Optimal treatment allocation in comparative biomedical studies

* Correspondence to: Weng Kee Wong, Department of Biostatistics, University of California, Los Angeles, CA 90095,U.S.A.s E-mail: [email protected]

Contract/grant sponsor: NIH; contract/grant number: R29 AR44177-01A1

CCC 0277}6715/2000/050639}10$17.50 Received March 1998Copyright ( 2000 John Wiley & Sons, Ltd. Accepted April 1999

STATISTICS IN MEDICINE

Statist. Med. 2000; 19 :639}648

Optimal treatment allocation in comparative biomedical studies

Wei Zhu1 and Weng Kee Wong2,*,s

1Department of Applied Mathematics and Statistics, State ;niversity of New >ork, Stony Brook, N> 11794, ;.S.A.2Department of Biostatistics, ;niversity of California, ¸os Angeles, CA 90095, ;.S.A.

SUMMARY

In clinical trials involving several treatment protocols, there are usually several objectives of di!erentimportance. A general method for "nding an optimal design which re#ects this practical need is discussed. Inparticular, we show that the graphical method of Cook and Wong can be applied when there are twoobjectives. Several illustrative examples are discussed, including multiple-objective designs for comparingseveral treatment groups with a placebo group. Copyright ( 2000 John Wiley & Sons, Ltd.

1. INTRODUCTION

A common problem in randomized clinical trials is the optimal assignment of patients totreatment protocols. The traditional optimal design is usually derived under a single criterion,although in reality there are usually several objectives in a clinical trial. These objectives typicallyvary in importance to the resercher (primary, secondary, etc.) and frequently represent di!erentoutcome assessments of the trial. The aim of this paper is to "nd designs which more adequatelyeet this practical aspect of the problem.

The current designs for comparing treatment protocols versus a control or comparing allpairwise di!erences among treatment group means usually assume that there is equal interest inall the comparisons. This may not be the case in practice. This issue was raised by Owen andBechhofer and Tamhane in their discussion of the paper by Hedayat et al. [1] on designs forstudying treatments versus a control group. Owen discussed the case when there is more than onecontrol, and the experimental treatment is tested against a placebo, the current commercialproduct and some of the competitors' products. In such a situation, there is di!erent interest inthe various controls and the design should account for this.

A more compelling argument for a similar situation is provided by Bechhofer and Tamhanewhere they emphasized that designs should satisfy the goals of the trial to the maximum extent. In

their case, that meant di!erent precision requirements for di!erent comparisons should be takeninto account when determining the optimal allocation of the experimental resources to di!erenttest treatments and the controls.

In this paper, we develop designs which are #exible and re#ect the needs of the biomedicalresearchers more adequately. These designs can incorporate various objectives of the trialsimultaneously such as di!erent precision requirements for di!erent sets of treatment compari-sons. In this work, we focus on developing optimal designs for comparing di!erent sets oftreatments with or without a control. In the simpler case, when there are two objectives, we showthat the optimal allocation scheme can be found graphically.

The next section provides the set-up and Section 3 demonstrates the method for comparingseveral treatment groups versus a plcebo group. Section 4 discusses design e$ciency versus thepower attained by the design in hypotheses testing and Section 5 concludes with a discussion ofthe method proposed here for other design problems.

2. BACKGROUND

In this paper, we view design as simply an assignment of a proprotion of a "xed number (N) ofpatients to each treatment protocol. The number N is typically determined at the onset of theconsultation by the principal investigator by asking the question &what is the number of patientsyou think you can recruit into the clinical trial?'. Once this is determined along with the numberof treatment groups to be involved, the design question is how to allocate the N number ofpatients optimally to these treatment protocols. Clearly, this depends on the objectives of the trialand whether these objectives are equally important. Typically more patients are allocated to themore important comparisons to ensure greater precision in the inference.

There is very little work on developing multiple-objective optimal designs for clinical trials.A few researchers have considered "nding multiple-objective designs for regression models;Stigler [2] conceptualized the ideas and LaK uter [3] laid the foundations for the construction ofthese designs, including an algorithm for generating special types of the multiple-objectivedesigns. Lee [4] found multiple-objective designs using the Lagrangian approach for the simpleand quadratic models, and Cook and Wong [5] proposed "nding multiple-objective designsusing e$ciency plots. We will demonstrate that the graphical method of Cook and Wong [5] canbe applied to "nd a multiple-objective design in a randomized clinical trial.

The method presented in this work can account for prognostic factors at the design stage aswell, but we will omit this complication here and refer the reader to Zhu [6] for details. Instead,we consider a simpler model

y"f T(x)b#e

where y is the response, b is a vector of unknown parameters and e is the error term assumed tohave mean 0 and variance p2. The vector f (x) is the treatment indicator functions and allobservations are assumed to be independent and normally distributed. Once the number oftreatment protocols is "xed, say k groups, the design problem consists of "nding the optimalproportion p*

iof patients to assign to treatment i, i"1, 2,2, k; the optimal design is the one

which assigns roughly Np*i

to treatment i, i"1, 2,2, k subject to

k+i/1

Np*i"N.

640 W. ZHU AND W. K. WONG

Copyright ( 2000 John Wiley & Sons, Ltd. Statist. Med. 2000; 19:639}648

In what is to follow, we will use m to denote design which assigns piproportion to treatment i (also

denoted as xi), i"1, 2,2, k.

Following convention, we measure the worth of a design m by its Fisher information matrix:

M (m)"k+i/1

pif (x

i) f (x

i)T

Because of the structure of f (x) in this paper, this matrix is diagonal. The optimality criterion ofinterest here is D

A-optimality de"ned by

'(m)"ln DATM(m)~1A D

where A is some user-selected matrix to re#ect the goals of the study. Some choices for A are givenin Section 22.4 of Atkinson and Donev [7], where they used the same criterion to "nd biased coindesigns for clinical trials for treatment comparisons. Designs which minimize the above criterionover all designs are called D

A-optimal design and will be denoted by m*. The D

A-optimal designs

have an appealing feature in that they minimize the volume of the con"dence ellipsoid for the setof parameters Ab. To verify if such a design m is D

A-optimal, we note that the criterion is convex in

m and a standard convex analysis argument yields m* is optimal provided the function

dA(x, m )"f T (x)M (m)~1AMATM(m)~1AN~1ATM(m)~1f (x) (1)

is bounded above by s for all possible values of x in the problem; here s is the rank of A, see Kieferand Wolfowitz [8] for details.

The DA-e$ciency of a design m is E (m)"MDATM(m* )~1A D/ DATM(m)~1A DN1@s. This number is

between 0 and 1 and its reciprocal measures the number of times the design m has to be replicatedto achieve the precision of m* for estimating Ab. Designs with high e$ciencies are sought becausethey require a smaller sample size to attain the same precision in the inference drawn.

There are two standard methods in the literature for constructing optimal designs to satisfymultiple objectives: constrained and compound optimal designs. For simplicity, suppose there aretwo objectives as implemented by the convex functions '

1and '

2, and suppose '

1represents the

more important objective. Let Ei(m) denote the design e$ciency of m relative to the '

i-optimal

design, i"1, 2. The constrained optimal design is the one which maximizes E2(m) subject to the

constraint E1(m)*e

1, where e

1is a user-de"ned constant. An alternative is to "nd the compound

optimal design m*j which minimizes a weighted average of the two criteria

'(mDj)"j'1(m)#(1!j)'

2(m)

where 0)j)1 is a user-selected constant. To avoid scaling problems, the two criteria areexpressed in terms of e$ciencies so that their values are comparable; see Cook and Wong [5] forexamples.

Lee [4, 9] was among the "rst few to propose and study constrained optimal designs forpolynomial regression models. He constructed examples and showed that they are di$cult to "ndeven for simple models [4]. On the other hand, the compound optimal design can be found byusing standard methods because for each "xed j, '(mDj) is also convex. Cook and Wong [5]showed that these two apparently di!erent methods are equivalent in the sense that everycompound optimal design is a constrained optimal design and vice versa. Since compound

OPTIMAL TREATMENT ALLOCATION IN BIOMEDICAL STUDIES 641


optimal designs are easier to generate, one could construct a series of compound optimal designsfor di!erent j's and afterwards obtain the required constrained optimal design graphically fromthe e$ciency plots [5, 10]. Procedurally, the researcher "rst speci"es the design e$ciency (e

1)

required for the more important criterion '1(m) and the e$ciency plot helps us "nd the right

value of j, say j*, to use in the compound optimal design problem. The desired constrainedoptimal design is then mj* .

Clyde and Chaloner [11] showed that constrained and compound optimal designs remainequivalent when there are three or more criteria in a non-linear Bayesian design problem. Part ofthis result will be used in Section 3 when we have more than two criteria.

3. THE PLACEBO}TREATMENT COMPARISON PROBLEM

When there are several objectives in a clinical trial, we need a design which is deemed adequate forall the objectives. Let '

i(m) denote the ith design criterion, i"1, 2,2, k and assume that each can

be expressed as convex function over the set of designs m. As before, we assume 'irepresents an

objective at least as important as 'i`1

. The compound optimal design m*j is the design whichminimizes the convex functional ' (mDj)"+ k

i/1ji'

i(m) where 0)j

i)1 and + k

i/1ji"1. Let

eT"(e1,2, e

k~1), where each e

iis a user-selected constant between 0 and 1. The constrained

optimal design m*e

is the design which maximizes Ek(m) subject to E

i(m)*e

i, 1)i)k!1. A large

value of eisigni"es that the ith criterion is more important in the study.

For our particular problem, we set

y"bi#e

where birepresents the average e!ect of treatment i observed from n

ipatients in the ith group. As

an example, suppose treatment 1 is the placebo group and our ith objective is to estimatebi`1

!b1

as precisely as possible, i"1,2, k!1, with possibly di!erent emphasis by choice ofj1,2, j

k~1. Let AT

i"(1, 0,2, 0,!1, 0,2, 0) where !1 is in the (i#1)th position, let n

i"Np

iand let

'i(m)"ln DAT

iM(m)~1A

iD"ln MN/n

1#N/(n

i`1)N"ln(1/p

1#1/p

i`1)

i"1,2, k!1. The optimality criterion is a weighted average of 'i(m), that is, + k~1

i/1ji'

i(m). Now

for "xed ji, i"1,2, k!1, set

H(p, w)"k~1+i/1

ji'

i(m)#w A

k+i/1

pi!1B

The optimal allocation proportions p*1,2, p*

kare found by solving

LH

Lp1

"

LH

Lpi`1

"

LH

Lw"0, 1)i)k!1

Straightforward algebra shows that this system of equations reduces to

k~1+i/1

JM(p*1)2#4j

ip*1N"2#(k!3)p*

1(2)



Table I. Compound optimal designs for comparing four treatments witha placebo group when there is unequal interest in the comparisons.

j1

j2

j3

j4

p*1

p*2

p*3

p*4

p*5

0.10 0.20 0.30 0.40 0.35 0.08 0.14 0.19 0.240.10 0.10 0.10 0.70 0.39 0.08 0.08 0.08 0.370.10 0.10 0.40 0.40 0.36 0.08 0.08 0.24 0.240.25 0.25 0.25 0.25 0.33 0.17 0.17 0.17 0.17

and for 1)i)k!1

2p*i`1

"JM(p*1)2#4j

ip*1N!p*

1. (3)

An analytical solution for the p*1

is not available and the optimal design has to be foundnumerically from (2) and (3). Table I lists compound optimal designs when k"5 for a few selectedvalues of j

i's.

Depending on the ei's stipulated by the constrained problem, the desired design is selected from

the set of such compound optimal designs. For example, if the most important goal is to comparetreatment 4 with the plcebo, followed by treatment 3 with the plcebo and so on, the e$ciencies ofthe compound optimal design in the "rst row of Table I for the four comparisons are 0.7111,0.6619, 0.6021 and 0.5080, respectively. On the other hand, when there is equal interest in all thefour comparisons, the design in the last row assures that each comparison with the placebo has ane$ciency of 0.6338. These e$ciencies are calculated using the optimal design for each comparisonbetween the placebo and the treatment group, that is, the design which allocates equal number toboth groups.

Note that in this treatment versus plcebo design problem, if k is arbitrary and we setji"1/(k!1), 1)i)k!1 in (2) and (3), the optimal solutions are w*"1 and

p*1"1/(1#J(k!1)) and p*

i"p*

1/J(k!1), 2)i)k. This agrees with the well known alloca-

tion rule stated in Fleiss [12] (p. 96) that when we are equally interested in all pairwisecomparisons with the placebo group, the optimal allocation ratio is J(k!1). It is interesting tonote from (2) that p*

1is invariant under permutation of the weights assigned to the k!1

comparisons. Furthermore, we observe from (3) that if jiincreases, then p*

i`1increases. Thus the

optimal design automatically assigns more patients to the comparison groups of greater interestto the researcher.

Extension of the above problem to the case alluded to by Owens and Bechhofer and Tamhanein Section 1 is straightforward. Suppose there are two sets of treatments to be compared with theplacebo with unequal interest. Assume that the "rst treatment represents placebo and the numberof treatments in each of the two treatment sets are k

1and k

2, respectively. Set k"1#k

1#k

2

'1(m)"

1

k1

k1`1+i/2

ln DATiM (m)~1A

iD

and

'2(m)"

1

k2

k+

i/k1`2

ln DATiM (m)~1A

iD .



Since each treatment within a set is to be compared with the plcebo with the same interest, theoptimal proportion of patients to be assigned to each treatment group within the same set will bethe same. The analogues of (2) and (3) are

k1J(p*2

1#4jp*

1/k

1)#k

2J(p*2

1#4(1!j)p*

1/k

2)"2#(k

1#k

2!2)p*

1

2p*i"J (p*2

1#4jp*

1/k

1)!p*

1, 2)i)k

1#1

2p*i"J(p*2

1#4(1!j)p*

1/k

2)!p*

1, k

1#2)i)i)k

Additionally, the e$ciencies of the compound optimal design are E1(m*j )"(1#Jk

1)2/

(1/p*1#1/p*

2) and E

2(m*j )"(1#Jk

2)2/(1/p*

1#1/p*

k1`2).

4. POWER AND DESIGN EFFICIENCY

In clinical trials, a popular design criterion is the attainment of adequate power in hypothesestesting. In our case, the null hypothesis for the ith comparison is that the di!erence between thetwo group means is zero. Under the normality assumption, it is straightforward to show that fora one-sided test, the power of such a test is

P (Z0'Za Dbi`1

!b1"*O0)"PAZ'Za!

*pJ(1/n

i`1#1/n

1)B

where Z0"(yN

i`1!yN

1)/pJ(1/n

i`1#1/n

1), Z is the standard normal random variable and Za is

the upper 100a-percentile of the standard normal distribution. This formula shows that, asexpected, the smaller the variance p2(1/n

i`1#1/n

1), the higher the power. Consequently, high

D-e$ciency implies high power and vice versa. However, the exact relationship between powerand e$ciency has to be worked out for each situation.

As an example, consider the situation when there are three treatment groups and the "rst groupreceives the placebo. The primary objective is to compare treatment 2 with the placebo and thesecondary goal is to compare treatment 3 with the placebo. The criterion of interest is'(mDj)"j'

1(m)#(1!j)'

2(m), where each criterion is a D

A-optimality criterion. The single-

objective DA-optimal design for objective 1 assigns half the subjects to the placebo group and half

to the group receiving treatment 1. Similarly, the DA-optimal design for objective 2 assigns

half the subjects to the placebo group and half to the group receiving treatment 2. Explicitanalytical solutions for the compound optimal designs can be derived from equations (2) and(3); for each "xed j, a direct calculation shows the optimal allocation rule is p*

1"

MJ(1#4j(1!j))!1N/(4j!4j2), p*2"M2j#1!J(1#4j(1!j))N/(4j) and p*

3"1!p*

1!p*

2.

Figure 1 shows how the three allocation proportions depend on the values of j.The relative e$ciencies of the compound optimal design mj* can be obtained from the formulae

at the end of Section 3 by setting k1"k

2"1. Doing so yields

E1(m*j )"4/(1/p*

1#1/p*

2)

"2M2j2!3j!1#(j#1)J(1#4j(1!j))N/Mj2MJ (1#4j(1!j))#1!2jNN



Figure 1. Proportions versus j.

Figure 2. E$ciency plot.

and E2(m*j )"E

1(m*

1~j). In practice, we plot these e$ciency functions, Ei(m*j ), i"1, 2, versus

values of j between 0 and 1 as shown in Figure 2. The desired j* is then read o! from the plotcorresponding to the value of e

1given in the constrained design problem. As an example, suppose

that it is required that the design should have at least 90 per cent e$ciency for the "rstcomparison. We set e

1"0.9 and draw a horizontal line at e

1"0.9 in the e$ciency plot (Figure 2)



Table II. Constrained optimal designs for comparing two treatment groups with a placebo group (group 1)

e1

e2

p*1

p*2

p*3

j Power (test 1) Power (test 2)

*/p"1 */p"2/3 */p"1/2 */p"1 */p"2/3 */p"1/2

0.50 0.82 0.43 0.18 0.39 0.25 0.864 0.573 0.394 0.969 0.755 0.5440.60 0.76 0.42 0.23 0.35 0.37 0.913 0.638 0.444 0.958 0.731 0.5160.69 0.69 0.414 0.293 0.293 0.50 0.941 0.689 0.484 0.941 0.689 0.4840.70 0.67 0.42 0.30 0.28 0.53 0.945 0.699 0.492 0.937 0.681 0.4760.80 0.54 0.43 0.38 0.20 0.71 0.966 0.745 0.536 0.885 0.600 0.4130.90 0.32 0.46 0.45 0.10 0.88 0.980 0.791 0.575 0.665 0.432 0.2950.95 0.18 0.48 0.47 0.05 0.95 0.984 0.810 0.595 0.475 0.277 0.206

and determine where this line meets the graph of e2. At the cross-point, a vertical line is drawn to

intersect the j-axis at j"0.88. It follows that the compound optimal design mj/0.88is the desired

constrained optimal design.The design e$ciency is invariant to sample size. It is a relative measure of the performance of

an arbitrary design relative to the optimal design with the same sample size. However, the powerof a test depends on the sample size. Assume that we have a total of N"60 patients in this trialand the two null hypotheses are H

0:b

i`1!b

1"0, i"1, 2. Additionally, let a"0.05 and

*/p"1 or 2/3 or 1/2 for each test.Table II lists the constrained optimal design sought in the "rst column and the corresponding

j for the compound optimal design in column 6. In between are the design e$ciency for thesecond objective (e

2) and the proportions of patients to be assigned to the three treatments

(p*i, i"1, 2, 3). The power of the two tests are listed in the last two blocks for three situations:

*/p"1, */p"2/3 and */p"1/2. We note that as j increases, both e1

and the power for test 1increase, while e

2and the power for test 2 decrease. This relationship is re#ected in the power plot

(Figure 3) where we plot the powers of the two tests for each compound optimal design assumingthat */p"1 for test 1, */p"2/3 for test 2, and the total sample size is N"60. We observe thatthe two tests achieve equal power of about 0.775 when j"0.16. In this case, we allocate about 45per cent of the patients to the placebo group, 12 per cent to treatment 1 group and the rest totreatment 2.

When j"0.5, we are equally interested in the two objectives and the optimal design coincideswith the one given by Fleiss [12] (p. 96). This optimal design assigns equal proportion1/(2#J2)"29.3 per cent of patients to each of the two treatment groups and J2 times thatproportion (41.4 per cent) to the placebo group. In this case, the design e$ciencies under bothobjectives are 0.69. If we set */p"1 for test 1 and */p"2/3 for test 2, the power of test 1 is 0.941and the power for test 2 is 0.689. When */p"2, both tests provide powers close to 1.

The method here provides considerable #exibility to the researcher for balancing the variousobjectives. For instance, if higher e$ciency for the "rst objective is desired, say, e

1*0.9, Table II

shows that the desired constrained optimal design allocates about 45 per cent of the subjects toeach of the "rst two groups and about 10 per cent to the third treatment group. Under thisallocation scheme, the "rst comparison has a power of 0.98 (*/p"1) and the second comiarisonhas a power of 0.43 (*/p"2/3). The e$ciency of the design for the second comparison is about0.32. As mentioned before, these e$ciencies do not depend on the sample size N but we canincrease the power of both tests by increasing the total sample size N.



Figure 3. Power plot (R"*/p).

5. DISCUSSION

The equal allocation rule seems to be a popular design strategy in the biomedical literature. Forexample, Loose et al. [13] conducted a double-blind placebo-controlled clinical trial where anequal number of rheumatoid arthritis patients were assigned per day 40 mg, 80 mg, 120 mg ofTenidap or placebo in an 8 week trial. Another example is the one in ¹he ¸os Angeles ¹imes on7 August 1996, where it was reported that the most optimistic "nding presented at the 11thInternational Conference on AIDS was a 73-week clinical trial in which a third were given ddC(Hivid), a third were given saquinavir and the remainder of the patients received both. A mainoutcome variable of interest here is the virus concentration in the blood after treatment. ThisAIDS trial can be viewed as a realization of the example given in Section 4 if we assume thecombination drug as the &placebo' and have ddC and saquinavir as the two treatments.

It is worth noting that Figure 1 shows that the equal allocation scheme is not a compound ora constrained optimal design because the three curves representing p*

i, i"1, 2, 3 do not meet.

More generally, it can be shown that the placebo}treatment comparison problem does not permitequal allocation (that is, p*

i"p, i"1, 2,2, k) as a solution in equations (2) and (3). A direct

calculation shows that in this case, the equal allocation rule is about 67 per cent e$cient for bothobjectives, and when the sample size is 60, the power is 0.936 (*/p"1) for test 1 and 0.678(*/p"2/3) for test 2. In this case, this design performs just about as well as the compoundoptimal design m*j/0.5

(Table II).We focus on the treatment}placebo comparison design problem when the errors are normally

distributed but the methodology is applicable to other design problems. The strength of themultiple-objective approach is particularly noticeable when the di!erent criteria are indi!erentunits or for di!erent models. For example, consider a dose}response study. Parameter estimationis frequently need in such study but it is seldom the end goal. Responses at higher and potentiallyrisky dose levels of the drug have to be modelled to understand the pharmacokinetics of the drug.



Since optimal designs for parameter estimation can perform poorly for extrapolation and viceversa, a multiple-objective optimal design which combines both these di!erent criteria is appeal-ing. Huang and Wong [14] provide details for this design problem. See also Zhu et al. [15], wherethey found Bayesian multiple-objective optimal designs for estimating several percentiles simulta-neously in a quantal dose}response study.

Multiple-objective designs have potential applications to design problems described inKrewski and Kovar [16], Portier and Hoel [17] and Krewski et al. [18]. The practical issues inthese papers were low-dose extrapolation design problems in non-linear problems. Current workin this direction is under way.

ACKNOWLEDGEMENTS

The authors would like to thank the referees for their helpful comments. The research of Wong is partiallysupported by an NIH research grant R29 AR44177-01A1.

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Optimal treatment allocation in comparative biomedical studies