Adaptive Design for Staggered-Start Clinical Trial

1 Introduction

In phase II and/or III clinical trial, there are several competing treatments to be assessed during the trial process. During the trial, each coming patient is allocated to one of the treatments according to some pre-specified rule, which is based on the current clinical knowledge of the treatment performances of the previously allocated patients up to that time. The aims are to assess the treatment performances well and minimize the the overall treatment losses to the patients. The rule for the allocation is called the design of a clinical trial and the minimal loss is often characterized by some optimality criteria. If the design involves unknown parameter(s) to be estimated and updated along with the trial progression, it is called adaptive design. Optimal adaptive clinical trial designs and their properties have been studied extensively in the literature, for example by Rosenberger and Lachin [1], Eisele [2], Melfi and Page [3], Rosenberger et al. [4], Bai et al. [5], Bai and Hu [6], Zhang et al. [7], Yuan and Chai [8], Zhu and Hu [9], Yuan et al. [10], among others. In all these existing trials, the treatments under investigation enter the trial at the same time. Recently in phase II (or phase III) clinical trials, a new type of trial, the staggered-start clinical trial has been proposed and studied by a number of researchers/experimenters [11–13] etc.). In this trial the treatments enter the trial at different times, as in the studies by Kublin [14], Cummings et al. [15], Hendrix et al. [16], and the experimental studies at the Fred Hunchinson Cancer Research Center. Staggered-start and delayed withdrawal designs were originally proposed by Leber and others as a way of providing evidence of disease modification without relying on biomarkers. In this study patients are randomized to drug or placebo at baseline and after a pre-specified period the placebo group is treated with the active agent. If the staggered-start group “catches up” with the group receiving treatment from the onset of the trial, no disease modification affect has been observed and the agent is deemed to have symptomatic benefits only. If the staggered-start group does not catch up with the group receiving treatment from the onset, then it is concluded that disease modification has occurred [14].

As an example for such trial, in the vaccine trial study [14], vaccine 1 and placebo are in the trial, after 12 months another new treatment vaccine 2 is available and joined the ongoing trial, and a third vaccine joined at a latter time. Thus, three vaccine regimens vs a shared placebo group are investigated in their trials. The trial is aimed to address the question: How should a lag in the availability of a vaccine product be accommodated in the research trial design? There are variety of other reasons for such type of trials, for example, by joining the new treatment(s) and the one(s) already under study, it can save the sample size and costs as compared to running separate trials. Also some times it is desirable to compare the behavior of the newly available treatment(s) to the ongoing one(s) in the same trial study.

In such staggered-start clinical trial there are some basic questions, such as how to allocate the coming patients in a reasonable way? how different this type trial compared to the existing ones in basic properties, will some properties be kept with this new trial and under what conditions? A simple way in this trial is to allocate all the coming new patients to the new treatment(s) and letting the existing one(s) in waiting until all the treatments have roughly the same number of patients, then one can study the treatments like in the standard trials. But this method is known to be undesirable, as the subjects in study is human being, any design should aim at allocating more patients to treatment with best efficacy and minimize the sample size as possible; also the deterministic assignment in the waiting period violates the basic principle of clinical trial. Since the existing treatments have been in the study for relatively long time, they can not be treated as initial observations in relation to the newly added treatments. In this type of trial, we want to know if it is still possible to allocate the coming patients to the treatments to achieve a given optimality criterion, or to minimize the treatment losses as at least in some extent. These pose new problems in clinical trial study.

For standard clinical trial, the adaptive design uses accumulating data to update aspects of the study as it continues without undermining the validity and integrity of the trial [3, 17, 18]; among others). The optimal design is to achieve some targeting objective criteria for the allocation proportions [2, 4, 7, 8, 19, 20]. These methods have various merits in application, but it seems none of the existing methods addressed the case of staggered-start trial.

Here we propose and study a class of adaptive designs for the mentioned type of clinical trials, in which weights are used in the assignment of patients to the treatments to balance the current allocation proportions, in such a way that the allocation proportions of the treatments will asymptotically achieve, under suitable conditions, some given optimality criterion as in the standard trial. This design is applicable to any type of responses and simple to use. Our initial findings for such design are that if the sample sizes of the ongoing treatments are relatively small as compared to those of the newly joined treatments, then asymptotically the trial can achieve the given optimality criterion as in the standard trial with all treatments starting at the same time; otherwise the optimality can only be partially achieved in some sense. In Section 2, we give a brief review of some related commonly used existing methods for standard adaptive clinical trials. Section 3 describes the proposed allocation rule for staggered-start trial; investigates its asymptotic optimality behavior in Section 4, by distinguish two cases: the initial sample sizes are not too large relative to the total sample size, so that asymptotic optimality can be achieved; the initial sample sizes has roughly the same magnitude as the total sample size and asymptotic optimality cannot be achieved. Lastly, in Section 5 we illustrate these methods by some examples and a simulation study. A short discussion is given at the end. The relevant technical proofs are given in the Appendix.

2 Brief review of existing methods for standard clinical trials

Assume a standard clinical trials with k treatments. The patients come to the trial sequentially. Let rn be the response (may be multiple) of the n-th patient under the assigned treatment, f(rn) be the summary score for this response. Without loss of generality we assume 0≤f(⋅)<∞, and bigger value of f(⋅) corresponds to better treatment effect. Let Ai be the event that the trial is under treatment i, μi=E(f(rn)|Ai) be the expected performance, or success rate, of the i-th treatment, σi2:=Var(f(r1)|Ai)<∞ be its variance (i=1,...,k). Denote μ=(μ1,...,μk) and σ2=(σ12,...,σk2). Denote the number of patients assigned to treatment i at time n by Ni(n), let N(n)=(N1(n),...,Nk(n)). The vector of allocation proportions N(n)/n is the main focus in phase III clinical trial studies, various methods are studied targeting the proportions to achieve, asymptotically, some specified optimality criteria.

For a vector x=(x1,...,xk), define |x|=x1+⋯+xk, and for x and y of the same length, define x−y=(x1−y1,...,xk−yk). The optimality criterion can be characterized by a specified object functional G(⋅,⋅) of the response distributions under the treatments, or of their means μ. The target allocation proportions is the vector v=(v1,...,vk) which achieves the given criterion

where ∇={v(⋅)=(v1(⋅),...,vk(⋅)):vj(⋅)>0(1≤j≤k),|v|=1.} is the set of all possible vectors of proportions. In the optimal adaptive design, a special randomization rule is constructed such that the design will achieve the asymptotic optimality

Some examples of optimality criterion G(μ,v(μ)) are given below.

Let Iji be the indicator that the j-th patient is assigned to treatment i. Since μ is unknown, we plug in its current estimate μn, with

More generally, we may consider the conditional distribution Fi(⋅) of the f(rn)|Ai’s. Let F(⋅)=(F1(⋅),...,Fk(⋅)). Their empirical versions are

where x=(x1,...,xk), and let Fn(x)=(Fn,1(x1),...,Fn,k(xk)), and χ(⋅) is the indicator function.

However, in many situations, we are also interested in optimizing the evaluations of the treatments along with the optimization of the assignment proportions. Let ξi=(ξi1,...,ξik) be the current evaluation of the k treatments if the coming patient is assigned treatment i, and Xn=(xn1,...,xnk) be the cumulative evaluations of the ξi’s for the k treatments up to time n, it is called the urn composition at time n. Denote |Xn|=∑j=1kxnj. For example, at time m, we can choose the ξi’s just be vm−1, independent of which treatment i is assigned at this time, see the Application section for the construction of the vm−1’s. More generally, we can only require the ξi be random with mean v, or to be of some more general nature. A joint optimal assignment for N(n)/n and Xn/n to achieve the same criterion v is a version of the generalized Pólya urn (GPU) design (for example [7, 8]). In this design, to assign the coming n-th patient to one of the treatments, a random variable is drawn from the multinomial distribution with probabilities Xn/|Xn|. If it is of type i, the patient is assigned to the i-th treatment, a random vector of masses ξi is added to the current urn compositions Xn, and the response rn is used to update the estimate of μ in the next step. Let Iξ=(ξij)i,j=1,...,k be the matrix representation of the ξij’s, and for each n, and Iξn be an i.i.d. version of Iξ. To simplify the expressions of the asymptotic variances to be derived later, we assume throughout this article that Iξ is independent of the response observations. The random vector ξi is termed the adding rule, and M=E(Iξ)=(mij) the design matrix with mij=E(ξij) (known). Recall that a left eigenvector of a matrix M corresponding to an eigenvalue λ is the row vector v such that vM=λv (in contrast a right eigenvector u is a column vector such that Mu=λu). The vector is normalized if |v|=1. The first eigenvalue λ of the design matrix and its normalized first left (row) eigenvector v plays a key role in the asymptotic properties of the GPU design. Many authors (for instance [21–23] studied asymptotic properties of Xn and N(n), and proved, under suitable conditions, that

This give a joint optimality of (Xn/n,N(n)/n). For a comprehensive review in this field, see Rosenberger and Lachin [1] and other related recent papers.

However, the existing GPU design can only achieve joint optimality of (Xn/n,N(n)/n) for the same criterion v. More generally, we may optimizing Xn/n and N(n)/n by different criteria, which will give us more flexibility. The compound adaptive GPU design [10] can achieve this goal, we won’t pursue it here.

3 The staggered-start trial and the proposed design

In this case k treatments enter the trial at different times. After treatment 1 under study with n1 patients, a new treatment 2 is added into the trial,..., after treatment (k−1) under study with nk−1 patients, the newest k-th treatment enter the trial, with some initial size nk,0. The previous treatments already under way for long times, and the sample sizes n1,...,nk−1 cannot be viewed as initial sample sizes. The questions are can the optimality be achieved in this trial? under what conditions? how to allocate the incoming patients to achieve the overall optimality? It is known that any deterministic assignment is not desirable, so we need a randomized design for this problem. Below we describe the GPU designs for two treatments, three treatments, ..., k treatments accordingly. The general description may seem involved, the simulation example in Section 5 will make it simple and clear with two treatments. For easy of understanding, we first describe the deign for two treatments, and then for the case of general k treatments.

4 Asymptotic properties

Now we study the asymptotic properties of this design for quantitative responses, the case of qualitative responses should be parallel. We distinguish two cases, first the case n1,...,nk−1 are large but nj/n→0 as n→∞; and secondly the case limnj/n>0(j=1,...,k−1). The asymptotic results are different for the two cases.

Case I. We first consider the case the initial sample sizes nj’s are not too big relative to the total sample size n, and show that optimality can be achieved as in the corresponding standard clinical trials.

Let →D denote convergence in distribution. For vectors a=(a1,...,ak) and b=(b1,...,bk) with bi≠0(i=1,...,k), define a/b=(a1/b1,...,ak/bk), and denote {a} the diagonal matrix for a. We impose the following conditions

1. 0<νij<∞, (1≤i,j≤k).

2. M1'=λ11'.

3. ςij(⋅,⋅) is continuous at (μ,y) for any y and there is an r>2 such that E||ξ1||r<∞ and supnE(||ζn||r|Fn−1)<∞, a.s.

4. M(μ)≥0(≠0) is differentiable, and there is a δ>0 such thatM(y)−M(μ)=∑j=1k∂M(y)∂yj|y=μ(yj−μj)+O(||y−μ||1+δ), as y→μ.

5. ∏j=1k−1nj=o(n1/2).

whereΩμ={σ2/v}, andΩF={F(x)(1−F(x))/v}.

(ii) Assume (B0)-(B4). Ifλ1>2Re(λ2), then

Remark 1) Whenλ1≤2Re(λ2), the convergence rate is generally slower thann1/2.

2). The asymptotic covariance matrixΩcan be made very simple by choice ofM, as in the Corollary in Yuan and Chai [8], as below. Let v1(⋅)be twice differentiable with|v1(⋅)|=1. SetM(⋅)=1'v1(⋅), Ξbe a constant matrix, then the conditions of Theorem 2(ii) are satisfied withλ1=1andλ2=0, andΩis simplified as

Case II. In this case, limni/n→αi>0(i=1,...,k−1), but since the stopping time n is unknown in advance, the proportions α1,...,αk−1 are only known when the trial terminates, otherwise the design can be made simpler by using the αj’s. We will see that in this case, the optimality criterion can only be partially achieved, in that the targeting limits are partially kept unchanged, as seen by comparing results of Theorems 3 and 4 vs those in Theorems 1 and 2.

In Theorem 3 (ii), (y,u)≠(v,v)). Thus, if the sample sizes nj’s are not small in relation to the total sample size n, full optimality cannot be achieved.

5 Application examples and simulation study

In practice v(⋅) should be chosen differentiable around μ≠0, and hence satisfies the required conditions. We will continue the three examples given in Section 2 and with two more examples in common applications, with a simulation study for the last example.