Consider a single-arm design with tumor response rate as the primary endpoint, where a binary outcome is defined as either “response” or “no response”. We want to test the hypotheses:
$$ {H}_0:p\le {p}_0 vs.\kern0.5em {H}_1:p>{p}_0 $$
with type I error rate α and type II error rate β. Here p denotes the true response rate, p0 is a fixed value that denotes the maximum response probability in order to terminate trial early. In practice, we will define p = p1 in the alternative hypothesis to represent the minimum response probability to warrant further studies in subsequent trials, therefore, the power of the test will be calculated at p = p1 > p0. If the null hypothesis is rejected, the study will be extended to phase III stage, given the warranted therapeutic efficacy. Otherwise, the study will be terminated, given the insufficiently promising efficacy.
Simon’s two-stage design
A most widely used two-stage design is proposed by Simon [2]. Two different two-stage designs are introduced that allow early trial termination for futility. Details are illustrated in Fig. 1. In the figure, we define.
n1, n2: the number of subjects in the first and second stage, respectively, and n = n1 + n2;
x1, x2: the number of responses observed in the first and second stage, respectively;
r1, r: the number of rejection points (under H0) in the first and second stage, respectively.
Thus, the probability of early termination (PET) at the end of first stage (under null hypothesis) is
$$ {PET}_{\mathrm{S}1}=B\left({r}_1;{n}_1,{p}_0\right) $$
where suffix S is used to represent the result of Simon’s design. Consequently, the probability of rejecting the treatment is
$$ {P}_{\mathrm{S}}(R)={PET}_{\mathrm{S}1}+\sum \limits_{x={r}_1+1}^{\min \left({n}_1,r\right)}b\left(x;p,{n}_1\right)B\left(r-x;p,{n}_2\right) $$
Here b(x;p,n) and B(x;p,n) are the probability mass and cumulative binomial distribution function, respectively [14].
For any pre-fixed values of p0, p1, α, and β, we can enumerate the candidate designs with different (n1, PETS1, EN) combinations, where EN is the expected sample size,., i.e.,
$$ {EN}_{\mathrm{S}}={n}_1+\left(1-{PET}_{\mathrm{S}1}\right){n}_2 $$
An optimal design is considered to minimize the expected sample size. Alternatively, a minimax design minimizes the maximum sample size n = n1 + n2, amongst these candidates designs. If there is more than one single candidate design with smallest n, the one with the smallest ENS (under null hypothesis) is chosen within all the possible minimax designs.
Fleming’s two-stage design
Unlike Simon’s two-stage design, Fleming’s design additionally allows early trial termination due to high successful response rate [8]. In Fleming’s two-stage design, one more character, a1, is required and it denotes a threshold of acceptance point (under H0) in the first stage. A single-arm two stage trial with both futility (a1) and superiority (r1) values in the first stage and a rejection value (r) in the second stage are described in Fig. 2.
Based on Fleming’s two-stage design, the probability of rejecting the treatment is
$$ {P}_{\mathrm{F}}(R)=B\left({a}_1;{n}_1,{p}_0\right)+\sum \limits_{x={a}_1+1}^{r_1}b\left(x;{n}_1,{p}_0\right)B\left(r-x;n-{n}_1,{p}_0\right) $$
where suffix F is used to represent the results of Fleming’s design [14]. The probability of early termination (PET) at the end of first stage (under H0) is
$$ {PET}_{\mathrm{F}1}=B\left({a}_1;{n}_1,{p}_0\right)+\left(1-B\left({r}_1;{n}_1,{p}_0\right)\right) $$
Thus, the expected sample size (EN) is
$$ {EN}_{\mathrm{F}}={n}_1+\left(1-{PET}_{\mathrm{F}1}\right){n}_2 $$
Although Fleming’s design ensures sample sizes no larger than the single-stage design, a limitation is that calculated critical values for accepting and rejecting the null hypothesis are based on pre-fixed sample sizes at stage 1 (n1) and stage 2 (n2), which may be undesirable for investigating and planning optimal designs. To remedy, Mander and Thompson extended Simon’s optimal and minimax criteria in Fleming’s two-stage design [10]. Such design will benefit from stopping early for either futility or efficacy, while preserve its simplicity and the small sample size.
Admissible design
Very often, the minimax design has a much smaller maximum sample size n than that of the optimal design, but it has an excessively large expected sample size EN. Similarly, optimal design requires a much smaller EN, but it suffers a considerably larger n as compares to the minimax design. In planning a phase II trial, we usually find ourselves in a dilemma when we must consider choosing one of the two designs by comparing the expected sample size and maximum sample size.
To overcome, it is desirable to search for a design between the optimal design and the minimax design such that it has EN close to that of the optimal design and n close to that of the minimax design. Jung et al. proposed an admissible adaptive design based on a Bayesian decision-theoretic criterion to compromise between EN and n [12, 13]. A design is called candidate design if it minimizes EN for a given n while satisfying the (α, β)-constraint. For pre-specified (p0, p1, α, β), let R denotes the space of all candidate designs satisfying the (α, β)-constraint, with n no more than an achievable accrued number of subjects N during the study period. For any given design d ∈ R, we consider its two outcomes n(d) in minimax design or EN(d) in optimal design. Let Q be a probability distribution defined over {n(d), EN(d)} as Q(n(d)) = q and Q(EN(d)) = 1-q, where q ∈ [0, 1].
Thus, for any design d ∈ R, the expected loss can be defined as
$$ \rho \left(q,d\right)=q\times n(d)+\left(1-q\right)\times EN(d), $$
and the Bayes risk is defined as
$$ {\rho}^{\ast}\left(\rho, d\right)={\displaystyle \begin{array}{c}\mathit{\operatorname{inf}}\kern0.5em \rho \left(q,d\right)\\ {}d\in R\end{array}} $$
Any design d ∈ R whose risk equals to the Bayes risk would be regarded as Bayes design, which will then be defined as admissible design against distribution Q. Note that q ∈ [0, 1] reflects the relative importance between maximum sample size and expected sample size in designing a phase II study. Thus, the minimax design is a special Bayes design with q = 1 and optimal design is a special Bayes design with q = 0. Conversely, for any q ∈ [0, 1], if no Bayes risk fits any design d ∈ R, the design would be defined as inadmissible.
Jung et al. [13] firstly proposed to apply admissible design to Simon’s two-stage design. In this article, we extend such admissible design to Fleming’s two-stage design, too.