RPW rule. As pointed out in Wei and Durham [33] and Wei [32], the PW rule is too deterministic and is not applicable when we have delayed responses from patients to treatments. Motivated as an extension to Zelen's [34] idea, Wei and Durham [33] introduced the following randomized play- the-Winner (RPW) rule for a two-arm clinical trial: We start with (a, a) balls (type 1 and 2 respectively) in the urn. If a type 1 6a// is drawn, a patient is assigned to the treatment 1; If a type 2 ball is drawn, a patient is assigned to the treatment 2. The ball is replaced and the patient response is observed. A success on the treatment 1 or a failure on the treatment 2 generates a type 1 ball in the urn; A success on the treatment 2 or a failure on the treatment 1 generates a type 2 ball in the urn. The RPW rule may be regarded as a generalized Polya urn (GPU) model (Wei [32]). Further,
let Yni (Yn2) be the number of balls of type 1 (2) after n stage. Prom the results of Athreya and Karlin [1], we have
Ym 92 , Nnl q2
T}——^7 > • a.s. and • a.s..
*nl + *n2 91+92 1 9l + 92
When P1+P2 < 1.5 (or Qi + q2 > 0.5), we have the following asymptotic normality:
and
where
ô 91+92
„2 _ 9i92[5-2(gi+g 2)]
^ " [2(91+92)-l](9i+92)2- ( 1 6 )
The asymptotic normality was first given in Smythe and Rosenberger [30].
When 9 1 + 9 2 < 0.5, the asymptotic distributions of both the urn com- position and the proportions of patients assigned to each treatments are unknown.
GPU model. One large family of randomized adaptive designs can be developed from the GPU model. Consider an urn containing balls of K types. Initially, the urn contains Y0 = ( l o i , - -- ,YQK) balls, where Yoi denotes the number of balls of type i, i = l,--- ,K. A ball is drawn at random from the urn. Its type is observed and the ball is then replaced.
At the n-th stage, following a type i drawn, Dij(n) balls of type j , for j = 1, • • • ,K, are added to the urn. In the most general sense, J5y(n) can be random and can be some function of a random process outside the urn process (in the case of adaptive designs, Dij (n) will be a random function of patient's response £n). A ball must always be added at each stage (in addition to the replacement), and the expectation of the total numbers of balls added in each stage is the same (say 7), so
P{Dij{n) = 0, for all j = 1, • • • , K} = 0, K
^ ) E { A j ( n ) | Jrn - i } = 7 , M = ! , • • • ,K,n = 1,2,-•• ,
where J-n is the sigma field generated by the events up to the stage n.
Without losing generality, we can assume 7 = 1. Let Hn be the matrix
comprising element { ^ ( n ) = E{Djj(n)|.Fn_i}}. We refer to D„ as the rules and Hn as the design matrices. If Hn = H for all n, the model is said to be homogenous. In general, it is assumed that Hn —ằ H. Let Yn = {Yn\, • • • , YnK), where Yni represents the number of balls in the urn of type i after n-th stage, and N„ = (N„i, ••• , Nnx), where Nni represents the number of times a type i ball drawn in the first n draws. In the clinical trials, Nni is the number of patients assigned to treatment i in the first n stages. Let v = (v\, • • • ,VK) be the left eigenvector corresponding to the largest eigenvalue of H with v\ + ... + VK = 1- Then Vi is just the limiting proportion of both the patients assigned to treatment i and the type i balls in the urn, i.e.,
N • Y •
L-<j=l 1n,
a.s. a n d —r? * v% o-s. (17) (c.f., Athreya and Karlin [1]). Smyth [29], Bai and Hu [3] [4] showed the
normality of Yn and N„. Bai, Hu and Zhang [7] and Hu and Zhang [18]
gives the strong approximation of Yn and Nn. Let Ai = 7 = 1, A2,..., XK be the eigenvalues of H, and A = max{ite(A2), • • •, RB(XK)}- Let Vgki(n) = Cov{£>,fc(n),£>,t(n)|J'n_i} and V „ , = (Vqki{n))£l=v q,k,l = 1,.. .,K.
T h e o r e m 3 . 1 . Assume that for some 0 < e < 1,
3 | D „ | |2 + e< C < o o , n = l , 2 , . . . , (18) n
fe=i
^ H H f c - H H o C n1- ' ) , fc=i
and let S i = diag{\) — v'v and S2 = Ylq=ivg^g- If X < 1/2, then on a suitably enlarged probability space with two independent K-dimensional standard Brownian motions Btj and Bt2, we can redefined the process ( Yn, Nn) without changing the distribution, such that for some K > 0,
Yn-EYn = Gn + o(n1/2~K) a.s.,
N , - £ Nn = Bn lS j/ 2 + / " SlZ. dx(I - l ' v ) + otn1/2-*) a.i Jo x
where Gt is a Gaussian process which is solution of the following type equa- tion:
Gt = Bt l£ }/ 2H + Bt 2£ 2/ 2+ / — ( H - l ' v ) ds, t > 0, G0 = 0. (19) Jo s
Furthermore
n
£Yn - nv = o(nx/2-K) + ]T v(Hfc - H),
k-l
n
£Nn - nv = oin1'2-*) + ]T v(Hfc - H).
fc=i
in particular, the asymptotic combining distribution of Yn and Nn can be obtained.
Example 3.1. (General R P W Rule) Suppose K = 2 and the adding rule matrices are denoted by
D = ( ^n l 1 ~ ^nX\
\ 1 _ £n2 £n2 / '
where {£„ = (£ni, £7,2), n = 1,2...} is an i.i.d. sequence with 0 < £ni < 1, Var(£ni) = of, E£„j = pj and g, = 1 — p, for i = 1,2. For this design, one has
H = E[D„] = fP l 9 lV
\ f t P2/
Ai = 1, A2 = pi + p2 - 1 and v = (52/(91 + ft), ft/(?i + ft))- Let a2 = ftft/Ozi + ft)2 and 62 = (cri^ + 0"2<?i)/(ft + ft)- Then there exist two independent one-dimensional standard Brownian motions Bt\ and Bt2 such that
Yni - nvi = A2cr1Gni + 0-2G„2 + o(n1 / 2~K) a.s., Nnl - nv1 = (TiGm + a2 f ^ ^ + o{n1'^) a.s.
Jo % whenever A2 < 1/2, where
Example 3.2. (Wei's Rule) Suppose K > 2 and the response sequence {£„} is an i.i.d. sequence with £„& = 0 or 1. £nfc = 1 or 0 corresponds to the response of the n patient on the treatment A: is a success or failure, respectively, i = 1 , . . . , K. Let the adding rule matrices D„ = (Dij(n))i . be denoted by Du(n) = £ni and Dij = (1 — t,ni)/(K — 1) for i ^ j . This design is proposed by Wei [32]. For this design, the design matrix H =
E[Dn] is the same as the one in (7), and v is the same as the one in (5). If A < 1/2, then
Y„ - nv = Gn + o(n1 / 2~K) a.s.,
N„ - nv = Bn lS J/ 2 + [ — dx(I - l ' v ) + o(n1/2-K) a.s.
Jo x
Designs with delay responses. Typically, clinical trials do not result in immediate outcomes, i.e., individual patient outcomes may not be imme- diately available prior to the randomization of the next patient. Con- sequently, the urn cannot be updated immediately, but can be updated only when the outcomes become available. Fortunately, it is verified that stochastic staggered entry and delay mechanisms do not affect the asymp- totic properties of both the urn composition Y„ and the sample fractions Nn for a wide class of designs defined by GPU (c.f., Bai, Hu and Rosen- berger [5] and Hu and Zhang [21]).
Designs dependent on estimated unknown parameters. Hu and Zhang [19]
considered a general model with rules of the type Dn = D ( 0n_ i , £n) and design matrices of the type Hn = H ( 0n_ i ) dependent on the estimated unknown parameter, where 0n_ i is the sample estimator of the unknown parameter 0 based on the responses of previous n — 1 stages. Strong consistency and the asymptotic normalities are established for both the urn composition and the number of patients assigned to each treatment.
For such design, we also have the following strong approximation.
Theorem 3.2. Suppose {£„} is an i.i.d. sequence, Q = E£n and 0n is defined as in Adaptive Design 2.1. Suppose the conditions in Theorem 2.2 are satisfied and the condition (18) in Theorem 3.1 is satisfied. If A < 1, then
• v a.s. and • v a.s., n n
also, on a suitably enlarged probability space with an K-dimensional stan- dard Brownian motion W ( , we can redefined the process ( Yn, Nn, £n) with- out changing the distribution, such that (12) holds for some K > 0.
If A < 1/2, then on a suitably enlarged probability space with three inde- pendent K-dimensional standard Brownian motions Bti , Bt2 and W t , we can redefined the process ( Yn, Nn, £n) without changing the distribution,
Gt = BtlEJ/2H + Bt2*r + f W*dxdia9( %..
JO X y/v{
+ f — ( H - l'v) ds, t > 0, Go - 0, Jo s
and F is defined in (11).
' ' y/VK
such that for some K > 0, (12) holds and Yn - nv = Gn + o(n1/2~K) a.s.,
Nn - nv = Bn lS }/ 2 + [ — dx(I - l'v) + o ^2- * ) a.s.
Jo x
where Gt is a Gaussian process which is solution of the following type equa- tion:
(20)
By comparing the two equations (19) and (20), one finds that (20) has a more term than (19) has. This term is generated by the randomness of Q„s.
Example 3.3. (General Wei Rule) Suppose K > 2 and the response sequence {£„} is the same as in Wei's rule, and 0 <pk <1, k = 1,...,K.
At the n-th stage, a success on the treatment fc generates a particle of type fc, and a failure on the treatment k generates 777 m~Va r par- tides of type j to the urn for all j ^ k, where 0 < a < 00, where Mm_ i ,Q = T,f=iPm-i,j> Pm-i,j = j v - ^ . + i . a n d Sm-i,j denotes the num- ber of successes of the j - t h treatment in all the Nm-i,j trials of previous n—1 stages, j = 1 , . . . , K. When a = 0, this design model is just Wei's rule, and when a = 1, it is the model proposed by Bai, Hu and Shen [6]. Write P m - i = (Pm-1,1, • • • ,Pm,K)- In this case, Hm = E[Dm|J"m-i] - H ( pm_ i ) , where H(x) is defined as in Example 2.4. Also,
Hm -> H = H<a>
and
* ô (a) j ^ " (a)
iv = vw and • v = vK > a.s., n n
where H ^ and v(Q) is defined as in (8) and (9), respectively. If A < 1/2, then the conclusions in Theorem 3.2 hold. In particular,
n1 / 2( Yn/ n - v W ) £ iV(0, A+) and n1 / 2( N „ / n - v<a)) 3 JV(0, A") for some A^ and A".