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Minimax and admissible adaptive two-stage designs in phase II clinical trials
- Guogen Shan^{1},
- Hua Zhang^{2} and
- Tao Jiang^{3}Email author
https://doi.org/10.1186/s12874-016-0194-3
© The Author(s) 2016
Received: 18 March 2016
Accepted: 22 July 2016
Published: 2 August 2016
Abstract
Background
Simon’s two-stage design is the most widely implemented among multi-stage designs in phase II clinical trials to assess the activity of a new treatment in a single-arm study. In this two-stage design, the sample size from the second stage is fixed regardless of the number of responses observed in the first stage.
Methods
We develop a new minimax adaptive design for phase II clinical trials, by using the branch-and-bound intelligent algorithm based on conditional error functions.
Results
We compare the performance of the proposed design and competitors, including Simon’s minimax design, and a modified Simon’s design that allows early stopping for futility or efficacy. The maximum sample size of the proposed minimax adaptive design is guaranteed to be less than or equal to those from other existing designs. When the proposed design has the same maximum sample size as others, it always has the smallest expected sample size. In addition to the minimax adaptive design, we also introduce admissible adaptive designs determined from a Bayesian perspective.
Conclusions
The proposed adaptive minimax design can save sample sizes for a clinical trial. The minimum required sample size is critical to reduce the cost of a project.
Keywords
Background
In phase II clinical trials, a new treatment or a new therapy is often assessed by measuring activity with dichotomized endpoints, responding ’yes’ or ’no’ to the intervention. For Oncology clinical trials, the response criteria may be determined by the Response Evaluation Criteria In Solid Tumours (RECIST) [1]. The traditional experiment in phase II Oncology trials is often conducted in a single arm study, which is also popular in other studies, such as AIDS. All patients enrolled in the study are treated with the same treatment, and their measurements are obtained at the end of the study and compared to the priori estimate from historical studies with the similar condition of experiment and patients. From ethical and economical considerations, a trial should be allowed to stop earlier after an interim analysis to better protect patients, especially in situations when the treatment is indeed ineffective. For this reason, a multi-stage design is often implemented, and among these designs the most popular is Simon’s two-stage design [2]. Simon [2] proposed two optimal designs: the optimal design with the expected sample size under the null (ESS_{0}) minimized, and the minimax design having the smallest ESS_{0} among the designs with the maximum sample size (MSS) minimized. Simon’s design allows early stopping in the first stage for futility only. Recently, Mander and Thompson [3] extended Simon’s design to allow stopping for efficacy or futility by introducing an additional design parameter that represents the stopping criteria for efficacy in the first stage. It is guaranteed that the MSS of the modified design is less than or equal to that of Simon’s.
In Simon’s design and the modified Simon’s design due to Mander and Thompson [3], the second stage sample size is always fixed and is not allowed to be modified as the result observed from the first stage. To make a design flexible and efficient, adaptive designs have been developed to allow the second stage sample size to depend on first stage responders. It is easy to show that Simon’s optimal design is a special case of adaptive designs, therefore, the expected sample size of an optimal adaptive design is always less than or equal to that of Simon’s design. Several optimal adaptive designs have been developed for phase II clinical trials, and the majority of them are based on the optimal criteria with the smallest ESS_{0}. Banerjee and Tsiatis [4] developed an optimal adaptive two-stage design by using a Bayesian decision-theoretic construct to minimize the expected loss through backward induction, with type I and II error rates respected. The sample size savings are small to modest when compared to Simon’s optimal design. Later, Englert and Kieser [5] developed an optimal adaptive two-stage design based on conditional error functions [6] and an efficient search strategy [7]. Although these adaptive designs guarantee the type I and II error rates, these designs suffer from a counter-intuitive feature that the second stage sample size may increase as the number of responses observed from the first stage increases. Very recently, Shan et al. [8] developed an optimal adaptive two-stage design with the monotonicity property respected; the second stage sample size is a non-increasing function of the first stage responders. This improvement makes it practical to use the optimal adaptive design.
In phase II clinical trials, it is desirable to achieve the primary goal of the study with the number of patients minimized, as the cost of the study highly depends on the number of patients. In addition, Institutional Review Boards approve proposed studies based on the maximum possible number of patients that are needed to address the scientific questions. Therefore, the minimax design is preferable by researchers with the smallest MSS as compared to the optimal design when the MSS difference between the two designs is not small. To our best knowledge, no adaptive design based on the minimax criteria has been proposed for use in practice. Due to the importance of such designs, we develop a new minimax adaptive design with the monotonic property respected in this article by using the branch-and-bound algorithm [7] based on conditional type I and II error rates.
Both minimax and optimal designs have been widely used in clinical trials. It is often the case that the expected sample size of the minimax design is much larger than that of the optimal design, although the minimax design has a smaller maximum sample size. To compromise between the maximum sample size and the expected sample size under the null, an admissible adaptive design was proposed by Jung et al. [9], which was implemented in Java language by them. By using the Bayes risk function as in Jung et al. [9], we propose a new admissible adaptive two-stage design that is between the minimax adaptive design and the optimal adaptive design.
The remainder of this article is organized as follows. In “Methods” Section, we introduce the detailed search method for the optimal adaptive design when the first stage and the MSS of the second stage sample size are fixed, then present the approach to find the minimax adaptive design. In “Results” Section we compare the MSS and the ESS_{0} of the proposed minimax adaptive design with competitors. A real clinical trial from a cancer study is used to illustrate the proposed design at the end of “Results” Section. Finally, we provide some remarks in “Discussion and conclusions” Section.
Methods
The null hypothesis is rejected for a large response rate. Let n_{1}, n_{2}, and n be the number of subjects enrolled in the first stage, the second stage, and both stages combined, respectively, and x_{1}, x_{2}, and x are the associated number of responses observed from the study.
In the clinical trial of the neoadjuvant therapy for urothelial cancer [10], Simon’s minimax design was used for sample size determination to achieve 80 % power (β=0.2) at the significance level of α=0.1 when the response rates were π_{ u }=35 % and π_{ a }=50 %. The design was calculated as: (r_{1}/n_{1},r/n)=(10/31,21/49) with the ESS_{0}=40.8. The trial was allowed to stop for futility at the first stage if the number of first stage responses x_{1}≤10 was observed from a total of n_{1}=31 patients. Otherwise, an additional n_{2}=n−n_{1}=49−31=18 patients were enrolled in the second stage, and at least 22 responses should be observed from total 49 patients, x≥22, in order to claim that the neoadjuvant therapy had sufficient activity. The MSS of the minimax design was 49. An alterntive to the minimax design is Simon’s optimal design whose ESS_{0} was the smallest among all designs that met the design criteria. The design parameters for the optimal design are: (r_{1}/n_{1},r/n)=(7/20,24/58) with the ESS _{0}=35.2. As expected, the ESS_{0} of the optimal design is less than that of the minimax design (35.2 agaisnt 40.8), but the MSS is much larger for the optimal design as compared to the minimax design (58 against 49).
The proposed adaptive minimax design for the urothelial cancer trial with the neoadjuvant therapy with (α,β,π_{ u },π_{ a })=(0.1,0.2,0.35,0.5)
S | n_{2}(S) | n(S) | r(S) |
---|---|---|---|
Minimax-EF design | |||
≤11 | 0 | 32 | 0 |
12 | 17 | 49 | 21 |
13 | 17 | 49 | 21 |
14 | 17 | 49 | 21 |
15 | 17 | 49 | 21 |
16 | 17 | 49 | 21 |
≥17 | 0 | 32 | 0 |
Minimax adaptive design | |||
≤9 | 0 | 28 | 0 |
10 | 21 | 49 | 21 |
11 | 21 | 49 | 21 |
12 | 21 | 49 | 21 |
13 | 21 | 49 | 21 |
14 | 19 | 47 | 20 |
15 | 18 | 46 | 20 |
≥16 | 0 | 28 | 0 |
where ESS\(_{0}=\sum _{s=0}^{n_{1}}[n_{1}+n_{2}(s)]\times b(s:n_{1},\pi _{u})\) is the expected sample size under the null for the design with n_{1} and max(n_{1}+n_{2}(s),s=0,1,…,n_{1}) as the first stage sample size and the MSS. The min in Eq. (1) is used in two folders. The function is first used to find all satisfied designs with the smallest MSS, \(\min _{\max (n_{1}+n_{2}(s),s=0,1,\ldots,n_{1})}\). The second is to identify the minimax adaptive design as the one from these in the previous step with the smallest ESS_{0}.
As the first stage sample size n_{1} increases, the size of this complete search space increases exponentially. Therefore, it is not feasible to conduct this naive search to identify the optimal design.
It is much more complicated to search for an optimal solution over a two-dimensional space than a one-dimensional space. For this reason, Englert and Kieser [5] suggested using the union of all type I conditional error functions and (0,1), referred to as Ω, as the parameter space. For each element in Ω, it contains the information of n_{2}(S) and r(S) as in the two-dimensional space. That said, it is equivalent to determine the conditional type I error value for S and (r(S),n_{2}(S)). It is still not feasible to conduct a grid search over the parameter space (a(n_{1}+1)−dimensional space) due to the fact that the size of the parameter space increases very quickly as n_{1} and n_{2,max} go up.
In order to overcome the computational burden, the branch-and-bound algorithm [7], an intelligent algorithm, is considered when searching for the optimal design over a one-dimensional space on each S. This algorithm can be used to search for the optimal design with or without constraints [5, 8]. The monotonicity restriction in the optimal adaptive design search by Shan et al. [8] is an important feature that makes a design usable in practice. The second stage sample size is a non-increasing function of the number of responses observed from the first stage: n_{2}(S_{1})≥n_{2}(S_{2}) when S_{1}<S_{2}. This monotonicity restriction is respected in the proposed minimax adaptive design.
As pointed out, it is time consuming to compute the actual type I and II error rates for each element in the parameter space, and the branch-and-bound algorithm is able to finish the design search in a timely manner by discarding elements that do not lead to the optimal design, which is the key idea of this intelligent algorithm. When the sample sizes (n_{1},n) are given, the ESS_{0} is the objective function. Two procedures are recursively utilized in the algorithm to identify the optimal design. The first procedure is the branching process that splits the problem into several complement problems. The conditional type I error functions are used in this step to split problems. Although it is not a requirement to sort the elements in Ω in the design search, it helps to reduce the computational intensity to sort them by n_{2}(S) in an ascending order, and P(S|π_{ u }) in an increasing order. The ordering of n_{2}(S) is used to meet the monotonicity feature of the proposed design.
These two constraints help to determine the set of feasible solutions, and discard the candidates that do not lead to the optimal design.
The minimax-EF design is a special case of the minimax adaptive design, therefore, the MSS of the minimax-EF design is the upper bound of the proposed adaptive design. For this reason, we start the search with the MSS, n_{ t }, which is the MSS of the minimax-EF design. For this given MSS, say n_{ t }, the possible number of subjects from the first stage, n_{1}, ranges between 1 and n_{ t }−1. The search for n_{1}=1 and n_{ t }−1 as the first stage sample sizes are excluded for practical reasons: it is not realistic to enroll only one patient to make a decision.
For each sample size configuration (n_{1},n_{ t }), the algorithm is applied for the design search. If the study is stopped for futility when S≤s−1, then we assign n_{2}(s,W_{ s })=n−n_{1} to guarantee that the MSS is exactly n_{ t }. It should be noted that the MSS could occur at multiple S values. The ascending order of n_{2}(S) for elements in parameter space Ω, is useful to meet the monotonic relationship between the n_{2}(S) and S in searching for the design. Among these obtained optimal adaptive designs, the one with the smallest ESS_{0} is the adaptive minimax design when the MSS is n_{ t }. From the relationship between the proposed design and the minimax-EF design, it is guaranteed that an optimal adaptive design will be obtained when MSS is n_{ t }. The MSS is then decreased by 1, and the optimal adaptive design is searched again with the MSS= n_{ t }−1. This procedure will be continued until no optimal design is obtained from three consecutive MMS values, say n^{∗}−1, n^{∗}−2, and n^{∗}−3. Then, n^{∗} is the minimum MSS, and the optimal design associated with n^{∗} is the final minimax adaptive two-stage design. It is obvious that n^{∗}≤n_{ t }.
Results
We compare performance of the proposed minimax adaptive design, Simon’s minimax design, the minimax-EF design, and the optimal adaptive design due to Shan et al. [8]. The first three designs are minimax designs, while the last one is under the optimal criteria. The first design and the last design are adaptive designs. To the best of our knowledge, we do not find a direct competitor in the category of adaptive two-stage designs under the minimax criteria. Simon’s minimax design is the most commonly used design under the minimax criteria, thus it is included in the comparison. The minimax-EF design stops for either futility or efficacy in the first stage. This stopping rule is also applied in the proposed design, thus, this design is also included in the comparison. Simon’s design only allows stopping of the trial at the first stage for futility, and the other three designs allows the stoppage for either futility or efficacy in the first stage.
Comparison between three optimal designs for expected sample size ESS_{0} at α=0.05 given π_{ a }−π_{ u }=0.2 and 0.15
Minimax | Optimal | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Simon | Minimax-EF | Adaptive | Adaptive | |||||||
π _{ u } | π _{ a } | Power | n | E S S _{0} | n | E S S _{0} | n | E S S _{0} | n | E S S _{0} |
π_{ a }−π_{ u }=20 % | ||||||||||
0.1 | 0.3 | 0.8 | 25 | 19.51 | 24 | 20.30 | 23 | 20.94 | 29 | 14.85 |
0.9 | 33 | 26.18 | 33 | 23.96 | 33 | 23.93 | 35 | 22.38 | ||
0.2 | 0.4 | 0.8 | 33 | 22.25 | 32 | 24.93 | 32 | 23.22 | 37 | 20.48 |
0.9 | 45 | 31.23 | 44 | 35.68 | 44 | 33.39 | 53 | 29.74 | ||
0.3 | 0.5 | 0.8 | 39 | 25.69 | 36 | 30.68 | 36 | 29.31 | 46 | 23.45 |
0.9 | 53 | 36.62 | 50 | 42.47 | 50 | 41.03 | 60 | 34.08 | ||
0.4 | 0.6 | 0.8 | 39 | 34.44 | 39 | 34.33 | 39 | 26.86 | 46 | 24.39 |
0.9 | 54 | 38.06 | 54 | 38.03 | 53 | 42.65 | 66 | 35.64 | ||
0.5 | 0.7 | 0.8 | 37 | 27.74 | 37 | 26.90 | 37 | 26.87 | 43 | 23.33 |
0.9 | 53 | 36.11 | 51 | 41.14 | 51 | 37.74 | 59 | 33.45 | ||
0.6 | 0.8 | 0.8 | 35 | 20.77 | 33 | 23.97 | 33 | 22.13 | 38 | 20.28 |
0.9 | 45 | 35.90 | 45 | 33.30 | 45 | 31.36 | 52 | 28.74 | ||
0.7 | 0.9 | 0.8 | 26 | 23.16 | 26 | 23.11 | 25 | 18.00 | 27 | 14.82 |
0.9 | 32 | 22.66 | 32 | 22.66 | 32 | 22.64 | 36 | 20.80 | ||
π_{ a }−π_{ u }=15 % | ||||||||||
0.1 | 0.25 | 0.8 | 40 | 28.84 | 38 | 33.94 | 38 | 28.87 | 43 | 24.49 |
0.9 | 55 | 40.03 | 53 | 47.87 | 53 | 41.29 | 62 | 36.45 | ||
0.2 | 0.35 | 0.8 | 53 | 40.44 | 53 | 40.41 | 53 | 40.33 | 63 | 34.87 |
0.9 | 77 | 58.42 | 76 | 66.51 | 74 | 59.58 | 87 | 50.80 | ||
0.3 | 0.45 | 0.8 | 65 | 49.63 | 64 | 51.32 | 64 | 48.08 | 77 | 41.33 |
0.9 | 88 | 78.51 | 88 | 78.45 | 88 | 68.29 | 104 | 59.96 | ||
0.4 | 0.55 | 0.8 | 70 | 60.07 | 69 | 54.17 | 69 | 49.84 | 82 | 44.05 |
0.9 | 94 | 78.88 | 94 | 76.30 | 94 | 74.20 | 106 | 63.84 | ||
0.5 | 0.65 | 0.8 | 68 | 66.11 | 68 | 66.05 | 67 | 58.41 | 81 | 43.01 |
0.9 | 93 | 75.00 | 93 | 72.20 | 93 | 69.84 | 109 | 61.87 | ||
0.6 | 0.75 | 0.8 | 62 | 43.79 | 62 | 42.89 | 61 | 45.26 | 69 | 38.53 |
0.9 | 84 | 73.20 | 84 | 73.13 | 84 | 64.00 | 97 | 54.99 | ||
0.7 | 0.85 | 0.8 | 49 | 34.44 | 49 | 34.36 | 49 | 33.00 | 59 | 29.78 |
0.9 | 68 | 48.52 | 65 | 50.46 | 65 | 48.78 | 78 | 42.60 |
Proposed optimal adaptive designs for π_{ a }=π_{ u }+0.2 at α=0.05. Simon’s minimax design (r_{1}/n_{1},r/n), and the minimax that stops for futility and efficacy ((r_{1},r_{2})/n_{1},r/n), are provided as reference
Power = 80 % | Power = 90 % | ||||||
---|---|---|---|---|---|---|---|
S | n_{2}(S) | n(S) | r(S) | S | n_{2}(S) | n(S) | r(S) |
π_{ u }=0.2 | |||||||
Simon: (4/18,10/33) | Simon: (5/24,13/45) | ||||||
Minimax-EF: ((2,6)/15,10/32) | Minimax-EF: ((4,9)/25,13/44) | ||||||
New: n_{1}=19 | New: n_{1}=23 | ||||||
≤4 | 0 | 19 | 0 | ≤4 | 0 | 23 | 0 |
5 | 13 | 32 | 10 | 5 | 21 | 44 | 12 |
6 | 13 | 32 | 10 | 6 | 21 | 44 | 13 |
7 | 13 | 32 | 9 | 7 | 21 | 44 | 13 |
8 | 13 | 32 | 10 | 8 | 21 | 44 | 13 |
9 | 11 | 30 | 10 | 9 | 21 | 44 | 13 |
≥10 | 0 | 19 | 0 | 10 | 15 | 38 | 11 |
≥11 | 0 | 23 | 0 | ||||
π_{ u }=0.3 | |||||||
Simon: (6/19,16/39) | Simon: (7/24,21/53) | ||||||
Minimax-EF: ((8,13)/27,15/36) | Minimax-EF: ((11,17)/37,20/50) | ||||||
New: n_{1}=20 | New: n_{1}=32 | ||||||
≤5 | 0 | 20 | 0 | ≤9 | 0 | 32 | 0 |
6 | 16 | 36 | 14 | 10 | 18 | 50 | 19 |
7 | 16 | 36 | 15 | 11 | 18 | 50 | 20 |
8 | 16 | 36 | 15 | 12 | 18 | 50 | 20 |
9 | 16 | 36 | 15 | 13 | 18 | 50 | 20 |
10 | 16 | 36 | 15 | 14 | 18 | 50 | 20 |
11 | 16 | 36 | 15 | 15 | 18 | 50 | 20 |
12 | 14 | 34 | 15 | 16 | 18 | 50 | 20 |
≥13 | 0 | 20 | 0 | 17 | 11 | 43 | 18 |
≥18 | 0 | 32 | 0 | ||||
π_{ u }=0.4 | |||||||
Simon: (17/34,20/39) | Simon: (12/29,27/54) | ||||||
Minimax-EF: ((17,19)/34,20/39) | Minimax-EF: ((12,19)/29,27/54) | ||||||
New: n_{1}=16 | New: n_{1}=35 | ||||||
≤6 | 0 | 16 | 0 | ≤14 | 0 | 35 | 0 |
7 | 23 | 39 | 20 | 15 | 18 | 53 | 26 |
8 | 23 | 39 | 20 | 16 | 18 | 53 | 27 |
9 | 23 | 39 | 20 | 17 | 18 | 53 | 27 |
10 | 23 | 39 | 20 | 18 | 18 | 53 | 27 |
11 | 23 | 39 | 21 | 19 | 18 | 53 | 26 |
12 | 22 | 38 | 20 | 20 | 17 | 52 | 26 |
13 | 16 | 32 | 18 | 21 | 17 | 52 | 26 |
14 | 9 | 25 | 16 | 22 | 17 | 52 | 26 |
15 | 5 | 21 | 15 | 23 | 17 | 52 | 27 |
16 | 3 | 19 | 16 | ≥24 | 0 | 35 | 0 |
π_{ u }=0.5 | |||||||
Simon: (12/23,23/37) | Simon: (14/27,32/53) | ||||||
Minimax-EF: ((10,15)/20,23/37) | Minimax-EF: ((17,23)/34,31/51) | ||||||
New: n_{1}=20 | New: n_{1}=28 | ||||||
≤10 | 0 | 20 | 0 | ≤14 | 0 | 28 | 0 |
11 | 17 | 37 | 23 | 15 | 23 | 51 | 30 |
12 | 17 | 37 | 23 | 16 | 23 | 51 | 31 |
13 | 17 | 37 | 23 | 17 | 23 | 51 | 31 |
14 | 17 | 37 | 23 | 18 | 23 | 51 | 31 |
15 | 15 | 35 | 22 | 19 | 23 | 51 | 31 |
≥16 | 0 | 20 | 0 | 20 | 23 | 51 | 31 |
21 | 21 | 49 | 29 | ||||
22 | 6 | 34 | 22 | ||||
≥23 | 0 | 28 | 0 | ||||
π_{ u }=0.6 | |||||||
Simon: (8/13,25/35) | Simon: (15/26,32/45) | ||||||
Minimax-EF: ((10,14)/17,24/33) | Minimax-EF: ((15,20)/25,32/45) | ||||||
New: n_{1}=15 | New: n_{1}=23 | ||||||
≤9 | 0 | 15 | 0 | ≤14 | 0 | 23 | 0 |
10 | 18 | 33 | 24 | 15 | 22 | 45 | 32 |
11 | 18 | 33 | 24 | 16 | 22 | 45 | 32 |
12 | 17 | 32 | 23 | 17 | 21 | 44 | 31 |
13 | 16 | 31 | 22 | 18 | 21 | 44 | 31 |
14 | 14 | 29 | 21 | 19 | 21 | 44 | 31 |
15 | 14 | 29 | 21 | 20 | 10 | 33 | 24 |
21 | 10 | 33 | 25 | ||||
22 | 8 | 31 | 24 | ||||
≥23 | 0 | 23 | 0 | ||||
π_{ u }=0.7 | |||||||
Simon: (19/23,21/26) | Simon: (13/18,26/32) | ||||||
Minimax-EF: ((19,20)/23,21/26) | Minimax-EF: ((13,18)/18,26/32) | ||||||
New: n_{1}=13 | New: n_{1}=18 | ||||||
≤9 | 0 | 13 | 0 | ≤13 | 0 | 18 | 0 |
10 | 12 | 25 | 21 | 14 | 14 | 32 | 26 |
11 | 12 | 25 | 21 | 15 | 14 | 32 | 26 |
12 | 12 | 25 | 20 | 16 | 14 | 32 | 26 |
13 | 7 | 20 | 16 | 17 | 14 | 32 | 26 |
≥17 | 3 | 21 | 18 |
Proposed optimal adaptive designs for π_{ a }=π_{ u }+0.15 at α=0.05. Simon’s minimax design (r_{1}/n_{1},r/n), and the minimax that stops for futility and efficacy ((r_{1},r_{2})/n_{1},r/n), are provided as reference
Power=80 % | Power=90 % | ||||||
---|---|---|---|---|---|---|---|
S | n_{2}(S) | n(S) | r(S) | S | n_{2}(S) | n(S) | r(S) |
π_{ u }=0.1 | |||||||
Simon: (2/22,7/40) | Simon: (3/31,9/55) | ||||||
Minimax-EF: ((4,6)/33,7/38) | Minimax-EF: ((6,8)/47,9/53) | ||||||
New: n_{1}=18 | New: n_{1}=33 | ||||||
≤1 | 0 | 18 | 0 | ≤3 | 0 | 33 | 0 |
2 | 20 | 38 | 7 | 4 | 20 | 53 | 8 |
3 | 20 | 38 | 7 | 5 | 20 | 53 | 9 |
4 | 19 | 37 | 6 | 6 | 20 | 53 | 9 |
5 | 19 | 37 | 6 | 7 | 18 | 51 | 8 |
6 | 18 | 36 | 6 | 8 | 17 | 50 | 8 |
≥7 | 0 | 18 | 0 | ≥9 | 0 | 33 | 0 |
π_{ u }=0.2 | |||||||
Simon: (6/31,15/53) | Simon: (8/42,21/77) | ||||||
Minimax-EF: ((6,13)/31,15/53) | Minimax-EF: ((13,18)/62,21/76) | ||||||
New: n_{1}=31 | New: n_{1}=47 | ||||||
≤6 | 0 | 31 | 0 | ≤9 | 0 | 47 | 0 |
7 | 22 | 53 | 15 | 10 | 27 | 74 | 20 |
8 | 22 | 53 | 15 | 11 | 27 | 74 | 20 |
9 | 22 | 53 | 15 | 12 | 27 | 74 | 20 |
10 | 22 | 53 | 15 | 13 | 27 | 74 | 20 |
11 | 22 | 53 | 15 | 14 | 26 | 73 | 20 |
12 | 21 | 52 | 15 | 15 | 26 | 73 | 20 |
≥13 | 0 | 31 | 0 | 16 | 26 | 73 | 20 |
17 | 25 | 72 | 20 | ||||
18 | 14 | 61 | 18 | ||||
≥19 | 0 | 47 | 0 | ||||
π_{ u }=0.3 | |||||||
Simon: (16/46,25/65) | Simon: (27/77,33/88) | ||||||
Minimax-EF: ((13,19)/43,25/64) | Minimax-EF: ((27,33)/77,33/88) | ||||||
New: n_{1}=32 | New: n_{1}=51 | ||||||
≤9 | 0 | 32 | 0 | ≤15 | 0 | 51 | 0 |
10 | 32 | 64 | 24 | 16 | 37 | 88 | 33 |
11 | 32 | 64 | 25 | 17 | 37 | 88 | 33 |
12 | 32 | 64 | 25 | 18 | 37 | 88 | 33 |
13 | 32 | 64 | 25 | 19 | 37 | 88 | 33 |
14 | 32 | 64 | 25 | 20 | 37 | 88 | 33 |
15 | 31 | 63 | 24 | 21 | 37 | 88 | 33 |
16 | 30 | 62 | 24 | 22 | 37 | 88 | 33 |
17 | 29 | 61 | 24 | 23 | 37 | 88 | 33 |
18 | 24 | 56 | 22 | 24 | 37 | 88 | 34 |
≥19 | 0 | 32 | 0 | 25 | 36 | 87 | 33 |
26 | 34 | 85 | 33 | ||||
27 | 34 | 85 | 33 | ||||
≥28 | 0 | 51 | 0 | ||||
π_{ u }=0.4 | |||||||
Simon: (28/59,34/70) | Simon: (24/62,45/94) | ||||||
Minimax-EF: ((16,23)/41,34/69) | Minimax-EF: ((21,31)/55,45/94) | ||||||
New: n_{1}=37 | New: n_{1}=52 | ||||||
≤15 | 0 | 37 | 0 | ≤20 | 0 | 52 | 0 |
16 | 32 | 69 | 33 | 21 | 42 | 94 | 44 |
17 | 32 | 69 | 34 | 22 | 42 | 94 | 45 |
18 | 32 | 69 | 34 | 23 | 42 | 94 | 45 |
19 | 32 | 69 | 34 | 24 | 42 | 94 | 45 |
20 | 32 | 69 | 34 | 25 | 42 | 94 | 45 |
21 | 31 | 68 | 33 | 26 | 42 | 94 | 45 |
22 | 31 | 68 | 33 | 27 | 42 | 94 | 45 |
23 | 31 | 68 | 33 | 28 | 42 | 94 | 45 |
24 | 21 | 58 | 29 | 29 | 42 | 94 | 45 |
≥25 | 0 | 37 | 0 | 30 | 42 | 94 | 45 |
31 | 39 | 91 | 43 | ||||
≥32 | 0 | 52 | 0 | ||||
π_{ u }=0.5 | |||||||
Simon: (39/66,40/68) | Simon: (28/57,54/93) | ||||||
Minimax-EF: ((39,40)/66,40/68) | Minimax-EF: ((30,38)/59,54/93) | ||||||
New: n_{1}=54 | New: n_{1}=55 | ||||||
≤28 | 0 | 54 | 0 | ≤28 | 0 | 55 | 0 |
29 | 13 | 67 | 39 | 29 | 38 | 93 | 54 |
30 | 13 | 67 | 40 | 30 | 38 | 93 | 54 |
31 | 13 | 67 | 40 | 31 | 38 | 93 | 54 |
32 | 13 | 67 | 40 | 32 | 38 | 93 | 54 |
33 | 13 | 67 | 40 | 33 | 38 | 93 | 54 |
34 | 13 | 67 | 39 | 34 | 38 | 93 | 54 |
35 | 13 | 67 | 39 | 35 | 38 | 93 | 53 |
36 | 13 | 67 | 39 | 36 | 38 | 93 | 54 |
37 | 9 | 63 | 38 | 37 | 38 | 93 | 53 |
≥38 | 0 | 54 | 0 | ≥38 | 0 | 55 | 0 |
π_{ u }=0.6 | |||||||
Simon: (18/30,43/62) | Simon: (48/72,57/84) | ||||||
Minimax-EF: ((16,22)/27,43/62) | Minimax-EF: ((48,53)/72,57/84) | ||||||
New: n_{1}=32 | New: n_{1}=58 | ||||||
≤19 | 0 | 32 | 0 | ≤37 | 0 | 58 | 0 |
20 | 29 | 61 | 42 | 38 | 26 | 84 | 57 |
21 | 29 | 61 | 42 | 39 | 26 | 84 | 57 |
22 | 29 | 61 | 42 | 40 | 26 | 84 | 57 |
23 | 28 | 60 | 42 | 41 | 26 | 84 | 57 |
24 | 28 | 60 | 42 | 42 | 25 | 83 | 57 |
25 | 28 | 60 | 42 | 43 | 25 | 83 | 57 |
26 | 27 | 59 | 41 | 44 | 23 | 81 | 56 |
27 | 27 | 59 | 41 | ≥45 | 0 | 58 | 0 |
28 | 15 | 47 | 34 | ||||
≥29 | 0 | 32 | 0 | ||||
π_{ u }=0.7 | |||||||
Simon: (16/23,39/49) | Simon: (33/44,53/68) | ||||||
Minimax-EF: ((16,21)/23,39/49) | Minimax-EF: ((29,35)/41,51/65) | ||||||
New: n_{1}=25 | New: n_{1}=37 | ||||||
≤18 | 0 | 25 | 0 | ≤26 | 0 | 37 | 0 |
19 | 24 | 49 | 39 | 27 | 28 | 65 | 51 |
20 | 24 | 49 | 39 | 28 | 28 | 65 | 51 |
21 | 24 | 49 | 39 | 29 | 28 | 65 | 51 |
22 | 23 | 48 | 38 | 30 | 28 | 65 | 51 |
23 | 8 | 33 | 26 | 31 | 27 | 64 | 50 |
≥24 | 0 | 25 | 0 | 32 | 27 | 64 | 50 |
33 | 24 | 61 | 48 | ||||
≥34 | 0 | 37 | 0 |
Probability of early termination at the first stage for the designs with π_{ a }=π_{ u }+0.2 and 80 % power
π _{ u } | Simon | Minimax-EF | New adaptive design |
---|---|---|---|
0.2 | 0.716 | 0.402 | 0.674 |
0.3 | 0.666 | 0.582 | 0.417 |
0.4 | 0.913 | 0.921 | 0.527 |
0.5 | 0.661 | 0.589 | 0.589 |
0.6 | 0.647 | 0.554 | 0.597 |
0.7 | 0.946 | 0.949 | 0.579 |
Admissible adaptive designs for (α,β,π_{ u },π_{ a })=(0.05,0.1,0.3,0.5)
Interval of q | n | ESS_{0} | Comment |
---|---|---|---|
[0.000,0.040] | 60 | 34.08 | Optimal design |
[0.040,0.105] | 59 | 34.12 | |
[0.105,0.132] | 57 | 34.36 | |
[0.132,0.468] | 54 | 34.81 | |
[0.468,0.580] | 53 | 35.69 | |
[0.580,0.721] | 51 | 38.45 | |
[0.721,1.000] | 50 | 41.03 | Minimax design |
Application
Discussion and conclusions
We develop a new minimax adaptive two-stage design for use in phase II clinical trials to assess the new treatment’s activity. The software program to implement the adaptive designs in this article is written in the statistical language, R [11–14], and it is available per request from the first author (guogen.shan@unlv.edu) or the corresponding author (jtao@263.net). We are also working together to develop a new R package to implement the adaptive minimax and admissible designs from this article and the adaptive optimal design by Shan et al. [8]. The proposed design allows the second stage sample size and its associated critical value to depend on the result from the first stage. The proposed design satisfies the monotonicity property of the relationship between the second stage sample size and the first stage responders, which is an important feature for a practical application.
The MSS of the proposed adaptive minimax design is always less than or equal to that of the minimax-EF design. We consider this as an important advantage of the minimax adaptive design to reduce the computational intensity as compared to adaptive designs based on the optimal criteria [8], where the upper bound of the sample size has to be set in the design search process. To reduce the computational time, one may use a backward search method as in this article, starting with the maximum sample size from the minimax-EF design. In addition, when the proposed design and other designs have the same MSS, the expected sample size under the null of the proposed design is always smaller than others.
The proposed adaptive design assumes a monotonic relationship between the second stage sample size and the first stage result. In practice, an investigator may want to accrue more patients in the second stage when the number of response from the first stage is large, to obtain as much information as possible from the clinical study. In this case, an additional constraint can be added during the design search to meet the investigator’s requirement: the second stage sample sizes are the same when S is above S_{ c }, where S_{ c } can be determined by the new constraint from the investigator. The new constraint added in the design search should be clinically meaningful.
The naive point estimate for the probability of response rate is calculated as the number of responses divided by the total number of patients, and it is well known that this estimate is biased. In the traditional Simon’s design, Jung and Kim [15] derived the uniformly minimum variance unbiased estimate for the probability of response based on the Rao-Blackwell theorem. To the best of our knowledge, no unbiased estimate for the probability of response has been proposed in an adaptive two-stage design setting. This may be due to the complexity of an adaptive design as compared to the traditional sample size fixed design.
Randomized clinical trials are used in clinical trials by comparing the new treatment or therapy to the best available treatment for disease. Randomized clinical trials are preferable in the majority of studies to reduce the selection bias and confounding effects, thus capturing the true effectiveness of the new treatment. The widely used two-stage design for a two-arm study with binary outcomes is the one due to Thall et al. [16], that does not allow the second stage sample size to change from the results of the first stage. We will extend the adaptive approach from the one-arm study to this two-arm study to develop a new adaptive two-stage design for a randomized clinical trial with dichotomous endpoints.
Abbreviations
ESS_{0}, the expected sample size under the null; MMS, maximum sample size
Declarations
Acknowledgment
The authors are very grateful to the Editor, and two referees for their insightful comments that help improve the manuscript.
Funding
Shan’s research is partially supported by grants from the National Institute of General Medical Sciences from the National Institutes of Health: P20GM109025, P20GM103440, and 5U54GM104944. Zhang’s work was supported by the Zhejiang Provincial Natural Science Foundation of China (grant no. LY15F020001) and the National Natural Science Foundation of China (grant no. 61170099).
Availability of data and materials
This is a manuscript to develop novel study designs, therefore, no real data is involved.
Authors’ contributions
The idea for the paper was originally developed by GS and TJ. GS and HZ computed and analysed the adaptive minimax and admissible two-stage designs in this paper. GS, HZ and TJ drafted the manuscript, revised the paper critically and approved the final version.
Competing interests
The authors declare that they have no competing interests.
Consent to publish
Not applicable.
Ethics approval and consent to participate
Not applicable.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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