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 1

• 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` olya 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. 2

• 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. • n A and n B : the number of allocations to the two competing treatments A and B , with n A + n B = n , and p k (= 1 − q k ) be the probability of success by treatment k , k = A, B . • Rosenberger et al. (2001): 3

min { q A n A + q B n B } subject to p B ) = p A q A + p B q B V ar ( � p A − � = K, (0.1) n A n B for a preassigned constant K . • R = n A /n B : the optimal proportion for treatment A , π A = R/ ( R + 1), comes out to be √ p A π A = . (0.2) √ p A + √ p B • sequentially estimate of p A and p B based on the available data, a plug-in estimate of π A to allocate any entering patient to treatment A with probability � π A . • Neyman allocation: ⋄ minimizes n A + n B subject to (0.1). ⋄ an optimal allocation which allocates proportional to stan- dard deviation of any treatment responses. ⋄ may not be ethical . 4

⋄ minimizes power (Rosenberger and Lachin, 2002). • Urn designs like the RPW or the DL: ⋄ limiting allocation 1 q A π A = . (0.3) q A + 1 1 q B ⋄ through the adaptation of the urn. • Any allocation design: minimize n A Ψ A + n B Ψ B , (0.4) subject to σ 2 + σ 2 = p A q A + p B q B A B = K, (0.5) n A n B n A n B for suitable Ψ k , k = A, B . • Now, based on this Ψ, the optimal allocation is √ p A q A √ Ψ A ρ Ψ = . √ p A q A √ p B q B √ Ψ A + √ Ψ B • In the RPW or DL rule (0.3), essentially one considers Ψ A = p A q 3 A . (0.6) 5

For the optimal rule (0.2) of Rosenberger et al. (2001), one considers Ψ A = q A . (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 1 q j π j = , (0.8) q A + 1 1 q B + 1 q C 6

or √ p j π j = , (0.9) √ p A + √ p B + √ p C for the j th 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 n A Ψ A + n B Ψ B + n C Ψ C , (0.10) subject to σ 2 σ 2 σ 2 p A q A p B q B p C q C A B C l A + l B + l C = l A + l B + l C = K, n A n B n C n A n B n C where n A + n B + n C = n , and Ψ k , k = A, B, C , is a function of p A , p B and p C such that Ψ A is decreasing in p A (for fixed p B and p C ), and Ψ A is positive. Similar interpretation holds for Ψ B and p C . 7

• Suppose n B /n A = R B and n C /n A = R C , and hence π A = n A 1 n = , 1 + R B + R C π B = n B R B n = , 1 + R B + R C π C = n C R C n = . 1 + R B + R C • Clearly, the problem (0.10) reduces to minimize n (Ψ A + R B Ψ B + R C Ψ C ) 1 + R B + R C subject to   σ 2 σ 2 1 + R B + R C B C  l A σ 2    = K. A + l B + l C n R B R C • After some routine steps, one needs to solve √ Ψ A + R C Ψ C √ l B p B q B R B = = F 1 ( R C ) , √ Ψ B � l A p A q A + l C p C q C R C √ Ψ A + R B Ψ B √ l C p C q C R C = = F 2 ( R B ) . (0.11) √ Ψ C � l A p A q A + l B p B q B R B • Note that, when l A = l B = l C , it immediately gives � p A q A Ψ A π j = , � p A q A � p B q B � p B q B Ψ A + Ψ B + Ψ B for j = A, B, C . 8

• For Ψ j = p j q 3 j , we get the allocation (0.8), and for Ψ j = q j we get the allocation (0.9). • Thus, for l A = l B = l C , the optimal allocation can be di- rectly extended from the corresponding two-treatment optimal allocation. • But, the situation will be different when the l j ’s are not same. • The implimentation of this optimal rule for unequal l j ’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 p A , p B and p C . ⋄ For the allocation of the ( i +1)st patient, i ≥ m , we calculate p Ai , � p Bi and � p Ci , the estimates (possibly MLE) of p A , p B and p C , � based on the first i observations. These are simply the proportion of successes to the corresponding treatments up to the first i patients. We treat these as true values at this stage, and plug-in 9

these into (0.11), and solve for R B and R C iteratively. ⋄ To solve for ( R B,i +1 , R C,i +1 ), the ( R B , R C )-values for the ( i + 1)st patient, one can take any reasonable value of R B and R C (say R (0) and R (0) C ) as the starting values for the it- B eration for the ( i + 1)st patient. A reasonable choice may be R (0) = R B,i and R (0) = R C,i , the R B and R C values for the B C i th patient. Let the values of R B and R C after convergence be R B,i +1 and R C,i +1 . Then, we allocate the ( i + 1)st patient to the three treatments with probabilities 1 / (1 + R B,i +1 + R C,i +1 ), R B,i +1 / (1 + R B,i +1 + R C,i +1 ) and R C,i +1 / (1 + R B,i +1 + R C,i +1 ), respectively. • Tables 1-2 give the π j ’s for different p j ’s (which might be the estimates at some stage). • Table 1 considers equal l j -values, where the results of Table 2 are obtained assuming unequal l j ’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 Ψ j = p j q 3 j , and (iv) our 10

optimal design with Ψ j = q j . • It is observed that the limiting allocation of (i) and (iii) are same for equal l j s, and those for (ii) and (iv) are also same for equal l j ’s. But, when the l j ’s are different, the limiting allocation of 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 Ψ k can be Ψ k = p k q 3 k , q k , 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 11

of covariates is of interest. 12

Table 1. Limiting allocation proportions for k = 3, l A = l B = l C , for different values of ( p A , P B , p C ). Design I: RPW rule for three treatments, Design II: Rosenberger et al.-type design for three treatments, Design III: optimal design with Ψ j = p j q 3 j , Design IV: optimal design with Ψ j = q j . ( p A , p B , p C ) ( π A , π B , π C ) 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