Optimal three-treatment response-adaptive designs for phase III - - PDF document

Optimal three-treatment response-adaptive designs for phase III clinical trials with binary responses Atanu Biswas Indian Statistical Institute, Kolkata, India atanu@isical.ac.in Saumen Mandal University of Manitoba, Winnipeg, Canada saumen mandal@umanitoba.ca

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• Response-adaptive designs are used in phase III clinical tri-

als to achieve some ethical goal by treating a larger number of patients by the better treatment arm.

• Several such adaptive designs are available for binary treat-

ment responses. ⋄ play-the-winner (PW) rule (Zelen, 1969) ⋄ randomied play-the-winner (RPW) rule (Wei and Durham, 1978) ⋄ generalized P`

• lya urn design (GPU) (Wei, 1979)

⋄ success driven design (Durham et al., 1998) ⋄ birth and death design (Ivanova et al., 2000) ⋄ drop-the-loser (DL) (Ivanova, 2003)

• These designs were suggested primarily from intuition, and

then some theoretical properties are illustrated.

• These designs allocate a larger proportion of patients to the

better treatment.

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• None of these designs are suggested from optimal view point,

yet some of them are quite popular.

• In fact, almost all the real applications available in the lit-

erature are based on the PW (Rout et al., 1993) and the RPW (Bartlett et al., 1985; Tamura et al., 1994; Biswas and Dewanji, 2004).

• Some of the designs can also be very easily extended for

more than two treatments (e.g. GPU, RPW, birth and death, DL).

• Recently, people are interested to derive optimal response-

adaptive designs for binary responses.

• nA and nB: the number of allocations to the two competing

treatments A and B, with nA + nB = n, and pk(= 1 − qk) be the probability of success by treatment k, k = A, B.

• Rosenberger et al. (2001):

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min{qAnA + qBnB} subject to V ar( pA − pB) = pAqA nA + pBqB nB = K, (0.1) for a preassigned constant K.

• R = nA/nB: the optimal proportion for treatment A, πA =

R/(R + 1), comes out to be πA = √pA √pA + √pB . (0.2)

• sequentially estimate of pA and pB based on the available

data, a plug-in estimate of πA to allocate any entering patient to treatment A with probability πA.

• Neyman allocation:

⋄ minimizes nA + nB subject to (0.1). ⋄ an optimal allocation which allocates proportional to stan- dard deviation of any treatment responses. ⋄ may not be ethical.

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⋄ minimizes power (Rosenberger and Lachin, 2002).

• Urn designs like the RPW or the DL:

⋄ limiting allocation πA =

1 qA 1 qA + 1 qB

. (0.3) ⋄ through the adaptation of the urn.

• Any allocation design: minimize

nAΨA + nBΨB, (0.4) subject to σ2

A

nA + σ2

B

nB = pAqA nA + pBqB nB = K, (0.5) for suitable Ψk, k = A, B.

• Now, based on this Ψ, the optimal allocation is

ρΨ =

√pAqA √ΨA √pAqA √ΨA + √pBqB √ΨB

.

• In the RPW or DL rule (0.3), essentially one considers

ΨA = pAq3

A.

(0.6)

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For the optimal rule (0.2) of Rosenberger et al. (2001), one considers ΨA = qA. (0.7)

• Popular urn designs like the RPW and DL are easy to extend

for three or more treatments. Unfortunately the above optimal designs are not quite easy to extend for more than two treat- ments, and no optimal design is available in the literature for more than two treatments and binary responses. The present paper attempts to fulfil that gap.

• Optimal design for three treatments:
• For simplicity, we illustrate our proposed design for three

treatments, A, B and C.

• One can easily extend (0.2) and (0.3) in an intuitive way,

and can suggest an allocation proportion of πj =

1 qj 1 qA + 1 qB + 1 qC

, (0.8)

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• r

πj = √pj √pA + √pB + √pC , (0.9) for the jth treatment, j = A, B, C.

• In fact, (0.8) is the limiting proportion of the urn designs

like the RPW or DL in a three-treatment scenario. But, we do not know whether the rules (0.8) and (0.9) are optimal or not, in some sense.

• Suppose we want to minimize

nAΨA + nBΨB + nCΨC, (0.10) subject to lA σ2

A

nA + lB σ2

B

nB + lC σ2

C

nC = lA pAqA nA + lB pBqB nB + lC pCqC nC = K, where nA + nB + nC = n, and Ψk, k = A, B, C, is a function

• f pA, pB and pC such that ΨA is decreasing in pA (for fixed pB

and pC), and ΨA is positive. Similar interpretation holds for ΨB and pC.

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• Suppose nB/nA = RB and nC/nA = RC, and hence

πA = nA n = 1 1 + RB + RC , πB = nB n = RB 1 + RB + RC , πC = nC n = RC 1 + RB + RC .

• Clearly, the problem (0.10) reduces to minimize

n 1 + RB + RC (ΨA + RBΨB + RCΨC) subject to 1 + RB + RC n

  lAσ2

A + lB

σ2

B

RB + lC σ2

C

RC

   = K.

• After some routine steps, one needs to solve

RB = √ΨA + RCΨC √lBpBqB √ΨB

• lApAqA + lCpCqC

RC

= F1(RC), RC = √ΨA + RBΨB √lCpCqC √ΨC

• lApAqA + lBpBqB

RB

= F2(RB). (0.11)

• Note that, when lA = lB = lC, it immediately gives

πj =

pAqA

ΨA

pAqA

ΨA +

pBqB

ΨB +

pBqB

ΨB

, for j = A, B, C.

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• For Ψj = pjq3

j, we get the allocation (0.8), and for Ψj = qj

we get the allocation (0.9).

• Thus, for lA = lB = lC, the optimal allocation can be di-

rectly extended from the corresponding two-treatment optimal allocation.

• But, the situation will be different when the lj’s are not

same.

• The implimentation of this optimal rule for unequal lj’s is

as follows. ⋄ The first m patients are treated with equal probability 1/3 to each treatment. After m responses are available, we have sufficient data to get an estimate of pA, pB and pC. ⋄ For the allocation of the (i+1)st patient, i ≥ m, we calculate

• pAi,

pBi and pCi, the estimates (possibly MLE) of pA, pB and pC, based on the first i observations. These are simply the proportion

• f successes to the corresponding treatments up to the first i
• patients. We treat these as true values at this stage, and plug-in

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these into (0.11), and solve for RB and RC iteratively. ⋄ To solve for (RB,i+1, RC,i+1), the (RB, RC)-values for the (i + 1)st patient, one can take any reasonable value of RB and RC (say R(0)

B

and R(0)

C ) as the starting values for the it-

eration for the (i + 1)st patient. A reasonable choice may be R(0)

B

= RB,i and R(0)

C

= RC,i, the RB and RC values for the ith patient. Let the values of RB and RC after convergence be RB,i+1 and RC,i+1. Then, we allocate the (i + 1)st patient to the three treatments with probabilities 1/(1 + RB,i+1 + RC,i+1), RB,i+1/(1 + RB,i+1 + RC,i+1) and RC,i+1/(1 + RB,i+1 + RC,i+1), respectively.

• Tables 1-2 give the πj’s for different pj’s (which might be

the estimates at some stage).

• Table 1 considers equal lj-values, where the results of Table

2 are obtained assuming unequal lj’s. We consider four designs for comparison, namely (i) the RPW rule for three treatments, (ii) Rosenberger et al. (2001) optimal allocation for three treat- ments, (iii) our optimal design with Ψj = pjq3

j, and (iv) our 10

• ptimal design with Ψj = qj.
• It is observed that the limiting allocation of (i) and (iii) are

same for equal ljs, and those for (ii) and (iv) are also same for equal lj’s. But, when the lj’s are different, the limiting allocation

• f the rules (iii) and (iv) change quite a bit, whereas those of (i)

and (ii) do not change.

• Keeping (0.6) and (0.7) in mind, the possible choices of Ψk

can be Ψk = pkq3

k, qk,

etc., for k = A, B, C.

• The convergence of the simultaneous equations (0.11) can

be guaranteed by a result from a result from Scarborough (1966,

• Ch. XII, p. 301), the conditions are satified in our set up.
• We have following generalizations in mind, and we are work-

ing on them. First, one may think of finding optimal designs with more than one constraint. Then, optimal design in the presence

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• f covariates is of interest.

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Table 1. Limiting allocation proportions for k = 3, lA = lB = lC, for different values

• f (pA, PB, pC). Design I: RPW rule for three treatments, Design II: Rosenberger et

al.-type design for three treatments, Design III: optimal design with Ψj = pjq3

j, Design

IV: optimal design with Ψj = qj. (pA, pB, pC) (πA, πB, πC) Design I≡Design III Design II≡Design IV (.8,.8,.8) (.333,.333,.333) (.333,.333,.333) (.8,.8,.6) (.400,.400,.200) (.349,.349,.302) (.8,.8,.4) (.429,.429,.142) (.369,.269,.262) (.8,.8,.2) (.444,.444,.111) (.400,.400,.200) (.8,.6,.6) (.500,.250,.250) (.366,.317,.317) (.8,.6,.4) (.545,.273,.182) (.389,.386,.275) (.8,.6,.2) (.571,.286,.143) (.423,.366,.211) (.8,.4,.4) (.600,.200,.200) (.414,.293,.293) (.8,.4,.2) (.632,.210,.158) (.453,.320,.227) (.8,.2,.2) (.666,.167,.167) (.500,.250,.250) (.6,.6,.6) (.333,.333,.333) (.333,.333,.333) (.6,.6,.4) (.375,.375,.250) (.355,.355,.290) (.6,.6,.2) (.400,.400,.200) (.388,.388,.224) (.6,.4,.4) (.428,.286,.286) (.380,.310,.310) (.6,.4,.2) (.462,.308,.230) (.418,.341,.241) (.6,.2,.2) (.500,.250,.250) (.464,.268,.268) (.4,.4,.4) (.333,.333,.333) (.333,.333,.333) (.4,.4,.2) (.364,.364,.272) (.369,.369,.262) (.4,.2,.2) (.400,.300,.300) (.414,.293,.293) (.2,.2,.2) (.400,.300,.300) (.414,.293,.293) 13

Table 2. Limiting allocation proportions for k = 3, lA = 1, lB = 0.5, lC = 0.25, for different values of (pA, PB, pC). Design I: RPW rule for three treatments (same as Table 1), Design II: Rosenberger et al.-type design for three treatments (same as Table 1), Design III: optimal design with Ψj = pjq3

j, Design IV: optimal design with Ψj = qj.

(pA, pB, pC) (πA, πB, πC) Design III Design IV (.8,.8,.8) (.453,.320,.227) (.453,.320,.227) (.8,.8,.6) (.511,.361,.128) (.467,.330,.203) (.8,.8,.4) (.534,.377,.089) (.485,.343,.172) (.8,.8,.2) (.546,.386,.068) (.511,.361,.128) (.8,.6,.6) (.624,.220,.156) (.489,.299,.212) (.8,.6,.4) (.658,.233,.109) (.509,.311,.180) (.8,.6,.2) (.676,.239,.085) (.537,.329,.134) (.8,.4,.4) (.713,.168,.119) (.540,.270,.190) (.8,.4,.2) (.735,.173,.092) (.571,.286,.143) (.8,.2,.2) (.768,.136,.096) (.624,.220,.156) (.6,.6,.6) (.453,.320,.227) (.453,.320,.227) (.6,.6,.4) (.490,.347,.163) (.473,.334,.193) (.6,.6,.2) (.511,.361,.128) (.501,.354,.145) (.6,.4,.4) (.554,.261,.185) (.504,.291,.205) (.6,.4,.2) (.581,.274,.145) (.536,.309,.155) (.6,.2,.2) (.624,.220,.156) (.589,.241,.170) (.4,.4,.4) (.453,.320,.227) (.453,.320,.227) (.4,.4,.2) (.480,.340,.180) (.485,.343,.172) (.4,.2,.2) (.525,.278,.197) (.540,.270,.190) (.2,.2,.2) (.453,.320,.227) (.453,.320,.227) 14