Extended Two-stage Adaptive Designs with Three Target Responses for Phase II Clinical Trials

Abstract

We develop a nature-inspired stochastic population-based algorithm and call it discrete particle swarm optimization (DPSO) to find extended two-stage adaptive optimal designs that allow 3 target response rates for the drug in a phase II trial. Our proposed designs include the celebrated Simon’s two-stage design and its extension that allows 2 target response rates to be specified for the drug. We show that DPSO not only frequently outperforms greedy algorithms, which are currently used to find such designs when there are only a few parameters; it is also capable of solving design problems posed here with more parameters that greedy algorithms cannot solve. In stage 1 of our proposed designs, futility is quickly assessed and if there are sufficient responders to move to stage 2, one tests one of the three target response rates of the drug, subject to various user-specified testing error rates. Our designs are therefore more flexible and interestingly, do not necessarily require larger expected sample size requirements than 2 stage adaptive designs. Using a real adaptive trial for melanoma patients, we show our proposed design requires one half fewer subjects than the implemented design in the study.

Introduction

Phase II clinical trials concern early exploration of efficacy effects and use the most recent results from a small group of patients to make decision for the next group of patients until some prefixed termination rule is met. Group sizes can range from 20–120 depending on the nature and seriousness of the disease. Kramar et al. (1996) provides a review on multistage designs for phase II clinical trials and statistical issues in cancer research and Brown et al. (2011) gives an overview of the role of phase II trials in oncology. Increasingly, these designs employ an adaptive approach, where design decisions for the next stage are made based on cumulative responses. Chow and Chang (2011) reviews adaptive randomization designs in clinical trials.

Simon’s two-stage design was developed under a framework that if the proportion of responders in stage 1 is small, the trial is terminated; otherwise, the trial goes on to stage 2 where the cumulative response rate is now used to test whether the efficacy rate is at a higher pre-specified level. In essence, two user-selected efficacy rates are posited p₀ and p₁ with p₀ < p₁, and p₀ is the uninspiring response rate. In stage 1, we test the null hypothesis H₀ : p = p₀ and if we fail to reject the null hypothesis, we terminate the trial; otherwise, we conclude that the drug is sufficiently promising to advance to stage 2, where more responses from more patients will be used to test p = p₁. Given pre-specified Type I and II error rates for the tests at the two stages, the design questions are the number of patients required in stage 1, number of responders required in stage 1, how many additional patients are required for stage 2 and the cumulative number of responders required at the end of stage 2. The design problem is to determine the optimal combination of these 4 numbers so that the expected number of patients treated with a drug of low activity under the null hypothesis is minimized. Such an optimal design may not be unique and other design criteria are possible.

Lin and Shih (2004) provided practical examples and showed that while it was relatively easy to specify the uninteresting rate p₀, the same was not true for p₁. To tackle the issue of uncertainty in targeting the hypothesis in stage 2, they proposed designs that allowed two specifications for p₁ and called one a skeptical choice and the other the optimistic choice. Their design problem now has 7 parameters to optimize in contrast to 4 in Simon’s two-stage design, subject to user-specified Type I and II error constraints. The additional 3 parameters beyond the 4 parameters required for Simon’s two-stage design are for testing the additional targeted efficacy rate. Both Simon and Lin and Shih employed a greedy search to find the optimal designs and the latter stated that they “did not extend the selection to more than two prefixed possible response rates mainly due to the complexity in the numerical solutions, and also because it is usually adequate for practitioners to contemplate between two (high/low) choices of the response rate p₁.”

Our work was motivated by a real problem from clients interested to conduct a single-arm two-stage phase II trial to the effect of head and neck cancer (HNC) on the incidence of obstructive sleep apnea (OSA). Our clients’ main goal was to ascertain accurately the incidence rate of OSA on HNC patients in a timely manner. Because of the potential huge beneficial impact on treating HNC patients, the clients wanted a relatively simple design that could terminate the study at stage 1 if the initial response (OSA incidence) rate was poor; otherwise, stage 2 of the design used the number of responders, who experienced OSA from stage 1 to recruit more patients to more accurately assess the incidence rate of OSA. Clearly, Simon’s two-stage design proposed in 1989 and the adaptive two-stage design proposed by Lin and Shih (2004) seemed useful.

Our clients were particularly interested to extend Lin and Shih (2004)’s approach to include an additional targeted alternative hypothesis for testing in the second stage. The main reason was the great uncertainty in the OSA incidence rate on HNC patients and the need to move a potentially promising impact further in the treatment of HNC patients expeditiously with a very good estimate of the true OSA incidence rate of HNC patients. Thus depending on the number of responders, who experience OSA in stage 1, more flexibility in specifying the incidence rate of OSA in stage 2 is required. In particular, the clients did not want to run another costly follow up trial to more accurately evaluate the incidence rate of OSA in HNC patients. Another motivation for having such a design was also recently made aware to the second author by personal communication from a senior researcher at the National Institutes of Health. She mentioned that in oncology, there is a great tendency for researchers to be too optimistic on the efficacy rate of a new drug. Our proposed designs should help address such an issue by allowing the clinician the flexibility to perform a test at stage 2 for one of the two lower efficacy rates should the drug efficacy rate be over-specified in the first place. In what is to follow, we discuss and construct extended two-stage adaptive designs for our OSA study and two other applications. The first one is Lin and Shih’s (2004) vinorelbine, bleomycin, and gemcitabine (VBG) study for patients with recurrent or refractory Hodgkin’s disease and the second one is a Phase II BREAK-2 study for melanoma patients.

We were able to formulate the optimization problem quickly as an extension of that from Lin and Shih (2004)’s paper, but anticipated the optimization burden for the new constrained optimization problem now with 10 parameters to optimize for the 3 user-selected targeted alternative hypotheses in stage 2 and subject to various user-specified Type I and II error rate constraints. Lin and Shih (2004) acknowledged computational difficulties in their work at that time and remarked that further extension of the problem would seem to be a too challenging or an impossible computational task. In what is to follow, we present an algorithm to solve this complex constrained optimization problem and show that our extended two-stage adaptive design has advantages over both Simon’s two-stage design and Lin and Shih’s designs in a number of ways.

There are many variations of strategies proposed for phase II designs since Simon’s two-stage design was proposed. We mention some here and refer to earlier cited papers for a comprehensive overview. Modifications and extensions of the likes of Simon’s two-stage design include having two binary outcomes (Kunz and Kieser (2011)) or finding designs that minimize the expected sample size or minimize the expected maximum sample size, not under the null hypothesis but under the alternative hypothesis. For example, Mander and Thompson (2010) and Mander et al. (2012) investigated novel designs which are optimal under the alternative hypothesis, that the tumor response rate is higher than the null hypothesis value, and also designs which allow early stopping for efficacy. Wason et al. (2011) considered reducing sample size for phase II trials with a continuous outcome, and Kwak and Jung (2014) proposed a two-stage adaptive optimal design for single arm trials with right-censored survival time that minimizes the expected sample size subject to Type I and II error rate specifications. Phase II designs with three stages were motivated and proposed in Ensign et al. (1994), Chen (1997), and Chen and Shan (2007). Schlesselman and Reis (2006) noted limits and benefits of phase II trials and Zhou (2004) gave some guidance on the choice of a design for an early phase trial.

Software for computing various types of phase II adaptive designs for such trials is available in commercial software statistical packages, like SAS, STATA and JMP. They are also codes on webpages that generate various adaptive optimal designs for early phase trials after the error rates are specified. One such site is at https://stattools.crab.org/Calculators/twoStage.htm. Because none of the current software can find our proposed designs, we have developed codes to run on a web browser using the R package shiny (http://shiny.rstudio.com) to generate our proposed designs. We call our R package ss2stagePSO and it is freely available at http://cansur.wayne.edu/. We also provide an option to compare competitive designs and user-specified designs relative to the optimal designs by reporting their relative efficiencies.

In the next section, we first describe extended two-stage adaptive designs for phase II clinical trials and technical background before discussing nature-inspired stochastic population-based algorithms for finding our proposed designs. There are many of them and, as an example, we focus on particle swarm optimization (PSO), which seem to be widely used. Motivated from our recent successes with this flexible algorithm, we modified it for our application at hand, because PSO was originally designed to solve optimization problems over a continuous domain, not optimization over a domain comprising of discrete positive integer numbers, and call it Discrete PSO (DPSO). In particular, we show how DPSO solves our extended two-stage adaptive design problems and related problems effectively that a greedy algorithm cannot. We then apply our algorithm to real applications and show our proposed designs require smaller sample sizes than those that were implemented in the trial. The paper is concluded with a summary.

Extended two-stage adaptive designs

Suppose p₀ is the maximum uninteresting response rate and there are three choices for the target response rates: p₁, p₂, p₃, where 0 < p₀ < p₁ < p₂ < p₃ < 1. Data from stages 1 and 2 will be used to test one of the null hypotheses in stage 2 depending on the number of responders in stage 1. Our adaptive design assumes that a total of n₁ patients are assigned at the first stage and the total number of patients and tests required in the entire trial will depend on the number of responses in the first and subsequent stages. The null hypothesis at the first stage is H₀: p ≤ p₀. According to the number of responses in the first stage, the corresponding alternative hypothesis will be one of three hypotheses: H₁₁: p > p₁, H₁₂: p > p₂ and H₁₃: p > p₃. Using similar notation in Lin and Shih (2004), our extended two-stage adaptive design has a total of 10 parameters given by θ = (s₁, r₁, q₁, n₁, s, l, r, m, q, n) and it operates as follows:

• Step I: Begin by recruiting n₁ patients in the first stage of our design and observe the number of the responses, x, from of n₁ patients.

• Step II:

  1. If x ≤ s₁, stop the trial with failure to reject H₀ (i.e., p ≤ p₀)
  2. If s₁ < x ≤ r₁, power the study at (1-β₁) for p = p₁ and enter l₂ = l − n₁ additional patients into the study. Reject the hypothesis that H₁₁: p > p₁ if the total number of responses ≤ s out of l patients.
  3. If r₁ < x ≤ q₁, power the study at (1-β₂) for p = p₂ and enter m₂ = m − n₁ additional patients into the study. Reject the hypothesis that H₁₂: p > p₂ if the total number of responses ≤ r out of m patients.
  4. If x > q₁, power the study at (1-β₃) for p = p₃ and enter n₂ = n − n₁ additional patients into the study. Reject the hypothesis that H₁₃: p > p₃ if the total number of responses ≤ q out of n patients.

By construction, the values of l, m, n are the total number of patients required for the entire trial corresponding to the alternative hypotheses, H₁₁: p > p₁, H₁₂: p > p₂, and H₁₃: p > p₃, respectively. The extended two-stage adaptive design has a total of 10 parameters denoted by θ = (s₁, r₁, q₁, n₁, s, l, r, m, q, n) that we wish to optimize given error rate constraints and the stipulated three response rates. If the true response probability is p, the probability of failing to reject H₀ is given by

$$\begin{array}{l}G(\theta |p)=B(s_1,n_1,p)+{\displaystyle \sum _{x=s_1+1}^{min(r_1,s)}b(x,n_{1}p)B(s-x,l_2,p)}\\ +{\displaystyle \sum _{x=r_1+1}^{min(q_1,r)}b(x,n_{1}p)B(r-x,m_2,p)}+{\displaystyle \sum _{x=q_1+1}^{min(q,n_1)}b(x,n_{1}p)B(q-x,n_2,p)},\end{array}$$ (1)

where b and B are the probability density function and cumulative density function of a binomial distribution, respectively, with l = n₁ + l₂, m = n₁ + m₂ and n = n₁ + n₂. It follows that when the true probability response rate is p, the expected sample size is

$$\begin{array}{l}E(N|p,\theta )=n_1+\left(B(r_1,n_1,p)-B(s_1,n_1,p)\right)l_2\\ +\left(B(q_1,n_1,p)-B(r_1,n_1,p)\right)m_2+\left(1-B(q_1,n_1,p)\right)n_2.\end{array}$$ (2)

Our design problem is to find a good choice $\widehat{\theta }\in \stackrel{\sim }{\Theta }$, the set containing all values of θ that satisfies four natural error constraints

$$G(\theta |p_0)\ge 1-\alpha ,G(\theta |p_1)\le {\beta }_1,G(\theta |p_2)\le {\beta }_2\mathit{\text{and}}G(\theta |p_3)\le {\beta }_3.$$ (3)

The goodness of $\widehat{\theta }$ may be determined by one of the following four optimality criteria:

• C1: $\widehat{\theta }={\text{arg min}}_{\theta \in \stackrel{\sim }{\Theta }}E(N|p_0,\theta )$;

• C2: $\widehat{\theta }={\text{arg min}}_{\theta \in \stackrel{\sim }{\Theta }}E(N|p_0,\theta )\mathit{\text{and}}\widehat{\theta }={\text{arg min}}_{\theta \in \stackrel{\sim }{\Theta }}\{max(l,m,n)\}$;

• C3: $\widehat{\theta }={\text{arg min}}_{\theta \in \stackrel{\sim }{\Theta }}\{{\max }_{i=0,1,2,3}E(N|p_i,\theta \}$;

• C4: $\widehat{\theta }={\text{arg min}}_{\theta \in \stackrel{\sim }{\Theta }}\{{\max }_i=0,1,2,3E(N|p_i,\theta \}\text{and}\widehat{\theta }={\text{arg min}}_{\theta \in \stackrel{\sim }{\Theta }}\left\{\max (l,m,n)\right\}$.

The optimality criteria C1–C4 are extensions of Lin and Shih (2004)’s criteria and, if there is only one target response, criteria C1 and C2 are exactly the same as Simon’s two optimality criteria. Our proposed extended 2-stage adaptive design is schematically described in Figure 1.

Figure 1.

Flowchart of the extended two-stage adaptive design for Phase II clinical trials (third column) with s₁ < r₁ < q₁. Simon’s two-stage design is obtained by setting r₁ = n₁ (first column) and Lin and Shih’s design is obtained by setting q₁ = n₁ (second column).

The above extension to three prefixed possible target responses may appear straightforward to implement, but in reality it is computationally impractical to find a solution by the use of greedy algorithms due to its complexity. Indeed, we formulate the optimization problem and show that greedy algorithms fail to find a solution for our 10-dimensional optimization problem after running it for 30 days. To this end, we briefly review a nature-inspired stochastic population-based algorithm, particle swarm optimization technique, and show that it not only provides solutions in a relatively short time but also facilitates performance comparison among different algorithms in the following sections.

Discrete Particle Swarm Optimization (DPSO)

Particle swarm optimization (PSO), a population-based global optimization method, was introduced by Kennedy and Eberhart (1995). It is an evolutionary algorithm and stochastically evolves a group of particles that mimic observational behavior from nature. PSO is motivated from observing how a flock of birds move in the sky and so is a member of the class of nature-inspired stochastic population-based algorithms. This class of algorithms has been gaining lots of recognition especially in the last decade or so for its ability to solve or nearly solve hard-to-optimize high dimensional problems in the real world. Whitacre (2011a,b) documented the meteoric rise in use of such algorithms in applied fields and increasingly in academia as well. Their main appeals are that they are simple to implement, assumption-free and tend to quickly converge to the optimum or get quickly to the vicinity of the optimum solution. Additionally, they are general-purpose optimization algorithms and so are adaptable to solve different types of optimization problems after the user inputs some tuning parameters to initiate the algorithm. Genetic algorithm, simulated annealing and PSO are examples of such algorithms.

Some recent applications of PSO to solve optimal design problems in statistics are Qiu et al. (2014), Chen et al. (2014), Phoa et al. (2016), and Wong et al. (2015). They tackled design problems in biomedical problems that ranged from finding D-optimal designs for several nonlinear models and mixture models with multiple constraints to finding minimax and standardized maximin type of optimal designs. The latter designs have non-differentiable optimality criteria that require a couple of nested levels of optimization and are notoriously difficult to find. Phoa et al. (2016) applied swarm intelligence to find an optimal supersaturated design in a high dimension problem that involves judicious and repeated exchanges of columns in the design matrix to minimize correlations among the columns via the E(s²) criterion. PSO has also successfully used for estimation in statistical problems. For example, Kim and Li (2011, 2014) applied PSO to estimate parameters in nonlinear mixed-effects pharmacokinetics models, and Kim et al. (2012) employed PSO to estimate efficacy of different lung cancer screening methods. In each of the above problems, we observed, as many others had, the flexibility of PSO and how it can be readily modified to solve an optimization problem at hand. In what is to follow, we first demonstrate yet again that PSO can be adapted to efficiently solve our adaptive design problem with a very different setup than those just mentioned above. Second, we show PSO can solve high dimensional problems with many parameters in our adaptive design problems that current greedy algorithms cannot and third, PSO can substantially outperform greedy algorithms in adaptive design problems with a small number of parameters.

Operationally, the user initiates PSO by first specifying the maximum of iterations allowed, say K and the flock size consisting of, say N, randomly generated particles for the search. Each particle represents a candidate solution to the problem and at any one time, each has a fitness value (i.e. the design criterion value). As it searches, it keeps track of its best value (personal or local best value) and communicates with the rest of the particles to determine the global best value, defined as the best personal best value among the flock up to that iteration. The particles updates these two values continuously as it moves across the search space in the direction guided by its most recent local best and the global best with a velocity determined stochastically by a combination of values of the tuning parameters, its personal best and the global best and the velocity with which it has arrived at the current position. Specifically, suppose the nth particle’s position vector is $x^n={\left(x_k^n\right)}_{k=1,\cdots ,K}^{\prime }$ and its updating velocity vector is $v^n={\left(v_k^n\right)}_{k=1,\cdots ,K}^{\prime }$, n = 1, ⋯, N. If its local best and global best are $x_{\mathit{\text{lbest}}}^n$ and x_gbest respectively, its velocity $v_{k+1}^n$ and the position $x_{k+1}^n$ at the (k + 1)th iteration are determined by

$$v_{k+1}^n=w_k{v}_k^n+c_1{r}_1\left(x_{\mathit{\text{lbest}}}^n-x_k^n\right)+c_2{r}_2\left(x_{\mathit{\text{gbest}}}-x_k^n\right);x_{k+1}^n=x_k^n+v_{k+1}^n.$$ (4)

Here w_k is the inertia weight (0 ≤ w_k ≤ 1) at the kth iteration, c₁ and c₂ are two positive constants called cognitive and social coefficient and, r₁ and r₂ two random variates in the range [0, 1]. The lower limit values of the constants c₁ and c₂ determine how far particles are allowed to wander beyond the target region before pulled back, and their upper limit values determine how sudden the particles should be moved back to the target region. Following convention, we set c₁ and c₂ to their default values equal to 2 in our simulation studies. At the kth iteration, the inertia weight w_k is $w_k=w_{\mathit{\text{max}}}-\frac{k}{K}(w_{\mathit{\text{max}}}-w_{\mathit{\text{min}}})$ where w_min and w_max are user-defined constants satisfying 0 ≤ w_K = w_min ≤ w_k ≤ w_max = w₀ ≤ 1.

During the search, the inertia weight adaptively controls the impact of the previous history of velocities on the current velocity and also influences the trade-off between global (wide-ranging) and local (nearby) exploration abilities of the particles as they move across the search space. A larger inertia weight facilitates global exploration (searching new areas) and a smaller inertia weight facilitates local exploration. When the inertia weight is suitably chosen, it can provide a balance between global and local exploration abilities and on average, require fewer iterations to find the global optimum (Shi and Eberhart (1998)). To exploit these properties of the inertia weight, we use a dynamic inertia weight to enable PSO escape from premature convergence when it is stagnated (Tao and Cai (2009)).

PSO was originally designed to solve optimization problems over a continuous domain, not optimization over a domain comprising of discrete positive integer numbers. Since PSO is a flexible algorithm, we modified it for our problem and call it Discrete PSO (DPSO). The task is to choose the right combination of several positive integers that would minimize the total sample size required under various hypothesis and at the same time meet the user-specified Type I and II error rates. The proposed DPSO has two different features from PSO: (i) the inertia weight $w_k^{\prime }$ is the nearest integer of PSO’s inertia weight w_k at the kth iteration, i.e. $w_k^{\prime }=\lfloor w_k+\frac{1}{2}\rfloor =\lfloor w_{\mathit{\text{max}}}-\frac{k}{K}(w_{\mathit{\text{max}}}-w_{\mathit{\text{min}}})+\frac{1}{2}\rfloor$, where ⌊x⌋ = arg max_m{m ∈ Z|m ≤ x} and Z is a set of integers, and (ii) the two random sequences are generated from a discrete uniform distribution.

Extended two-stage adaptive design with 3 target responses

We consider two optimization approaches to find $\widehat{\theta }$, the optimal value for θ = (s₁, r₁, q₁, n₁, s, l, r, m, q, n) for our adaptive design problem. One is the greedy search and the other is DPSO proposed in this study. The greedy search adopts the strategies used in Simon’s or Lin and Shih (2004)’s algorithms for the extended 2-stage adaptive design. Specifically, we first calculate the required sample sizes n_i, i = l, m, and n, for a single-stage design, where n_l, n_m, and n_n are the required sample sizes corresponding to the target response rates p₁, p₂, and p₃, respectively, and constrain l, m, and n to be in the range of ⌊0.85 · n_i⌋ ≤ i ≤ ⌊1.5 · n_i + 1⌋, where i = l, m, and n. For each value of l, m, and n, and each of n₁ in [1,min(l, m, n) − 1], s₁ in [0, n₁], r₁ in [s₁ + 1, n₁], and q₁ in [r₁ + 1, n₁], we then find the set of feasible solutions $\stackrel{\sim }{\Theta }$

that satisfies the error constraints (3) for q in [q₁ + 1, n], r in [r₁ + 1, m], and s in [s₁ + 1, l] and determine the optimum according to one of optimality criteria C1–C4.

To implement DPSO, we specify the lower and upper boundaries for each component of the parameter vector θ = (s₁, r₁, q₁, n₁, s, l, r, m, q, n). Similar to the greedy search, we use the sample sizes n_l, n_m, and n_n, required for a single-stage design corresponding to the target response rates p₁, p₂, and p₃, respectively and set

$$\begin{array}{r}{L}_i=\lfloor 0.85\cdot n_i\rfloor ,i=l,m,n;\\ U_i=\lfloor 1.5\cdot n_i+1\rfloor ,i=l,m,n;\\ U=min\{U_i;i=l,m,n\}.\end{array}$$ (5)

The lower and upper boundaries for each component of θ are then obtained as follows:

$$\begin{array}{r}{s}_1\in \left[0,\lfloor \frac{U_m}{2}+1\rfloor \right];r_1\in \left[0,\lfloor U\cdot p_1+1\rfloor \right];q_1\in \left[0,\lfloor U\cdot p_2+1\rfloor \right];n_1\in [1,U];\\ s\in \left[0,\lfloor U_l\cdot p_1+1\rfloor \right];l\in [L_l,U_l];r\in \left[0,\lfloor U_m\cdot p_2+1\rfloor \right];m\in [L_m,U_m];\\ q\in \left[0,\lfloor U_n\cdot p_3+1\rfloor \right];n\in [L_n,U_n],\end{array}$$

and used in DPSO to find the optimal solution under each of the C1–C4 optimality criteria among the feasible solutions in $\stackrel{\sim }{\Theta }$ that satisfy the error constraints (3).

The standard PSO is usually not sensitive to the initial values of the tuning parameters and can find an optimal solution using the default values. However, in our DPSO, a lot of iterations is required for convergence, especially when there are two or three target responses and the initial values were not among the feasible solutions in $\stackrel{\sim }{\Theta }$. For this reason, we consider two approaches (G-DPSO and D-DPSO) to find an appropriate initial set of values among the feasible solutions in $\stackrel{\sim }{\Theta }$ to boost the speed of convergence of DPSO.

In the first approach, the initial value is found by a greedy search over a smaller range of values for n_i and not over the entire range, where i = l, m, and n. Specifically, the required sample sizes n_i, i = l, m, and n, are in the range of n_i − 1 ≤ i ≤ n_i + 1, where i = l, m, and n. Within this smaller domain, we searched for an appropriate set of initial values using the same strategy as the greedy search did for the rest of the parameters. By shrinking the range of the parameters n_i, i = l, m, and n, we potentially reduced the computational burden. We call DPSO with this first approach G-DPSO. In particular, we observed the consistency of the computation time with a short amount of time, when G-DPSO was applied to the case with one target response (see Table 1 and Supplementary Information Table S1). However, similar to the greedy search, the computation time of G-DPSO increases as the number of target response s increases, especially for two or three target responses (for example, see Tables 2 and 3 and Supplementary Information Tables S2 and S3). Therefore, we devise the second approach that does not depend on the number of target responses for the cases with two or three target responses.

Table 1.

Various adaptive 2-stage optimal designs with one target response when α = 0.05 and β = 0.20.

p0	p1	Optimal criteria	Method	s1/n1	s/n	1 − α	β	E(N|p0)	E(N|p1)
0.05	0.20	C1	GS	0/10	3/29	0.953	0.199	17.624	26.960
			G-DPSO	0/10	3/29	0.953	0.199	17.624	26.960
		C2	GS	0/11	3/28	0.956	0.199	18.330	26.540
			G-DPSO	0/11	3/28	0.956	0.199	18.330	26.540
		C3	GS	0/11	3/28	0.956	0.199	18.330	26.540
			G-DPSO	0/11	3/28	0.956	0.199	18.330	26.540
		C4	GS	0/11	3/28	0.956	0.199	18.330	26.540
			G-DPSO	0/11	3/28	0.956	0.199	18.330	26.540
0.20	0.35	C1	GS	5/22	19/72	0.951	0.200	35.368	63.855
			G-DPSO	5/22	19/72	0.951	0.200	35.368	63.855
		C2	GS	3/21	15/53	0.950	0.200	41.148	51.941
			G-DPSO	3/21	15/53	0.950	0.200	41.148	51.941
		C3	GS	6/31	15/53	0.950	0.198	40.436	51.983
			G-DPSO	6/31	15/53	0.950	0.198	40.436	51.983
		C4	GS	3/21	15/53	0.950	0.200	41.148	51.941
			G-DPSO	3/21	15/53	0.950	0.200	41.148	51.941
0.55	0.70	C1	GS	15/26	48/76	0.952	0.195	42.021	69.735
			G-DPSO	15/26	48/76	0.952	0.195	42.021	69.735
		C2	GS	20/35	43/67	0.953	0.200	45.802	64.662
			G-DPSOa	20/35	43/67	0.953	0.200	45.802	64.662
		C3	GS	20/35	43/67	0.953	0.200	45.802	64.662
			G-DPSO	20/35	43/67	0.953	0.200	45.802	64.662
		C4	GS	20/35	43/67	0.953	0.200	45.802	64.662
			G-DPSOb	20/35	43/67	0.953	0.200	45.802	64.662

Table 2.

Various adaptive 2-stage optimal designs with two target responses when α = 0.05, β₁ = 0.20 and β₂ = 0.10.

p0	p1	p2	Optimal criteria	Method	s1/r1/n1	s/m	r/n	1 − α	β1	β2	E(N|p0)	E(N|p1)	E(N|p2)
0.05	0.20	0.25	C1	GS	0/1/9	3/30	4/41	0.957	0.199	0.094	17.548	33.383	36.119
				G-DPSO	0/1/10	3/28	3/31	0.953	0.199	0.088	17.481	27.940	29.254
				D-DPSO	0/1/8	4/41	3/36	0.952	0.200	0.107	18.821	32.980	34.532
			C2	GS	0/2/16	3/29	3/22	0.955	0.199	0.082	22.978	24.097	23.249
				G-DPSO	0/2/14	3/29	3/21	0.954	0.200	0.084	21.444	23.925	22.982
				D-DPSO	0/2/14	3/29	3/21	0.954	0.200	0.084	21.444	23.925	22.982
			C3	GS	0/4/9	4/41	5/22	0.960	0.174	0.084	20.831	36.333	37.668
				G-DPSO	0/5/11	3/28	6/25	0.956	0.199	0.083	18.330	26.505	27.179
				D-DPSO	0/5/11	3/28	6/25	0.956	0.199	0.083	18.330	26.505	27.179
			C4	GS	0/2/16	3/29	3/22	0.955	0.199	0.082	22.978	24.097	23.249
				G-DPSO	0/2/15	3/28	3/23	0.957	0.197	0.078	21.796	24.533	24.007
				D-DPSO	0/2/15	3/28	3/23	0.957	0.197	0.078	21.796	24.533	24.007
0.20	0.35	0.40	C1	G-DPSO	4/9/20	17/62	10/36	0.952	0.199	0.069	35.487	53.869	53.499
				D-DPSO	5/10/22	19/72	11/36	0.951	0.199	0.078	35.311	60.004	60.179
			C2	G-DPSO	8/12/37	16/57	13/41	0.950	0.196	0.059	42.913	46.949	44.252
				D-DPSO	5/10/29	16/56	11/36	0.951	0.196	0.055	43.094	46.415	42.638
			C3	G-DPSO	6/11/27	16/58	13/43	0.950	0.200	0.063	35.833	51.406	50.886
				D-DPSO	3/10/21	15/53	11/36	0.950	0.200	0.058	41.131	50.629	49.683
			C4	G-DPSO	8/12/38	16/56	13/43	0.952	0.200	0.057	43.830	47.337	45.224
				D-DPSO	3/10/21	15/53	11/36	0.950	0.200	0.058	41.131	50.629	49.683
0.55	0.70	0.75	C1	G-DPSO	15/20/26	48/76	27/39	0.951	0.200	0.053	41.807	63.720	61.521
				D-DPSO	15/20/26	48/76	27/39	0.951	0.200	0.053	41.807	63.720	61.521
			C2	G-DPSO	24/28/41	47/73	30/45	0.951	0.194	0.039	48.872	55.463	50.268
				D-DPSO	11/17/24	45/70	30/45	0.951	0.199	0.043	57.956	59.756	54.718
			C3	G-DPSO	13/20/25	43/67	25/42	0.951	0.194	0.038	47.730	62.880	61.206
				D-DPSO	14/20/26	43/67	23/39	0.950	0.199	0.041	45.163	59.975	56.926
			C4	G-DPSO	25/30/44	44/68	32/49	0.950	0.197	0.037	51.862	56.540	52.457
				D-DPSO	10/16/21	43/67	28/41	0.950	0.200	0.044	51.907	60.628	57.152

Table 3.

Various adaptive 2-stage optimal designs with three target responses when α = 0.05, β₁ = 0.20, β₂ = 0.10 and β₃ = 0.05.

p0	p1	p2	p3	Optimal criteria	Method	s1/r1/q1/n1	s/l	r/m	q/n	1 − α	β1	β2	β3	E(N|p0)	E(N|p1)	E(N|p2)	E(N|p3)
0.05	0.20	0.25	0.30	C1	G-DPSO	0/1/4/10	3/28	3/31	5/28	0.953	0.200	0.088	0.037	17.481	27.841	29.020	29.593
					D-DPSO	0/1/2/9	4/36	3/34	3/31	0.959	0.199	0.093	0.044	18.816	30.463	31.375	31.691
				C2	G-DPSO	0/1/2/13	3/31	3/28	3/20	0.951	0.200	0.088	0.036	21.158	23.725	22.613	21.636
					D-DPSO	0/1/2/14	4/36	3/33	3/19	0.952	0.198	0.094	0.043	24.391	24.899	22.847	21.245
				C3	G-DPSO	0/2/5/11	3/28	3/28	6/21	0.956	0.200	0.084	0.033	18.330	26.458	27.042	27.116
					D-DPSO	0/3/4/11	3/29	5/29	5/22	0.953	0.198	0.085	0.034	18.761	27.101	27.437	27.172
				C4	G-DPSO	0/1/2/15	3/28	3/27	3/24	0.958	0.198	0.077	0.026	21.698	24.904	24.615	24.354
					D-DPSO	0/1/2/14	4/32	3/30	3/22	0.960	0.198	0.079	0.028	22.675	25.189	24.130	23.260
0.20	0.35	0.40	0.45	C1	G-DPSO	5/9/10/24	17/61	10/36	11/32	0.952	0.200	0.064	0.017	36.401	48.570	45.349	40.823
					D-DPSO	5/11/13/22	19/72	12/36	14/32	0.951	0.200	0.078	0.028	35.355	62.126	63.957	61.554
				C2	G-DPSO	5/9/10/29	17/60	13/44	11/34	0.951	0.197	0.061	0.015	44.649	45.199	40.615	37.117
					D-DPSO	3/6/8/19	19/67	11/38	9/31	0.950	0.194	0.060	0.016	43.149	47.817	43.504	39.094
				C3	G-DPSO	4/9/13/22	16/57	10/39	16/36	0.951	0.196	0.059	0.014	37.889	50.724	49.244	46.353
					D-DPSO	3/8/9/17	17/62	12/36	11/28	0.951	0.197	0.070	0.024	37.230	54.484	54.003	50.929
				C4	G-DPSO	5/9/10/26	16/57	13/44	11/34	0.950	0.197	0.059	0.013	38.717	46.656	43.284	39.542
					D-DPSO	4/8/11/24	18/64	11/36	12/28	0.952	0.193	0.068	0.026	44.584	48.279	42.940	37.622
0.55	0.70	0.75	0.80	C1	G-DPSO	15/20/21/26	48/76	29/43	23/31	0.951	0.199	0.052	0.010	41.812	63.491	60.658	51.947
					D-DPSO	10/15/16/19	46/72	23/39	18/29	0.951	0.198	0.047	0.008	44.911	62.696	60.681	54.260
				C2	G-DPSO	17/22/23/32	47/73	32/48	24/35	0.950	0.195	0.041	0.005	51.994	54.810	46.623	39.493
					D-DPSO	8/13/14/17	46/72	27/39	21/31	0.951	0.195	0.048	0.009	52.800	62.503	58.360	51.268
				C3	G-DPSO	15/20/21/27	46/72	27/41	24/35	0.951	0.192	0.041	0.005	44.743	59.648	54.638	46.497
					D-DPSO	12/17/18/22	47/74	26/39	24/32	0.950	0.197	0.054	0.014	44.314	63.001	60.007	52.356
				C4	G-DPSO	17/22/23/32	47/73	32/48	24/35	0.950	0.195	0.041	0.005	51.994	54.810	46.623	39.493
					D-DPSO	7/12/14/16	45/70	26/39	19/31	0.951	0.200	0.044	0.006	55.302	60.783	56.534	50.252

The second approach abbreviated as D-DPSO is used when the number of target responses is two or more. D-DPSO finds the initial set of values using an optimal set of values decided by the case with the one less number of target response. In other words, when the number of target responses is k, the initial value is decided using the sample sizes determined by the case when the number of target responses is k − 1, where k = 2, 3. By doing so, we could make the second approach not depend on the number of target responses in terms of computation time. The specific details for implementing D-DPSO to search for any one of the four types of optimal design when there are two and three target responses are as follows:

The number of target responses is two

We first obtain two sets of sample sizes, $E_1=\{s_1^1,n_1^1,s^1,n^1\}$ and $E_2=\{s_1^2,n_1^2,s^2,n^2\}$ from the target responses p₁ and p₂, respectively, based on the G-DPSO method when there is one target response. Then, using the two sets of sample sizes E₁ and E₂, the initial value is set to ${\stackrel{\sim }{\theta }}_2=({\stackrel{\sim }{s}}_1,{\stackrel{\sim }{r}}_1,{\stackrel{\sim }{m}}_1,\stackrel{\sim }{s},\stackrel{\sim }{m},\stackrel{\sim }{r},\stackrel{\sim }{n})=({\mathit{\text{rank}}}_3\{s_1^1,n_1^1,s_1^2,n_1^2\},s^1,n^1,s^2,n^2)$ where rank_kS is the top k elements of the set S arranged in ascending order. The first three values $({\stackrel{\sim }{s}}_1,{\stackrel{\sim }{r}}_1,{\stackrel{\sim }{m}}_1)$ correspond to the null response p₀ (i.e., the first stage), so the sample sizes $(s_1^1,n_1^1,s_1^2,n_1^2)$, which correspond to the first stage of E₁ and E₂, are used. Since $(\stackrel{\sim }{s},\stackrel{\sim }{m})$ and $(\stackrel{\sim }{r},\stackrel{\sim }{n})$ correspond to the target values p₁ and p₂ (i.e., the second stage), respectively, the values (s¹,n¹) and (s²,n²), which correspond to the second stage of E₁ and E₂, are used.

The number of target responses is three

We first obtain two sets of sample sizes $E_1=\{s_1^1,r_1^1,m_1^1,s^1,m^1,r^1,n^1\}$ and $E_2=\{s_1^2,r_1^2,s^2,n^2\}$ using the D-DPSO with the target responses (p₁, p₂) and the G-DPSO with the target response p₃, respectively. Then, the initial value is set to ${\stackrel{\sim }{\theta }}_3=({\stackrel{\sim }{s}}_1,{\stackrel{\sim }{r}}_1,{\stackrel{\sim }{q}}_1,{\stackrel{\sim }{n}}_1,\stackrel{\sim }{s},\stackrel{\sim }{l},\stackrel{\sim }{r},\stackrel{\sim }{m},\stackrel{\sim }{q},\stackrel{\sim }{n})=({\mathit{\text{rank}}}_4\{s_1^1,r_1^1,m_1^1,s_1^2,n_1^2\},s^1,m^1,r^1,n^1,s^2,n^2)$. The first four values $({\stackrel{\sim }{s}}_1,{\stackrel{\sim }{r}}_1,{\stackrel{\sim }{q}}_1,{\stackrel{\sim }{n}}_1)$ correspond to the null response p₀ (i.e., the first stage), so the sample sizes $(s_1^1,r_1^1,m_1^1,s_1^2,n_1^2)$, which correspond to the first stage of E₁ and E₂, are used. Since $(\stackrel{\sim }{s},\stackrel{\sim }{l},\stackrel{\sim }{r},\stackrel{\sim }{m})$ and $(\stackrel{\sim }{q},\stackrel{\sim }{n})$ correspond to the target values (p₁, p₂) and p₃ (i.e., the second stage), respectively, the values (s¹, m¹, r¹, n¹) and (s², n²), which correspond to the second stage of E₁ and E₂, are used.

Simulation studies

We use simulation studies to evaluate the performance of extended two-stage adaptive designs. We also generate Simon’s two-stage design and Lin and Shih (2004)’s adaptive two-stage design using DPSO and compare results when the number of target response is one and two, respectively. We refer to Simon’s two-stage design as an adaptive two-stage design with one target response, Lin and Shih (2004)’s adaptive two-stage design as an adaptive two-stage design with two target responses, and our extended two-stage adaptive design as an adaptive two-stage design with three target responses.

Our simulation covers one set of Type I (α) and Type II (β) error rates with (α, β₁, β₂, β₃) = (0.05, 0.20, 0.10, 0.05) and three sets of target response rates, which are (p₀, p₁, p₂, p₃) ∈ {(0.05, 0.20, 0.25, 0.30), (0.20, 0.35, 0.40, 0.45), (0.55, 0.70, 0.75, 0.80)}. Note that we empirically selected these combinations for null and target response rates to reflect the general performances of each method. The optimal design under each of the four optimality criteria C1–C4 was found using two optimization methods: a greedy search (GS) and discrete PSO (DPSO). GS looks for all possible and feasible cases to find the optimal solution, while DPSO stochastically searches for the optimal solution using Equations (1) and (2). The number of particles used to optimize each design parameter was 10 and the population size was set to 10000 in our DPSO. The total number of iterations allowed was 50. The two methods described in the previous section were applied to obtain initial values for G-DPSO and D-DPSO.

Tables 1–3 display optimal designs found from GS, G-DPSO and D-DPSO methods when there are one, two, and three target responses, respectively. Details of the computation times in minutes to evaluate computational burden for the different algorithms are available in the Supplementary Information Tables S1–S3. Note that the computation times given in Supplementary Information Tables S1 to S3 might not accurately reflect the true computational complexity of each method and we display them to give some insights on how efficient each method is. All simulations were rendered up to 30 days using a high performance Grid enabled computing system at Wayne State University. The Grid currently has the combined processing power of 4,420 cores: 1,200 Intel cores, 3,220 AMD cores, with over 12TB of RAM and over half a petabyte of disk space.

We make two remarks before we present our results. First, to ensure a fair comparison, we implemented all the GS, G-DPSO and D-DPSO algorithms using the statistical software R. Some of our G-DPSO results are a bit different from published results found by the GS method. This is not surprising because of the stochastic nature of the G-DPSO algorithm and the results from G- or D-DPSO depend on the algorithm convergence criterion and the tuning parameters, which for simplicity, we had set a priori to be same for all cases. Consequently, optimal designs found by the G- and D-DPSO algorithm may vary each time we ran. For instance, when p₀ = 0.55 and p₁ = 0.70, our simulation settings for PSO produced different results for criteria C2 and C4 than those reported in Simon’s paper found by a greedy search. Some such discrepancies are indicated in Table 1 by superscripts 1 and 2 but they can be resolved using a different set of tuning parameters for the algorithm. For example, in these instances, when the number of particles was increased from 10 to 20 and the population size was increased from 10000 to 70000, our DPSO algorithm converged to the same optimal design with s₁/n₁ = 20/35 obtained in Simon’s paper in a small fraction of the time required by GS. Second, our codes repeatedly call on IMS functions for computing Binomial probabilities every single time it is needed, which is unlike the codes in Simon’s paper where they were predefined upfront. Consequently, our CPU times may appear several factors longer than those reported in earlier work, but are in fact substantially faster than the computing time required by GS.

Adaptive two-stage design with one target response

When there is one target response rate, the adaptive two-stage design is the same as Simon’s two-stage design. Table 1 compares the results between GS and G-DPSO when p₁ − p₀ = 0.15. As stated before, we applied D-DPSO only when the number of target response is either two or three. In all cases, both optimization methods give the exactly same optimal design with the same expected sample sizes. Table S1 in the Supplementary Information shows G-DPSO can be up to 100 times faster than GS except when p₀ = 0.05. One reason that G-DPSO is slower than GS is that the size of the solution domain when p₀ = 0.05 is small and the computational time of GS can be highly dependent on the size of the solution domain. As such, the computational time required by GS can be highly variable. In contrast, the time required by G-DPSO to find the optimum does not depend on the size of the solution set once the initial values are obtained. Consequently, we observe a consistent computational time range required by DPSO for all the cases. The table also shows that the sample size requirements for optimality criteria C2-C4 are smaller than those required by criterion C1.

Adaptive two-stage design with two target responses

We now apply GS and DPSO (G-DPSO and D-DPSO) to generate the Lin and Shih (2004)’s adaptive two-stage designs and Table 2 displays them. Each optimization was allowed to run up to 30 days. GS reports the optimal designs only when p₀ = 0.05 because the size of the solution area increases exponentially with the number of target responses when p₀ is close to 0.5. As a result, GS could not reach out to the solutions for other cases. Interestingly, as shown in Supplementary Information Table S2, the computation time of GS when p₀ = 0.05 is much longer (at least 68 times longer) than those of both G-DPSO and D-DPSO, even though the determined sample sizes are comparable with or slightly smaller than those of G-DPSO and D-DPSO. As expected, the required sample sizes of G-DPSO and D-DPSO are similar even though the computation time for G-DPSO and D-DPSO are different; the former depends on the size of the solution set but the latter does not, hence their computing times are different. For example, when (p₀, p₁, p₂) = (0.55, 0.70, 0.75), G-DPSO takes 23 times longer than D-DPSO to determine the optimal design, but when the size of the solution set is small as when (p₀, p₁, p₂) = (0.05, 0.20, 0.25), the computation time required by G-DPSO is shorter than that required by D-DPSO, which consistently ranges from 5.07 to 6.10 minutes and independent of the solution size.

Adaptive two-stage design with three target responses

Table 3 displays the extended 2-stage adaptive designs with three target response rates. Within 30 days run, GS always failed to determine the optimal design, but both G-DPSO and D-DPSO were able to find the optimal designs with similar results. Interestingly, when (p₀, p₁, p₂, p₃) = (0.55, 0.70, 0.75, 0.80), the computation time of G-DPSO is almost 30 days, but that of D-DPSO is about 6.5 minutes, demonstrating that D-DPSO is much more time efficient that G-DPSO even in complicated situations and computation burden, see Table S3 in the Supplementary Information.

At the request of an anonymous reviewer, we also performed another simulation with α = 0.01. Table S4 to S6 in the Supplementary Information display the results. Due to the expensive computation time, we included the results only from D-DPSO when there are three target responses. The overall performances of each method with α = 0.01 are close to the case when α = 0.05.

Figure 2 compares expected sample size requirements from the 3 types of 2-stage designs discussed here for criteria C1–C4. We considered 3 testing scenarios with various hypothesized efficacy rates for the drug in the designs and investigated which method estimates the smallest sample size among the 3 types of 2-stage designs. Interestingly, we observe that among the 3 types of adaptive designs, there is no clear winner that consistently requires the smallest expected sample sizes. We further compared expected sample size requirements under various alternative hypotheses shown in Figures S1 to S3 in the Supplementary Information. We observe that the adaptive designs with two or three targets are comparable similar to the null hypothesis, but the 2-stage design with one target requires the largest sample sizes. Our proposed extended 2-stage adaptive designs are very flexible and yet surprisingly do not necessarily require larger expected sample sizes than the other two types of adaptive designs and sometimes smaller expected sample sizes, especially for criterion 3. Even when our proposed designs have larger expected sample size requirements, the additional numbers may not be large and seem manageable. Clearly, compounding the comparison problem is that the expected sample sizes also crucially depend on the various hypothesized rates in a complicated manner. The summary here is that our proposed designs offer practitioners a good alternative, especially when it is difficult to ascertain the drug efficacy rate at the onset.

Figure 2.

Expected sample sizes under the null hypothesis for the 4 criteria C1–C4 with 1, 2 or 3 target alternatives estimated by D–DPSO for 3 scenarios (from left to right): (i) p₀ = 0.05, p₁ = 0.20, p₂ = 0.25, p₃ = 0.30, (ii) p₀ = 0.20, p₁ = 0.35, p₂ = 0.40, p₃ = 0.45, and (iii) p₀ = 0.55, p₁ = 0.70, p₂ = 0.75, p₃ = 0.80. Error rates were set at α = 0.05, β₁ = 0.20, β₂ = 0.10 and β₃ = 0.05

Applications

Obstructive sleep apnea (OSA) study

Our clients did not allow us to discuss the entire nature of their study and so we minimally describe the study design. This study aims to assess the effect of HNC on the incidence of OSA compared to healthy patients. The literature suggests the maximum incidence rate of snoring and sleep apnea on healthy patients is 16.5%, resulting in the null hypothesis of 16.5% (i.e., p₀ = 0.165). There was neither historical nor preliminary data available, except that the incidence rate of OSA will be higher in HNC patients. Our clients provided an empirical range of the target response rates, which is from 24.38% to 39.00%. If Simon’s two-stage design were used for sample size calculation with 80% power and 5% significance level, the required sample sizes range from 30 to 197, using the optimality and not the minimax criterion. This shows the impact of the great uncertainty of the OSA incidence rate on HNC patients has on the sample size requirements. So there is high risk of under or over powering study if one mis-specifies the response rate. Due to wide range of the target response rates, Lin and Shih’s approach will not be able to cover the great uncertainty. For our application, one may use the extreme limits in the range for HNC patients as two target rates in the alternative hypotheses with the third that targets the midpoint between the extreme limits in the presumed range. Table 4 shows the D-DPSO generated sample size requirements for this scenario with three target responses (p₁ = 24.38, p₂ = 31.69%, and p₃ = 39.00%) and various error rates specifications (β₁ = 0.20, β₂ = 0.15, and β₃ = 0.10).

Table 4.

Sample sizes from D-DPSO for the 4 optimality criteria for OSA, Lin and Shih’s VBG, and the Phase II BREAK-2 studies.

Optimality criteria	s1/r1/q1/n1	s/l	r/m	q/n	E(N|p0)	E(N|p1)	E(N|p2)	E(N|p3)
				OSA
C1	2/8/9/21	39/188	13/55	10/39	136.92	167.28	157.91	124.73
C2 and C4	1/11/12/25	34/161	20/72	13/36	152.07	158.99	154.07	135.42
C3	1/8/9/17	37/177	12/58	10/35	144.40	166.73	167.70	153.87
			Lin and Shih’s VBG
C1	20/29/35/49	83/183	33/75	39/77	101.45	157.83	153.53	129.21
C2	20/28/29/54	81/175	42/87	31/65	125.76	134.87	107.31	81.83
C3	20/31/33/51	77/168	40/76	37/68	107.64	154.27	150.26	129.11
C4	19/28/31/54	79/170	39/83	34/66	134.72	136.06	109.57	84.89
			BREAK-2
C1	4/10/11/19	26/80	14/44	12/34	51.522	72.216	65.961	58.879
C2 and C4	4/9/12/22	27/83	16/43	13/31	62.079	65.691	50.196	43.063
C3	2/7/9/13	25/77	15/44	14/39	55.526	70.032	66.463	62.186

Lin and Shih’s VBG study

This study is adapted from Lin and Shih’s work and is to investigate the efficacy of the combination therapy of vinorelbine, bleomycin, and gemcitabine (VBG) on patients with recurrent or refractory Hodgkin’s disease (HD). The null hypothesis was set at 40% based on the single agents’ response rate, but the target response rates ranges from 0.5 to 0.6. So, Lin and Shih applied their design to cover this uncertainty with two target response rates, p₁ = 55% and p₂ = 60%, under β₁ = 0.20 and β₂ = 0.15 at a 5% significance level. Using our proposed design, three target responses p₁ = 50%, p₂ = 55%, and p₃ = 60% are considered with β₁ = 20%, β₂ = 15%, and β₃ = 5% and the optimal designs estimated are available in Table 4. The D-DPSO’s required sample size in stage 1 (49 to 54) is a little larger than those of Lin and Shih’s (39), because the smallest target rate in D-DPSO (p₁ = 0.50) is smaller than that in Lin and Shih’s (p₁ = 0.55). However, the required sample sizes of both approaches in stage 2 are comparable to each other.

Phase II BREAK-2 study

We now use a real trial from the literature and discuss advantages of our proposed extended two-stage adaptive designs. In particular, we show our proposed optimal design can reduce the sample size used in the real trial by one-half if it were implemented.

A multicenter, international, single-arm, phase II study (BREAK-2) was carried out to assess the overall response rate of dabrafenib from patients with BRAF^V600E mutation-positive metastatic melanoma (Ascierto et al. (2013)). The null hypothesis was set at p₀ = 0.25 and the alternative hypothesis was set at p₁ = 0.40. The trial wanted to recruit at least 85 patients and the plan was to declare the treatment a success if at least 29 patients responded. The Green-Dahlberg 2-stage design (Green and Dahlberg (1992)) was employed with an interim analysis planned after the first 30 patients with BRAF^V600E mutation-positive metastatic melanoma were enrolled. The expectation was that at least 7 patients were needed to respond at the interim analysis to continue the study when Type I and II error rates were set at 0.05 and 0.10, respectively. The study BREAK-2 is ongoing, but not recruiting participants as of December 14, 2015 (http://www.clinicaltrials.gov, NCT01153763). Green and Dahlberg’s two-stage design (also known as Southwest Oncology Group [SWOG] two-stage design) is similar to Simon’s two-stage design but uses a simple and uniform (i.e., fixed) significance to estimate the sample sizes for each stage with 0.02 level in stage 1 and 0.055 level in stage 2, resulting in less sensitive (or more robust) to the discrepancy between the actual sample size and the planned one. For example, as stated in Green and Dahlberg (Green and Dahlberg (1992)), multicenter studies are not easy to control over the number of patients accrued. In addition, it often occurs for multicenter studies that some patients who entered the study are ineligible after the accrual is suspended. To address these potential issues, BREAK-2 used Green and Dahlberg’s approach and, in fact, ended up a total of 76 patients which differ from the planned sample size of 85. Under the design parameters, using Green and Dahlberg’s approach and notation, the sample sizes for each stage are n₁ = 45 and n₂ = 40 (i.e., n = 85). If Simon’s two-stage design is used, the sample sizes are n₁ = 57 and n₂ = 13 (i.e., n = 83) with the minimax constraint and n₁ = 37 and n₂ = 10 (i.e., n = 99) with the optimal constraint.

The efficacy results show that 76 patients with BRAF^V600E mutation-positive metastatic melanoma were enrolled and 45 patients (59%) had a confirmed response. Although its parent phase I study (Falchook et al. (2012); http://www.clinicaltrials.gov, NCT00880321) showed the same type of patients had a response rate of 50%, this phase II study chose the response rate of 40% as an alternative hypothesis by lowering the response rate of phase I study. However, based on the phase I study, it would be of benefit if the higher response rate was explored in addition to 40% because the final response rate of the phase II study was 59%. Examining the benefit of applying the adaptive 2-stage design, we determined the optimal design of this phase II study with p₀ = 0.25, p₁ = 0.40, p₂ = 0.50, and p₃ = 0.55 and with α = 0.05, β₁ = 0.15, β₂ = 0.10, and β₃ = 0.05 using D-DPSO.

Table 4 shows the two-stage adaptive optimal designs found from D-DPSO. The same DPSO setting of parameters was used as the simulation studies did. Comparing to the original BREAK-2 study (i.e., when p₁ = 0.4), all the sample sizes of the adaptive design (s/l in Table 4) are comparable but smaller than that of BREAK-2 (29/85), while the sample size of the stage 1 of BREAK-2 (7/30) is larger than those of the adaptive design (s₁/n₁ in Table 4), although the Type II error of the adaptive design is larger than that of BREAK-2 (i.e., 0.15 vs. 0.10). Given the response rate of 59% (i.e., the final results of BREAK-2), the probabilities of not terminating the experiment at the end of the first stage are more than 99% for all three sample sizes in the table. The probabilities of choosing the sample sizes corresponding to either p₂ = 0.5 or p₃ = 0.55 in the second stage are 79%, 97% and 75%, respectively, for C1, C2 and C4, and C3, given the response rate of 59%, implying that only a half of the planned number of patients would be required for the BREAK-2 study if the adaptive design were used. Furthermore, the probabilities of choosing the sample sizes corresponding to p₃ = 0.55 in the second stage are 63%, 74% and 33%, respectively, for C1, C2 and C4, and C3, given the response rate of 59%.

Conclusions

We proposed a novel and effective nature-inspired stochastic population-based algorithm called discrete particle swarm optimization (DPSO) to find extended two-stage adaptive designs. This design terminates the trial if there are too few responders in the first stage and tests 1 of 3 preselected hypothesized response rates of a drug at the second stage, where the choice depends on the number of responders in the first stage. The design controls user-selected Type I error rate in the first stage and both Type I and II error rates at the second stage. Our results show that algorithms based on a greedy search invariably failed to find extended two-stage adaptive designs and an improved version of DPSO, called D-DPSO finds the optimum. When the problem is simplified to one or two target response rates, we also showed that D-DPSO outperformed their peers by a wide margin. Additionally, we employed simulation studies and showed all D-DPSO generated designs were able to meet pre-specified Type I and II error requirements and demonstrated how such designs can reap benefits if they were implemented in a real melanoma adaptive trial.