = 0.65$ be used. When MTI is large, a fixed biased coin probability will be too small when the imbalance is close to MTI and too large when the imbalance is close to zero. Both the block urn design and the asymptotic maximal procedure use a biased coin probability adapted to the value of the imbalance. For example, with 1:1 allocation and $MTI = 3$ , the asymptotic maximal procedure uses biased coin probability of 0.586 or 0.707, respectively, when imbalance is 1 or 2. In the same scenario, the block urn design uses $3 / 5 = 0.6$ and $3 / 4 = 0.75$ as the biased coin probability. The difference between these respective biased coin probabilities used by the asymptotic maximal procedure and the block urn design explains the performance differences between the two designs shown in Figure 4. When applied to unequal allocations, the asymptotic maximal procedure offers more options for MTI than the block urn design does. For example, for a 3:2 allocation, the block urn design must have a block size as a multiple of 5. As the treatment imbalance is defined as $2 n_{A} - 3 n_{B}$ , the MTI has to be a multiple of 6. As shown in Table 4, the asymptotic maximal procedure can have any positive integers greater than 4 as MTI. This feature makes the asymptotic maximal procedure more flexible to meet the allocation randomness needs of the actual trial.

The equal probability for all feasible allocation sequences is a desirable property of the maximal procedure, especially when a randomization test is considered. This property is inherited by the asymptotic maximal procedure. For example, consider an asymptotic maximal procedure for a two-arm balanced allocation trial with a sequence length $n = 6$ . Under the restriction of $MTI = 3$ , there are three possible terminal nodes, i.e. (4,2), (3,3), and (2,4). As shown in Figure 1, there are 15, 20, and 15 feasible sequences from the start node (0,0) to these three possible terminal nodes, respectively. Based on the conditional allocation probability provided in Table 4, all feasible sequences from (0,0) to these three terminal nodes have the same probability of 0.01777, 0.02513, and 0.01777, respectively. Consider a two-arm trial with unequal allocation 3:2 and $MTI = 6$ . Let the allocation sequence length $n = 6$ . Possible terminal nodes after six assignments are (4,2) and (3,3). Based on Table 4, each of the 14 sequences landed at (4,2) has the same probability of 0.04119 being selected. Each of the 16 sequences landed at (3,3) has the same probability of 0.02646 being selected. In other words, given the observed terminal node, which is not necessarily balanced, all feasible allocation sequences landed on this node will have the same probability being selected by the asymptotic maximal procedure. Further works are needed to provide the more rigorous analytical proof of the equal sequence probability.

The asymptotic maximal procedure is not perfect. First, it cannot accurately target unequal allocations involving irrational numbers, such as $\sqrt{2} : 1$ .¹⁷ Second, it does not preserve the unconditional allocation probability for each treatment assignment, which is a desirable statistical feature for a randomization design.¹⁸ Our simulation study indicates that for 2:1 allocation with $MTI \geq 5$ , the unconditional allocation probability in the asymptotic maximal procedure varies within the range of $(0.663, 0.677)$ . For 3:2 allocation with $MTI \geq 7$ , the unconditional allocation probability varies within the range of $(0.573, 0.628)$ . In practice, the occurrence of any selection bias due to these small differences in the unconditional allocation probability is unlikely. However, a small MTI will have unconditional allocation probability drifts that are not negligible, and therefore, should not be used. For trials with balanced allocations, the asymptotic maximal procedure preserves the unconditional allocation probability, due to the symmetric nature between the treatment arms.

With the superior allocation randomness, the Markovian property and the conditional allocation probability provided in Table 4 allowing easy implementation, and the desirable equal allocation sequence probability, the asymptotic maximal procedure is an attractive randomization design.

Footnotes

Acknowledgement

The authors thank the anonymous reviewers for their careful review and great comments for this manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Wenle Zhao’s research is partly supported by the NIH/NINDS grants U01NS0059041 (NETT), and U01NS087748 (StrokeNet). Zhenning Yu’s research is supported by the NIH/NINDS grant U01NS087748 (StrokeNet).

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