Fixed Width Confidence Interval of P ( X < Y ) under a Data Dependent Adaptive Allocation Design

Abstract: The present article is related to a nonparametric fixed-width confidence interval estimation of the parameter θ = P (X < Y ) = ∫ F (y)dG(y), whereF andG are two unknown continuous distribution functions. The estimation procedure is based on a sample obtained under some non-iid adaptive situation. We provide various asymptotic results related to the proposed procedure and compare it with a non-adaptive procedure.


Introduction
Suppose a clinical trial is conducted for comparing two competing treatments, say A and B.Here each entering patient (subject) is to receive one of the treatments once by using a data-dependent adaptive allocation design.Such an allocation is sequential in nature.It can be seen that the design is balanced when the two treatment effects are identical, and it becomes skewed if there is a treatment difference and a larger number of patients is expected to be treated by the better treatment in course of this allocation.Let Z i be the response of the i-th entering patient.We assume that Z i ∼ F or G according as the i-th patient receives treatment A or B using the adaptive design, where F and G are two unknown continuous distribution functions (d.f.'s).Obviously, the Z i 's are neither independently nor identically distributed.Here, using a data dependent adaptive allocation design, we consider a fixed-width confidence interval estimation of θ = P (X < Y ) = F (y)dG(y).Here, this type of inference would be worthwhile to compare the remission times by the two drugs A and B. Specifically, it is intended to make an inference about the probability of requiring lower remission time by one drug than the other.This type of inference is, however, very common in clinical trial setting.
With increasing popularity of adaptive designs in phase III clinical trials, the real life applications of such designs are gradually increasing (see e.g. Bartlett et al., 1985, Tamura et al., 1994, Ware, 1989, Rout et al., 1993, Muller and Schafer, 2001, and Biswas and Dewanji, 2004).In order to have a better understanding, different theoretical properties of Austrian Journal of Statistics, Vol. 36 (2007), No. 3, 189-205 such designs were studied and examined by a number of research workers.These are mostly on binary responses.But our present work is related to continuous response adaptive design.In an adaptive design, we have a sequence of indicator variables δ 1 , δ 2 , . . ., such that δ i = 0 or 1 according as A or B is used to treat the i-th entering patient, and δ n+1 is allowed to depend on {(δ i , Z i ), i = 1, 2, . . ., n} or {δ i , i = 1, 2, . . ., n}.The present framework is related to the first case only.We write We set δ 1 = 1, δ 2 = 0 and, for each n ≥ 2, we find, respectively, the numbers of patients treated by A and B up to the n-th stage as and counting the number of times an X-observation is smaller than a Y -observation up to the n-th stage, where We note that for any n N An + N Bn = n .
Then an appropriate estimator of θ is given by Suppose θn is strongly consistent for θ.Then, for given d > 0 there exists a stopping variable (finite with probability one) defined by which can easily be related to a sequential fixed-width confidence interval of θ based on θn .
Many researchers worked on the variables of the type (2) under various non-adaptive situations.These are e.g.Hjort and Fenstad (1992) and Ghosh et al. (1997).In the present situation, assuming continuous responses, our work is also related to (2) by considering an adaptive design which allows δ n to depend on all the previous allocations and observations in order to achieve some ethical gain in terms of a larger proportion of allocation to the better treatment.In this connection, one can also go through the work by Rosenberger and Sriram (1997) for an application of adaptive design to the variable of the type (2) using binary responses.
The rest of the paper is organized as follows.In Section 2, we describe our adaptive allocation rule along with some results.Some asymptotics related to the random variable N a (d) are studied in Section 3. In Section 4, we briefly discuss some asymptotics related to the variable of the type (2) under non-adaptive equal allocation scheme.Two natural sequential fixed-width confidence intervals are constructed in Section 5. Section 6 contains some numerical computations for comparing the two schemes.Finally some concluding remarks are given in Section 7.

The Design and The Related Results
There are several adaptive designs primarily for phase III clinical trials.These are, e.g., play-the-winner rule (Zelen, 1969), biased coin design (Efron, 1971), randomized playthe-winner rule (Wei and Durham, 1978), generalized Polya's urn design (Wei, 1979), and the success driven design (Durham et al., 1998).Also Hu and Zhang (2004) worked with the doubly adaptive biased coin design.These designs are for binary responses of the study variables.The binary response trials are also used by Rosenberger et al. (2001) in connection with an adaptive design.The designs by Rosenberger (1993Rosenberger ( , 2002) ) and Bandyopadhyay and Biswas (2004) are for continuous responses of the study variables.
In the present study, we work with the allocation rule described by Bandyopadhyay and Biswas (2004).The rule is a generalization of randomized play-the-winner (RPW) rule for continuous responses.We describe the rule in the following.The Rule: Suppose we have a sequential chain of study subjects and we are to allocate them to either of the treatments A and B. We start with allocating the treatment B to the first incoming subject and the treatment A to the second incoming subject such that δ 1 = 1 and δ 2 = 0.At the n-th (n > 2) allocation we make use of an urn which has generated α + βT n and α + β(N An N Bn − T n ) balls of kinds B and A, respectively, yielding a total of 2α + βN An N Bn balls in the urn, where α and β are some positive integers.We draw a ball from the urn and allocate the entering subject by the treatment identified by the drawn ball.Then we add β(T n+1 − T n ) and β(N An+1 N Bn+1 − N An N Bn − T n+1 + T n ) balls of kinds B and A to the urn.This process is continued, and hence, the conditional probability of {δ n+1 = 1} given the earlier data is where Z (n) = (Z 1 , . . ., Z n ) and δ (n) = (δ 1 , . . ., δ n ) .More formally, denoting I{.} as an indicator function, we have where U n , n ≥ 1, are independently and identically distributed according to the uniform(0, 1) distribution and are independent of (X n , Y n ), n ≥ 1.Now we prove some propositions related to the above adaptive allocation design.
Proposition 2.1: As n → ∞, almost surely, Proof: We establish the result for k = B only and the other follows in a similar way.For this we note that, for any n > 2, Also, for every m, we find under the event {N Bn = N Bm for all n > m} say.Then we have It is easy to see that, for every m, ∞ n=1 u n (m) is divergent.Hence, using the same technique as in the proof of the Borel-Cantelli lemma (see Laha and Rohatgi, 1979, p.72), we get the required result.
Proposition 2.2: Let, for each n > 2 and under the proposed adaptive set-up, F N An (x) be the sample d.f.based on X-observations.Then, writing we have, for any > 0, lim Proof: Suppose, for each n, there are fixed positive integers ) is non-decreasing, and Then, by Glivenko-Cantelli's lemma, we can find, for given , δ > 0, a positive integer M such that P sup where D * (n A ) is given by ( 4) under a non-adaptive set-up using the fixed pairs (n A , n B ).
By virtue of (ii) and (iii), we can also find, for given M , a positive integer ν 0 such that n A (ν 0 ) = M and n A (ν) ≥ M for all ν > ν 0 .Hence, (5) implies By Proposition 2.1 we can find, for given M and δ, a positive integer which by ( 5) and ( 6) is less than or equal to δ for all ν exceeding ν * = max{ν 0 , ν 1 } and hence the required result follows.
Theorem 2.1: For every > 0, be two sets of positive, increasing, integer valued, almost surely finite random variables with P (α i = β j ) = 0 for all i, j.Then, we have Hence, using Theorem 2.1 of Melfi and Page (2000), X α i 's are iid with d.f.F and are independent of Y β j 's, which are iid with d.f.G. Thus, we can re-write θn as Austrian Journal of Statistics, Vol. 36 (2007), No. 3, 189-205 So, for every positive , The random variables V 1 , V 2 , . . .are iid with mean 0 and finite variance.Hence, using Theorem 3.1 of Melfi and Page (2000), the second term of the right hand member of (8) converges to zero as n → ∞.Then, by applying Proposition 2.2, the required result follows.
Note: From (3) we write The proof, using the above note, directly follows from Theorem 1 of Melfi, Page, and Geraldes (2001).
3 Asymptotic Results Related To N a (d) In this section we study asymptotic behaviors of N a (d).For this we have for every υ > 0 where r is the smallest integer larger or equal υ/d 2 , and υ 0 = rd 2 satisfies υ 0 − d 2 < υ ≤ υ 0 .Thus, we have to study the limiting distribution of √ r sup n≥r | θn − θ|.Let θ n be given by ( 1) under a non-adaptive set-up using the fixed pairs (n A , n B ) as defined in the proof of Proposition 2.2.Then, as in Sen (1981), we have almost surely as n → ∞, where (α 1 , α 2 , . . ., α n A ) and (β 1 , β 2 , . . ., β n B ) are defined in Section 2, and Ḡ(x) = 1 − G(x).Hence, by the same technique as in the proof of Proposition 2.2, it follows that Thus, the right hand member of (13) converges to zero in probability, provided the random variables are bounded with probability 1.Hence, writing and using (10), (11), and (12), we get that sup n≥r { √ r| θn − θ|} and sup n≥r { √ r|θ * n − θ|} asymptotically behave the same.
To study the asymptotic distribution of sup n≥r { √ r|θ * n − θ|}, we follow Hjort and Fenstad (1990) under martingale set-up.For this, we set for each real (c 1 , c 2 ), and Also for each c > 1 we define the stochastic process W a rc = {W a rc (t), 1 ≤ t ≤ c} as which belongs to D[1, c] equipped with the Skorokhod topology.
Theorem 3.1: Let W = {W (t), t ≥ 0} be a standard Brownian motion process.Then The proof of Theorem 3.1 depends on the following propositions.Proposition 3.1: For any in distribution, where Proof: For any real vector l = (l 1 , l 2 , . . ., l d ) , we consider say, where with t 0 = 0. Then it can be easily shown that r ≥ 1} form a martingale difference array from a zero-mean-square-integrable martingale.Hence, as in Wei et al. (1990) (see also Theorems 3.13 and 2.13 of Hall and Heyde (1980), we have after some routine calculations T a r → N (0, η 2 ) in distribution, where (T a r ) 2 converges in probability to as r → ∞.Here N (µ, σ 2 ) represents a random variable having normal distribution with mean µ and variance σ 2 .Hence, by Cramer-Wold device the required result follows.
Proposition 3.2: The sequence {W a rc } is tight.Proof: Take any 1 ≤ s ≤ t ≤ u ≤ c.Then, using the martingale theory as used in the proof of Proposition 3.1, we have Hence, by using Theorem 15.6 of Billingsley (1968), the required result follows.Then, for every fixed c ≥ 1 there exists a non-negative integer k such that 2 k ≤ c ≤ 2 k+1 , and hence (14) The second member of the right hand side (r.h.s.) of ( 14) equals which by the same technique as in Wei et al. (1990) tends to 198 Austrian Journal of Statistics, Vol. 36 (2007), No. 3, 189-205 Using Hall and Heyde (1980, p.22) the first member on the r.h.s. of ( 14) is for some K > 0 So, combining ( 15) and ( 16) we get Hence we get the required result.
Proof of Theorem 3.1: Using Propositions 3.1 and 3.2 we get on in distribution and since the distribution of is the same as that of sup c −1 ≤t≤1 |W (t)|, we get by using Proposition 3.3, as in Theorem 4.1 of Billingsley (1968), Hence the required result follows.
U. Bandyopadhyay and R. Das 199

Asymptotic Results in Non-adaptive Equal Allocation Design
In connection with the fixed-width interval estimation of θ, it would be quite natural to compare the adaptive allocation design with a non-adaptive equal allocation design, where the treatments A and B are equally randomized to the experimental units.For this, we briefly describe the non-adaptive 50:50 allocation rule along with the related asymptotic results.Suppose, the allocation indicators δ i 's are iid Bernoulli variables with success probability 1/2.Then the resulting design becomes non-adaptive equal allocation.Hence, we have the observations which is a strongly consistent estimator of θ.Using θn we can also define a stopping variable N e (d), say, as in (2), which also admits expression (9) after replacing θn in place of θn .Now, setting we introduce for every c > 0 and integer r > 0 a stochastic process W e rc = {W e rc (t), 1 ≤ t ≤ c} defined by Then by the same technique as used in Section 3 it is easy to show that in distribution as r → ∞.

Fixed-Width Confidence Intervals
Now we construct two sequences of fixed-width confidence intervals of θ based on the asymptotic results derived in Sections 3 and 4.These intervals are determined in the following way.
In an adaptive allocation design, we have Austrian Journal of Statistics, Vol. 36 (2007), No. 3, 189-205 which by ( 9) and Theorem 3.1 tends to as r → ∞, where From Sen (1981, p.42) (see also Anderson, 1960) ψ s (w a ) can be computed as where Φ(x) represents the d.f. of a standard normal random variable.Similarly, in case of equal allocation design it is easy to find that where w e is given by Let a α be such that ψ s (a α ) = 1 − α for given 0 < α < 1.Then, for given (α, d) we find the following stopping rules corresponding to the adaptive and equal allocation designs, respectively where σkn and σkn , k = 1, 2, are the consistent estimators of σ 1 and σ 2 in adaptive and equal allocation designs, respectively.The forms of the estimators are The estimators σ2 kn , k = 1, 2, have similar forms, and hence are omitted.Using martingale convergence concept it can be checked by laborious but straightforward computations that U. Bandyopadhyay and R. Das 201 in probability as r → ∞.Again, since ψ s (•) is a continuous function of (σ 1 , σ 2 ), we have as in the previous section Hence, the sequences of fixed-width confidence intervals for θ of length 2d with confidence coefficient 1 − α are ( θn − d, θn + d), n ≥ νa , in the adaptive design, and ( θn − d, θn + d), n ≥ νe in the equal allocation design.In the next section we carry out various numerical computations to judge the performance of the adaptive allocation relative to that of the equal allocation.

Numerical Study
Here we give a numerical comparison between the adaptive allocation design and its nonadaptive counterpart in terms of the minimum sample sizes required to obtain the fixed width confidence intervals of θ.The true values of these minimum sample sizes are, respectively, given by So we compute ν a and ν e at different choices of (F, G, d).In case of the adaptive design we also compute the proportion of allocation of the observations to the better treatment.
with 0 < p < 1 as the mixing proportion.The value of τ is varied to see the effect of different shapes in the behaviors of ν a and ν e .In particular, we take τ = 0.5, 1, 2, δ = 0.25, 0.5, 1, d = 0.05, 0.1, and p = 0.8, 0.9.Here, treatment B is better for the above choices of δ.Denoting the proportion of the observations on treatment B by prop B , we see that the true value of prop B is equal to θ.So we compute prop B for the above parametric combinations.The whole computation is done by taking α = 0.05 for which we note that a α = 2.242.The results are reported in Table 1.There it is observed that for given (δ, τ ) the sample size ν a for the adaptive design is larger than the sample size ν e corresponding to a non-adaptive equal allocation design.Also the proportion of allocation prop B to the better treatment for the adaptive design always exceeds 1/2 which is the value of the proportion corresponding to a 50:50 allocation design.In an adaptive design it is also noted that for any τ , prop B increases with δ.That means, the larger the deviation in the locations of F and G, the higher is the ethical gain measured in terms of the proportion of allocations to the better treatment.The sampling becomes skewed in the presence of ethical gain.However, the skewness is inversely proportional to the value of τ .
Table 1 shows that as τ becomes larger the difference ν a − ν e becomes insignificant along with the gradual decrease of prop B .It indicates that the adaptive design performs equivalently with the equal allocation design.But for smaller values of τ the sample size ν a of the adaptive design is slightly larger than the sample size ν e of 50:50 allocation design.Simultaneously, the proportion of allocations to the better treatment takes the higher values.At the cost of drawing extra ν a − ν e (which is very small except very few cases) observations, a considerable amount of ethical gain can be achieved by using the proposed adaptive design in place of the non-adaptive equal allocation design while constructing the fixed width confidence intervals of θ.

Concluding Remarks
The efficiency of the proposed adaptive allocation design relative to the non-adaptive 50:50 allocation design can also be assessed by where W max = sup 0≤t≤1 |W (t)|.Hence, we get .
Thus, one can easily determine the value of E * r for given F and G.But such a derivation depends on the conditions related to the uniform integrability of d 2 N a (d) and d 2 N e (d).
Techniques from Hjort and Fenstad (1992) would be appropriate, but we are not going to pursue this.
d→0 E(N a (d)) E(N e (d)) = lim d→0 E(d 2 N a (d)) lim d→0 E(d 2 N e (d))provided the expectations converge.Now, from the convergence ofd 2 N a (d) are d 2 N e (d)in distributions discussed in Sections 3 and 4, respectively, we expect that as d → 0