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Biometrika Trust

Optimum Biased Coin Designs for Sequential Clinical Trials with Prognostic factorsAuthor(s): A. C. AtkinsonSource: Biometrika, Vol. 69, No. 1 (Apr., 1982), pp. 61-67Published by: Biometrika TrustStable URL: http://www.jstor.org/stable/2335853 .

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Biornetrika1982), 69, 1, pp. 61-7 61Printed n GreatBritain

Optimum iasedcoindesignsfor equential linicaltrialswithprognostic actorsBY A. C. ATKINSONDepartment f Mathematics,mperialCollege, ondon

SUMMARYThe theory f DA-optimum esignofexperimentss used to provide designs of thebiased cointypefor equentialclinicaltrials n thepresence, r absence, of prognosticfactors.The designs void many arbitrary eatures f earlier chemes.Somekeywords:Balance forprognostic actors;Biased coin design; DA-optimum xperimentaldesignSequential clinical rial.

1. INTRODUCTIONPatients na clinical rialarrive equentially nd are assigned o one of t treatments.This assignment hould maintain a balance between the numbers receiving eachtreatment, et should be sufficientlyandom to avoid any suspicion of conscious orunconscious heating.To balance theserequirements fron 1971) introducedbiasedcoin designs or he comparison ftwo treatmentsnwhich llocationofthe treatmentsdetermined robabilistically,ut witha bias towards heunderrepresentedreatment.Onedisadvantage fEfron's cheme s that t does not ncludebalance overcovariatesor prognostic actorswhichmay affect he responseof the patientto the treatment.Biased coin schemeswhichdo force alance over bothtreatments ndprognostic actorsaregivenbyPocock & Simon 1975)andEfron 1980).Thepropertiesf hedesignshavebeenelucidatedbynumerical tudies ndthey re nowbeing ncreasinglysed nclinicaltrials. Reviews of the literature nd discussionsof the practical implications f thedesigns regivenbyPocock (1979) and Simon 1979). However, hedesigns ufferromthedisadvantagethatthey relyon arbitrary unctions o achieve the desiredbalance.The procedures hus ack a firm heoretical ramework.

An alternative pproach nthepresence fprognostic actorsBegg& Iglewicz,1980)uses optimumdesign theory o suggest deterministicesign criterion, hich s thenmodified or omputational onvenience.n thispaperI use optimum esigntheory oprovide procedure fthe biasedcointypefor n arbitrary umber ftreatmentsnthepresence,or absence, of prognosticfactors.This has the theoretical dvantage ofobviatingdependence n a seriesofarbitraryunctions. he necessary ptimum esigntheorys presentedn ?2 and, in ? 3, appliedto biased coinexperiments. he simplestcase, that oftwotreatmentsn the absence ofprognostic actors,s studied n ?4. Theextension o threeor more treatmentss in ? 5, followed,n ? 6, by the allowance forprognosticactors.Although,n the biased coindesigns, llocationoftreatmentss madeprobabilistically,he results fthepapercan be used for hesequentialconstruction fDA-optimum esigns nwhichrandomizations notintroduced ytheexperimenter.All the designsmentioned o far are sequential,but not adaptive. In the usefuldistinctionmade by Pericchi (1981), they are data-dependent but not outcome-dependent: ach allocation depends on the previous llocationsand prognostic actors,

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62 A. C. ATKINSONbutnot on theresponses.This is a crucial differencerom he designsof Louis (1975)whenallocation s made on the basis of earlierresponses.

2. D- AND DA-OPTIMUMDESIGNSFor the linear modelE( Y) = xTflwith ndependent observations of variance a2, thevarianceofthe least squaresestimateof B svar (j) = U2(XTX) - 1

whereXT X is thep xp dispersionmatrix, ssumedto be offull ank,which esults romthe nobservations. he fitted alue at x is y'(x)= PTxwithvar y-(x)} a2 XT(XT X) -1 X.The optimumdesignof experimentss concernedwith the choice ofX to minimizevariousfunctionsfthematrixvaluedvarianceof1?. succinct ccountofthetheorysgiven by Silvey (1980). Both in the theoryand in the construction f optimumexperimental esigns t is convenient o replacethe n point designbya measure overthedesignregionX. For an exactn trialdesign, hat s one forwhich heweights t theexperimental oints reintegermultiples f /n, he measure s denoted n. If the ntegerrestrictions removed he design srepresentedy 4. Thedispersionmatrix fthedesignis calledM(4) whenM(4") = n l XT X). It is also convenient,nstead of var{y(x)}, toconsider he standardizedvariance

d(x, ) = XTM-1(4)x (1)Onedesign riterion hichhas beenmuch tudied s that ofD-optimalityn which hedeterminant etM(4) is maximized.This minimizes he generalizedvariance of theparameter stimates.Another riterion, -optimality,s concernedwith he varianceofthepredicted esponse. fwe let

d(4) = sup d(x,4)xexa G-optimum esign s one whichminimizes (4), that is it minimizes he maximumvariance of the predictedresponseover the design region.The celebratedGeneralEquivalenceTheorem Kiefer& Wolfowitz, 960)statestheequivalenceofthefollowingthreerequirementsn theoptimummeasure

(i) 4* maximizes detM(4);(ii) 4* minimizes (4);(iii) d(c*) = p, the number f inearly ndependent arametersn the model.This equivalenceofD- and G-optimumesigns s used in thesequentialconstructionfD-optimum esigns.Givenan ntrialdesign, he n + I)st trial s added at thepointwhered(x,Xn) is a maximum. he computational implicityf this equentialconstruction asignoredby Begg & Iglewicz 1980).The criterion fD-optimalitys appropriate fall theparametersn a model are ofinterest. ut in a clinicaltrial nterest s often n contrastsbetween reatment ffects.Supposethe contrasts re s linearcombinationswhich re elements f the vectorAT/,whereA is an s xp matrixofrank s < p. The covariancematrixof the least squaresestimateAT is proportional o ATM '() A and the analogue ofD-optimality s tomaximizedet AT M- '(c) } ' . This criterion,alledDA-optimalityy Sibson 1974), sa specialcase of S-optimalityLaiuter, 976).

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Optimum iased coindesigns 63The Equivalence Theorem orDA-optimalitys describedby Silvey 1980,p. 44). Theanalogue ofthevariance 1) is thequantity

dA(x, 8) = XTM- 1 E) A ATM- '(E) Al}-1AT M-'(E,) x (2)Sequential DA-optimumdesigns are generatedone trial at a time by adding anobservation t the x where 2) is a maximum.For a DA-optimum esign

supdA(x, *) =s (3)xexIn thispaper our nterestwillbe indesigns or stimating reatment ontrasts, o that weshall consider A-optimum esigns.A brief omparisonwithD-optimum esigns sgivenin ?4.

3. A BIASED COIN DESIGNThe design regionX consistsoft points,the ith ofwhichcorrespondso allocatingtreatment to the nextpatient. Let thecorrespondingalue ofdA(x, ) be dA(i, (). Thesequential construction f the optimumdesignallocates the (n+ 1)st patientto thetreatment orwhichdA(i,Xn) s a maximum.At theoptimumdesign,which llocates afractionf ofthepatients o treatment, all dA(i, *) areequal. A largevalue ofdA(i, n)indicates treatmentwhich s underrepresented.In trialswith everalcentres nd clinicians thas been arguedthat random llocationis not necessary incethe order farrival fpatientsprovides ufficientandomization oguardagainst election ias. If this s the case thesequentialoptimum esign anbe useddirectly. ut ifrandomizationsrequiredwepropose, s the basisof biasedcoindesign,choosing reatment withprobabilityPi= {f dA(i, Jn)}j{Efi dA(i,Xn)

This has thepropertyhatfor heoptimum esignwhenXn and (3) is satisfied,achtreatments selectedwithprobability i. If thedesigndepartsfrom heoptimum, heallocationattempts o reduce thedeparture.The values oftheoptimum ractionsf dependuponthe matrixA. If interests inthet- 1 dimensional pace ofall contrasts rthogonal o themean,theDA-optimum esignconsistsof allocatingan equal numberof patientsto each treatment.Thus, for alltreatments, equals l/t,which s also theD-optimum esign.Thebiasedcoin rulethensimplifieso choosing reatment withprobabilityPi= {dA(i, ~X)}/{ dA(i,Xn)} (4)

When Xn= * and the design is balanced, each treatment s selected with equalprobability.Anarbitrary eature fthesedefinitionsfpi is thatdA(i, ) could be replacedby anymonotone unctionf tself.n theory he monotone unction ouldbe chosen obalancethe oss due to selectionbias against the oss due to imbalance n treatments llocation.This possibility s not explored n the presentpaper.

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64 A. C. ATKINSON4. TwO TREATMENTSWithtwo treatmentsnd no prognostic actors he model s E( Y) = f31i = 1,2). Ifinterest s in the differencen treatment ffects l1/2, AT = (1 -1), so that s = 1.If, of the n patients, n1 have received treatment1 and n2 treatment2 then

M(4,) = diag {nj/n},a diagonal matrix. Substitution in (2) shows that forthe criterionofDA-optimalitydA(l, n) = n2/n1, dA(2, n) = nl/n2.

Fromthe biased coinrule 4) theprobability fselecting reatment isn2Pi = + (5)

withthe complementaryxpression or 2.This DA-optimumesignhas beenderivedon theassumption hatonlythedifferencein treatmentss of nterest,f the meantreatmentffects also relevant heD-optimumdesign hould be used. From (1) d(i, n) = n/ni i = 1,2). If these variancesare used asthe basis of a biased coin rule

p1 _ n2/n, p' nl/n. (6)For both D- and DA-optimumesigns heoptimum llocation s to have equal numbersofpatientsreceiving ach treatment, hat is n1= n2 -2n. The equivalencetheoremsconfirm his obvious result sincedA(i, d*) = 1 = s and d(i,4*) = 2 = p, the numberofparametersn the linearmodel.Somenumerical aluesofthe allocationprobabilities 1 andp' aregiven nTable 1forvaluesofn and ofthe imbalance' defined s n-nl. Both rulesderivedfrom ptimumdesigntheoryprovidea 'restorative orce'which ncreases s the designmoves awayfrombalance. This is in contrast o the original uggestion f Efron 1971) in which,ifn2 > n1, treatment1 is chosen with probabilitygreaterthan -, with2/3Efron'sfavourite alue. The lack ofresponse fEfron's cheme o themagnitude fn2-n1 ledWei (1977) to proposean urn designwhich, ikeour scheme, oes respond o increasingimbalance.The results fTable 1show thatthe designbasedonDA-optimalityespondsmuch more to imbalance than does the D-optimumbased design. For n = 10 animbalanceof 1 gives a valueof 1 > 2/3but, as n increases,arger mbalances reneededto achievevalues comparablewithEfron'srule, resultwhichfollows n expressing ias a function f theration2/n1.

Table 1. Probability f ssigning reatmentunder hree chemes ortwo reatmentsn theabsence fprognosticactors;n is thenumber ftrialsDA-OPtimUm D-optimumP1 equation (5) p'1 equation (6)Imbalance Numberoftrials Numberoftrials Efron2'nn 10 20 50 100 10 20 50 100 nunspecified1 0 692 0 599 0 540 0 520 0 6 0 55 0 52 0 51 0 6672 0'845 0'692 0579 0'540 07 060 054 052 06673 0941 0775 0618 0560 08 065 056 053 06674 0 988 0 845 0 656 0 579 0 9 0 70 0 58 0 54 0 667.5 1 0 900 0 692 0'599 1 0 75 0 60 0 55 0 6676 0-941 0727 0618 080 062 056 0667

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Optimumbiased coin designs 655. THREE OR MORE TREATMENTSWith two treatments, he formof the matrixAT is obvious. With threeor moretreatments henall contrasts etween hetreatmentsre of equal interest he form fAT is arbitraryprovidedthat the s = t-I dimensionalspace of the constrasts s

orthogonal o the overallmean. n the absenceofprognostic actorsA is irrelevantncethe valuesofdA(i, Xn)have been obtained.However, n explicitform f A is needed forthecalculationof designswithprognostic actors nd one possibleform s given n theAppendix.The calculationsoutlined here ead to the result hatdA(i, ln)= (n-ni)/ni= ri-1,

whereri= n/ni, he reciprocal f theproportion fpatientsreceiving reatment .The DA-optimumdesign again, of course, consists of equal replication anddA(i, 4*) = t 1. The biased coin designtherefore as probabilities ivenbyPi= (ri-1)/E ri-t)

which s the appropriategeneralization f 5) to morethantwotreatments.

6. PROGNOSTIC FACTORSWe now extend the discussionto includeprognostic actors.The linear model iswrittennpartitioned ormE(Y) = XT: = XT1+X2, '(7)

where l is the vector f ndicator ariablesfor hetreatments ndx2 sthevector fp - tprognostic actors. incethe nuisance parametersf2are not of nterestwe require heDA-optimum esignfor helinear combination T ,BwhereLT = (AT 0), (8)

withA as before dentifyinghe t 1 contrastsof interest.We are thus designingexperimentso estimateATf,3 as precisely s possible.This is a generalization fthecriterionfDs-optimalitybtainedwhenAT is the dentitymatrix Silvey,1980,p. 45).To calculatethedesignwe needto partition hedispersionmatrix n accord with 7)and let

Umll m12]

Wecannownolonger btainanalytic xpressions or hepi. Howeverthecalculation fdA(i,X")can be simplified.or thecontrasts ivenby (8) thegeneral xpression2)becomes dA(x,4 =XTM11BM"1 +2XTMM"BM12X +XTM21BM12X (9)whereB = A(ATMllA)-AT.Calculation fthequadraticform9) togetherwith earching verthedesignregiontofind he t valuesof A(i, Xn) is simple.The threekernels fthequadraticform ependuponpast history nd so can be calculated before he arrival of the next patient.Thevectorx2 is the set of prognostic actors or he newpatient. Once these areknown 9)

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66 A. C. ATKINSONreduces o a quadraticform nx1,a linearterm nd a constant.But x1 is the ndicatorvariable for the treatment elected. Thus searchingover X is achieved by pickingsuccessive diagonal elements of M" BM" and successive terms of the vectorMl BM12 x2. With the increasing vailability of cheaper and ever more powerfulmicroprocessors,hese calculationsdo not seem unduly onerous. Once the t valuesdA(i,Qfl)have been calculated, t onlyremains o find he probabilities i given n ? 3 andto performhebiased coin allocation.

7. DiscussioNThe theory ftheoptimum iased coin designs escribedherefalls ntotwo parts. Thefirsts the use of optimum esign heory o provide measureof mbalance.Unliketheprocedures fEfron 1980),Pocock & Simon 1975) and Wei (1978) thedesignsdo notdependupon arbitrarymeasures f mbalance. f all contrasts etween reatment ffectsare of nterest A-optimum esigns reappropriate:fonly few ontrastsreof nterestthese can be reflectedn the choice of the matrix A. Similarly he structure f thecovariatesor of the treatments an be elaboratedwithout ny complications rising,other hanthe needto calculatetheappropriate alues ofthedA(i, n. The method husprovides n exampleof justhow usefulmathematics s inthetheory fdesign' Kiefer,1959, 1972) free from any of the ironies and sarcasms which marred the earlierappearances of the phrase.The second part of thetheory oncerns heuse of 4) to provide biasedcoindesign.The results fTable 1 showthat forn = 10theDA-optimumesign espondsmuchmorequicklyto imbalancethan does Efron'sdesign.This may carrywith t an increasedpossibilityof selectionbias since the next treatmentmightbe evident with highprobability. imilarly oo small a response o imbalancecould lead to nearlyrandomexperimentation, itha lack ofbalance in the completed xperiment. ne possibilitythatcouldbe exploredwould be toreplace 4) bya normalized umofpowers fdA(i, Xn)with, s a furtherlaboration, hepower function fn. But before hedesigns escribedinthepaper,or thesegeneralizations,anhopefor cceptance n clinical rials,numericalcomparisonswill have to be made withtheprocedures f Efron 1980) and Pocock &Simon 1975).

I am very grateful o Henry Wynnwho insistedon the need, in ? 3, to allow foroptimumdesignswithunequal replication.APPENDIXCalculation fdA(i, n)for three r more reatmentsLet 1 -I 0 ... 0

AT= I 0 -1 ... 0

1 0 0 ... -1so that we consider the set of t- 1 contrasts # fi (i = 2, ..,t). If fi= ni/nthedispersionmatrix s M(Ln)= diag{f1}.f further .= I/f.= n/n..

ATM-l(4)A = r1J+diag{r2,...,rt},

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Optimumbiased coin designs 67whereJ is (t- 1) x (t- 1) with all elements ne. The identity

(C+UVT)-l= C-1_ C-1>jCwhereu and v are vectors, eads to the result

f2(1 -f2) -f2 f3 ** -f2ft{A TM 1(4)A}'= -f2 f3 f3(1-f3) *** -f3 ft

-f2ft -f3ft *.. ft(l-ft)fromwhich tfollows hatdA(i, n) = (n ni)/ni-ri-

REFERENCESBEGG,C. B. & IGLEWICZ,. (1980).A treatment llocationprocedure or equential linical rials.Biometrics36, 81-90.EFRON, B. (1971). Forcing sequentialexperiment o be balanced. Biometrika 8, 403-17.EFRON,B. (1980). Randomizing nd balancing complicated equentialexperiment. n Biostatistics ase-book, d. R. G. Miller,B. Efron,B. W. Brown and L. E. Moses,pp. 19-30. New York: Wiley.KIEFER, J. (1959). Optimum xperimental esigns withdiscussion).J. R. Statist. oc. B 21, 272-319.KIEFER, J. (1972). Contributiono discussion f papers by H. P. Wynn and P. J. Laycock. J. R. Statist.Soc. B 34, 177-8.KIEFER,J. & WOLFOWITZ,. (1960). The equivalenceof two extremum roblems.Can. J.Math.12, 363-6.LAUTER, . (1976). Optimal multipurpose esignsforregressionmodels.Math.Oper.u. Statist. , 51-68.Louis, T. A. (1975). Optimal allocation in sequential tests comparingthe means of two Gaussianpopulations.Biometrika2, 359-69.PERICCHI, L. R. (1981). A Bayesian approachto transformationso normality. iometrika8, 35-43.POCOK, S. J. (1979). Allocationofpatientsto treatmentsn clinicaltrials.Biometrics5, 183-97.POCO0K, S. J. & SIMON, . (1975). Sequentialtreatment ssignmentwithbalancingforprognostic actors nthe controlled linicaltrial. Biometrics1, 103-15.SIBSON, R. (1974). DA-optimalitynd duality.In Progress n StatisticsProc. 9th European MeetingofStatisticians,Budapest, 1972),Ed. J. Gani,K. Sarkadi and I. Vineze, Vol. 2, pp. 677-92. Amsterdam:NorthHolland.SILVEY,S. D. (1980). OptimalDesign.London: Chapman& Hall.SIMON,R. (1979). Restricted andomization esigns n clinicaltrials.Biometrics5, 503-12.WEI, L.-J. (1977). A class ofdesignsfor equentialclinicaltrials.J. Am. Statist.Assoc. 72, 382-6.WEI,L.-J. 1978).Anapplication f n urnmodelto thedesign f equential ontrolled linical rials.J. Am.

Statist. Assoc. 73, 559-63.[Received uly1981.RevisedOctober981]