Accepted 18 January 2014

Published online 18 February 2014 in Wiley Online Library

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

Shift-invariant target in allocation problems Saumen Mandala * † and Atanu Biswasb We provide a template for finding target allocation proportions in optimal allocation designs where the target will be invariant for both shifts in location and scale of the response distributions. One possible application of such target allocation proportions is to carry out a response-adaptive allocation. While most of the existing designs are invariant for any change in scale of the underlying distributions, they are not location invariant in most of the cases. First, we indicate this serious flaw in the existing literature and illustrate how this lack of location invariance 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 is location invariant, it might lead to a completely irrelevant and useless target for allocation. Then we discuss how such location invariance can be achieved for general continuous responses. We illustrate the proposed method using some real clinical trial data. We also indicate the possible extension of the procedure for more than two treatments at hand and in the presence of covariates. Copyright © 2014 John Wiley & Sons, Ltd. Keywords:

clinical trial; constraints; ethical allocation; location invariance; minimization; response-adaptive design

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 randomized interventional trial. We wanted to design an allocation ratio for a trial on cataract surgery by two popular 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 astigmatism induced by the cataract surgery that is measured in angles. After suitable transformation, those angles can be converted to distances in a linear scale. These distances are our responses where a smaller distance is preferred. Our objective was to frame an allocation ratio among two surgical procedures. Surprisingly, we observed that the existing optimal allocation ratios for such purpose might provide poor result in terms of skewing the allocation in favour of the better treatment, unless we have prior idea on the locations of the response distributions. Hence, we had to restrict ourselves for a traditional randomized 50:50 trial for that time-bound cataract surgery study. Most of the existing allocation designs are not location invariant (LI). But this motivated us for a detailed 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 several experimental problems including clinical trials. The primary statistical task is to prefix a target allocation proportion for each treatment that might as well be a function of the unknown parameters.

a Department

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of Statistics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada 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] b Applied

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In a two-treatment problem, the most trivial target is the 50:50 allocation, where each experimental unit is being randomized among the two competitive treatments by tossing a fair coin. With some specific objective in mind, the allocation should be driven by some suitable optimality criterion, and the resultant allocation design (i.e. target allocation proportion) might be parameter dependent. If A and B be the means and A2 and B2 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 essentially minimizes the total sample size n subject to Var .O A O B / D

2 A2 C B DK nA nB

(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 nA C nB D n. Clearly n D n.K/, a function of K. This allocation design depends on the ratio 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 lower response is preferred, in order to skew the allocation in favour of the better treatment, Bandyopadhyay and Biswas [2] (denoted by BB design) suggested the ad-hoc target allocation A D ˆ

B A T

(1.2)

where ˆ./ is the cumulative distribution function of a standard normal random variable and T is a tuning constant that may depend on A2 and B2 . An optimal target will combine this ethical allocation with some optimality criterion. 1.3. Optimal allocation proportion: existing works Several allocation proportions and allocation designs are available in the literature for skewing the allocation in favour of the better performing treatment, mostly from intuitive considerations (e.g. the randomized play-the-winner rule [3], generalized Pòlya urn design [4], randomized Pólya urn model [5] and drop-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], continuous version of randomly reinforced urn (RRU) [10] and drop-the-loser design [11] for continuous responses). 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 proportions using some optimality criterion include the works of Rosenberger et al. [16] (extending the approach of Hayre [17]) for binary responses, and Biswas and Mandal [18] (denoted by BM design) for continuous responses. 1.3.1. Optimizing ethics subject to variability. As a general approach, Biswas et al. [19] (denoted by BBZ design) proposed to minimize nA ‰A C 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 loss that is to be minimized. The optimal target allocation proportion to treatment A comes out to be p A ‰ B A D p p : A ‰ B C B ‰ A

(1.4)

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The works of Rosenberger et al. [16] and Biswas and Mandal [18] are special cases of the BBZ design for specific choices of ‰A and ‰B . Suppose the random variables YA and YB denote the potential responses from treatments A and B, where we assume YA F .x A /, YB G.x B /, with A and B are measures of location of the distributions F and G, respectively. That means that A is mean or median of F . Here ‰A and ‰B should be functions of F , G, A , B and all other parameters in F and G. In general, A2 =nA is the variance of O A based on nA samples, where O A is the maximum likelihood estimate of A and A is the scale parameter of F . Copyright © 2014 John Wiley & Sons, Ltd.

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Clearly, if F N A ; A2 and G N B ; B2 , (1.4) reduces to the design of Biswas and Mandal [18] for ‰k D P .Yk > c/ D ˆ..k c/=k /, and r A ˆ BBc A D r r : A ˆ BBc C B ˆ AAc

(1.5)

Note that ˆ..k c/=k / is the probability of a response greater than the threshold c by treatment k. Thus, here (1.5) minimizes the expected number of responses greater than the threshold c when nk patients 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 to the 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 p A B A D p p : A B C B A

(1.6)

It is important to note that here, for normally distributed responses, A and B can take any value in the real line, positive or negative. If, at any stage, the estimate of A or B becomes negative, the ZR design crashes down. The ZR design is applicable for positive responses (like the exponential) or after suitable adjustment. Jennison and Turnbull [21] (p. 328) (denoted by JT) considered ‰A D amaxfA B ;0g=ı and ‰B D amaxfB A ;0g=ı for some a and ı. The JT formulation does not enjoy any elegant interpretation like 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 designs can be available in Hu and Rosenberger [22], Tymofyeyev et al. [23], Biswas and Bhattacharya [24] and Zhang et al. [25], among others. 1.3.2. Optimizing combined ethics and variability. In a slightly different approach of finding optimal target, Atkinson and Biswas [9] (denoted by AB design) combined ethics and variability and provided an 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 randomness by forcing the allocation towards a prefixed quantity. The parameter provides balance between the two. Atkinson and Biswas [9] expressed UV as UV D A A C .1 A /B where k is the measure of information from applying treatment k (Atkinson and Biswas [9] described this in terms of DA -optimality). The expression of UR , defined in terms of entropy, is

A UR D A log ‰A

1 A C .1 A / log ‰B

:

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

‰A exp.A =/ : ‰A exp.A =/ C ‰B exp..1 A /=/

(1.7)

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Atkinson and Biswas [9] proposed ˆ..B A /=T / as the choice of ‰A (in the set-up where a lower response 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 is also changed (such as in a conversion from degrees centigrade to degrees Fahrenheit). The variances of the two treatment responses are multiplied by the same constant with the change in scale, and the targets take ratio of the standard deviations (SDs) of the two treatment responses. But the optimal allocation will 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 the responses also. In Figure 2 (which are described later in detail), we illustrate how a shift h in all the responses affects the designs by BM and ZR, where normal responses are considered with B D 0. For example, with A D 2, B D 0, A D B D 1, the proportion of allocation to treatment A by the BM rule is 0.801, and it becomes only 0.577 if a common constant 2 is added to all the observations. In the case 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 some design that will be location invariant. In this connection, it is worthwhile to mention that the choice of ‰k ’s by JT are LI, but as already mentioned, the JT formulation lacks suitable interpretation of the objective function. Note that the ad-hoc target of BB in (1.2) is LI but not driven by any optimality criterion. BB design results in A D 0:5 when A D B , irrespective of the values of .A ; B /. Thus BB design targets skewed allocation on the basis of A B only, while the skewness due to unequal k s are not induced. It is also clear from (1.4) that location invariance of the target depends on the choice of ‰k ’s. In this paper, we restrict our discussion mostly on the type of optimality discussed in (1.3) and (1.1). The same choice 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 treatment response set-up using the optimality criterion (1.3) and (1.1). We numerically compute and compare the allocation proportions of the proposed LI design with the existing competitors in Section 3. Specifically, we compare (a) the LI design with (b) the BM and (c) (modified version of) ZR designs. In some situation, 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. In Section 5, we study the influence of change in location to the allocation probability for BM and ZR rules by using some influence function. Note that this influence function is zero for our proposed LI rule. In Section 6, we introduce optimality criterion-based response-adaptive LI design for more than two treatments 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 through the choices of ‰A and ‰B . To achieve an LI target, one should choose ‰A and ‰B as functions of treatment 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 possible to extend them for more than two treatment scenarios. Here we formulate invariant designs on the basis of some optimality criterion with a logical interpretation of ‰A and ‰B , for general distributions F and G, 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 of the medians or means k s, we should consider the designs that consider the difference of k s instead of the individual k s.

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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 patient by treatment A if YA < YB (this is a conceptual formulation as both YA and YB cannot be observed for a Copyright © 2014 John Wiley & Sons, Ltd.

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

n X

ıi I.YAi > YBi / C

i D1

n X

.1 ıi /I.YAi < YBi /

i D1

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 ‰A C nB ‰B with Z ‰A D

Z G.x C A B /dF .x/; ‰B D 1

G.x C A B /dF .x/:

For normal responses, we immediately have 0

1

0

1

B A B C B B A C ‰A D ˆ @ q A ; ‰B D ˆ @ q A: A2 C B2 A2 C B2 Thus, we set the LI design by minimizing the ‘expected number of wrong allocations’ (ENWA), which is 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 we are comparing them. In fact, we are considering their distributions, in a probabilistic way. Thus what we really consider is ‰A and ‰B , which are functions of parameters only. The aforementioned formulation of 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 considered all possible domain of YA and YB , leading towards ‰A C ‰B D 1. But we need not consider that restriction. We may penalize an allocation that is possibly too much damaging; that is, we may penalize an allocation to treatment A when YA > YB C for any suitable > 0. Consequently, we may define Z ‰A D P .YA > YB C / D

G.x C A B /dF .x/ Z

‰B D P .YB > YA C / D

F .x C B A /dG.x/:

Clearly, ‰A C ‰B D P .jYA YB j > / < 1. We interpret the corresponding fnA ‰A C nB ‰B g as the ‘expected number of -wrong allocations’ (EN-WA). For normal responses, these reduce to 0

1

0

1

B A B C B B A C ‰A D ˆ @ q A ; ‰B D ˆ @ q A: 2 2 A C B A2 C B2

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

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‰A D P .YA > B C / D 1 F . C B A / ‰B D P .YB > A C / D 1 G. C A B /: Copyright © 2014 John Wiley & Sons, Ltd.

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The corresponding fnA ‰A C nB ‰B g can be interpreted as the expected number of ‘-above the other median’ responses. Thus, here any response by a treatment that is .> 0/ more than the median response from 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 ‰B g is interpreted as the expected 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 0:01g and fYB D YA C 10g are taken to be the same. As a more sensible approach, we may penalize more for a wrong response. One possibility is to consider ˇ allocation ˇ ! with more extreme wrong ! ˇ ˇ ˇ ˇ ‰A D E qYA2YB2 ˇ YA > YB and ‰B D E qYB2YA2 ˇ YA < YB . For general responses, A CB ˇ A CB ˇ Z 1 .x y/dF .x A /dG.y B /: ‰A D q A2 C B2 x>y For normal responses, it reduces to ! A B q 2 C 2 A B

A B ‰A D q C A2 C B2 ˆ

!

A B q 2 2 A CB

where .x/ D .2/1=2 exp.x 2 =2/, the probability density function of the standard normal distribution. ˇClearly, this ! also results in an LI design. We may further generalize to ‰A D ˇ ˇ E qYA2YB2 ˇ YA > YB C . A CB ˇ q In all the aforementioned formulations, should be treated as D 0 A2 C B2 to retain scale invariance. In general, for any suitable formulation, we denote ‰A and ‰B by ‰A ./ and ‰B ./. The optimal solution for A is given by A ./ D

p A ‰A ./ p A ‰A ./

C

p B ‰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/. p For A B D 0, we p

‰

A > p BpA if and only if have A D A =.A C B / whatever be . Note that p‰ ‰CBp ‰A B ‰B A C ‰A B B A p p ‰B =‰A > ‰B =‰A . Hence, we have the following theorem.

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

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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 proportion is invariant of any change in origin and scale (as in the example of degrees centigrade and Fahrenheit). Copyright © 2014 John Wiley & Sons, Ltd.

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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 LI design, 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 is every chance of a ‘breakdown’ of the design due to negative estimates at any stage. We also considered a detailed numerical comparison with some other response-adaptive designs for continuous responses like the link function based-design of BB and the RRU design for continuous responses [10], although they are not driven by any optimality criterion. The BB design is LI but non-optimal. The RRU is not even LI. We are not going to provide the details of the results for the sake of brevity. We present our results only in comparison with the BM and ZR, and for the BB design.

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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 the location, whereas the proposed LI design is invariant of all these location changes. We carry out 10 000 simulations with a total sample size n D 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. We generate the data from normal populations, where B , the mean of treatment B, is always taken as 0. We consider different possible values of A , where we always take A 6 B (lower value indicates better performance). 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). From the 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 is added to all the data, then the allocation becomes close to 50:50 irrespective of the treatment difference. The reason is that ˆ..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, and their 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 to notice how the allocation proportions change with the shift h, even if the treatment difference is kept fixed. We observe a similar situation for other combinations of .A ; B /. For example, for A < B , the allocation should be skewed in favour of treatment B (to get more information on that treatment) even if A D B . We observe exactly the same picture from our computational results. A similar feature for A < B is observed for different shifts h. Also, we observe remarkably varying proportions of allocations for different h. Exactly the same occurs for A > B , but the allocation proportion is skewed in 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 modification 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 chances of 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 Mandal design versus location-invariant (LI) design; middle row: modified Zang and Rosenberger design versus LI design; bottom row: Bandyopadhyay and Biswas design versus LI design. For Biswas and Mandal design and modified 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 symbols correspond 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 allocation is skewed in favour of the better treatment for small shifts, but the allocation tends to 50:50 for larger positive shifts. For negative shifts, the chances of getting negative estimates are more, and we stick to the initial 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 sticks to 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 ! ˆ

B A q 2 C 2 A B

A B q 2 C 2 A B

and ‰B D

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

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do not depend on the shifts h. The allocation proportions increase with treatment difference. Note that the SDs are reasonably small although not reported. For .A ; B / D .1; 1/, the allocation proportions start at 50% for treatment A for A D B , and it is more than 50% for A < B . For A < B , it starts Copyright © 2014 John Wiley & Sons, Ltd.

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at less than 50% for A D B , and for A > B , it starts at more than 50% for A D B . These are desirable results. Most importantly, as it does not depend on the shifts h, the LI design is preferred over the 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 this figure. 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 the proposed 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 criterion ENWA 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 for the 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,

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4

h

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 (large Cve h)

Tends to 50:50

Tends to 50:50

Invariant

Invariant

A (large ve h)

Highly skewed (> 50%) but high variability 50:50

Tends to 50:50

Invariant

Invariant

50:50

50:50

50:50

< 50%

< 50%

50:50

< 50%

> 50%

> 50%

50:50

> 50%

Minimum

—

—

—

ETR

—

Minimum

—

—

ENWA

—

—

—

Minimum

Varies a lot

Varies a lot

—

Invariant

A for A D B (for A D B ) A for A D B (for A < B ) A for A D B (for A > B ) ETF

Sample size (against h)

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

S. MANDAL AND A. BISWAS

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 treatment difference of D 0:5 for a test at level 5%. Clearly, the required sample size varies remarkably in the ZR and BM formulations with the shift h in the responses, whereas the LI formulation remains invariant in this context also. So, in practice, it is not easy to find the target sample size for ZR or BM rules, unless the 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 Dworkin et 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 treatment of postherpetic neuralgia. There were n D 173 patients of which 84 received the standard therapy placebo and 89 were randomized to pregabalin. The primary efficacy measure was the mean of the last seven daily pain ratings, as maintained by patients in a daily diary using the 11-point numerical pain rating scale (0 D no pain, 10 D worst possible pain) and, therefore, a lower score (response) indicates a favourable situation. After the 8-week duration of the trial, it was observed that pregabalin-treated patients experienced a higher decrease in pain score than patients treated with placebo. We use the final mean scores, that is, 3.60 (with SD D 2:25) for pregabalin, and 5.29 (with SD D 2:20) for placebo as the 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-adaptive trial for n D 173 patients with N.3:60; 2:252 / distribution for pregabalin and N.5:29; 2:202 / distribution for placebo. We update allocation probabilities according to the rule considered. Figure 4 gives the result. The BM procedure has an expected proportion of allocation to A as 0.512 (SD D 0:055), without considering the shift in the data. If we carry out the ZR procedure, the probability of getting negative response at least once is 0.0154. Using the modified version of the ZR rule (as described in simulation section), we have that the expected proportion of allocation to A as 0.549 (SD D 0:053). A shift in the data is quite practical in this scenario. For example, one could record the data in an alternative 11-point numerical scale f1; ; 11g (having 1 no pain, 11 worst possible pain), or f0; ; 10g or may be fR1 ; ; R11 g in general, where R1 < < R11 are suitable 11 numerical scores. With possible shifts in the data, allocation scenarios for BM and modified ZR designs change. For BM design, a positive shift makes the allocation proportion closer to the 50% mark, and a negative shift makes the allocation skewed up to certain extent and then revert back towards the 50% mark. But the SD increases remarkably. For modified ZR design, the allocation comes out closer to the 50% mark for large positive shift. For large

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

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S. MANDAL AND A. BISWAS

negative shifts, the allocation sticks to the 50% mark, the adaptive flavour of the design is lost, and the allocation is mostly based on the initial balanced allocation, driven by the fact of negative estimates of 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 any change in shifts.

5. Influence of change in location Here we provide an account of change in allocation proportion for a change in location. Suppose any allocation design of the form (1.4) is implemented, and an allocation proportion of A0 occurs in favour of treatment A for some .A0 ; B0 / (given A and B ). Thus, p A ‰B0 A0 D p p A ‰B0 C B ‰A0 where ‰k0 D ‰k .A0 ; B0 /; k D A; B. Now suppose that instead of Yk , we observe Yk C h; k D A; B, and thus a shift h in location occurred without changing the scale. Expanding ‰k around k , after some routine steps, we obtain that A .h/, the allocation proportion after the location shift, is p A ‰B .A C h; B C h/ A .h/ D p p A ‰B .A C h; B C h/ C B ‰A .A C h; B C h/ 1 h @‰B @‰B C D A0 1 C 2 ‰B0 @A0 @B0 9 3 @‰B @‰B @‰A @‰A pA pB = C C C @ @ @ @ ‰B0 ‰A0 A0 B0 A0 B0 C O.h2 /5 p p ; .A ‰B0 C B ‰A0 / where

@‰k @j 0

D

@‰k @j

A DA0 ;B DB0

, k; j D A; B.

Now we define influence of location change (IoLC) as A .h/ A0 h A0 h!0 8 1< 1 @‰B @‰B D C 2 : ‰B0 @A0 @B0

IoLC D lim

pA ‰B0

@‰B @A0

C

@‰B @B0

C

pB ‰A0

@‰A @A0

p p .A ‰B0 C B ‰A0 /

C

@‰A @B0

9 = ;

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

) ( p p A ˆA0 C BA A0 ˆB0 1 B0 B B0 D p p p 2 B ˆB0 ˆA0 ˆB0 .A ‰B0 C B ‰A0 /

c c where k0 D k0 and ˆk0 D ˆ k0 , k D A; B. k k For ZR design, the IoLC reduces to

p p A A0 C B B0 1 1 ; p p p 2 B0 A0 B0 .A B0 C B A0 /

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IoLCZR D

S. MANDAL AND A. BISWAS

Figure 5. Plot of influence of location change function for different values of A0 when B0 D 4 and c D 4 (for Biswas and Mandal rule). The black lines are for Biswas and Mandal rule, green lines are for Zhang and Rosenberger rule and the red is for location-invariant rule. For black and green, the solid lines correspond 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 by LI, we have IoLCJ T 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 A and B as the IoLCZR does not exist for negative A and/or B . The black lines are for BM rule, the green 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 be extended for designs with three or more treatments at hand. For simplicity, here we illustrate that with three 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 ‰A C nB ‰B C nC ‰C

(6.1)

A2 2 2 C B C C DK nA nB nC

(6.2)

subject to

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say, where nA C nB C nC D n, and ‰A , ‰B and ‰C are some cost functions associated with the three treatments that are functions of the parameters. For an LI design, ‰A , ‰B and ‰C should be invariant of any change in location of the variables. Note that treatment A is the best if A < B ; C . Thus, we need a ‰A , which is increasing in both A B and A C (i.e. ‰A decreases as A B or A C or both decreases). Denoting by YA , YB and YC the potential responses from the three treatments, we consider Copyright © 2014 John Wiley & Sons, Ltd.

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S. MANDAL AND A. BISWAS

0

1

A C C B A B ‰A D P .YA > min.YB ; YC // D ˆ @ q ;q A A2 C B2 A2 C C2 where ˆ .; / is the CDF of a N2

0 0

1 ;

D q

1 A2 2

A2 C B

(6.3)

distribution with

A2 C C2

:

As an alternative to (6.3), we can as well use 0

1

0

1

B A B C B A C C ‰A D P .YA > YB /P .YA > YC / D ˆ @ q A ˆ @q A A2 C B2 A2 C C2

(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 expected number of wrong allocations (ENWA), which was considered by Biswas and Mandal [27], as follows. Theorem 2 The optimal allocation proportion for the optimization problem (6.1) subject to (6.2) is A D

pA ‰A

C

pA ‰A pB ‰B

C

pC ‰C

:

Similarly for B and C . We carried out detailed numerical study for three treatments along with the three-treatment versions of 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 covariate vector X with p components influences the response. We assume that the model (may be after suitable transformation) can be presented as Yk jX N k C X T ˇk ; k2 ; k D A; B; and X Np .X ; ˙X /: Then 0

1 A B C TX .ˇA

ˇB / B C ‰A D P .YA > YB / D ˆ @ q A: A2 C B2 C .ˇA ˇB /T ˙X .ˇA ˇB / Subsequent development is similar.

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

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(a) We showed the need for an LI design and then provided such a design with different interpretations. (b) Consequently, we provided a multi-treatment generalization and covariate-adjusted version of the design.

S. MANDAL AND A. BISWAS

While (b) is also a non-trivial work, we feel that point (a) is very important, both in identifying a very serious problem that is present in almost all the existing designs and then in suggesting an elegant solution 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 existing response-adaptive designs for continuous responses using optimality criteria, namely the BM design and the modified ZR design, are too undesirable for large shifts (either positive or negative) in the data (in addition 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 this context. We could perform an exercise of calculation of powers of a test of equivalence for different designs and compare them. This is a routine work, and hence, we skipped that except Figure 3, where we provide sample sizes required to achieve a prefixed power. We did not pay our attention to calculate trivia like expected number of failures and follow-up inferences. The reason is that we wanted to provide a flexible target with logical interpretation that is invariant of any shift in the data. But the fact that the existing designs (BM and ZR) perform so poorly with respect to change in locations of the data keeps the 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 pointed that unless a target allocation is ‘location invariant’, it might lead to a completely irrelevant and useless target for allocation. Finally, we focussed our present work on ‘targets’, which is applicable not only in clinical 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 suggestions that led to some improvement over an earlier version of the manuscript. A Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada supported the research of S. Mandal.

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