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A class of adaptive designsUttam Bandyopadhyay a & Atanu Biswas ba Department of Statistics , University of Calcutta ,35 B. C. Road, Calcutta, 700019, INDIAb Applied Statistics Unit , Indian StatisticalInstitute , 203 B. T. Road, Calcutta, 700035, INDIAPublished online: 29 Mar 2007.

To cite this article: Uttam Bandyopadhyay & Atanu Biswas (2000) A class of adaptivedesigns, Sequential Analysis: Design Methods and Applications, 19:1-2, 45-62, DOI:10.1080/07474940008836439

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A CLASS OF ADAPTIVE DESIGNS

U t t a n ~ I3andyopadhj.q Atanu Biswas Department of Statistics Applied Statistics Unit 1Jnivcrsity of Calcutta Indian Statistical Institute 35 13. C. Road 203 H. T. Road Calcutta - 700 019, 1Sl)lA Calcutta - 700 033, IKDlA

S o m e K e y iC.ords and Phrnses: u r n model; mndomzzed play-the-u.inner rule; delayed r e s p o n ~ c .

ABSTRACT

For sequential entrance of patients in a clinical trial several adaptive designs are given by different authors from intuitive considerations. The object of such designs is to allocate the entering patients among two or more competing treatments. IIere ~ v c provide a unified approach to d e ~ i v e a broad class of such designs through a re- cursion relation of the allocation probabilities of the successive entering patients. A spccial class of adaptive designs including the standard randon~ized play-the-u-inner rule is obtained f ron~ the relation. Here both the instantaneous and delayed re- sponse cases are covered.

1. INTRODUCTION

For comparing two or more treatments in clinical trials with sequential entrance of study subjects, data-dependent allocations are becon~ing popular in the recent days, although many experts are concerned about the ethical consequence of such

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46 BANCYOPADHYAY A X E BiSWAS

adaptive designs (see, for example, Byar et al. (1976): Begg (1990), Royall (1991)). In a recent discussion, Zelen and Wei (1995) describe a clinical trial reported by Connor et al. (1994) to evaluate the hypothesis that the antiviral therapy AZT reduces the risk of matrrnal-trrinfant HIV transmission. A standard randomization technique was used resulting 238 pregnant women receiving AZT and 238 receiving placebo. It was observed that 60 newborns were HIV-positive in the placebo group and 20 newborns werc HIV-positive in the AZT group. Yao and LVei (1996) have shown that i f a sl~itable rando~nizcd play-thewinner rule (an adaptive design, to bc dcscribed later) were adopted it could result the allocation pattern in a 300:176 allocation with the greater allocation to AZT and, in this process, the lives of 11 newborns could have been saved.

Robhins (1952) is perhaps the forerunner of adaptive design followed by Anscon~l ,~ (1963) and Colton (1963). Based on Robbins' idea, Zelen (1969) intro- duced his popular concept of play-thewinner (PW) rule. Later A'ci and Durham (1978) and Wei (1979) introduced thc randomized play-the-winner (RPW-) rule t?\. modifying Zcler~'s PLY rulc. ' I h theory of lW\V rulc is further enriched by LVei (1988), B a ~ ~ d y o a d h a and Biswi~s (1996, 1997a. 199%). Rosenberger (1993); h l i ~ t t h e w ar~d I<oscntmgcr (1!)97) (.I(.. LYri (1 988) tlrsrri hrd an exarl permutation trst t)wcvi on some real lifr data on cxtracorporcal ~ncmbrane oxygenation (EC310) to trcm so111c newhorns with respiratory failure (with reference to Hartlett rt al (1985), Cornell, Landenbrrger and Hnrtlct t (1986). \.Pare and Epstein (1985)). A recent usc of an IWW rule in an anti-depression trial of fluoxetine is reported by Tamura, Fhries, Andersen and I-Ieiligenstein (1994). A detailed history of adaptive design is available in Hardwick (1989) and Rosenberger and Lachin (1993), where the later paper discusses many ethical and logistical aspects. Faries e t al. (1993) rightly fee! that. the draxxitic ad~vances in the computer technology and data access in the past decade makes the logistics of the adaptive trials much more feasible. This continuing advancements force the clinical trials more adaptive in all respects and statisticians should actively influence the design of such trials.

It is to be noted that ahlost all the existing adaptive designs available in the literature are initially introduced from intuitive considerations. In the present pa- per our main object is to derive a class of such designs satisfying some basic. re- quirements. We cover two-treatment (instantaneous and dclayed responses) and ~nulti-treat ment cases.

2. GENERAL FORM OF ADAPTIVE DESIGNS

Suppose there is a srqnential ci~ain of patient's entrance in a clinical trial and let 'D ' be an allocation rule by which each patient is treated by either of the two

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CLASS OF ADAPTIVE DESIGNS 47

treatments A and B with success probabilities p~ and p~ respectively. Then, if l r ~ i is the probability of treating the i-th entering patient by A, we aysume that our 'Dl should be such that the sequence {lrA, , i 2 1) is regular. In most of the standard urn designs available in literature ( T . ~ ; } is found to be nionotonic (either increasing or decreasing). For some details, see the appendix. Obviously, in such cases { r A i } converges to some T A In the present context our design 'D' is derived from an urn model as follows : We consider an urn which initially contains a total of 2a balls of both kinds and after the assignment and response of each patient 0 balls are suitably added to the urn and hence, after the i-th st.age, the urn contains (20 + ip) balls. The way these ,tl balls are added a t every stage is the design. Here we assume instantaneous patient's response, i.e., a patient's response is either immediate or i t is obtained before the entrance of the next patient. 'l3elayed response' can also be incorporated in this design and this is considered in section 4 (see also Bandyopadhyay and Biswas (1996)).

Now, in order to get an allocation rule for the successive entering patients, we define a pair (6,. 2;] fol- the i-th entering patient as : 6i (real-:slued) is the indi- rat,or of allocation. It may be vector-valued i f we have more than two treatnients. iVe discuss it in section 5 . 2, (real-wlurd) is the indicator of response. I t may be vector-~alued for Livariatc response. 'I'hen, upto thc i-th entering patien!, we define a statistic T, = T ( d ( , ) , Z( ,)) based on all the earlier assignments b( i j -- (4, . . . , d,)

and all the earlier responses Z(,) = (2'). . . . , Z,), where T,' = (1, Tli , T2;, . . . ,7pi) is sufficient to get inforniation about t h ~ nurnber of allocations and the nurnber of successes/failures to each treatment. lIere we assunle that the conditional proba- bility that 6i+i = 1 given the previous assignments and responses Z(i) is a linear

function of T i ' s , j = l(l)p, i.e., for some u' = (uo, ul, . . . , up ) , we have

It should be noted that, for a balanced ' D ' , TA, = Vi whenever p~ = p ~ , and thus it is reasonable to assume that, for p~ + pg, T A ~ = + d,, where di 2 ( or 5)O V i according as p~ > p~ or p~ < PB with dl = 0. The other d,'s can be obtained from some recursion relation to be described in the next section.

3. ADAPTIVE DESIGNS IN TWO-TREATMENT SET-UP

Here for the i-th entering patient n e define a pair {6 i , Zi} of indicator variables as : 6, = 1 or 0 according as the i-th entering patient is treated by treatment A or B, and Zi = 1 or 0 according as the i-th entering patient results a success or a failure. Then upto the i-th patient we get the following accuntulatcd information :

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CLASS OF ADAPTWE DES!GNS 49

for different cl, c:,. c3, providing a large class of adaptive designs which covers most of the existing and some new designs. In the present paper we will focus our attention to the above choice only. Note that for the RPW rule the recursive relationship was first irltroduced by Wei and 1)urhanl (1978) in their introductory paper on RPW lulc. 'l'llcy liav(1 also developed the f o r ~ ~ ~ u l a ful the l in~iting proportion of patients on each treatment group for RPW rule.

From ( 2 . 1 ) . taking expert ations. we get

Kow we discuss the fblloiving cases :

Case 1 : d, = 0 b i ( a ~ ( 6 , = 1) = 1 ~ i ) .

Then (3.2) gives an identity. Sol\.ing this we have u = ( n . 0 , 0 , 0 , ~ / 2 ) ' , and hence ( 2 . 1 ) gives

which corresponds to the following sampling srheme : Start 1cit11 a n u r n having tlco types of balls A a7~d B, n balls of each type. For a n enterzng patient Ire treat hzm by drauing a ball from the urn with replacement and we add P / 2 balls of each kind after each allocation without paying at tent ion to i ts response. Thus this is the case of complete randomization.

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CLASS OF ADAPT!VE DES!GNS

and hcrlcc the adapt i\.r tlcsign can be taken as . A.c rnrlier, the inil iul ur.rr corn- position is a bnl/s o j each type. Pntient.: are treated by drawing a ball from the u rn with replaccrrtcrrt. H e w s u c ~ c . ~ s by a t reatment results t he addi t ion of T O ball? of the .come kind nlong rl.ith ( 1 - 1)3 balls of the opposite k ind . 011 t l ~ c o ther lrnnd, jni1ur.c by n treofrne~ct rrsu1t.s the addit ion of ( I - T ) O ball.9 of the .cnrnr Xirr t l c ~ , t r ! r.7 bnl l .~ o j f hc oppo.-ite kind. Ilerc T should br 1 i , otherwise the better treatment will have smaller allocation.

Special Case ( i ) : -r = giws complete rar~do~rlizatiot~ (Cast. 1)

Special Case (ii) : T = I l-(.sult~

'I'hus thr acinpti\.rdc~sigrl i s : Stn1.t rrillr urr t / r 3 r r hav ing tu.o types of bn1l.s. n brill.? o j cart) typc. Fbr (111 crrter,ircy pnt i t~nt we trrnt trirn b y drau:iny n ball fr.orn tire c~rn ~ r i t h rc:placen!cnt. I! tlrc r r s p o r m is a Frlccess we add 0 1 1 addit ional fll balls ~ ? f ?!!e a n ? e k i n d Or1 thc o ther Irtrrrti. if thc. reppon.se is o jailr~rc crc atltl n n ntlditionnl 0 boll? of tlrc oppo~ i t r . kirrd. Cltlar-13. this is the st:lndard HI'R'(o.3) schernc of sa t~~pl i r~g .

Case 3 : l'ake

IIcrc, iks in casc 2, d l , 1 , i 2 1 , car1 be expressed as :

and it ran be secn that, as i -, co, d , - l ( p d - p B ) / ( 2 ( 2 - ~ p . . , - r p D ) ) (as in case 2). Using (3.5) in (3.2), and solving the identity we get u = ( 0 , 2 ~ 0 , - 0 , -TO,O) ' , and from (2.1) . we get

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CLPSS OF ADAPTIVE DESIGNS 53

and hence the adaptive design can be taken as : At the outset the u r n conta ins N balls of each kind. IVe treat a patient by drawing a ball f rom the u r n wi th replacement. Success by a t reatment results the addit ion o f p balls of t he same k ind . O n the o ther hand , a failure by a t reatment results the addit ion of all the rp balls of the opposite kind along wi th ( 1 - T)? balls of the s a m e k ind . Thus here we are more strict about a success. If T = 1, we get the standard RPiV rule. Note that whenever 0 < T 5 1, the limiting d u e s is free of T and it is same as that in the case of RPLV. But the exact proportion of allocation depends on T .

Table 1 shows the computations of the exact (with sample size=20 and cu = 8) and limiting proportion of allocation for all the above designs. Greater value of T

implies larger allocation to a better treatment (and hence ethical gain) with some loss of efficiency of the related inference.

4. ADAPTIVE DESIGN FOR DELAYED RESPONSE

In se~.eral real life situatiorls the responses may be instantaneous. l o incorporate such possible dela). in lesponscs we iritruduce a set { t ] , , ( 2 , . . . . , c , I , ) uf indicato~ \.arinbles in addition to 10,. Z,} as follo~\.s : r l , = 1 or U according as the response of t hc, j-t t i p i t i ru t is o l , ~ :liricd or 11ot bcfo~cl ttiv elit? uf the i-th patient, j = l (1) i - 1. Here it is easy to observe that

= total number of successes by treatment A whose

responses are obtained upto the i-th stage, 1

n - i ( i ) = x ~ ~ 1 + l 6 ~ ]= I

= total number of allocation to treatment A whose

responses are obtained upto the i-th stage,

S t ( i ) = x ej,+:Z, ]=I

= total number of successes of the responses upto the i-th stage,

= total number of responses upto the i-th stage.

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54 BAiu'Di'OPADiiYAi' AND BIS'YVAS

T a b l e 1 : Exact (ivith sample size = 20 and a = /3) and limiting proportion of allocation to treatment A in different designs

P A P B T Caw 2 Exact I Limiting

0.4 0.3 0.25

Caw 3 Case 4

and i are sufficient to get information about all types of allocations and suc- cess/failures whose results are obtained. Also irrespective of the response status (whether it is obtained or not), P balls are added to the urn after the assignment of each patient. Then, in (2.1), using so:ne more conditioning variables ~ 1 , + 1 , . . . , ~ , ; + l , we take ZL = (uO, U , , uz. 210, uq, u;)' and Ti = (1, S i ( i ) , Ari(i), SC(i), i t , i)'. Now we assume that

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CLASS OF ADAPTIVE DESIGNS 55

with {xt , t 2 1) as a non-decreasing sequence satisfying x, 4 1 as t + m, and for j # j'. Clearly, elk's being independent of any other 6, and Z;, the set {elk, k 2 j + 1 ) is distributed independently of {clrk, k 2 j l + l ) . From ( 2 . 1 ) , taking expectations as in ( 3 . 2 ) , we get

1Iere we consider the recursion relation of d,'s as :

d,,, = a ( i ) + x b ( l . j ) d j . ]= I

.,;.here a ( i ) ' s 2nd h ( i , j ) ' ~ ([GI. 5xcd j ) arc both :::cnotcnic (in the same directions) and have t h e limits. I f we take

then d i , I . i > 1 : can he expressed as :

and, as i 4 co: d , 4 (p,.l - p u ) / ( 2 ( 2 - p.1 - p B ) ) . Clearly such d,'s satisfy the assumptions on x .4 , '~ . L'sing ( -1 .2) in ( 4 . 1 ) : and solving the identity a.e get u = ( a , 2 P , -0, -P , 0 1 2 , ~ / 2 ) ' , and hence from ( 2 . 1 ) , we get

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BANDYOPADHYAY AND BISWAS

which may be interpreted as the following urn design: Star t with a n urn having two types of balls A and B, a balls of each type. When a patient enters i n the system it is assigned to n treatment by drawing a ball from the urn with replacement. When a patient is assigned to a treatment we add P/2 balls of each kind. Krhen the response is obtained we withdraw those b/2 balls of each kind from the urn and add ,O balls of the same (opposite) kind on which the patient was assigned if the response is a success (failure). This is an RPW rule for delayed response, and is given by Bandyopadhyay and Biswas (1996). Note that in case of possible dclaycd responses, Wei and Durham (1978) suggested to add new balls only when t,he response is obtained. Wei (1988) also indicated this rriodel. Although this model can be applicable in practice, from a mathematical point of view it is difficult to handle as the total number of balls in the urn at any stage

(= ,Oz;=, cji+l) is random Here the denominator of the conditional probability

P(bi+ 1 = . . . , di, Z1,. . . , Zi ,c l ;+ l , . . . , E ~ ~ + ~ ) depends on t l i ~ l ~ . . . , t i i - 1 and the explicit expression of n.4i's are not readily obtained. To overcome this difficulty Bandyopadhyay and Biswas (1996) have introduced the delayed response model (4.3). Biswas (1999), while revisiting the model suggested by Bandyopadhyay and Biswas (1996), has shown that asymptotically the two models are equivalent, and in the small sample case also the difference in perfor~nances is negligible.

5. ADAPTIVE DESIGN IN MULTI-TREATMENT SET-UP

An obvious generalization of the two-treatment k t -up is the possibility of k(2 2) treatments, although not much works are available in the adaptive clinical trial literature in this direction. Different attempts are due to Wei (1979), Li (1995), Anderscn, Faries and Tamura (1994) and Bandyopadhyay and Biswas (1999) in this connection.

First we consider I((> 2) treatments A1, A2, . . . , AIc with unknown success prob- abilities pl,pz, e . . where pk E ( 0 , l ) V k . Suppose there is a sequential chain of patient's entrance, and corresponding to the i-th entering patient we define a set {&i, b2,, . . . , blci, 2,) of indicator variables as follows : bki = 1 or 0 according as treatment Ak is applied or not, k = l(l)IC, and Zi = 1 or 0 according as success or failure occurs for the i-th patient. We denote

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CLASS OF ADAPTIVE DES!GNS

SAL(i) = x 6 k j ~ j j=l

= total number of successes by treatment A k , k = 1 (1 )K ,

!\-,'ik (i) = -i- 6k L I ] = I

= total number of allocation to treatment A k , k = l ( l ) K

It is easy to observe that here ( ( S A k ( i ) > &,(i)) k = l ( 1 ) K ) is sufficient to get information about all types of allocations and success/failures. Initially the urn contains KO balls and we add ( I ( - 1)0 balls for every patient. Thus (2.1) holds here with u = ( uo ,u l , v l , yn,uz, . . . , u ~ { , u ~ ( ) ' and T, =

(1, S ( i ) ( z ) ( i ) ( i ) , S ( 2 ) ( i ) ) . LCk assume that . 7 r ~ , , = P(Ok, = 1 ) = & + dh, where dkl = 0 and d,, = 0 V i , as c::, 6si = 1. Here thc scqurrlcc { d l i ! i.; runnotonic and tm~undcd for ?\.Pry k and a c c n r d i r ~ ~ l ~ has a

limit. Fro111 (2 .1 ) , taking expectations. ive have

Here we consider the following recurrence relation :

with v;(i) = vk ( i ) - ul(i). Clearly, this is of the form (3.1). In particular, we take

Then dk,;i, i 2 1 , can be expressed as

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and it can be shown that , as i - m,

Using (5.2) in (5 .1) and solving the identity using the fact that zfLl ds l , Lve get 1 4 = cr,uk = P(K - 1 ) , u , = - P , s j f k, uk = 0, us = 8 , s # k , and hence from ( 2 . 1 ) we get

and thus the sampling scheme can be interpreted as : S tar t wi th a n u r n having K tvpes of balls A 1 , A 2 , . . . . AIc, a balls of each type. A n enter ing patient i s treated by drawing a ball f rom the u r n w i th replacement. If success occurs we add a n addit ional ( K - 1 ) P balls of the s a m e kind i n the u r n . O n the o ther hand , if failure occurs we add B balls o f each o f the remaining ( K - 1 ) k inds . For K = 2 , it reduces to the standard RPFV rule. For a general I ( , it is called multi- treatment randomized play-thewinner (hIRPtV) rule. This u.a~ first introduced by \Vei (1979) . Bandyopadhyay and Biswas (1999) discussed this scheme of sampling in a decision theoretic frame work.

APPENDIX

Result : d, -' d = (7 - )) (pr - yc ) / (2r + ( 1 - 27)(p,4 + p B ) ) as i - m,

where d, is given by the recursion relation (3 .4) .

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CLASS OF ADAPTIVE DESIGNS 59

Proof : It is easy to note that for ( p 1 , p 2 ) : 1 ) ~ > (the other case is similar),

we have

u ( r - 1) (PA - Y * ) d l = 0 and dz = > 0

2 a + P

From ( 3 . 3 ) , we get

Suppose d 2 5 d , i I? , 3 ( 2 r - 1 ) ( 1 - p~ - p n ) 5 2 0 IIere ( A . 2 ) implies

iCow we obserr.e that

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BANDYOPGDHYAY AND RISWAS

Using (A.2) and ( A 3) , {d,. x 2 2) is monotonically increasing and bounded by d Similarly the case when dl, > d can be tackled and it can be shown that

and hence {d,. i 2 2) is monotonically decreasing and bounded below by d Hence the result follours.

The authors wish to thank the referee and thc editor for their valuable suggestions ... L!.L 1.3 L A ~ \ I I I L I L leu LU an improi.emeni of an earlier version of the manuscript.

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