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Biometrics & Biostatistics International Journal

Research Article Volume 11 Issue 5

An analytical study of the critical values of response rate in single-arm phase II clinical trial designs

Yi Liu,1 Sin-Ho Jung2

1Department of Statistics, North Carolina State University, USA
2Department of Biostatistics and Bioinformatics, Duke University, USA

Correspondence: Sin-Ho Jung, Department of Biostatistics and Bioinformatics, Duke University, USA

Received: November 07, 2022 | Published: December 30, 2022

Citation: Yi L, Sin-Ho J. An analytical study of the critical values of response rate in single-arm phase II clinical trial designs. Biom Biostat Int J. 2022;11(5):178-183. DOI: 10.15406/bbij.2022.11.00374

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Abstract

A single-arm phase II clinical trial is usually conducted for finding an appropriate dose-level and testing toxicity for an experimental cancer therapy in comparison to some historical controls, and is usually the most doable trial type due to the feasibility under limited budget, patient pool and medical conditions that can be met. We considered the standard setting of a single-arm two-stage phase II clinical trial, and investigated the patterns of critical values and sample sizes at both two stages of minimax and optimal designs under different design parameters, i.e., under different response rates of control and treatment, significant level of the test, and statistical power. We provided analytic derivations to the patterns we are interested under large sample approximation, and investigated them under finite ans small sample via a numerical study by considering extensively possible design parameters over a fine grid. We finally concluded that the critical values at different stages of the test are related to the sample sizes at different stages in a similar way for both optimal and minimax designs, but they also reveal some differences and reflected the nature of these two types of design.

Keywords: single-arm two-stage trial, optimal design, minimax design, critical value, response rate

Introduction

A phase II clinical trial is often conducted to find an appropriate dose-level and test toxicity for experimental cancer therapies in compare to some historical controls. This kind of trials usually requires small sample sizes due to ethical considerations, before proceeding to the subsequent phase of the trial for assessing the efficacy, outcomes and adverse effect in a larger group of subjects.1,2 At the same time, a single-arm design is often the most doable trial due to the feasibility under limited budget, patient pool and medical conditions that can be met. The number of written requests reported on 2021 of single-arm trials issued by the Food and Drug Administration (FDA) of the United States is an essential volume.3

Statistically, the most simple and traditional single-arm phase II clinical trial design-the single-stage design, is specified as follows. Given the values of design parameters (α*,1β*,p0,p1)(α,1β,p0,p1) , that is, statistical significance level, power of the test, null response rate, and treatment response rate, respectively, we want to find a pair of positive integers n,an,a  to specify the sample size and rejection boundary (or critical value, for rejecting H0H0 ) of the phase II trial under some constraints, and usually we also have a upper bound of nn  specified subjectively, such as 25502550  to reflect the small-sample characteristic of a phase II trial.4 Then, starting from nn0>0nn0>0  for a fixed n0n0  (the minimal sample size needed), we can search over a grid the smallest integer aa  such that based on level α*α , we reject H0H0  if X>aX>a  and otherwise fail to reject H0H0 , that is, α=1B0(a;n)α*α=1B0(a;n)α  where B0(;n)B0(;n)  is the CDF of Bin(n,p0)Bin(n,p0) . Then if the power 1β=1B1(a,n)1β1β=1B1(a,n)1β  where B1(;n)B1(;n)  is the CDF of Bin(n,p1)Bin(n,p1) , the pair n,an,a  is said to be the optimal design. If there is no aa  meet these conditions simultaneously, we continue to n+1n+1 . Thus, the optimal design minimizes the sample size among all designs satisfies the conditions. It is also easy to know that the optimal single-stage design under a parameter setting is unique.

In some scenario, two-stage designs are more likely to be considered. In a two-stage design, the phase II clinical trial is conducted via two stages sequentially, and whether to proceed to the second stage depends on the results from the first stage. A desired design under the specifications of some parameters is given by finding appropriate sample sizes and critical values of the response rate of both two stages. The two-stage design is more economical and ethical, which makes the trial stop earlier if there is no efficacy tested, and the data collected is more informative than that from a single-stage trial if we proceed to phase III.4 Similar to single-stage design, the two-stage design is a mathematically easy problem since it can be done over a finite grid once the parameters are specified. In this paper, the problem is stated and discussed under the design of a single-arm two-stage phase II clinical trial.

The remainder of the paper is organized as follows. Section 4 reviewed the statistical setup of the single-arm two-stage phase II trials and described the scientific question of interest. We reviewed the classical optimal and minimax designs and discussed an issue we are concerned about involved in these two designs, and briefly stated our hunch. Section 5 introduced a numerical analysis to confirm our hunch. Section 6 concluded the paper.

Single-Arm two-stage Phase II clinical trials

Setup

Consider the test H0:p=p0H0:p=p0  vs. H1:p=p1H1:p=p1 , where p0p0  is the response rate of a historical therapy, p1p1  is the response rate of the experimental therapy needs to be tested, and under our context, p1=p0+δp1=p0+δ  for some δ>0δ>0  as a clinical meaningful difference. We conduct the test by two stages. Denote n=n1+n2n=n1+n2  with njnj  as the sample size of the test at stage j(j=1,2)j(j=1,2) . Denote X1Bin(n1,pi)X1Bin(n1,pi)  and X2Bin(n2,pi)X2Bin(n2,pi)  under Hi,i=0,1Hi,i=0,1 . X1X1  and X2X2  are independent.

We consider two types of stopping criteria when the phase II trial needs to be stopped after stage 1. First, we consider the futility (lower) stop only, which means we stop the trial after stage 1 only if we find the treatment is ineffective based on the result from stage 1, in other words, this happens when we find the positive response of stage 1 x1<a1x1<a1  for a critical value a1a1  that requires us to reject the null hypothesis H0H0  under specified design parameters. Second, we consider both futility and superiority (upper) stops. The superiority stop means we also stop the trial when x1<b1x1<b1  for a critical value b1b1  that specifies the enough effectiveness of the treatment, i.e., because of the enough positive responses to the treatment, we can stop and claim the effectiveness of the treatment, and there is no need to conduct stage 2.

Specifically, the framework of finding specific designs of a single-arm two-stage phase II clinical trial is specified as follows. Under futility stop with design parameters (α*,1β*,p0,p1)(α,1β,p0,p1) :

  1. Calculate α=P0(X1>a1,X1+X2>a)α=P0(X1>a1,X1+X2>a) , 1β=P1(X1>a1,X1+X2>a)1β=P1(X1>a1,X1+X2>a)
  2. Define probability of early termination (PET) under H0H0 :P0(X1a1)P0(X1a1) and expected sample size (EN) under H0H0 : n1×PET+n×(1PET)n1×PET+n×(1PET)  
  1. Among designs with αααα and 1β1β*1β1β :

 - Optimal design (a1,n1,a,n)(a1,n1,a,n)  minimizes EN

 - Minimax design (a1,n1,a,n)(a1,n1,a,n)  minmizes n=n1+n2n=n1+n2  

where PiPi  is the probability measure under Hi(i=0,1)Hi(i=0,1) . The significant level and power can be calculate by Pi(X1>a1,X1+X2>a)=n1x1=a1+1bi(x1;n1){1Bi(ax1;n2)}Pi(X1>a1,X1+X2>a)=n1x1=a1+1bi(x1;n1){1Bi(ax1;n2)}  where i=0i=0  for αα  and i=1i=1  for 1β1β , bi(;n)bi(;n)  is the probability mass of Bin(n,pi)Bin(n,pi)  and Bi(;n)Bi(;n)  is the corresponding CDF, i=0,1i=0,1 . The algorithm of finding an optimal or minimax design is easy to implement via greedy searching over possible choices of (n,n1,a1,a)(n,n1,a1,a) . Starting from a nn0nn0 , for n1[1,n1],a1[0,n1],a[a1+1,n]n1[1,n1],a1[0,n1],a[a1+1,n] , the probabilities and EN aforementioned can be calculated, and so the corresponding designs can be found.

Under both futility and superiority stops with design parameters (α*,1β*,p0,p1)(α,1β,p0,p1) :

  1. Calculate 1α=P0(X1a1)+P0(a1<X1<b1,X1+X2a),β=P1(X1a1)+P1(a1<X1<b1,X1+X2a)1α=P0(X1a1)+P0(a1<X1<b1,X1+X2a),β=P1(X1a1)+P1(a1<X1<b1,X1+X2a)
  2. Define probability of early termination under   Hi:PET(pi)=Pi(X1a1)+Pi(X1b1)Hi:PET(pi)=Pi(X1a1)+Pi(X1b1) , and expected sample size (EN) under Hi:EN(pi)=n1×PET(pi)+n×(1PET(pi))Hi:EN(pi)=n1×PET(pi)+n×(1PET(pi)) , i=0,1i=0,1 , and then EN=12[EN(p0)+EN(p1)]EN=12[EN(p0)+EN(p1)]  
  3. Among designs with αα*αα and 1β1β*1β1β :

 - Optimal design (a1,n1,a,n)(a1,n1,a,n)  minimizes EN

 - Minimax design (a1,n1,a,n)(a1,n1,a,n)  minmizes n=n1+n2n=n1+n2  

A user-friendly software for finding optimal and minimax designs of two-stage phase II clinical trials can be found at http://www2.cscc.unc.edu/impact7/CTDSystems.

What are we looking for?

Following the notations in Section 4.1, we are interested if there is any pattern(s) among the a1,b1a,n1,na1,b1a,n1,n  values we found from the desired designs given the constrains on level, power, etc. In fact, Jung4 found a phenomenon that an2(p0+p1)an2(p0+p1) ,a1n1p0a1n1p0  and b1n1p1b1n1p1  by a few examples from the output of optimal and minimax trial designs, but has not investigated this in more details. We wanted to provide a formal analysis in the relationship of these numbers that an optimal or minimax design returns, given any specific setting of the design parameters, which we believe can provide some prior knowledge about what we can expect from a specific design, help us to understand more on the difference and similarity of minimax and optimal designs, and look at them from a new perspective.

First, we proved that under some regularity conditions and constraints by phase II clinical trials, and using large sample approximation, a1n1=p0+O(n1/21)a1n1=p0+O(n1/21) , b1n1=p1+O(n1/21)b1n1=p1+O(n1/21) , and an=λp1+(1λ)p0+O(n1/2)an=λp1+(1λ)p0+O(n1/2)  for some λ[0,1]λ[0,1] , and λλ  is a parameter that can be related at least to the response rates of control and treatment, i.e., p0p0  and p1p1 . In other words, we may expect that the rejection criteria, i.e., the critical values a1,b1a1,b1  and aa  of the test are mainly determined by the response rates of control and treatment when the sample size is large. Details can be found in Appendix 7.1 for single-stage design and 7.2 for two-stage designs. Second, we used an extensive numerical study in Section 5 to confirm our hunch when the sample size is finite and small, and based on results from optimal and minimax designs.

Numerical analysis

Setup

We investigated the patterns of a1,b1,a,n1,na1,b1,a,n1,n  over a fine grid of parameters (p0,p1,α,1β)(p0,p1,α,1β) . We set p0p0  ranged from [0.05,0.7][0.05,0.7]  with a small increment 5×1035×103 , and p1=p0+δp1=p0+δ , where δ=0.2,0.25δ=0.2,0.25 . We consider α{0.05,0.1}α{0.05,0.1}  and 1β{0.8,0.85,0.9}1β{0.8,0.85,0.9}  for all of these p0p0  and p1p1  values. We restricted the maximum of the total sample size nn  by 55 to reflect the small-sample-size characteristic of phase II trials. Finally, we considered both optimal and minimax designs under two stop criteria, i.e., (1) with futility stop only; (2) with both futility and superiority stops.

Under a parameter setting, we searched the (minimax or optimal) design (a1,a,n1,n)(a1,a,n1,n)  if the trial stops at futility stop of stage 1, (a1,b1,a,n1,n)(a1,b1,a,n1,n)  if the trial is allowed to stop at both futility and superiority stops of stage 1.

Based on the theoretical results in Appendix 7, we investigated: (1) under futility stop only, the trends of a1/n1p0a1/n1p0  (and a1n1p0a1n1p0 ) for stage 1, and a/n(p0+p1)/2a/n(p0+p1)/2  (and an(p0+p1)/2an(p0+p1)/2 ) for stage 2, i.e. we choose λ=1/2λ=1/2  only in our numerical experiment (based on some previous experience and we would like to confirm how if 1/21/2  is a good candidate for λλ ); (2) under both futility and superiority stops, the trends of a1/n1p0a1/n1p0  (and a1n1p0a1n1p0 ) and b1/n1p1b1/n1p1  (and b1n1p1b1n1p1 ) for stage 1, and a/n(p0+p1)/2a/n(p0+p1)/2  (and an(p0+p1)/2an(p0+p1)/2 ) for stage 2.

Results

For simplicity and because of the similarity of our findings in multiple cases, we only present the results under (α,1β,δ)=(0.05,0.8,0.2)(α,1β,δ)=(0.05,0.8,0.2) , where δ=p1p0δ=p1p0 , as a representative, under both stop criteria and both designs. The complete results under all parameters we considered can be found in Appendix 8.

Figures 1& 2 show that for different p0p0 , the trends of ratio differences we are interested, i.e., a1/n1p0a1/n1p0 , b1/n1p1b1/n1p1  and a/n(p0+p1)/2a/n(p0+p1)/2 . They indicate that overall, these differences are close to 0, and the difference is smaller on stage 2, i.e., a/n(p0+p1)/2a/n(p0+p1)/2 , under both stopping criteria, which means (p0+p1)/2(p0+p1)/2  is a good approximation to a/na/n  over the p0p0  we considered under this parameter setting. At the same time, we can observe that, the other two differences are also close to 0, but the differences under optimal design are overall closer to 0 than those of minimax design. The results for all other cases in Figure 3 to Figure 12 in Appendix 8.1 and 8.2 are similar to this case. We plotted all trends including differences in ratio and frequency. They also result in the same conclusion.

Figure 1 Trends of a1/n1a1/n1  and a/na/n  under both optimal and minimax designs, when (α*,1β*,δ)=(0.05,0.8,0.2)(α,1β,δ)=(0.05,0.8,0.2)  and there is futility stop only.

Figure 2 Trends of a1/n1a1/n1 ,b1/n1b1/n1 and a/na/n under both optimal and minimax designs, when (α*,1β*,δ)=(0.05,0.8,0.2)(α,1β,δ)=(0.05,0.8,0.2) and there are both futility and superiority stops.

Figure 3 The trends of a1n1p0a1n1p0 over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there is futility stop only.

Table 1 calculates the summary statistics of the trends in Figures 1 &2 over different p0  (in total, there are 131 p0 ’s in each figure). We found from the mean that, these differences are very close to 0 on average, and thus the approximation we use is good. We found from the interquartile range (IQR) and standard deviation (SD) that they are larger for minimax design than optimal design for a same difference, thus the variation of the approximation on optimal design is smaller. Table 2 further provides all numerical results under this setting, i.e. (α*,1β*,δ)=(0.05,0.8,0.2) , including all design outputs, ratios and differences of interest. We present results over p0[0.05,0.7]  but with increment 5×102  compared to 5×103  for the figures in order to maintain the concision and information provided by the table.

Concluding remarks

In this paper, we focused on single-arm two-stage phase II clinical trials, which is widely needed in reality. We specifically take the trends of critical values for rejecting null hypothesis at both two stages in a phase II clinical trial into consideration. Using both theoretical derivations under large-sample and numerical studies under finite and small sample, we confirmed that the critical values can be approximated and dominated by the total sample sizes and response rates of different stages. We also found that, this approximation is more obvious in optimal design than that in minimax design under the same design parameter setting. Our finding indicates that although minimax design provides less total sample size needed for a phase II trial, the property of an optimal design is closer to what we expect under large sample, and overall the optimal designs have more stable patterns on their relationships among critical values, sample sizes of both stages and response rates under null and alternative hypotheses.

Appendix: technical proofs

The numerical study in Section 5 confirms our hunch in Section 4.2 under finite (and actually small) sample. In this section, we provide some analytic proofs about the relationship between n1,a1,b1,n,a,p0  and p1  when the total sample size n  is large, using normal approximation, based on some regular assumptions (specified later).

Large sample approximation of single-stage phase II trials

In a single-arm single-stage trial, given a large n , we can find the approximate value of a  using large sample approximation. We assume that there exist ε  and δ>0  such that 0<εp0<p1=p0+δ1ε<1 , which means the response rates satisfy the “positivity”, i.e., bounded away from 0 and 1. In reality, this means we assume that a treatment cannot be effective or ineffective to all subjects in the target population. Next, denote ˉX=X/n , then under Hi (i=0,1) :

n(ˉXpi)dN(0,piqi)   (7.1.1)

as n , where qi=1pi , i=0,1 . Denote P(|p=pi)  by Pi(), i=0,1 . Given (α*,1β*) , n  and a  satisfy α*=P0(X>a)  and 1β*=P1(X>a) . So, under a large sample,

a/n=p0+z1α*p0q0/n    (7.1.2)

 and

a/n=p1z1β*p1q1/n   (7.1.3)

where zγ  is the γ -quantile of the standard normal distribution. From (7.1.2) and (7.1.3), we know

p1=p0+δp0+z1α*p0q0/n=p1z1β*p1q1/n

or

n=z1α*p0q0+z1β*p1q1p1p0

By plugging this in (7.1.2) or (7.1.3), we have

an=λp0+(1λ)p1   (7.1.4)

with   λ=z1β*p1q1z1α*p0q0+z1β*p1q1$ .

Note that both z1α*  and z1β*  are finite, and usually, in phase II clinical trials, we choose α*<β*  so that z1α*>z1β* . Hence, if p0q0<p1q1 , the weights in (7.1.4) are closer to 1/2 .

Large sample approximation of two-stage phase II trials

In the case of two-stage trials, we investigated in Section 5 by numerical studies under finite and small sample. Now, we provide an analytic proof assuming some regularity conditions about  under large sample. Following notations in Section 4.1, note that under Hi (i=0,1) :

nj(ˉXjpi)dN(0,piqi)   (7.2.1)

as nj , where qi=1pi, ˉXj=Xj/nj, j=1,2 .

Under futility stop only

Under a large n=n1+n2  and assume n1/nr(0,1) , a constant, as n , the condition P0(X1>a1,X1+X2>a)=α*  is equivalent to

α*=a1/n1S0;ˉX2(n1n2[an1ˉx1])f0;ˉX1(ˉx1)dˉx ,

where f0;ˉX1  is the density of ˉX1 , and S0;ˉX2  is the survival function of ˉX2  under H0 . Note that

S0;ˉX2(n1n2[an1ˉx1])=1Φ(n1n12(a/n1ˉx1)p0p0q0/n2)

where Φ  is the CDF of a standard normal distribution. Let

τ=a1/n1Φ(n1n12(a/n1ˉx1)p0p0q0/n2)f0;ˉX1(ˉx1)dˉx1 ,

Note that there exists an ε>0  such that 0<ετ1ε<1  since the lower limit of the above integral a1/n1>0 .

Then,

P0(X1>a1)=a1/n1f0;ˉX1(ˉx1)dˉx1=τ+α*   (7.2.2)

Therefore, by (7.2.1) and (7.2.2), we know

a1n1=p0+z1τα*p0q0/n1    (7.2.3)

where zγ  is the γ -quantile of the standard normal distribution. Thus, if we choose α*1τ  (achievable usually since α*  should be small in practice), then z1τα*  is finite, and thus a1n1p0  as n .

Next, we show the relationship among a, n, p0  and p1 . Let X=X1+X2  and by independence, X~Bin(n,pi)  under Hi (i=0,1) . Thus, (7.1.1) holds for ˉX=X/n . We know P0(X1>a1,X>a)=α*  and P1(X1>a1,X>a)=1β* .

One the one hand, P1(X>a)=1β*+P1(X>a,X1a1) , and let

τ1=P1(X>a,X1a1) =a1/n10S1;ˉX2(n1n2[an1ˉx1])f1;ˉX1(ˉx1)dˉx1

τ1=P1(X>a,X1a1) =a1/n10S1;ˉX2(n1n2[an1ˉx1])f1;ˉX1(ˉx1)dˉx1

On the other hand, P0(X>a)=α*+P0(X>a,X1a1)  and let

τ0=P0(X>a,X1a1) =a1/n10S0;ˉX2(n1n2[an1ˉx1])f0;ˉX1(ˉx1)dˉx1

S0;ˉX2(an2)a1/n10f0;ˉX1(ˉx1)dˉx1

We note that there exist an ε>0  such that Si;ˉX2(an2)1ε<1  for both i=0,1 , thus the right hand side of the above two probabilities are conservatively equal to 1ε  under large sample. Now,

a/n=p0+z1α*τ0p0q0/n=p1z1β*+τ1p1q1/n

Thus,

an=λp1+(1λ)p0   (7.2.4)

with λ=z1α*τ0p0q0z1α*τ0p0q0+z1β*+τ1p1q1 . By choosing α*,1β*<ε , we have τ0<1α*  and τ1<β* , so both z1α*τ0  and z1β*+τ1  are finite. Whether the weights in (7.2.4) are closer to 1/2 is a more complicated problem here than that in Section 7.1.

Under both futility and superiority stops

Following the same notations and assumptions made in Section 7.2.1, note that similar to the thoughts in Section 7.2.1, condition 1α*=P0(X1a1)+P0(a1<X1<b1,X1+X2a)  is equivalent to

P0(X1>a1)=a1/n1f0;ˉX1(ˉx1)dˉx1=τ+α*

where

τ=b1/n1a1/n1S0;ˉX2(n1n2[an1ˉx1])f0;ˉX1(ˉx1)dˉx1

Thus, by choosing α*<1τ , we can similarly show that

a1n1=p0+z1α*τp0q0/n1   (7.2.5)

 Thus, a1n1p0 .

Second, we note that P1(X1a1)+P1(a1<X1<b1,Xa)=P1(X1a1,Xa)+P1(X1a1,X>a)+P1(a1<X1<b1,Xa)=P1(X1<b1,Xa)+P1(X1a1,X>a)=P1(X1<b1)P1(X1<b1,X>a)+P1(X1a1,X>a) . Hence, the condition β*=P1(X1a1)+P1(a1<X1<b1,X1+X2a)  is equivalent to P1(X1<b1)P1(a1<X1<b1,X>a)=β* , thus

P1(X1>b1)=b1/n1f1;ˉX1(ˉx1)dˉx1=1β*+τ

where

τ=b1/n1a1/n1S1;ˉX2(n1n2[an1ˉx1])f1;ˉX1(ˉx1)dˉx1

Then, by choosing β*>τ , we know that

b1n1=p1z(1β*+τ)p1q1/n1   (7.2.6)

 Thus, b1n1p1 .

Finally, denote X=X1+X2 , and we have that β*=P1(X1a1)+P1(a1<X1<b1,Xa) ,

and 1α*=P0(X1a1)+P0(a1<X1<b1,Xa) . Let

τ0=P0(a1<X1<b1,Xa)=b1/n1a1/n1Φ(n1n12(a/n1ˉx1)p0p0q0/n2)f0;ˉX1(ˉx1)dˉx1

and

τ1=P1(a1<X1<b1,Xa)=b1/n1a1/n1Φ(n1n12(a/n1ˉx1)p1p1q1/n2)f1;ˉX1(ˉx1)dˉx1

where Φ  is the CDF of the standard normal distribution, and thus choosing α*<1τ0  and β*>τ1 , similar to the proof in Section 7.2.1, we have

an=λp0+(1λ)p1   (7.2.7)

where λ=z1α*τ0p0q0z1α*τ0p0q0+z1β*+τ1p1q1 .

Appendix: complete results of the numerical study

We provide figures of two types of trends. That is, trends in frequency (Figures 3 & 4) for investigating the trends of  and , and trends in difference of ratio (Figures 5 & 6) for investigating the trends of a1/n1  and a/n .

Figure 4 The trends of an(p0+p1)/2 over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there is futility stop only.

Figure 5 The trends of a1/n1p0 (ratio) over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there is futility stop only.

Figure 6 The trends of an(p0+p1)/2 (ratio) over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there is futility stop only.

Table 3– Table 8 present all numerical results, including the returns of designs, ratios, and differences of interest under all parameter settings in this numerical study. Specifically, we only present under p0[0.05,0.7]  with an increment 5×102  compared to 5×103  in the figures, for maintaining the information and concision provided by the tables simultaneously. Each table represents a case of (α,1β) .

Results under both futility and superiority stops

Same as Section 8.1, we provide figures of trends in frequency (Figure 7-Figure 9) for investigating the trends of a1 , b1  and a , and trends in difference of ratio (Figure 10-Figure 12) for investigating the trends of a1/n1 , b1/n1  and a/n .

Figure 7 The trends of a1n1p0 over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there are both futility and superiority stops.

Figure 8 The trends of b1/n1p1 over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there are both futility and superiority stops.

Figure 9 The trends of an(p0+p1)/2 over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there are both futility and superiority stops.

Figure 10 The trends of a1/n1p0 (ratio) over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there are both futility and superiority stops.

Figure 11 The trends of b1/n1p1 over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there are both futility and superiority stops.

Figure 12 The trends of an(p0+p1)/2 (ratio) over different choices of α*,1β*,δ=p1p0 , under both optimal and minimax designs, when there are both futility and superiority stops.

Table 9-Table 14 present all numerical results, including the returns of designs, ratios, and differences of interest under all parameter settings in this numerical study. Specifically, we only present under p0[0.05,0.7]  with an increment 5×102  compared to 5×103  in the figures, for maintaining the information and concision provided by the tables simultaneously. Each table represents a case of α,1β .

Acknowledgments

None.

Conflicts of interest

The authors declared that there are no conflicts of interest.

References

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