Shift-invariant target in allocation problems

Research Article

Received 20 July 2013, Accepted 18 January 2014 Published online 18 February 2014 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/sim.6110

Shift-invariant target inallocation problemsSaumen Mandala*† and Atanu Biswasb

We provide a template for finding target allocation proportions in optimal allocation designs where the target willbe invariant for both shifts in location and scale of the response distributions. One possible application of suchtarget allocation proportions is to carry out a response-adaptive allocation. While most of the existing designsare invariant for any change in scale of the underlying distributions, they are not location invariant in most of thecases. First, we indicate this serious flaw in the existing literature and illustrate how this lack of location invari-ance makes the performance of the designs very poor in terms of allocation for any drastic change in location,such as the changes from degrees centigrade to degrees Fahrenheit. We illustrate that unless a target allocation islocation invariant, it might lead to a completely irrelevant and useless target for allocation. Then we discuss howsuch location invariance can be achieved for general continuous responses. We illustrate the proposed methodusing some real clinical trial data. We also indicate the possible extension of the procedure for more than twotreatments at hand and in the presence of covariates. Copyright © 2014 John Wiley & Sons, Ltd.

Keywords: clinical trial; constraints; ethical allocation; location invariance; minimization; response-adaptivedesign

1. Introduction

1.1. Motivation

Bakshi [1] performed a study at Disha Eye Hospital and Research Centre, India, over a period of 2 years(2008–2010). It included 37 eyes of 37 patients. The study was a comparative prospective randomizedinterventional trial. We wanted to design an allocation ratio for a trial on cataract surgery by two pop-ular approaches, namely small incision cataract surgery (SICS) with snare technique (one treatment)and SICS with vectis technique (other treatment). The primary response variable is the unwanted astig-matism induced by the cataract surgery that is measured in angles. After suitable transformation, thoseangles can be converted to distances in a linear scale. These distances are our responses where a smallerdistance is preferred. Our objective was to frame an allocation ratio among two surgical procedures. Sur-prisingly, we observed that the existing optimal allocation ratios for such purpose might provide poorresult in terms of skewing the allocation in favour of the better treatment, unless we have prior idea on thelocations of the response distributions. Hence, we had to restrict ourselves for a traditional randomized50:50 trial for that time-bound cataract surgery study.

Most of the existing allocation designs are not location invariant (LI). But this motivated us for adetailed study on the need of LI designs and then to derive such a class of designs.

1.2. Allocation problem

Allocation designs for two or more competitive treatments are an important design problem in severalexperimental problems including clinical trials. The primary statistical task is to prefix a target allocationproportion for each treatment that might as well be a function of the unknown parameters.

aDepartment of Statistics, University of Manitoba, Winnipeg, MB, R3T 2N2, CanadabApplied Statistics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India*Correspondence to: Saumen Mandal, Department of Statistics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada.†E-mail: [email protected]

Copyright © 2014 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 2597–2611

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S. MANDAL AND A. BISWAS

In a two-treatment problem, the most trivial target is the 50:50 allocation, where each experimentalunit is being randomized among the two competitive treatments by tossing a fair coin. With some spe-cific objective in mind, the allocation should be driven by some suitable optimality criterion, and theresultant allocation design (i.e. target allocation proportion) might be parameter dependent. If �A and�B be the means and �2A and �2B be the variances of the responses from the two treatments A and B,say, the Neyman allocation allocates �A D �A=.�A C �B/ proportion to treatment A, which essentiallyminimizes the total sample size n subject to

Var . O�A � O�B/D�2AnAC�2BnBDK (1.1)

a prefixed constant, and O�k is the estimate of �k , k D A;B , on the basis of a random target allocation.nA; nB/ with nAC nB D n. Clearly nD n.K/, a function of K. This allocation design depends on theratio of variability of the two treatments only and does not consider the means.

In contrast, the allocation can also be location dependent in some situations. For example, if a lowerresponse is preferred, in order to skew the allocation in favour of the better treatment, Bandyopadhyayand Biswas [2] (denoted by BB design) suggested the ad-hoc target allocation

�A Dˆ��B ��A

T

�(1.2)

where ˆ.�/ is the cumulative distribution function of a standard normal random variable and T is a tun-ing constant that may depend on �2A and �2B . An optimal target will combine this ethical allocation withsome optimality criterion.

1.3. Optimal allocation proportion: existing works

Several allocation proportions and allocation designs are available in the literature for skewing theallocation in favour of the better performing treatment, mostly from intuitive considerations (e.g. the ran-domized play-the-winner rule [3], generalized Pòlya urn design [4], randomized Pólya urn model [5] anddrop-the-loser rule [6] for binary treatment responses; and the linear rank test statistic-based design [7],link function-based design [2], Wilcoxon score-based design [8], utility-based design [9], continu-ous version of randomly reinforced urn (RRU) [10] and drop-the-loser design [11] for continuousresponses). Real life applications of adaptive designs for group comparisons are due to Bartlett et al. [12],Rout et al. [13], Tamura et al. [14], Biswas and Dewanji [15], among others. Optimal allocation pro-portions using some optimality criterion include the works of Rosenberger et al. [16] (extending theapproach of Hayre [17]) for binary responses, and Biswas and Mandal [18] (denoted by BM design) forcontinuous responses.

1.3.1. Optimizing ethics subject to variability. As a general approach, Biswas et al. [19] (denoted byBBZ design) proposed to minimize

nA‰AC nB‰B (1.3)

subject to (1.1), where ‰A and ‰B are functions to a subset of RC. Here (1.3) should reflect ethical lossthat is to be minimized. The optimal target allocation proportion to treatment A comes out to be

�A D�Ap‰B

�Ap‰B C �B

p‰A

: (1.4)

The works of Rosenberger et al. [16] and Biswas and Mandal [18] are special cases of the BBZ design forspecific choices of‰A and‰B . Suppose the random variables YA and YB denote the potential responsesfrom treatments A and B, where we assume YA � F.x � �A/, YB � G.x � �B/, with �A and �B aremeasures of location of the distributions F and G, respectively. That means that �A is mean or medianof F . Here ‰A and ‰B should be functions of F , G, �A, �B and all other parameters in F and G.In general, �2A=nA is the variance of O�A based on nA samples, where O�A is the maximum likelihoodestimate of �A and �A is the scale parameter of F .

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Clearly, if F � N��A; �

2A

�and G � N

��B ; �

2B

�, (1.4) reduces to the design of Biswas and

Mandal [18] for ‰k D P.Yk > c/Dˆ..�k � c/=�k/, and

�A D

�A

rˆ��B�c�B

�

�A

rˆ��B�c�B

�C �B

rˆ��A�c�A

� : (1.5)

Note that ˆ..�k � c/=�k/ is the probability of a response greater than the threshold c by treatmentk. Thus, here (1.5) minimizes the expected number of responses greater than the threshold c when nkpatients are treated by treatment k, k D A;B . If a response greater than c is considered to a ‘failure’,BM essentially minimized ‘expected total failures’ (ETF). For ‰k D �k , k D A;B , (1.4) reduces tothe design of Zhang and Rosenberger [20] (denoted by ZR design) where the ‘expected total response’(ETR) is minimized subject to (1.1), and the solution is

�A D�Ap�B

�Ap�B C �B

p�A

: (1.6)

It is important to note that here, for normally distributed responses, �A and �B can take any value in thereal line, positive or negative. If, at any stage, the estimate of �A or �B becomes negative, the ZR designcrashes down. The ZR design is applicable for positive responses (like the exponential) or after suitableadjustment. Jennison and Turnbull [21] (p. 328) (denoted by JT) considered ‰A D amaxf�A��B ;0g=ı and‰B D a

maxf�B��A;0g=ı for some a and ı. The JT formulation does not enjoy any elegant interpretationlike the ZR or BM formulation. Hence we do not consider the JT formulation in our discussion.

Recent developments in the context of optimal allocation proportions and the corresponding designscan be available in Hu and Rosenberger [22], Tymofyeyev et al. [23], Biswas and Bhattacharya [24] andZhang et al. [25], among others.

1.3.2. Optimizing combined ethics and variability. In a slightly different approach of finding optimaltarget, Atkinson and Biswas [9] (denoted by AB design) combined ethics and variability and providedan optimal target by maximizing the utility

U D UV � �UR

where the contribution of UV is to provide estimates with low variance, whereas UR provides random-ness by forcing the allocation towards a prefixed quantity. The parameter � provides balance betweenthe two. Atkinson and Biswas [9] expressed UV as

UV D �A�AC .1� �A/�B

where �k is the measure of information from applying treatment k (Atkinson and Biswas [9] describedthis in terms of DA-optimality). The expression of UR, defined in terms of entropy, is

UR D �A log

��A

‰A

�C .1� �A/ log

�1� �A

‰B

�:

Maximizing UV (when � ! 0) implies allocation with probability one to the treatment providinghighest information, where maximizing �UR (when � !1) implies �A D ‰A. For a � 2 .0;1/, thetarget allocation proportion �A comes out to be

�A D‰A exp.�A=�/

‰A exp.�A=�/C‰B exp..1� �A/=�/: (1.7)

Atkinson and Biswas [9] proposedˆ..�B��A/=T / as the choice of‰A (in the set-up where a lowerresponse is preferred). However, one may consider any other choice as in BM, ZR or JT for ‰A.


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1.4. Do we need a location-invariant target?

Although the allocation proportions of both the BM and (suitably modified) ZR designs seem applicable,there is a basic flaw in these targets. Suppose that a constant is added to all observations and the scale isalso changed (such as in a conversion from degrees centigrade to degrees Fahrenheit). The variances ofthe two treatment responses are multiplied by the same constant with the change in scale, and the targetstake ratio of the standard deviations (SDs) of the two treatment responses. But the optimal allocationwill change because of change in origin, although there is no real change in the underlying problem.So we certainly need an optimal target that will be invariant with respect to any change of origin of theresponses also. In Figure 2 (which are described later in detail), we illustrate how a shift h in all theresponses affects the designs by BM and ZR, where normal responses are considered with �B D 0. Forexample, with �A D�2, �B D 0, �A D �B D 1, the proportion of allocation to treatment A by the BMrule is 0.801, and it becomes only 0.577 if a common constant 2 is added to all the observations. In thecase of the ZR rule, the proportion of allocation to treatment A for this parameter combination is 0.502,and it becomes 0.735 if the common constant 2 is added to all the observations. Thus, we need somedesign that will be location invariant. In this connection, it is worthwhile to mention that the choiceof ‰k’s by JT are LI, but as already mentioned, the JT formulation lacks suitable interpretation of theobjective function.

Note that the ad-hoc target of BB in (1.2) is LI but not driven by any optimality criterion. BB designresults in �A D 0:5 when �A D �B , irrespective of the values of .�A; �B/. Thus BB design targetsskewed allocation on the basis of �A��B only, while the skewness due to unequal �ks are not induced.

It is also clear from (1.4) that location invariance of the target depends on the choice of ‰k’s. In thispaper, we restrict our discussion mostly on the type of optimality discussed in (1.3) and (1.1). The samechoice of ‰k’s can be used in the optimality (1.7).

1.5. Sketch of the paper

In the next section, we propose some LI optimal response-adaptive designs in the general treatmentresponse set-up using the optimality criterion (1.3) and (1.1). We numerically compute and comparethe allocation proportions of the proposed LI design with the existing competitors in Section 3. Specifi-cally, we compare (a) the LI design with (b) the BM and (c) (modified version of) ZR designs. In somesituation, we bring (d) the BB design also for comparison.

We illustrate and compare all the designs using some existing real clinical trial data in Section 4. InSection 5, we study the influence of change in location to the allocation probability for BM and ZR rulesby using some influence function. Note that this influence function is zero for our proposed LI rule. InSection 6, we introduce optimality criterion-based response-adaptive LI design for more than two treat-ments having continuous responses, and we also discuss possibility of covariates in the model. Finally,we provide some concluding remarks in Section 7.

2. Location-invariant optimal target

From (1.4) and (1.7), we observe that the lack of location invariance in �A may be induced only throughthe choices of ‰A and ‰B . To achieve an LI target, one should choose ‰A and ‰B as functions oftreatment differences, such that ‰A is a decreasing function of the treatment difference �A � �B and‰B is increasing in �A � �B . The choices of JT and AB were like that, although functional forms of‰A and ‰B in the formulation of JT were heuristic and not easily interpretable. Again, it is not possibleto extend them for more than two treatment scenarios. Here we formulate invariant designs on the basisof some optimality criterion with a logical interpretation of ‰A and ‰B , for general distributions F andG, and also extend the approach for more than two treatments, in Section 6.

Because the response-adaptive designs skew the allocation pattern according to the relative values ofthe medians or means �ks, we should consider the designs that consider the difference of �ks instead ofthe individual �ks.

Minimization of expected number of wrong allocation:As �A < �B indicates that treatment A is better than B, and a smaller response is preferred, we would

like to maximize nA in such a situation, on an average. Here we would like to treat a particular patientby treatment A if YA < YB (this is a conceptual formulation as both YA and YB cannot be observed for a

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particular patient, only one of them is observed after treating the patient, but we can always assume theconceptual existence of both these potential responses). Writing YAi and YBi as the potential responsesfrom treatments A and B, respectively, and denoting the indicator of assignment for the i th patient ıi(1 or 0 according as the i th patient is treated by treatment A or treatment B), we find that the total num-ber of wrong allocations (that is treatment A is used when YAi > YBi , and treatment B is used whenYAi < YBi ) among the n patients is

W D

nXiD1

ıiI.YAi > YBi /C

nXiD1

.1� ıi /I.YAi < YBi /

where I.E/ is the indicator variable taking values 1 or 0 according as the event E occurs or not. Clearly,

E.W /D nA‰AC nB‰B

with

‰A D

ZG.xC�A ��B/dF.x/; ‰B D 1�

ZG.xC�A ��B/dF.x/:

For normal responses, we immediately have

‰A Dˆ

0B@ �A ��Bq

�2AC �2B

1CA ; ‰B Dˆ

0B@ �B ��Aq

�2AC �2B

1CA :

Thus, we set the LI design by minimizing the ‘expected number of wrong allocations’ (ENWA), whichis the same as to maximize the ‘expected number of correct allocations’ with the allocation .nA; nB/.Note that here YA and YB are potential responses from treatments A and B, typically unobserved, and weare comparing them. In fact, we are considering their distributions, in a probabilistic way. Thus what wereally consider is ‰A and ‰B , which are functions of parameters only. The aforementioned formulationof W and E.W / is a latent justification of this specific choice of ‰A and ‰B .

This approach can further be generalized. In the aforementioned formulation of ‰A and ‰B , we con-sidered all possible domain of YA and YB , leading towards‰AC‰B D 1. But we need not consider thatrestriction. We may penalize an allocation that is possibly too much damaging; that is, we may penalizean allocation to treatment A when YA > YB C � for any suitable � > 0. Consequently, we may define

‰A D P.YA > YB C �/D

ZG.x � �C�A ��B/dF.x/

‰B D P.YB > YAC �/D

ZF.x � �C�B ��A/dG.x/:

Clearly, ‰AC‰B D P.jYA�YB j> �/ < 1. We interpret the corresponding fnA‰ACnB‰Bg as the‘expected number of �-wrong allocations’ (EN�-WA). For normal responses, these reduce to

‰A Dˆ

0B@�A ��B � �q

�2AC �2B

1CA ; ‰B Dˆ

0B@�B ��A � �q

�2AC �2B

1CA :

Minimization of expected number of �-above the other median:Another choice of ‰A and ‰B may be

‰A D P.YA > �B C �/D 1�F.�C�B ��A/

‰B D P.YB > �AC �/D 1�G.�C�A ��B/:


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The corresponding fnA‰A C nB‰Bg can be interpreted as the expected number of ‘�-above the othermedian’ responses. Thus, here any response by a treatment that is � .> 0/more than the median responsefrom the other treatment is considered to be damaging and is penalized.

Minimization of expected number of �-above median of B:A fourth choice of ‰A and ‰B may be to consider those potential responses that are above �B C �

(considering treatment B as the standard/placebo). In this case, fnA‰A C nB‰Bg is interpreted as theexpected number of responses ‘�-above the median of treatment B’. Here ‰A D 1� F.�B � �A C �/and ‰B D 1�G.�/.

Penalizing more extreme allocations:In all the aforementioned formulations, we consider any response greater than a preset threshold to

be equally damaging. For example, in minimization of ENWA, the weights of fYB D YA C 0:01gand fYB D YA C 10g are taken to be the same. As a more sensible approach, we may penal-ize more for a wrong allocation with more extreme wrong response. One possibility is to consider

‰A DE

YA�YBq�2AC�2

B

ˇ̌̌ˇ̌YA > YB

!and ‰B DE

YB�YAq�2AC�2

B

ˇ̌̌ˇ̌YA < YB

!. For general responses,

‰A D1q

�2AC �2B

Zx>y

.x � y/dF.x ��A/dG.y ��B/:

For normal responses, it reduces to

‰A D�A ��Bq�2AC �

2B

C

�

�A��Bq�2AC�2

B

!

ˆ


B

!

where �.x/ D .2�/�1=2 exp.�x2=2/, the probability density function of the standard normaldistribution. Clearly, this also results in an LI design. We may further generalize to ‰A D

E

YA�YBq�2AC�2

B

ˇ̌̌ˇ̌YA > YB C �

!.

In all the aforementioned formulations, � should be treated as �D �0q�2AC �

2B to retain scale invari-

ance. In general, for any suitable formulation, we denote‰A and ‰B by‰A.�/ and ‰B.�/. The optimalsolution for �A is given by

�A.�/D

�Ap‰A.�/

�Ap‰A.�/

C �Bp‰B .�/

:

For specific functional forms of ‰A.�/ and ‰B.�/, we may plot �A.�/ against � for given �A � �B ,�A, �B , to get optimal �, which gives a desired allocation proportion ��A.�/ for a given �A � �B .For illustration, we consider minimizing EN�-WA. Figure 1 gives the plot of �A.�/ against � for�A � �B D �6;�5;�4;�3;�2;�1; 0 and .�A; �B/ D .1; 1/; .1; 2/; .2; 1/. For �A � �B D 0, we

have �A D �A=.�A C �B/ whatever be �. Note thatp‰B�Ap

‰B�ACp‰A�B

>

p‰�B�Ap

‰�B�ACp‰�A�B

if and only ifp‰B=‰A >

p‰�B=‰

�A. Hence, we have the following theorem.

Theorem 1For any .�A; �B/, the target allocation �A will be maximized for some .‰A; ‰B/ that maximizes‰B=‰A.

Also for any ‰A and ‰B , the allocation �A is an increasing function of �A=�B .In reality, the experimenter may consider � depending on the prior idea on the parameters. Without

any prior knowledge, one may consider �D 0 to implement the design. Clearly, the allocation proportionis invariant of any change in origin and scale (as in the example of degrees centigrade and Fahrenheit).

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Figure 1. Allocation proportions to treatment A against �; the symbols correspond to different values of�A ��B : [ı] �6, [4] �5, [C] �4, [�] �3, [Þ] �2, [5] �1, [�] 0.

3. Numerical computations

In this section, we carry out detailed simulation study to assess the performance of the proposed LIdesign, and we compare its performance with some of the existing designs such as the BM and ZR.Note that the ZR rule, although we kept in the comparisons, may not be applicable in practice as there isevery chance of a ‘breakdown’ of the design due to negative estimates at any stage. We also considered adetailed numerical comparison with some other response-adaptive designs for continuous responses likethe link function based-design of BB and the RRU design for continuous responses [10], although theyare not driven by any optimality criterion. The BB design is LI but non-optimal. The RRU is not evenLI. We are not going to provide the details of the results for the sake of brevity. We present our resultsonly in comparison with the BM and ZR, and for the BB design.

Allocation proportions:Specifically, we obtain the proportion of allocation (and its SD, the standard deviation). We obtain

these for different shifts in location and illustrate how the other designs are affected by these shifts in thelocation, whereas the proposed LI design is invariant of all these location changes. We carry out 10 000simulations with a total sample size nD 100. We present the simulation results in Figure 2.

In the top row of Figure 2, we provide the performance of the BM design for various shifts of data. Wegenerate the data from normal populations, where �B , the mean of treatment B, is always taken as 0. Weconsider different possible values of �A, where we always take �A 6 �B (lower value indicates betterperformance). We consider different shifts in data (in all the data from A and B), namely h D �5, �2,0, 2 and 5. Also, we consider different combinations of .�A; �B/, namely (1,1), (1,2) and (2,1). Fromthe first row of Figure 2, we observe that for .�A; �B/ D .1; 1/ and h D 0 (that is no shift in the data),the allocation is roughly 50:50 for �A D �B , and it is skewed in favour of treatment A if �A < �B .The allocation is more and more skewed with larger treatment differences. But if a large positive shift isadded to all the data, then the allocation becomes close to 50:50 irrespective of the treatment difference.The reason is thatˆ..�k�c/=�k/, k D A;B , becomes close to 1 for both the treatments in this situation.But for large negative shift, ˆ..�k � c/=�k/, k D A;B , becomes close to 0 for both the treatments, andtheir ratio varies a lot. This results an ethical allocation but at a cost of high variability (SD). However,we do not intend to present the SDs in tabular form, for the sake of brevity. Again, it is remarkable tonotice how the allocation proportions change with the shift h, even if the treatment difference is keptfixed.

We observe a similar situation for other combinations of .�A; �B/. For example, for �A < �B , theallocation should be skewed in favour of treatment B (to get more information on that treatment) evenif �A D �B . We observe exactly the same picture from our computational results. A similar featurefor �A < �B is observed for different shifts h. Also, we observe remarkably varying proportions ofallocations for different h. Exactly the same occurs for �A > �B , but the allocation proportion is skewedin favour of treatment A even when �A D �B .

For the simulations from ZR design, in the second row of Figure 2, we make the following modi-fication of the design to get rid of the negative estimates. In fact, for any negative estimate of �A or�B , we used the allocation probability for the earlier patient. Here, for large positive shifts, the chancesof negative estimates become less. Thus, the allocation is more driven by the treatment difference. As


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Figure 2. Allocation proportions to treatment A against �A (keeping �B D 0). Top row: Biswas and Mandaldesign versus location-invariant (LI) design; middle row: modified Zang and Rosenberger design versus LIdesign; bottom row: Bandyopadhyay and Biswas design versus LI design. For Biswas and Mandal design andmodified Zang and Rosenberger design (the first two rows), the symbols correspond to [ı] h D 0, [4] h D 2,[C] h D 5, [�] h D �2 and [Þ] h D �5; and [5] corresponds to the LI design. For bottom row, the symbolscorrespond to [ı] .�A; �B/ D .1; 1/, [4] .�A; �B/ D .1; 2/, [C] .�A; �B/ D .2; 1/, and these three are for LI

design; [�] T D 1, [Þ] T D 2, [5] T D 5, and these three for Bandyopadhyay and Biswas design.

the ratio of the estimate of �A to that of �B ultimately matters, for .�A; �B/ D .1; 1/, the allocationis skewed in favour of the better treatment for small shifts, but the allocation tends to 50:50 for largerpositive shifts. For negative shifts, the chances of getting negative estimates are more, and we stick to theinitial balanced state of art more and more, and the allocation tends to 50:50. For other combinations of.�A; �B/, the allocation proportions have the similar tendencies as that of BM design for positive shifts,but for negative shifts, there are more and more negative estimates, and the allocation proportion sticksto nearly 50:50. Note that, without this modification of the ZR rule, the rule will crash down.

For illustration of the results from LI design, we consider ‰A D ˆ


B

!and ‰B D

ˆ

�B��Aq�2AC�2

B

!. However, for LI design, as in the bottom row of Figure 2, the allocation proportions

do not depend on the shifts h. The allocation proportions increase with treatment difference. Note thatthe SDs are reasonably small although not reported. For .�A; �B/ D .1; 1/, the allocation proportionsstart at 50% for treatment A for �A D �B , and it is more than 50% for �A < �B . For �A < �B , it starts

2604



at less than 50% for �A D �B , and for �A > �B , it starts at more than 50% for �A D �B . These aredesirable results. Most importantly, as it does not depend on the shifts h, the LI design is preferred overthe existing BM or ZR design.

We also compare the allocation proportion to treatment A by the BB design for T D 1; 2; 5 in thisfigure. The allocation proportions by BB design do not depend on �A and �B . Here �A D 0:5 when�A D �B whatever be .�A; �B/. The target allocation increases when �A��B < 0. Thus, we prefer theproposed LI design over this ad-hoc BB design as LI design is optimal and depends on .�A; �B/ as well.

ETF, ETR and ENWA:We also computed the three characteristics, ETF, ETR and ENWA, for the three formulations—BM,

ZR and LI. We skip the numerical values for the sake of brevity. It is interesting to note that the criterionENWA is invariant of any shift h. We base our proposed LI design on this invariance criterion. However,ETR is minimum for ZR formulation, ETF is minimum for BM formulation and ENWA is minimum forthe LI formulation, as these formulations minimize the respective criterion.

Sample size:One reasonable hypothesis to test, following allocation, may be the equivalence of the two treatments,

that is, H0 W �A D �B , against the one-sided alternative of the superiority of treatment A, that is,

−4 −2 0 2 4

500

1000

1500

2000

h−4 −2 0 2 4

h−4 −2 0 2 4

h

Sam

ple

size

500

1000

1500

2000

Sam

ple

size

500

1000

1500

2000

Sam

ple

size

σA=1, σ

B=1 σ

A=1, σ

B=2 σ

A=2, σ

B=2

Figure 3. Sample sizes required for different shifts (h) for different designs: [ı] Biswas and Mandal design, [4]modified Zhang and Rosenberger design and [C] location-invariant design.

Table I. Comparative characteristics of different designs.

BM Modified ZR BB LI

�A (largeCve h) Tends to 50:50 Tends to 50:50 Invariant Invariant

�A (large �ve h) Highly skewed (> 50%) Tends to 50:50 Invariant Invariantbut high variability

�A for �A D �B 50:50 50:50 50:50 50:50(for �A D �B )�A for �A D �B < 50% < 50% 50:50 < 50%(for �A < �B )�A for �A D �B > 50% > 50% 50:50 > 50%(for �A > �B )ETF Minimum — — —

ETR — Minimum — —

ENWA — — — Minimum

Sample size Varies a lot Varies a lot — Invariant(against h)

BM, Biswas and Mandal design; ZR, Zhang and Rosenberger design; BB, Bandyopadhyayand Biswas design; LI, location invariant design; ETF, expected total failures; ETR, expectedtotal response; ENWA, expected number of wrong allocations.


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H1 W �A < �B . A left-tailed test can be carried out on the basis of the ultimate estimate of �A � �B ,that is O�A � O�B . Figure 3 reports the sample size required to achieve 90% power to detect a treatmentdifference of � D 0:5 for a test at level 5%. Clearly, the required sample size varies remarkably in theZR and BM formulations with the shift h in the responses, whereas the LI formulation remains invariantin this context also. So, in practice, it is not easy to find the target sample size for ZR or BM rules, unlessthe shift h is exactly known.

We present features of different designs discussed in this section in a tabular form in Table I.

4. Illustration with real data

To illustrate the need of the adaptive procedures, we consider the real clinical trial conducted by Dworkinet al. [26]. This data set was also used in ZR and BBZ designs, for illustration. It was a randomized,placebo-controlled trial with an objective to evaluate the efficacy and safety of pregabalin the treat-ment of postherpetic neuralgia. There were nD 173 patients of which 84 received the standard therapyplacebo and 89 were randomized to pregabalin. The primary efficacy measure was the mean of the lastseven daily pain ratings, as maintained by patients in a daily diary using the 11-point numerical painrating scale (0 D no pain, 10 D worst possible pain) and, therefore, a lower score (response) indicatesa favourable situation. After the 8-week duration of the trial, it was observed that pregabalin-treatedpatients experienced a higher decrease in pain score than patients treated with placebo. We use the finalmean scores, that is, 3.60 (with SD D 2:25) for pregabalin, and 5.29 (with SD D 2:20) for placebo asthe true ones for our purpose with an appropriate assumption regarding the distribution for pain scores.

We obtained the results in the following by simulations with 10 000 repetitions of a response-adaptivetrial for n D 173 patients with N.3:60; 2:252/ distribution for pregabalin and N.5:29; 2:202/ distri-bution for placebo. We update allocation probabilities according to the rule considered. Figure 4 givesthe result.

The BM procedure has an expected proportion of allocation to A as 0.512 (SD D 0:055), withoutconsidering the shift in the data. If we carry out the ZR procedure, the probability of getting negativeresponse at least once is 0.0154. Using the modified version of the ZR rule (as described in simulationsection), we have that the expected proportion of allocation to A as 0.549 (SD D 0:053). A shift in thedata is quite practical in this scenario. For example, one could record the data in an alternative 11-pointnumerical scale f1; � � � ; 11g (having 1 � no pain, 11 � worst possible pain), or f0; � � � ; 10g or may befR1; � � � ; R11g in general, where R1 < � � � < R11 are suitable 11 numerical scores. With possible shiftsin the data, allocation scenarios for BM and modified ZR designs change. For BM design, a positive shiftmakes the allocation proportion closer to the 50% mark, and a negative shift makes the allocation skewedup to certain extent and then revert back towards the 50% mark. But the SD increases remarkably. Formodified ZR design, the allocation comes out closer to the 50% mark for large positive shift. For large

Figure 4. Allocation proportions to pregabalin for different shifts (h) for different designs: [ı] Biswas andMandal design, [4] modified Zhang and Rosenberger design, [C] location-variant design.

2606



negative shifts, the allocation sticks to the 50% mark, the adaptive flavour of the design is lost, and theallocation is mostly based on the initial balanced allocation, driven by the fact of negative estimatesof mean(s).

However, by the LI rule, the expected proportion of allocation to A is 0.610 (SD D 0:061). Thus,the LI procedure is best so far the allocation proportion is concerned. Moreover, it is unaffected by anychange in shifts.

5. Influence of change in location

Here we provide an account of change in allocation proportion for a change in location. Suppose anyallocation design of the form (1.4) is implemented, and an allocation proportion of �A0 occurs in favourof treatment A for some .�A0; �B0/ (given �A and �B ). Thus,

�A0 D�Ap‰B0

�Ap‰B0C �B

p‰A0

where ‰k0 D‰k.�A0; �B0/; k D A;B . Now suppose that instead of Yk , we observe YkCh; k D A;B ,and thus a shift h in location occurred without changing the scale. Expanding ‰k around �k , after someroutine steps, we obtain that �A.h/, the allocation proportion after the location shift, is

�A.h/D�Ap‰B.�AC h; �B C h/

�Ap‰B.�AC h; �B C h/C �B

p‰A.�AC h; �B C h/

D �A0

�1C

h

2

1

‰B0

�@‰B

@�A0C@‰B

@�B0

�

�

�Ap‰B0

�@‰B@�A0

C @‰B@�B0

�C �Bp

‰A0

�@‰A@�A0

C @‰A@�B0

�.�Ap‰B0C �B

p‰A0/

9=;CO.h2/

35

where @‰k@�j0

D�@‰k@�j

��AD�A0;�BD�B0

, k; j D A;B .

Now we define influence of location change (IoLC) as

IoLC D limh!0

�A.h/� �A0

h �A0

D1

2

8<: 1

‰B0

�@‰B

@�A0C@‰B

@�B0

��

�Ap‰B0

�@‰B@�A0

C @‰B@�B0

�C �Bp

‰A0

�@‰A@�A0

C @‰A@�B0

�.�Ap‰B0C �B

p‰A0/

9=;

which is an indicator of the (undesired) relative change in allocation proportion to treatment A. Note thatthis IoLC depends on �A and �B , and also varies for different true values of (�A; �B ).

For the BM design, the IoLC is

IoLCBM D1

2

(�B0

�BˆB0�

�A�B�B0pˆA0C

�B�A�A0pˆB0

pˆA0ˆB0.�A

p‰B0C �B

p‰A0/

)

where �k0 D ��k0�c�k

�and ˆk0 Dˆ

��k0�c�k

�, k D A;B .

For ZR design, the IoLC reduces to

IoLCZR D1

2

1

�B0�

�Ap�A0C �B

p�B0

p�A0�B0.�A

p�B0C �B

p�A0/

;


2607


Figure 5. Plot of influence of location change function for different values of �A0 when �B0 D 4 andc D 4 (for Biswas and Mandal rule). The black lines are for Biswas and Mandal rule, green lines are forZhang and Rosenberger rule and the red is for location-invariant rule. For black and green, the solid lines cor-respond to .�A; �B/ D .1; 1/, the broken lines correspond to .�A; �B/ D .2; 1/ and the dotted lines are for

.�A; �B/D .1; 2/.

and for the LI designs like those of JT, AB and our general formulation of the present paper, denoted byLI, we have

IoLCJT D IoLCAB D IoLCLI D 0:

In Figure 5, we plot IoLC for the BM and ZR designs for different values of �A0, where we take.�A; �B/ D .1; 1/; .2; 1/; .1; 2/; c D 4 (for BM design), �B0 D 4. We also indicate the horizontal line(corresponding to zero) for the IoLC of the proposed LI design. We deliberately plot it for positive �Aand �B as the IoLCZR does not exist for negative �A and/or �B . The black lines are for BM rule, thegreen ones are for ZR rule and the horizontal red line corresponds to any LI rule.

6. Multiple treatments and covariates

6.1. More than two treatments

This idea of minimizing the expected number of �-wrong allocations or the other formulations can beextended for designs with three or more treatments at hand. For simplicity, here we illustrate that withthree treatments. For three treatments A, B and C, to find the optimal allocation proportions �A, �B and�C D 1� �A � �B , Biswas and Mandal [27] proposed to minimize

nA‰AC nB‰B C nC‰C (6.1)

subject to

�2AnAC�2BnBC�2CnCDK (6.2)

say, where nA C nB C nC D n, and ‰A, ‰B and ‰C are some cost functions associated with the threetreatments that are functions of the parameters. For an LI design,‰A,‰B and‰C should be invariant ofany change in location of the variables. Note that treatment A is the best if �A < �B ; �C . Thus, we needa ‰A, which is increasing in both �A � �B and �A � �C (i.e. ‰A decreases as �A � �B or �A � �Cor both decreases). Denoting by YA, YB and YC the potential responses from the three treatments,we consider

2608



‰A D P.YA >min.YB ; YC //Dˆ��

0B@ �A ��Bq

�2AC �2B

;�A ��Cq�2AC �

2C

1CA (6.3)

where ˆ��.�; �/ is the CDF of a N2

��0

0

�;

�1 ��

�� 1

��distribution with

�� D�2Aq�

�2AC �2B

� ��2AC �

2C

� :As an alternative to (6.3), we can as well use

‰A D P.YA > YB/P.YA > YC /Dˆ

0B@ �A ��Bq

�2AC �2B

1CAˆ

0B@ �A ��Cq

�2AC �2C

1CA (6.4)

as a sensible working choice. Both (6.3) and (6.4) will work well in practice. Similar expressions for‰B and ‰C can be used. Here the optimization problem can be considered as minimization of expectednumber of wrong allocations (ENWA), which was considered by Biswas and Mandal [27], as follows.

Theorem 2The optimal allocation proportion for the optimization problem (6.1) subject to (6.2) is

�A D

�Ap‰A

�Ap‰AC �Bp

‰BC �Cp

‰C

:

Similarly for �B and �C .

We carried out detailed numerical study for three treatments along with the three-treatment versionsof the other designs (like BM design). The results are as expected: the LI optimal design performs well,whereas the other optimal designs perform very poorly for a shift in location.

6.2. Covariates

The proposed methodology can be easily extended for the presence of covariates. Suppose a covariatevector X with p components influences the response. We assume that the model (may be after suitabletransformation) can be presented as

YkjX �N��k CX

Tˇk; �2k

�; k D A;B; andX �Np.�X ; ˙X /:

Then

‰A D P.YA > YB/Dˆ

0B@ �A ��B C�

TX .ˇA � ˇB/q

�2AC �2B C .ˇA � ˇB/

T˙X .ˇA � ˇB/

1CA :

Subsequent development is similar.

7. Concluding remarks

We feel that there are two important contributions of our paper.

(a) We showed the need for an LI design and then provided such a design with different interpreta-tions.

(b) Consequently, we provided a multi-treatment generalization and covariate-adjusted version of thedesign.


2609


While (b) is also a non-trivial work, we feel that point (a) is very important, both in identifying a veryserious problem that is present in almost all the existing designs and then in suggesting an elegantsolution with logical interpretation. We are now ready to apply such LI designs for real clinical trials.

In this paper, we proposed the LI designs for two or more treatments. We illustrated that the existingresponse-adaptive designs for continuous responses using optimality criteria, namely the BM design andthe modified ZR design, are too undesirable for large shifts (either positive or negative) in the data (inaddition to the fact that the ZR design crashes down with the possible negative estimates of the mean(s)).While the BB design is LI, it is non-optimal. We observe that the LI design really performs well in thiscontext. We could perform an exercise of calculation of powers of a test of equivalence for differentdesigns and compare them. This is a routine work, and hence, we skipped that except Figure 3, wherewe provide sample sizes required to achieve a prefixed power. We did not pay our attention to calculatetrivia like expected number of failures and follow-up inferences. The reason is that we wanted to providea flexible target with logical interpretation that is invariant of any shift in the data. But the fact that theexisting designs (BM and ZR) perform so poorly with respect to change in locations of the data keepsthe proposed LI design much ahead of the existing designs.

We believe that we successfully addressed a very important issue (that the need of location invariance)in finding optimal ‘target’ allocation that was simply overlooked in the existing works. We also pointedthat unless a target allocation is ‘location invariant’, it might lead to a completely irrelevant and uselesstarget for allocation. Finally, we focussed our present work on ‘targets’, which is applicable not only inclinical trials but also in many areas of statistics and biostatistics.

Acknowledgements

The authors wish to thank the associate editor and a referee for their careful reading and constructive suggestionsthat led to some improvement over an earlier version of the manuscript. A Discovery Grant from the NaturalSciences and Engineering Research Council (NSERC) of Canada supported the research of S. Mandal.

References1. Bakshi PK. Evaluation of various surgical techniques in brunescent cataracts. DNB (Diplomate National Board) Thesis,

National Board of Examinations (Ophthalmology), India, 2010.2. Bandyopadhyay U, Biswas A. Adaptive designs for normal responses with prognostic factors. Biometrika 2001;

88:409–419.3. Wei LJ, Durham S. The randomized play-the-winner rule in medical trials. Journal of the American Statistical Association

1978; 73:840–843.4. Wei LJ. The generalized Polya’s urn design for sequential medical trials. Annals of Statistics 1979; 7:291–296.5. Durham SD, Flournoy N, Li W. A sequential design for maximizing the probability of a favorable response. Canadian

Journal of Statistics 1998; 26:479–495.6. Ivanova A. A play-the-winner type urn model with reduced variability. Metrika 2003; 58:1–13.7. Rosenberger WF. Asymptotic inference with response-adaptive treatment allocation designs. Annals of Statistics 1993;

21:2098–2107.8. Bandyopadhyay U, Biswas A. An adaptive allocation for continuous response using Wilcoxon-Mann-Whitney score.

Journal of Statistical Planning and Inference 2004; 123:207–224.9. Atkinson AC, Biswas A. Bayesian adaptive biased coin design for sequential clinical trials. Biometrics 2005; 61:118–125.

10. Beggs AW. On the convergence of reinforcement learning. Journal of Economic Theory 2005; 122:1–36.11. Ivanova A, Biswas A, Lurie A. Response adaptive designs for continuous outcomes. Journal of Statistical Planning and

Inference 2006; 136:1845–1852.12. Bartlett RH, Roloff DW, Cornell RG, Andrews AF, Dillon PW, Zwischenberger JB. Extracorporeal circulation in neonatal

respiratory failure: a prospective randomized trial. Pediatrics 1985; 76:479–487.13. Rout CC, Rocke DA, Levin J, Gouw’s E, Reddy D. A reevaluation of the role of crystalloid preload in the prevention of

hypotension associated with spinal anesthesia for elective cesarean section. Anesthesiology 1993; 79:262–269.14. Tamura RN, Faries DE, Andersen JS, Heiligenstein JH. A case study of an adaptive clinical trial in the treatment of

out-patients with depressive disorder. Journal of the American Statistical Association 1994; 89:768–776.15. Biswas A, Dewanji A. Inference for a RPW-type clinical trial with repeated monitoring for the treatment of rheumatoid

arthritis. Biometrical Journal 2004; 46:769–779.16. Rosenberger WF, Stallard N, Ivanova A, Harper CN, Ricks ML. Optimal adaptive designs for binary response trials.

Biometrics 2001; 57:909–913.17. Hayre LS. Two-population sequential tests with three hypotheses. Biometrika 1979; 66:465–474.18. Biswas A, Mandal S. Optimal adaptive designs in phase III clinical trials for continuous responses with covariates. In

mODa 7 - Advances in Model-Oriented Design and Analysis, Bucchianico AD, Lauter H, Wynn HP (eds). Physica-Verlag:Heidelberg, 2004; 51–58.

2610



19. Biswas A, Bhattacharya R, Zhang L. Optimal response-adaptive designs for continuous responses in phase III trials.Biometrical Journal 2007; 49:928–940.

20. Zhang L, Rosenberger WF. Response-adaptive randomization for clinical trials with continuous outcomes. Biometrics2006; 62:562–569.

21. Jennison C, Turnbull BW. Group Sequential Methods with Applications to Clinical Trials. Chapman and Hall/CRC: BocaRaton, Florida, 2000.

22. Hu F, Rosenberger WF. The Theory of Response-Adaptive Randomization in Clinical Trials. John Wiley and Sons:Hoboken, New Jersey, 2006.

23. Tymofyeyev Y, Rosenberger WF, Hu F. Implementing optimal allocation in sequential binary response experiments.Journal of the American Statistical Association 2007; 102:224–234.

24. Biswas A, Bhattacharya R. Near efficient target allocations in response-adaptive randomization. Statistical Methods inMedical Research 2014; in press, DOI: 10.1177/0962280212468378.

25. Zhang LX, Hu F, Cheung SH, Chan WS. Asymptotic properties of covariate-adjusted adaptive designs. Annals of Statistics2007; 35:1166–1182.

26. Dworkin RH, Corbin AE, Young JP, Sharma U, LaMoreaux L, Bockbrader H, Garofalo EA, Poole RM. Pregabalin for thetreatment of posther-petic neuralgia. A randomized, placebo-controlled trial. Neurology 2003; 60:1274–1283.

27. Biswas A, Mandal S. Optimal three-treatment response-adaptive designs for phase III clinical trials with binary responses.In mODa 8 - Advances in Model-Oriented Design and Analysis, Lopez-Fidalgo J, Rodriguez-Diaz JM, Torsney B (eds).Physica-Verlag: Heidelberg, 2007; 33–40.


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