Wide brick tunnel randomization - an unequal allocation procedure that limits the imbalance in treatment totals

Research Article

Received 21 May 2013, Accepted 3 November 2013 Published online 3 December 2013 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/sim.6051

Wide brick tunnel randomization – anunequal allocation procedure that limitsthe imbalance in treatment totalsOlga M. Kuznetsovaa*† and Yevgen Tymofyeyevb

In open-label studies, partial predictability of permuted block randomization provides potential for selectionbias. To lessen the selection bias in two-arm studies with equal allocation, a number of allocation proceduresthat limit the imbalance in treatment totals at a pre-specified level but do not require the exact balance at theends of the blocks were developed. In studies with unequal allocation, however, the task of designing a random-ization procedure that sets a pre-specified limit on imbalance in group totals is not resolved. Existing allocationprocedures either do not preserve the allocation ratio at every allocation or do not include all allocation sequencesthat comply with the pre-specified imbalance threshold. Kuznetsova and Tymofyeyev described the brick tunnelrandomization for studies with unequal allocation that preserves the allocation ratio at every step and, in thetwo-arm case, includes all sequences that satisfy the smallest possible imbalance threshold. This article intro-duces wide brick tunnel randomization for studies with unequal allocation that allows all allocation sequenceswith imbalance not exceeding any pre-specified threshold while preserving the allocation ratio at every step.In open-label studies, allowing a larger imbalance in treatment totals lowers selection bias because of the pre-dictability of treatment assignments. The applications of the technique in two-arm and multi-arm open-labelstudies with unequal allocation are described. Copyright © 2013 John Wiley & Sons, Ltd.

Keywords: unequal allocation; open-label studies; brick tunnel randomization; imbalance in treatment totals;wide brick tunnel

1. Introduction

In clinical trials, patients are commonly assigned to one of the studied treatment regimens using ran-domization [1]. Randomization is employed to reduce bias and make the treatment groups more similarwith respect to baseline covariates. Permuted block (PB) randomization [2] became a standard alloca-tion procedure due to its simplicity and wide-spread availability, flexibility to support equal and unequalallocation to two or more treatment arms, good control of the imbalance in the treatment group sizesthroughout the enrollment (at least when the block size is small), and easy adaptation for multi-centertrials and stratified randomization.

However, in open-label studies with PB randomization, the investigator who knows the sequence ofall previous allocations (as is the case in single-center studies or studies with randomization stratified bycenter) can sometimes deduce the next treatment assignment. This predictability is caused by the prop-erty of PB randomization to reach the targeted allocation ratio at the end of each block. Predictability ofupcoming assignments, in turn, might lead to a selection bias and thus, biased study results.

To lessen the selection bias in open-label studies, randomization procedures other than PB can beemployed. Complete randomization [1], where each patient is assigned independently to one of the treat-ment arms in a pre-specified allocation ratio, is absolutely unpredictable and thus, largely eliminates theselection bias. However, it can result in treatment groups being allocated in ratios very different from

aLate Development Statistics, Merck & Co., 126 E. Lincoln Avenue, PO box 2000, Rahway, NJ 07065-0900, U.S.A.bQuantitative Sciences, Janssen RD of Johnson & Johnson, TE2-3 1125 Trenton-Harbourton Road, Titusville, NJ 08560,U.S.A.

*Correspondence to: Olga M. Kuznetsova, Late Development Statistics, Merck & Co., 126 E. Lincoln Avenue, PO box 2000,Rahway, NJ 07065-0900, U.S.A.

†E-mail: [email protected]

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Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1514–1530

O. M. KUZNETSOVA AND Y. TYMOFYEYEV

the targeted one throughout the enrollment. It makes this procedure susceptible to an accidental biasassociated with the time trend; also, in small studies, the total numbers of patients enrolled in each groupcould quite differ from the target sizes, which can negatively impact power or other study design needs.

Biased coin (BC) randomization designed for two-group studies with 1:1 allocation [3] provides agood balance in treatment assignments throughout the enrollment at the price of some selection bias.However, it does not fully control the imbalance in treatment assignments, as there is a small probabilityof a relatively large imbalance in moderate size samples [4].

Before presenting existing allocation procedures designed to control the imbalance in treatmentassignments let us introduce some notation that will be used throughout the paper.

Consider a two-group study with C1 W C2 allocation to treatments A and B, where C1 and C2 arepositive integers that have no common divisors. Let us call S D C1CC2 the block size. For PB random-ization, S is the smallest block size that corresponds to the allocation ratio C1 W C2, but we will use theterm ‘block size’ for other allocation procedures as well.

In this paper, we will visualize an allocation sequence in a two-group study with C1 W C2 allocationto treatments A and B as a path along the integer grid in the two-dimensional space as described in [5].The horizontal axis represents allocation to treatment A, while the vertical axis represents allocation totreatment B. The allocation path starts at the origin and with each allocation moves either one unit tothe right (Treatment A assignment) or one unit up (Treatment B assignment). After i allocations, theallocation path ends up at the node with coordinates .NAi ; NBi /, where NAi is the number of treatmentA allocations within the first i allocations, and NBi is the number of treatment B allocations within thefirst i allocations. We will call the sequence of first i treatment assignments ‘i-allocation’ sequence.

The set of nodes .NAi ; NBi / ; i D 0; 1; 3; : : : ; that can be realized with given allocation procedureforms its allocation space. We will call the nodes .NAi ; NBi / that can be realized after i allocationsthe nodes of generation i . We will call the probability for an allocation sequence to reside in the node.NAi ; NBi / after i allocations the resident probability of the node. The sum of resident probabilitiesacross the nodes of the same generation is 1. The probability of Treatment A allocation from the node.NAi ; NBi / will be called transition probability from .NAi ; NBi / to .NAi C 1;NBi /. The probabilityof treatment B allocation from the node .NAi ; NBi / is the transition probability from .NAi ; NBi / to.NAi ; NBi C 1/. If only one of the transition probabilities is positive, the allocation from the node.NAi ; NBi / is deterministic.

We will call the line AR D .C1u;C2u/ ; u > 0, with the slope equal to the allocation ratio ‘the allo-cation ray’. The absolute imbalance in treatment assignments after i allocations is commonly definedas jNBi �NAi � C2=C1j (or proportional to this difference) [6, 7]. For equal allocation .C2 D C1/, theabsolute imbalance reduces to jNBi � NAi j. The absolute imbalance is low if the node .NAi ; NBi / isclose to the allocation ray. The absolute imbalance is 0, and the observed allocation ratio is equal to thetargeted one if the node .NAi ; NBi / lies exactly on the allocation ray. For example, for PB allocationwith block size S , the imbalance is 0 at the end of each block, that is for i DmS;mD 1; 2; 3 : : :

A number of allocation procedures developed for 1:1 allocation to two treatment arms can limit theimbalance in treatment assignments at a pre-specified level. Among these procedures are the replace-ment randomization [8], modified replacement randomization [9], maximal procedure [5], Soares andWu’s big stick design [10], Chen’s BC design with imbalance tolerance [11], Ehrenfest urn design [12],and Baldi Antognini and Giovagnolli’s [13] adjustable BC design (with limited allowed imbalance). Allthese procedures restrict the set of allowed allocation sequences to those for which the absolute imbal-ance in treatment assignments never exceeds pre-specified threshold b W jNBi � NAi j 6 b. Thus,the allocation space for these procedures is a strip of height b around the allocation ray .u; u/; u > 0.However, these procedures differ in how they assign the probabilities to the allowed sequences.

The replacement randomization [8] and the maximal procedure [5] assign equal probabilities to allallowed sequences, which determines the conditional probability of treatment A allocation to the (i+1)-th subject given the existing group totals NAi and NBi after i allocations. The big stick design [10]keeps conditional probability of treatment A allocation at 0.5 whenever the allocation to both treatmentsis permissible. Chen’s BC design with imbalance tolerance [11] follows the BC rule in assigning theunderrepresented treatment with probability p > 1=2 when allowed; however, if the imbalance thresh-old is reached, the underrepresented treatment is assigned with certainty. With Ehrenfest urn design [12],the allowed imbalance is limited to a pre-specified threshold, and the probability to assign an underrepre-sented treatment grows when the imbalance grows. Further flexibility is added to the procedure in BaldiAntognini and Giovagnolli’s [13] adjustable BC design where the probability of treatment A allocationis a general function of the imbalance in the treatment totals. This function can be defined in a way that


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limits or does not limit the imbalance in treatment totals; it can also make the allocation sequences tostay closer to the perfect balance in treatment assignments or be more spread within the allowed space.Due to symmetry of the equal allocation algorithms with respect to A and B treatment assignments, theallocation ratio is 0.5 for each allocation.

In studies with unequal allocation, however, the task of designing a randomization procedure thatallows all allocation sequences that comply with the pre-specified imbalance in group totals whilepreserving the allocation ratio at every step is not resolved.

Salama, Ivanova, and Quaqish [6] generalized the maximal procedure for unequal allocation to twotreatment groups in the following way. They allowed all allocation sequences for which the absoluteimbalance never exceeds a pre-specified threshold b:

jNBi �NAi �C2=C1j6 b: (1)

Requirement (1) means that the allocation space is the strip ˙ b in height around the allocation ray.Salama, Ivanova, and Quaqish assigned equal probabilities to all permissible sequences.

Kuznetsova and Tymofyeyev [14, 15] pointed out the following problem with this approach: havingequiprobable sequences leads to the variations in the allocation ratio from allocation to allocation. Forexample, in a study with 2:3 allocation to treatments A and B and b D 2, the patients allocated first, sec-ond, fourth, and fifth, will be allocated in the 3:5 allocation ratio, while the patient allocated third will beallocated in the 1:1 ratio [14,15]. Such variations in the allocation ratio from allocation to allocation areundesirable, as they provide a potential for selection and evaluation bias even in double-blind studies;they also provide a potential for accidental bias associated with a time trend, in particular, in multicen-ter studies [14–18]. Variations in the allocation ratio also lead to problems with the randomization test[16–21].

To eliminate variations in the allocation ratio in a study with unequal allocation while keeping allo-cation sequences close to the allocation ray, Kuznetsova and Tymofyeyev offered the brick tunnel (BT)randomization for studies with allocation to k > 2 treatment arms in C1 W C2 W : : : W Ck ratio [14, 15].The BT randomization allows all allocation sequences that, when depicted as paths on the k-dimensionalinteger grid, are constrained within the chain of k-dimensional unitary cubes pierced by the allocationray AR D .C1u; C2u; : : : ; Cku/ ; u > 0. The key property of the BT randomization is that the transi-tion probabilities are derived in a way that preserves the allocation ratio at every allocation [14, 15]. Anexample of the allocation space for 2:3 BT randomization to treatments A and B is provided in Figure 1(solid lines).

Figure 1. Example 1: Adding two nodes to the 2:3 brick tunnel randomization in the fifth generation using theswitch of the fifth and sixth treatment assignments. Solid lines – 2:3 brick tunnel allocation space; blue dashed

lines – newly added allocation segments; double line – allocation ray.

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In the two-group case, the BT randomization has the same set of allowed allocation sequences as theSalama, Ivanova, and Quaqish [6] randomization with b D bBT , where bBT is the height of the BT, buthas no variations in the allocation ratio. It can be shown that for C2 > C1; bBT D .C2 � 1/ =C1C 1.

Two-group BT randomization has two nodes .NAi ; NBi / in each generation i except at the endof the blocks .i DmS;mD 1; 2; 3 : : :/, where it has a single node with coordinates .mC1; mC2/(Figure 1). For one of the two nodes in generation i ¤ mS , the treatment assignment is deterministic,while allocations to both arms are possible from the other node [14, 15]. The allocation schedule for BTrandomization can be easily generated in SAS or another programming language.

The BT randomization sequences ensure that the observed allocation ratio remains very close to thetargeted one throughout the enrollment. This is very useful in studies where the allocation ratio leadsto a large block size and thus, PB randomization can lead to considerable deviations from the targetedallocation ratio within a block.

However, to reduce predictability of the next treatment assignment in a two-group open-label studywith unequal allocation, one might want to expand the allocation space to cover strip (1) around the allo-cation ray wider than the one with b D .C2 � 1/ =C1 C 1. The ways of doing that while preserving theallocation ratio at every allocation have not been previously described. Zhao and Weng [22] describedthe Block Urn allocation procedure whose allowed space consists of a sequence of overlapping PBs ofat least twice the minimal size with the lowest corner at .mC1; mC2/. When the block size is large, theallocation space is too wide for most applications. Block urn allocation preserves the allocation ratioat every step, but other than in 1 W C2 randomization, it does not cover the whole strip (1) around theallocation ray.

In this paper, we describe the randomization technique for studies with unequal allocation that allowsall allocation sequences that comply with pre-specified imbalance threshold while preserving the allo-cation ratio at every allocation. The technique works by adding new nodes to a two-treatment BTrandomization. We will call this procedure Wide BT (WBT) randomization. The sequences of the WBTrandomization could be made to stay closer to the allocation ray or to be spread more within the strip. AWBT allocation schedule can be easily generated in most programming languages, in particular, in SASor R.

In Section 2, we will demonstrate the switch technique that allows expanding the BT randomization toa wider set of allocation sequences while preserving the allocation ratio at every allocation. In Section 3,we will describe the iterative use of the switch technique that adds layers of bricks to the tunnel until theallowed space (1) is filled. We will lay out the details of the implementation of the method in Section 4.We will describe properties and applications of the technique, including those in multi-arm trials withequal or unequal allocation, in Section 5. Discussion completes the paper.

2. Adding nodes to the brick tunnel randomization space through theswitch technique

An easy way to add a new node to the allowed space of an allocation procedure is by switching twoconsecutive treatment assignments with certain probability. Kuznetsova [23] described an example ofsuch a switch to reduce predictability of the PB randomization at the end of the block. When the lasttreatment assignment in a PB is switched with the first treatment assignments of the next PB, new allo-cation sequences that do not result in targeted allocation ratio at the end of each block are added to theallocation space. Thus, the treatment assignments at the mS allocations (where S is the block size) willno longer be completely predictable.

If the original randomization procedure preserved the allocation ratio at every step, the expanded pro-cedure where allocations iand .i C 1/ are allowed to be switched with probability 0 < ı < 1, willalso preserve the allocation ratio. By varying the probability of a switch 0 < ı < 1, we can change theprobability for an allocation sequence to go through the newly added nodes.

Let us illustrate the switch technique with the example of adding two nodes at the end of a block ofa BT randomization. As Example 1, consider 2:3 BT randomization to treatments A and B .S D 5/,whose allocation space will be expanded to add two more fifth generation nodes. The allowed space forthe 10-allocation BT randomization is depicted by solid lines (Figure 1). For BT sequences, we will usethe following notation: we will denote the two nodes above and below the allocation ray in generationi ¤ mS by X i1 and X i�1, respectively. We will denote the single node that lies exactly on the allocationray in generation i DmS by X i0.


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Let us add two more nodes to the fifth generation – node X5�1 below AR with coordinates (3, 2)and node X51 above AR with coordinates (1, 4). This could be done by generating a BT randomizationsequence and then allowing the fifth and sixth treatment assignments to switch places with probability0 < ı < 1.

Indeed, the two-step allocation path segment that started in node X41 and had the fifth and sixth allo-cations to A and B, respectively, as allowed by the BT randomization, will become, after a switch of thefifth and sixth allocations, a BA segment. Thus, the new path (the dashed line going through the nodeX51 in Figure 1) will pass through the node X51 in the fifth generation. Similarly, the segment BA thatstarted at the node X4�1 will become, after a switch, AB segment and thus will pass through the nodeX5�1 in the fifth generation (the dashed line going through the node X5�1 in Figure 1).

The switch does not alter the resident probabilities in the sixth generation. Indeed, the new and alteredtwo-step sequences arrive at the same node in the sixth generation, while they split the probability of theoriginal two-step allocation sequence.

The new transition probabilities in the fifth and sixth generations as well as the new resident probabil-ities across the fifth generation nodes X5�1; X

50 , and X51 could be easily derived by listing all two-step

allocation segments (original and reversed) from the fourth generation nodes and their conditional proba-bilities (Table I). The original 2:3 BT resident probabilities are taken from [15]. Of note, for the allocationsequences with the same fifth and sixth treatment assignments, the original and reversed segments arethe same.

As can be seen from Table I, from the node X4�1 in generation 4, the allocation path will go to thenodeX5�1 (allocation to A) with probability 2=5�ı and to the nodeX50 (allocation to B) with probability1 � 2=5 � ı. Similarly, the transition probabilities from the node X41 to the nodes X51 (allocation to B)and X50 (allocation to A), are 3=5 � ı and 1 � 3=5 � ı, respectively. Thus, the resident probabilities inthe fifth generation nodes X5�1; X

51 , and X50 are

R5�1 DR51 D 6=25� ı;

R50 D 1� 12=25� ı

By varying the probability of the switch ı, the resident probabilities of the new nodes X5�1 and X51 canbe made as high as 6/25 (when ı D 1) or as low as 0 (when ı D 0).

The transition probabilities from the fifth to the sixth generation are the following. From nodeX5�1, the allocation path follows to X6�1 (allocation to B) with probability 1; from X51 , the allocationpath follow to X61 (allocation to A) with probability 1. From X50 , the probability of A allocation is.10� 6ı/ = .25� 12ı/, and the probability of B allocation is .15� 6ı/ = .25� 12ı/.

Table I. Example 1: Original and reversed two-step allocation segments from the nodes in the fourthgeneration to the nodes in the sixth generation.

Original Conditionalor reversed probability

Fourth two-step Two-step of the two-step Fifth Sixthgeneration Resident allocation allocation allocation generation generationnode probability segment segment segment node node

X4�1 3/5 original BA 2=5� .1� ı/ X50 X6�1

X4�1 3/5 reversed AB 2=5� ı X5�1 X6�1

X4�1 3/5 original BB 3=5� .1� ı/ X50 X61

X4�1 3/5 reversed BB 3=5� ı X50 X61

X41 2/5 original AA 2=5� .1� ı/ X50 X6�1

X41 2/5 reversed AA 2=5� ı X50 X6�1

X41 2/5 original AB 3=5� .1� ı/ X50 X61

X41 2/5 reversed BA 3=5� ı X51 X61

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3. Iterative expansion of the brick tunnel in layers of bricks

The BT randomization can be expanded to cover all the nodes within the strip˙ b in height surroundingthe allocation ray.

Consider the allocation space for the 2:3 BT randomization depicted in Figure 2 (pale blue solidcells). The allocation space includes the squares on unitary grid pierced by the allocation ray AR D.2u; 3u/; u > 0 (slanted double line). Within each block of five allocations, the BT consists of twocolumns, each of two bricks in height. Let us expand the BT by placing one brick on top of each columnand also attaching one brick at the bottom of each column (starting with the second column, as the allo-cation space is restricted to the positive quadrant of the grid). We will call newly added bricks (shadedwith red horizontal stripes) the layer 1 of bricks; we will call the original BT the layer 0. Each of thelayer 1 bricks shares two sides with the bricks of the original BT.

Together, the original BT and the first layer of bricks cover the nodes of the original BT, the BT shiftedone step up, and the BT shifted one step down (within the first quadrant). Because the original BT coversall nodes within the strip of height ˙ bBT around the allocation ray, where bBT D 2, the expanded setcovers all nodes within the strip of height ˙ .bBT C 1/ D ˙ 3 around the allocation ray. We will callthis space a 2:3 WBT of height ˙ 3 and denote it WBT2W3(3).

Similarly, the second layer of bricks can be added to the allocation space by placing one brick on topof each layer 1 brick and attaching one brick below each layer 1 brick (within the first quadrant of thegrid). The layer 2 bricks are shaded by green diagonal stripes in Figure 2. Together, the original BT andthe two layers of bricks cover all nodes within the strip ˙ .bBT C 2/ D ˙ 4 around the allocation ray,or WBT2W3(4) in our notation.

Adding the full third layer of bricks will cover all nodes within the strip ˙5 around the allocationray, that is, WBT2W3(5). However, it is possible to cover a strip wider than WBT2W3(4), but narrowerthan WBT2W3(5) by adding an incomplete third layer. For example, to cover a strip of height b D 4:5, athird layer brick is placed on top of the second column, but not the first column, to form WBT2W3(4.5)).Figure 2 presents WBT2W3(4.5) that contains two complete and the third incomplete layers of bricksadded to the 2:3 BT. Of note, WBT2W3(4.5) allocation space is wide enough to include the allocationspace for the PB randomization with block size 5, but is narrower than the allocation space for the PBrandomization with block size 10.

We will denote the nodes within a WBT as follows. Within the generation i , the nodes above theallocation ray will be numbered X i1; X

i2; X

i3; : : : , in the direction away from the allocation ray, while

Figure 2. Allocation space for WBT2W3.4:5/. Brick tunnel allocation space: shaded blue solid; layer 1: shadedwith red horizontal stripes, new nodes: red dots; layer 2: shaded with diagonal green stripes, new nodes: green

triangles; layer 3: shaded with brown checkers, new nodes: brown pentagons.


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the nodes below the allocation ray will be numbered X i�1; Xi�2; X

i�3; : : :, in the direction away from the

allocation ray.

3.1. Example 2: Generating wide brick tunnel sequences that end up in exactly targetedallocation ratio

Now let us explain how WBT randomization sequences can be generated through iterative switching ofthe allocations on the original BT randomization sequence. We will illustrate the process with Example 2of expanding the 10-allocation BT randomization to treatments A and B in 2:3 ratio within WBT2W3.4:5/depicted in Figure 3.

The first layer of bricks brings the new nodesX2�2; X32 ; X

5�1; X

51 ; X

7�2, andX82 (marked by red dots in

Figure 3) into the allowed space. The node X2�2 is added to the allowed space by switching the secondand third treatment assignments of a BT allocation sequence that starts with ABA. Of note, if any otherBT sequence has its second and third treatment assignments switched, the new sequence will remainwithin the original BT allocation space. Similarly, the first layer node X32 is added to the allowed spaceby switching the third and fourth treatment assignments of a BT allocation sequence that starts withBBA, and so on.

The second layer of bricks contributes nodes X4�2; X42 ; X

6�2, and X62 (marked by green triangles in

Figure 3). The third, incomplete, layer of brick contributes the two nodes: X3�2 and X72 (marked bybrown pentagons). If layer j contributes a new node in the i-th generation to the allocation space, thenode is added by switching the i-th and the (i+1)-th treatment assignments in the respective allocationsequence that belongs to the allocation space formed by the original BT and all earlier added layers (upto layer .j � 1/).

A 10-allocation WBT2W3(4.5) allocation sequence is generated in the following way. First, a10-allocation BT allocation sequence is generated as described in [15]. Then, the switches of consecutivepairs of treatments that correspond to addition of a new layer 1 node are executed with probability ı, inthe order of allocation. After that, the switches are executed for layer 2, and after that, for layer 3. Theorder of the switches is presented in Table II.

It should be noted that when the switches are applied to the 10-allocation BT sequence (Example 2),not all the nodes within the 10 generations inside the strip ˙ 4.5 in height surrounding the allocationray are added to the allowed space. While all the nodes in generations 1 through 7 within the strip are

Figure 3. Example 2: Expanding 10-allocation 2:3 brick tunnel randomization to WBT2W3.4:5/ by addinglayers of bricks. Brick tunnel allocation space: shaded blue solid; layer 1: shaded with red horizontal stripes,new nodes: red dots; layer 2: shaded with diagonal green stripes, new nodes: green triangles; layer 3: shaded with

brown checkers, new nodes: brown pentagons.

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Table II. Example 2: Treatment assignments that are switched in thethree added layers to generate a 10-allocation WBT2W3.4:5/ sequencethat ends in exactly 2:3 allocation ratio.

Treatment assignmentthat is switched with

the next one withOrder of execution Layer being added probability ı

1 1 22 1 33 1 54 1 75 1 86 2 47 2 68 3 39 3 7

WBT, wide brick tunnel.

included in the allowed space, there are nodes in generations 8, 9, and 10 that are left out. Indeed, byswitching the existing allocations, we only obtain the 10-allocation sequences that end up at the node(4, 6) and thus provide exactly the 2:3 allocation ratio at the end of the sequence.

3.2. Example 3: Generating WBT sequences that are not required to end up in exactly targetedallocation ratio

Reaching the exact allocation ratio at the end of the allocation sequence might be needed in some cases– for example, when a small sample at a study center needs to be split exactly in the targeted ratio.However, in other cases lowering predictability takes higher priority and it might be better to allow anallocation sequence to end up anywhere within the strip after 10 patients are randomized. To fill allthe nodes within the strip across the 10 generations, one should start with the BT sequence that hasmore than 10 allocations – for example, 15 allocations – and proceed with the switches. The resulting15-allocation sequences will fill all the nodes inside the strip ˙ 4.5 in height surrounding the allocationray across the first 10 generations. Table III presents the switches that could be executed within the threelayers of the WBT in this example (Example 3).

In the remainder of this paper, we will explore the approach where the allocation sequences can endat any node within the WBT and not necessarily result in the exact targeted allocation ratio.

Example 3 also illustrates the problem that can arise when there are two consecutive allocations inthe same layer j that are to be switched – the first one at the upper border of the allocation space andthe second one at the lower border of the allocation space. For some allocation sequences with segmentsthat go along the upper border of layer j , if both switches are executed, the new sequence falls outsidelayers 0 through .j C 1/.

Indeed, consider the 10-allocation sequence that follows the upper border of the second layer of bricks– the sequence BBBBA BABBA. Suppose, the two consecutive switches in the third layer – the switchof the seventh and eighth treatment assignments and the switch of the eighth and ninth treatment assign-ments are applied to this sequence. The switch of the seventh and eighth treatment assignments willturn this sequence into the sequence BBBBA BBABA that follows the upper border of the third layerof bricks. When the eighth and ninth treatment assignments of the new sequence are switched next, theresulting sequence BBBBA BBBAA exits the fourth layer and crosses the border of the allowed spaceabove the allocation ray.

Thus, whenever two consecutive switches can take the sequence outside the designated layer, we willallow only one of the switches to be executed. If the prespecified probability of a single switch ı > 0:5,we will execute one of the two switches chosen at random. Thus, the probability of either switch will be0.5. If ı < 0:5, we will choose one of the two switches at random with probability 0.5 and then executeit with probability 2ı. Thus, the probability of either switch will be ı.

Not every pair of consecutive switches will take the allocation path two layers above its original layer.In particular, a switch of the two allocations at the bottom border followed by the switch of the two


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Table III. Example 3: Treatment assignments that are switched in thethree added layers to generate a 10-allocation WBT2W3.4:5/ sequencethat is not required to end in exactly 2:3 allocation ratio.

Treatment assignmentthat can be

switched with thenext one with

Order of execution Layer being added probability ı

1 1 22 1 33 1 54 1 75 1 86 1 107 1 128 1 139 2 410 2 611 2 912 2 1113 3 314 3 715 3 816 3 12

WBT, wide brick tunnel.

allocations on the top border will carry no such danger. An example is provided by the two consecutiveswitches in the first layer in Example 3 of the second and third assignments (at the bottom border) andof the third and fourth assignments (at the top border). Thus, these two switches could be executedindependently, each with probability ı.

We will discuss how to determine when two consecutive switches in a row present a danger of takinga sequence out of its designated layer in the section on implementation details (Section 4).

4. Implementation details

This chapter lays out the steps required to generate a WBTC1WC2.b/ randomization sequence that allo-cates patients to treatments A and B in C1 W C2 ratio with the imbalance threshold b. The set of all suchsequences fills in the strip (1).

In the succeeding text, we will provide the derivations needed to generate an allocation sequence.Consider the allocation space for the C1 W C2 BT randomization pictured on a two-dimensional unitarygrid (shaded in solid blue in Figure 4). It can be visualized as the set of vertical columns of width 1,with C1 columns per block. The m-th column is located above the horizontal segment Œm� 1;m�, withits top at

Ytop.m/D ceil .m�C2=C1/ (2)

and its bottom at

Ybot.m/D f loor ..m� 1/�C2=C1/ : (3)

Here ceil.x/ is the smallest integer > x and floor.x/ is the highest integer 6 x. The number ofcomplete layers in WBTC1WC2.b/ is

LD f loor .b � bBT /D f loor .b � .C2 � 1/ =C1 � 1// : (4)

WBTC1WC2.b/ might or might not also include some nodes from the incomplete (L+1)-th layer, asLemma 1 describes.

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Figure 4. Illustration to Lemma 2 proof. Solid squares: first j layers of WBTC1WC2.b/. Dashed blue lines: layer.j C 1/ nodes added after the first switch in a pair of consecutive switches. Dotted red line: a switch of thetwo top border allocations followed by the switch of the two bottom border allocations generates a node above

layer .j C 1/.

Lemma 1The node

�m� 1; Ytop.m/CLC 1

�above the m-th column, where Ytop.m/ is defined by (2), belongs to

the .LC 1/-th layer of WBTC1WC2.b/ iff

Ytop.m/CLC 1� .m� 1/�C2=C1 6 b: (5)

The node .m; Ybot.m/�L� 1/ below the m-th column, where Ybot.m/ is defined by (3) and Ybot.m/ >LC 1, belongs to the .LC 1/-th layer of WBTC1WC2.b/ iff

m�C2=C1 � .Ybot.m/�L� 1/6 b: (6)

The proof is provided in Appendix A.When j layers are added to the WBT, the m-th column has its top at Yj;top.m/D Ytop.m/C j and its

bottom at Yj;bot.m/Dmax .Ybot.m/� j; 0/.The layer j node

�m;Yj;top.m/

�above the m-th column is at an inner corner of the WBT (Figure 4).

When allocation A on the top border of the WBT before this node is switched places with the allocationB after the node, the layer .j C 1/ node

�m� 1; Yj;top.m/C 1

�is added to the allocation space. This

node is added by switching the allocations

mC Ytop.m/C j and mC Ytop.m/C j C 1 (7)

of the WBT sequence.For m such that Ybot.m/ > j , the layer j node

�m� 1; Yj; bot.m/

�at the bottom of the m-th column is

at an inner corner node of the WBT. When the two allocations on both sides of this node on the bottomborder of the WBT (B and A) are switched places, the layer .j C 1/ node

�m;Yj;bot.m/� 1

�is added to

the allocation space. This node is added by switching the allocations

m� 1C Ybot.m/� j and mC Ybot.m/� j (8)

of the WBT sequence following the border segment.

Lemma 2Following the top border switch of the

�mC YtopC j

�-th and

�mC Ytop.m/C j C 1

�-th treatment

assignments with the bottom border switch of the�mC YtopC j C 1

�-th and

�mC Ytop.m/C j C 2

�-th


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treatment assignments of j layers of WBTC1WC2.b/ adds an allocation segment above layer .j C 1/, ifand only if the difference in height between the m-th and .mC 1/-th columns of the BT is at least two:

ceil ..mC 1/�C2=C1/� ceil .m�C2=C1/> 2: (9)

The pair of consecutive switches where the switch at the bottom border of the allocation space of jlayers of WBTC1WC2.b/ is followed by the switch at the top border of the allocation space will keep theallocation sequence within the layers 0 through .j C 1/ of the WBTC1WC2.b/.

The proof is provided in Appendix A.With all the essentials of the method derived previously, the WBTC1WC2.b/ sequence is generated

using the following steps:

(1) BT allocation sequence is generated per Kuznetsova & Tymofyeyev [15](2) The number of complete layers L is derived per (4).(3) The nodes in the last, incomplete, (L+1)-th layer, if any, are identified through (5) and (6)(4) Repeat for layers 1 through L+1:

(a) the nodes to be switched at the top border of the allocation space within the layer are identifiedthrough (7)

(b) separately, the nodes to be switched at the bottom border of the allocation space within thelayer are identified through (8)

(c) the allocations at which either top or bottom switches (or both) are possible are orderedsequentially

(d) all consecutive pairs of switches where the switch at the top border is followed by the switchat the low border and condition (9) is met are identified (‘bad pairs’ of switches); only oneswitch in such pair can be executed.

(e) Execute switches sequentially, where

(i) each switch that is not a part of a ‘bad pair’ is executed with probability ı(ii) for a ‘bad pair’ of switches, do the following

(1) if ı > 0:5, execute one of the two switches chosen at random with probability 0.5.(2) if ı < 0:5, choose one of the two switches at random with probability 0.5 and then

execute it with probability 2ı.

To add further flexibility to the method, probability of a switch ı can vary from layer to layer.When the probability of a switch increases in the outer layers, the resident probability in these nodesalso increases.

5. Properties and applications of the technique

In this section, we will describe the properties of the WBT randomization – its closeness to the alloca-tion ray and asymptotic behavior of its resident probabilities. We will also describe its applications intwo-arm and multi-arm studies.

5.1. Impact of the switch probability on the resident probabilities within a wide brick tunnel

We examined the impact of the switch probability ı on the resident probabilities of the WBT in Example3 settings through simulations.

The resident probabilities of 10-allocation WBT2W3.4:5/ (Example 3) derived from 250,000 simula-tions of a longer (15-allocation) sequence are presented in Table 4. We considered two scenarios: withlow probability of a switch .ı D 0:2/ and high probability of a switch .ı D 0:8/. As expected, thesimulations show that when the probability of a switch is low, the resident probabilities remain closer tothose of the BT, being high around the allocation ray and low away from the AR (Table IV).

Table IV also provides the probabilities of treatment A allocation for each node of the WBT2W3.4:5/for the two aforementioned scenarios. We can see that while for low ı.ı D 0:2/, the probability of allo-cation to some of the remote nodes is very low (which makes the next allocation almost predictable),it is sufficiently large for ı D 0:8 to reduce predictability. For example, the probability of treatment 1allocation from the node X3�1 is only about 0.015 with ı D 0:2 and is about 0.212 with ı D 0:8.

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Table IV. Example 3: Resident and transition probabilities for 10-allocation WBT2W3.4:5/ sequence with noexact balance at the end of the sequence.

Probability of treatment A allocationResident probability from the node

Generation Node BT WBT with ı D 0:2 WBT with ı D 0:8 BT WBT with ı D 0:2 WBT with ı D 0:8

0 X0 1 1 1 2/5 0.40060 0.40060

1 X1�1 0.4 0.40060 0.40060 0 0.05015 0.19952

X11 0.6 0.59940 0.59940 2/3 0.63311 0.53496

2 X2�2 0 0.02009 0.07993 — 0.00299 0.20198

X2�1 0.8 0.75999 0.64133 1/4 0.28510 0.36514

X21 0.2 0.21992 0.27874 1 0.83641 0.53494

3 X3�2 0 0.00006 0.01614 — 0.00000 0.00000

X3�1 0.2 0.23671 0.29796 0 0.01501 0.21237

X31 0.8 0.72726 0.55627 1/2 0.49529 0.44607

X32 0 0.03598 0.12963 — 0.98610 0.67224

4 X4�2 0 0.00361 0.07942 — 0.00000 0.00000

X4�1 0.6 0.59336 0.48282 0 0.07461 0.23174

X41 0.4 0.40253 0.39528 1 0.88100 0.62254

X42 0 0.00050 0.04249 — 1.00000 1.00000

5 X5�1 0 0.04788 0.19131 — 0.00869 0.16024

X50 1.0000 0.90372 0.61700 2/5 0.39229 0.40401

X51 0 0.04840 0.19169 — 0.92785 0.62896

6 X6�2 0 0.00042 0.03066 — 0.00000 0.00000

X6�1 0.4 0.40199 0.40993 0 0.08660 0.21792

X61 0.6 0.59410 0.48829 2/3 0.60705 0.51710

X62 0 0.00349 0.07112 — 0.98282 0.79562

7 X7�2 0 0.03523 0.11999 — 0.00091 0.08401

X7�1 0.8 0.72782 0.57309 1/4 0.27556 0.33643

X71 0.2 0.23689 0.29238 1 0.84528 0.62841

X72 0 0.00006 0.01454 — 1.00000 1.00000

8 X8�2 0.2 0.00003 0.01008 — 0.00000 0.00000

X8�1 0.8 0.23576 0.30271 0 0.01413 0.23431

X81 0 0.72750 0.56402 1/2 0.49631 0.44247

X82 0 0.03671 0.12318 — 0.98431 0.65612

9 X9�2 0 0.00336 0.08101 — 0.00000 0.00000

X9�1 0.6 0.59349 0.48135 0 0.07499 0.23211

X91 0.4 0.40257 0.39528 1 0.88303 0.62351

X92 0 0.00058 0.04236 — 1.00000 1.00000

10 X10�1 0 0.04787 0.19274

X100 1 0.90447 0.61608

X101 0 0.04766 0.19118

BT, brick tunnel; WBT, wide brick tunnel.


1525


a) b) c)

Figure 5. Example 3: Bubble plot of resident probabilities for 10-allocation WBT2W3.4:5/. (a) ı D 0:2; (b)ı D 0:8; (c) ı D 0:4; 0:6, and 0.8, for first, second, and third layers, respectively.

Figure 5 depicts the resident probabilities of the nodes of the WBT2W3.4:5/ through the bub-ble plot, where the radius of the circle around the node is proportional to its resident probability.We can see that the bubbles in Figure 5a that represents WBT2W3.4:5/ with ı D 0:2 are largeraround the AR and smaller in outer layers compared with the bubbles in Figure 5b that representsWBT2W3.4:5/ with ı D 0:8. Figure 5c illustrates the use of ı changing with layers: it presents theresident probabilities in WBT2W3.4:5/ with ı D 0:4, 0.6, and 0.8 for the first, second, and third layers,respectively.

5.2. Asymptotic behavior of the resident probabilities

As simulations show, in a long WBT randomization sequence, the resident probabilities very soonapproach their asymptotic values. After a few initial allocations, the placement of the WBT nodeswith respect to the allocation ray is periodic with the period equal to the block size S . For eachj D 0; : : : ; S � 1, the resident probabilities of the nodes in Sm C j generation approach their ownasymptotic probabilities as m!1.

Figure 6 presents the resident probabilities for WBT2W3.4:5/ allocation of 100 patients based on100,000 simulations. The five panels of Figure 6 present the allocation numbers grouped by the remain-der of the allocation number with respect to the block size 5 Wmod.AN; 5/D 0; : : : ; 4. Except for the firsttwo generations, there are three nodes in generations that are multiples of 5 (mod(AN, 5)D0), and fournodes in all other generations. The nodes are assigned indexes from �2 to 2 following the conventionused in Table IV.

As Figure 6 shows, the resident probabilities in the considered example approach their asymptoticvalues very early in allocation, perhaps by the end of the second block.

5.3. Use of wide brick tunnel in two-arm studies with large block size

Wide brick tunnel is especially useful in studies with a large block size, as in the example of a two-armstudy with 7:10 allocation. The allocation sequences of WBT7W10(3) randomization fill in the narrowstrip around the allocation ray with each generation having two or three nodes. The resident probabili-ties of the WBT nodes farther removed from the allocation ray are low. Thus, throughout the enrollment,the allocation ratio with WBT7W10(3) randomization stays close to the targeted ratio – much closer thanwith PB randomization (a standard choice in clinical trials). At the same time, the allocation space iswider than the space for the BT randomization [15] and thus, the allocation sequence is less predictable.

While the use of the WBT randomization in open-label two-arm studies with large block size is seenas its most common application, Section 5.4 describes how such randomization can serve as a foundationof the allocation schedule for multi-arm studies with large block size.

1526



Figure 6. Resident probabilities for 100-allocation WBT2W3.4:5/ randomization with ı D 0:8.

5.4. Generalization to more than two treatments

All of the equal allocation two-arm randomization procedures that limit the absolute difference in thenumber of patients allocated to the two arms at any point [5, 8–13] can be easily generalized to equalallocation to k treatment arms. In the case of k arms, the range of the numbers of patients allocatedto k arms is not allowed to exceed a pre-specified integer threshold b W range .N1; : : : ; Nk/ 6 b.The allocation space for such procedures consists of the sequence of overlapping k-dimensional cubeswith the sides equal to b. The m-th cube has the lowest (in all dimensions) corner on the diago-nal point .m � 1;m � 1; : : : ; m � 1/. The same allocation space is used by a k-arm equal allocationblock urn allocation procedure [22]. It is also the allocation space for within-center treatment assign-ments with k-arm equal allocation modified Zelen’s approach used in multi-center trials [2, 18, 24, 25].Different allocation procedures assign different probabilities to the allocation sequences within theallowed space.

With b > 2, the allocation space is wider than the sequence of PBs of size 2k, which makes theallocation sequences less predictable in the open-label studies, but also less balanced in time.

The use of the switch technique allows building an allocation procedure for equal allocation trials thatis not based on the threshold set for the range of treatment assignments to k arms. Indeed, consider anexample of a six-arm study with equal allocation. If the study uses PBs allocation with the block size k,there are more than one treatment option for all allocations except the last one in the block, thus they can-not be predicted with certainty. The last allocation in the block, however, is fully predictable. To avoidthis predictability, the sixth and seventh allocations are switched. Similarly, all treatment assignmentsat the ends of the blocks are switched with the next treatment assignment. This allocation procedureensures that the first five treatment assignments will all be different; also, in subsequent blocks, secondthrough fifth assignments will be different. The allocation sequence will stay close to the allocation raywhile avoiding the predictability of the last allocation in the block.

In open-label studies with unequal allocation to k > 2 arms the need to stay close to the targeted allo-cation ratio is often stronger than the need to lower predictability of an upcoming treatment assignment,especially when the block size is large. With more than 2 arms to randomize to, there are often more thanone option for the next treatment assignment. Thus, a randomization procedure that closely approximatesthe targeted allocation ratio throughout the enrollment such as BT randomization [14,15] or constrainedrandomization [23, 26–29] can often be used in open label studies. Additional protection from complete


1527


predictability can be added at the ends of the blocks by switching the two treatment assignments – thelast in the block and the first in the following block.

If further reduction of predictability of the next treatment assignment in a multi-arm study withunequal allocation is needed, the following approach based on a two-arm WBT randomization couldbe used.

Consider an example of a three-arm open-label study where two formulations of a new drug arecompared with a control. The most efficient design in this case is achieved with the allocation ratio ofp2 W 1 W 1 to the control, formulation 1 and formulation 2 arms [30, 31], which is close to a more con-

venient allocation ratio of 7:5:5. The allocation procedure that has sufficient unpredictability of the nexttreatment assignment can be built by first generating the WBT randomization for two arms randomizedin 7:10 ratio – for example, a WBT7W10(3). Here the two arms – formulation 1 and formulation 2 arms –are pooled into a single ‘active’ arm. After the allocation sequence for control and ‘active’ is generated,the second step is to randomly split ‘active’ allocations between formulation 1 and formulation 2 arms.It could be done by using the PB randomization with block size 2 or 4 or by using another technique(for example, a procedure that limits the imbalance between formulation 1 and formulation 2 allocationsto 2).

The allocation in multi-arm studies where all allocation ratios are different could be treated in a sim-ilar way. For example, in a study with 9:7:3 allocation to treatment 1, treatment 2, and treatment 3, theWBT9W10(3) sequence for randomization to treatment 1 versus treatments 2 and 3 combined can be gen-erated first. It can then be followed by the BT randomization of the treatments 2 and 3 assignments in7:3 ratio.

6. Discussion

In this paper, we offered a solution (WBT) for previously unresolved problem of designing the unequalallocation procedure for a two-arm study that sets a pre-specified limit for allowed imbalance in treat-ment totals while preserving the allocation ratio at every step. Existing allocation procedure by Salamaet al. [6] includes all sequences that comply with the imbalance threshold, but does not preserve theallocation ratio at every step, while the block urn randomization procedure by Zhao and Weng [22] pre-serves the allocation ratio at every step, but does not include all randomization sequences that satisfy themaximum imbalance condition.

The BT randomization proposed by Kuznetsova and Tymofyeyev earlier [14, 15] generates the allo-cation procedure that preserves the allocation ratio at every step and complies with the smallest possibleimbalance threshold. WBT randomization increases the imbalance threshold to a required level andthus, is useful in open-label studies where it reduces the selection bias due to predictability of the nextassignment. We have not performed simulations to quantify the selection bias or average predictabilityof the treatment assignments in this paper – our goal was to introduce a new technique and describe itsproperties and applications.

The allocation sequences of the WBT can be made to stay closer to the allocation ray or to be spreadmore around it by changing the probability of a switch – possibly, from layer to layer. While using highprobability of switch ı increases the resident probabilities in the nodes removed from the allocation ray,they remain low even for ı close to 1. This typically matches the goals of WBT randomization: keep theallocation sequence close to the allocation ray, but allow some restricted deviations to lessen the selectionbias in open-label studies. Simulations of long sequences of WBT show that the resident probabilities(and, therefore, the transition probabilities) converge very soon to some stable values.

As shown in Section 5, the switch technique can be used in multi-arm studies with equal allocation toreduce the predictability of the last assignment in a block. Additionally, the WBT procedure can be usedas a part of randomization algorithm in multi-arm studies with unequal allocation.

Wide brick tunnel allocation sequences are easy to generate. A computer program written in R thatimplements WBT allocation is available from the authors.

Appendix A

Proof of Lemma 1

Ytop.m/CLC 1� .m� 1/�C2=C1

1528



is the distance from the node�m� 1; Ytop.m/CLC 1

�to the allocation ray. Thus, the node belongs to

the strip (1), if (5) holds.Similarly, m�C2=C1 � .Ybot.m/�L� 1/ is the distance from the node .m; Ybot.m/�L� 1/ to the

allocation ray. Thus, the node belongs to the strip (1), if (6) holds. �

Proof of Lemma 2.Suppose the difference in heights of the m-th and .m C 1/-th columns of the C1 W C2 BT ran-

domization is less than 2, then the same holds for the m-th and .m C 1/-th columns of the firstj layers of WBTC1WC2.b/ W Yj; top.m C 1/ � Yj; top.m/ < 2. Because C2 > C1, this means thatYj; top.mC 1/� Yj; top.m/D 1.

The layer j node�m;Yj; top.m/

�above the m-th column is at the inner corner of the WBT (Figure 4).

When allocation A on the top border of the WBT before this node (allocation m C Yv.m/ C j / isswitched places with the allocation B after the node (allocation mCYtop.m/C j C 1/, the layer .j C 1/node

�m� 1; Yj; top.m/C 1

�is added to the allocation space. The new node and the two new allocation

segments form an outer corner of the allocation space (blue dashed lines in Figure 4) of the WBT thatincludes layers 0 through j .

As we can see from Figure 4, now the height of the m-th column with layer .j C 1/ brick addedis the same as the height of the m-th column of the BT. Thus, if the switch m C Ytop.m/ C j $m C Ytop.m/ C j C 1 is followed by the switch of the subsequent allocations, m C Ytop.m/ C j C 1and mC Ytop.m/C j C 2, no new segments above layer .j C 1/ will be added.

Now, suppose for column n, Yj; top.nC1/�Yj; top.n/> 2. Then, after the layer .j C1/ brick is addedon top of the n-th column (blue dashed lines), the height of the n-th column Yj; top.n/C 1 is still belowthe height of the next column Yj; top.nC 1/. Thus, the switch of the new

�nC Ytop.n/C j C 1

�-th and

the existing (nC Ytop.n/C j+2)-th allocations of the border sequence (red dotted lines) will generate anew node

�n� 1; Yj; top.n/C 2

�above the .j C 1/-th layer.

Consider column k such that Ybot.k/ > j . The layer j node�k � 1; Yj;bot.k/

�at the bottom of the k-th

column is at inner corner node of the WBT. When the two allocations on both sides of this node on thebottom border of the WBT, the .k � 1C Ybot.k/� j / -th allocation to B and the .kC Ybot.m/� j / -thallocation to A, are switched places (blue dashed lines), the layer .jC1/ node

�k; Yj;bot.k/� 1

�is added

to the allocation space. The new node and the two new allocation segments form an outer corner of theWBT that includes layers 0 through .j C 1/.

As can be seen from Figure 4, if now the subsequent allocations .kC Ybot.m/� j / and.kC Ybot.m/� j / C 1 are switched places, the allocation sequence following the bottom border can-not exit the first layer space, as both allocations are to treatment B. Thus, the undesirable border exit canonly occur in a pair of consecutive switches where the switch at the top border of the allocation space isfollowed by the switch at the bottom border of the allocation space, but not the other way around. �

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Wide brick tunnel randomization - an unequal allocation procedure that limits the imbalance in treatment totals