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, *p*_{0} is a fixed value that denotes the maximum response probability in order to terminate trial early. In practice, we will define *p* = *p*_{1} 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 = p*_{1} > *p*_{0}. 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.

*n*_{1}, *n*_{2}: the number of subjects in the first and second stage, respectively, and *n* = *n*_{1} + *n*_{2};

*x*_{1}, *x*_{2}: the number of responses observed in the first and second stage, respectively;

*r*_{1}, *r*: the number of rejection points (under *H*_{0}) 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 *p*_{0}, *p*_{1}, *α*, and *β*, we can enumerate the candidate designs with different (*n*_{1}, *PET*_{S1}, *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* = *n*_{1} + *n*_{2}, amongst these candidates designs. If there is more than one single candidate design with smallest *n*, the one with the smallest *EN*_{S} (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, *a*_{1}, is required and it denotes a threshold of acceptance point (under *H*_{0}) in the first stage. A single-arm two stage trial with both futility (*a*_{1}) and superiority (*r*_{1}) 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 *H*_{0}) 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 (*n*_{1}) and stage 2 (*n*_{2}), 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 (*p*_{0}, *p*_{1}, *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.