Spatial Two-stage Designs for Phase II Clinical Trials

Abstract

A common endpoint in a single-arm phase II study is tumor response as a binary variable. Two widely used designs for such a study are Simon’s two-stage minimax and optimal designs. The minimax design minimizes the maximal sample size and the optimal design minimizes the expected sample size under the null hypothesis. The optimal design generally has the larger total sample size than the minimax design, but its first stage’s sample size is smaller than that of the minimax design. The difference in the total sample size between two types of designs can be large and so both designs can be unappealing to investigators. We develop novel designs that compromise on the two optimality criteria and avoid such occurrences using the spatial information on the first stage’s required sample size and the total required sample size. We study properties of these spatial designs and show our proposed designs have advantages over Simon’s designs and one of its extensions by Lin and Shih. As applications, we construct spatial designs for real-life studies on patients with Hodgkin disease and another study on effect of head and neck cancer on apnea.

1. Introduction

Our work concerns phase II trials that use information from the phase I trials to explore efficacy effects of the drug and further confirmation of its toxic effects. Simon’s two stage design, reviewed briefly in Section 2, is one of the best known designs for this purpose.¹ There are many variations of Simon’s two stage designs that address various clinical issues with new design criteria.^2–9 For example, some addressed missing data issues, some proposed 3-stage designs, some modified the design for targeted therapies,¹⁰ some recommended to minimize the median sample size rather than the expected sample size when the true response rate is poor as postulated in the null hypothesis H₀: p = p₀,¹¹ and yet some had argued that the sample sizes in the two stages should be computed under the alternative hypothesis H₁: p = p₁ (> p₀) and not under the null hypothesis. Recent modifications include using ideas from Simon’s design to construct adaptive designs for precision medicine and incorporating optimality criteria in the design construction.¹²

A common critique of Simon’s design is that it is not clear how to specify the value of p₁ in the alternative hypothesis in advance. This motivated Lin and Shih⁵ to propose adaptive two-stage designs for testing one of two pre-specified alternative hypotheses with different error rates. Each alternative hypothesis has a different value of p₁ and only one of them is tested in stage 2, depending on the evidence of the results from stage 1. In their paper, Lin and Shih noted that their problem tested the limits of computational power at that time. Kim and Wong⁷ extended their work to a more flexible framework where they allowed three alternative hypotheses in stage 2 and only one needs to be tested based on results from stage 1. The resulting problem was a big computational challenge as it required solving an optimization problem involving 10 integer-valued variables with multiple nonlinear constraints. An innovative swarmbased metaheuristic algorithm was then used to solve the design problem successfully.

Simon’s minimax design minimizes the maximal sample size and Simon’s optimal design minimizes the expected sample size under the null hypothesis. The minimax designs generally have a smaller total sample size than the optimal designs and typically a larger sample size for stage 1 than that of the optimal design. Thus, if an investigator expects a positive outcome at the first stage and/or anticipated accrual rate is feasible, the minimax design would be a better choice. On the other hand, the optimal design will be a preferred choice if it is desired to know the interim outcome earlier, such as when the side effects are serious or the cost of trial failure is high and/or the accrual rate is expected to be slow as when we are studying a rare disease. However, although the optimal design produces the smaller stage 1’s sample size, the total sample size is often substantially larger than that of the minimax design. Consequently, both the minimax and optimal designs can be unappealing to investigators. For example, when p₀ = 0.55 and p₁ = 0.70 with the Type I error rate of 5% and power of 80%, Simon’s optimal and minimax designs have the stage 1’s sample sizes of 26 and 35, respectively, and the total sample sizes of 76 and 67, respectively. The optimal design requires 9 (= 35–26) fewer patients in stage 1 than the corresponding number in the minimax design, but requires 9 (= 76–67) more patients in the whole trial than the total sample size required for the minimax design. Other parameter settings may also give rise to greater discrepancy in these numbers.

Jung et al.⁴ introduced a subjective, heuristic graphical search to find a design that is a compromise between the optimal and minimax designs and subsequently developed admissible two-stage designs⁹ using a Bayesian decision-theoretic criterion. Admissible designs have the smallest expected loss defined by a weighted average between the total sample size and the expected sample size corresponding to the pre-specified weight. These approaches aim to compromise between the null expected sample size (i.e., the optimal design) and the maximal sample size (i.e., the minimax design).

Our aim in this paper is to develop designs that reduce the stage 1’s sample size, the total sample size, and/or the null expected total sample size using the Euclidean distance. In the above example, a desired design may be one having a stage 1’s sample size between 26 and 35 and/or the total sample size between 67 and 76. We propose designs using one-, two-, and three-dimensional spatial criteria and compare their performances with Simon’s two-stage designs¹, Jung’s admissible design⁹, and Lin and Shih’s adaptive two-stage designs.⁵ Unlike the designs found by Jung et al.^4,9, we search for optimal designs under one or more analytical distance-based criteria, using the domain between the stage 1’s sample size (n₁) and the total sample size (n), which may be useful for practical applications. We call our designs spatial designs because of the use of spatial criteria.

Throughout, we denote p as the unknown response rate and each of the tests of interest has one Type I error rate and one or more Type II error rates. Type I and Type II error rates are user-specified and we denote them by α and β, respectively, with appropriate subscripts when there are multiple of these rates in the test problem.

The rest of the paper is organized as follows. In Sections 2 and 3, we briefly review Simon’s two-stage designs and Lin and Shih’s adaptive designs. Section 4 discusses our spatial designs and Section 5 compares operating characteristics of the spatial designs with Simon’s, admissible designs, and Lin and Shih’s designs. In Section 6, we implement spatial designs for two clinical applications; one study is for patients with recurrent or refractory Hodgkin’s disease and another study is on obstructive sleep apnea (OSA). Section 7 concludes with discussion.

2. Simon’s two-stage designs

Simon’s designs¹ can be succinctly described as follows when the two error rates are given:

Step I:

Recruit n₁ patients in the first stage of the design and observe the number of the responses, x, from the n₁ patients.

Step II:

• iIf x ≤ r₁, stop the trial with failure to reject H₀ (i.e., p ≤ p₀)
• iiIf x > r₁, enter n₂ = n − n₁ additional patients into the study. Fail to reject the hypothesis that H₀ : p ≤ p₀ if the total number of responses ≤ r out of n patients.

Simon’s two-stage designs have a total of four parameters denoted by θ = (r₁, n₁, r, n) to optimize given the error rate constraints. If the true response probability is p, the probability of failing to reject H₀ is given by

$$G(\theta |p)=B\left(r_1,n_1,p\right)+\sum _{x=r_1+1}^{min\left(n_1,r\right)}b\left(x,n_1,p\right)B\left(r-x,n_2,p\right),$$ (1)

where b and B are the probability density function and cumulative density function of a binomial distribution with n = n₁ + n₂, respectively. When the true probability response rate is p, the expected sample size is

$$E(N|p,\theta )=n_1+\left(1-B\left(r_1,n_1,p\right)\right)n_2.$$ (2)

The goal is to find a good choice $\widehat{\theta }\in \tilde{\Theta }$, where $\tilde{\Theta }$ is the set of θ that satisfies the two error constraints:

$$G\left(\theta |p_0\right)\ge 1-\alpha ;G\left(\theta |p_1\right)\le \beta$$ (3)

and the goodness of $\widehat{\theta }$ is determined by one of the following two optimality criteria:

• O1: $\widehat{\theta }={\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}E\left(N|p_0,\theta \right)$;

• O2: $\widehat{\theta }={\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left\{n_{\theta }\right\}$,

where n_θ in O2 is the total sample size required for testing the alternative hypothesis with p = p₁ for a choice $\theta =\tilde{\Theta }$. The search strategy for $\tilde{\Theta }$ is described in the original Simon’s design, which was essentially a grid search.¹ The optimal designs found under O1 and O2 are Simon’s optimal and minimax designs, respectively.

3. Lin and Shih’s adaptive two-stage designs

Lin and Shih⁵ provided practical examples and showed that while it was relatively easy to specify the uninteresting rate p₀, the same was not true for specifying the value of p₁. To tackle the uncertainty in targeting the postulated value of the response rate in the alternative hypotheses in stage 2, they proposed designs that allowed users to specify two possible values for the response rates under the alternative hypotheses, but depending on results from stage 1, only one of them is tested provided the trial continues. The smaller value is called the skeptical choice and the other one is called the optimistic choice. Hereafter, we refer to Lin and Shih’s adaptive designs as simply adaptive designs for easy reference. Specifically, let the response rate p under the null be p₀, which, in practice, is selected to be the maximum uninteresting response rate and any response rate smaller than p₀ implies the drug is not worthy for further investigation. The two choices for the target response rates in the alternative hypotheses are p₁ and p₂, where 0 < p₀ < p₁ < p₂ < 1. The adaptive design assumes that a total of n₁ patients are assigned at the first stage and the total number of patients required for the entire trial depends on the number of responses in the first and second stages. The null hypothesis at the first stage is H₀: p ≤ p₀. Depending on the number of responses in the first stage, one of the two alternative hypotheses will be tested: H₁₁: p > p₁ or H₁₂: p > p₂. The steps for constructing the adaptive design proposed by Lin and Shih are as follows:

Step I:

Recruit n₁ patients in the first stage and observe the number of the responses, x, from the n₁ patients.

Step II:

• iiiIf x ≤ s₁, stop the trial with failure to reject H₀ (i.e., p ≤ p₀).
• ivIf s₁ < x ≤ r₁, recruit m₂ = m − n₁ additional patients into the study. Fail to reject the hypothesis that H₀:p ≤ p₀ if the total number of responses is s or fewer out of the m patients.
• vIf x > r₁, recruit n₂ = n − n₁ additional patients into the study. Fail to reject the hypothesis that H₀: p ≤ p₀ if the total number of responses is r or fewer out of the n patients.

By construction, the values of m and n are the total number of patients required for the entire trial under the alternative hypotheses, H₁₁: p > p₁ and H₁₂: p > p₂, respectively. The adaptive design has a total of seven parameters given by θ = (s₁, r₁, n₁, s, r, m, n) that we wish to optimize given error rate constraints and the stipulated 2 response rates. If the true response probability is p, the probability of failing to reject H₀ is given by

$$G(\theta |p)=B\left(s_1,n_1,p\right)+\sum _{x=s_1+1}^{min\left(r_1,s\right)}b\left(x,n_1,p\right)B\left(s-x,m_2,p\right)+\sum _{x=r_1+1}^{min\left(n_1,r\right)}b\left(x,n_1,p\right)B\left(r-x,n_2,p\right),$$ (4)

where b and B are the probability density function and cumulative density function of a binomial distribution, respectively, with m = m₁ + m₂ and n = n₁ + n₂. It follows that when the true probability response rate is p, the expected sample size is

$$E(N|p,\theta )=n_1+\left(B\left(r_1,n_1,p\right)-B\left(s_1,n_1,p\right)\right)m_2+\left(1-B\left(r_1,n_1,p\right)\right)n_2.$$ (5)

The adaptive design problem then is to find a good choice $\widehat{\theta }\in \tilde{\Theta }$, where $\tilde{\Theta }$ is the set of θ that satisfies the three error constraints:

$$G\left(\theta |p_0\right)\ge 1-\alpha ;G\left(\theta |p_1\right)\le {\beta }_1;G\left(\theta |p_2\right)\le {\beta }_2,$$ (6)

and the goodness of $\widehat{\theta }$ may be determined by one of the following four optimality criteria. Let m_θ and n_θ be the total sample sizes required for testing the alternative hypotheses H₁₁: p > p₁ and H₂₂: p > p₂, respectively, corresponding to a choice of $\theta \in \tilde{\Theta }$.

• C1: $\widehat{\theta }={\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}E\left(N|p_0,\theta \right)$, where one selects the design with the smallest n if ties are present;

• C2: $\widehat{\theta }={\mathit{\text{argmin}}}_{\theta \in \tilde{\Psi }}E\left(N|p_0,\theta \right)$ and $\tilde{\Psi }=\left\{\theta |\theta ={\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left\{\mathit{\text{max}}\left(m_{\theta },n_{\theta }\right)\right\}\right\}$;

• C3: $\widehat{\theta }={\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left\{\underset{i=0,1,2}{\mathit{\text{max}}}E\left(N|p_i,\theta \right)\right\}$, where one selects the design with the smallest n if ties are present;

• C4: $\widehat{\theta }={\mathit{\text{argmin}}}_{\theta \in \tilde{\Psi }}\left\{\underset{i=0,1,2}{\mathit{\text{max}}}E\left(N|p_i,\theta \right)\right\}$ and $\tilde{\Psi }=\left\{\theta |\theta ={\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left\{\mathit{\text{max}}\left(m_{\theta },n_{\theta }\right)\right\}\right\}$;

The search strategy for the best $\theta \in \tilde{\Theta }$ for any one of the above criteria is also essentially a grid search and details are in Lin and Shih.⁵ Simon’s optimal (O1) and minimax (O2) designs are special cases of the criteria C1 and C2, respectively.

4. Spatial design

Spatial designs are found by incorporating the spatial information, which is based on the Euclidean distance from an origin, from three sources: the stage 1’s sample size, the total sample size and the expected sample size under H₀. The main idea of spatial designs is to find a design that has the shortest Euclidian distance from an origin in the domain constructed by the two or three sources. These designs are motivated by the fact that investigators, in practice, favor a design that has a small sample size for stage 1 and a small total sample size. We initially considered controlling both the stage 1’s sample size and the total sample size because these two design parameters are directly related to the designs that have the smaller stage 1’s and total sample sizes. However, we have observed that the stage 1’s sample size is occasionally underestimated and so unrealistically small, especially when adaptive designs are used; see, for example, when p₀ = 0.55 in Table 3. This problem can be alleviated by using the expected sample size as another parameter for searching a good design.

Table 3.

Spatial designs and adaptive designs for α = 0. 05, β₁ = 0. 20 and β₂ = 0.10.

p0	p1	p2	s1/r1/n1	s/m	r/n	E(n|p0)	E(n|p1)	E(n|p2)	Criteria
0.10	0.25	0.30	2/4/21	8/44	5/29	28.30	32.80	31.35	C2
			2/4/26	7/38	6/35	31.54	35.24	35.14	C4
			1/3/18	7/38	6/37	28.90	36.52	36.88	L1,M2
			1/3/18	6/37	7/38	28.54	36.94	37.57	C3,M4
			1/3/15	7/40	6/39	26.22	37.46	38.42	M1,H1,H2
			2/3/20	6/38	7/40	26.08	37.91	39.15	M3
			2/3/18	6/38	8/49	24.40	42.93	45.99	C1,L3
			1/3/12	8/52	8/50	25.59	44.96	47.58	L2
			1/3/13	7/43	8/51	24.63	42.52	45.73	M5,M6
0.15	0.30	0.35	4/7/27	12/51	8/34	35.48	39.57	37.27	C2
			3/6/19	12/55	10/46	30.22	47.20	48.20	C1,L3
			6/9/38	11/46	10/46	40.73	45.71	45.94	C3,C4,L1
			1/4/15	11/47	10/47	36.81	45.87	46.55	M2
			3/4/22	10/46	11/48	32.65	46.04	47.27	M3,M4
			2/3/18	10/46	11/48	33.13	45.99	47.18	H2
			1/5/13	11/49	12/49	34.66	46.71	47.94	M1,H1
			1/3/11	12/55	12/54	33.27	49.60	51.76	M5
			2/3/15	13/57	12/57	31.62	51.67	54.41	M6
			1/2/10	14/65	13/62	34.52	54.94	58.06	L2
0.35	0.50	0.55	17/20/43	38/87	21/45	50.66	56.09	50.70	C2
			19/26/50	29/66	29/62	54.36	63.71	63.36	C4
			15/22/42	29/66	29/65	51.41	64.62	65.20	C3
			10/11/33	30/65	29/66	54.07	65.40	65.87	L1
			1/5/9	30/68	29/68	60.86	66.85	67.46	M1,M2
			12/13/33	30/68	30/69	45.79	66.00	68.10	M3,M4
			3/4/13	26/60	31/71	52.42	67.36	69.28	H1
			8/9/24	29/68	31/71	45.81	67.20	69.62	H2
			3/4/12	25/57	34/79	50.57	71.45	74.94	M5
			10/11/27	30/69	34/80	43.10	72.37	76.97	C1,L3
			1/4/6	35/82	35/80	57.71	73.47	76.41	L2
			7/8/20	36/84	34/80	44.58	72.59	76.81	M6
0.55	0.70	0.75	22/26/38	48/74	27/41	47.83	53.98	47.79	C2
			22/30/43	42/65	31/48	56.78	57.09	52.46	C4
			22/25/39	42/65	41/64	48.52	62.96	63.86	C3
			3/6/10	41/64	42/65	58.76	64.08	64.59	L1,M2
			14/18/24	52/83	41/66	41.43	70.10	72.60	C1
			2/3/7	39/61	43/67	56.39	64.69	65.88	M1
			14/15/26	38/60	43/67	44.27	64.07	66.19	M3,M4
			12/13/23	38/60	43/67	45.07	64.14	66.16	H2
			6/10/13	44/69	44/68	49.02	65.30	67.31	H1
			5/7/11	45/71	46/72	49.18	66.88	69.65	M5
			13/15/23	47/74	46/73	41.43	67.26	71.12	L3
			12/14/21	55/88	46/73	42.42	69.85	73.08	M6
			1/2/4	47/74	47/74	57.10	68.14	70.45	L2
0.65	0.80	0.85	23/26/33	47/63	27/36	39.13	46.29	41.28	C2
			22/28/34	41/55	30/38	43.32	49.33	44.79	C4
			12/15/18	49/67	40/54	35.09	56.96	58.71	M6
			15/16/24	39/53	41/55	39.96	53.77	54.79	C3,M4
			1/2/5	37/50	41/55	51.39	54.41	54.77	L1,M1,M2
			11/13/18	42/56	42/57	39.05	54.77	56.43	H2
			16/17/24	42/57	43/58	36.01	54.87	57.29	M3
			1/2/4	39/53	44/59	50.18	56.58	57.76	L2
			15/18/22	47/64	44/61	34.62	57.42	60.73	C1,L3
			5/6/9	45/61	45/61	40.66	56.55	59.24	M5,H1
0.75	0.90	0.95	17/21/22	33/39	23/25	27.47	36.57	34.40	C3,C4,M3,M4
			19/21/24	42/49	22/26	29.25	33.89	28.52	C2
			1/2/4	30/35	33/39	36.38	38.68	38.93	L1,M1,M2
			4/6/7	33/39	33/39	31.21	38.18	38.88	H2
			6/7/9	35/42	34/40	28.22	38.70	39.87	H1
			10/11/13	40/48	36/43	24.01	40.21	42.82	C1,L3
			1/2/3	36/42	36/43	36.33	41.64	42.57	L2
			3/4/5	38/45	36/43	29.84	40.56	42.55	M5
			9/10/12	37/44	36/43	24.34	39.79	42.49	M6

We now propose and construct spatial designs for two cases:the first case is when there is one target response rate as in Simon’s and the second case when there are two target response rates as in adaptive designs. The first step to constructing spatial designs is to generate a set of feasible solutions, $\tilde{\Theta }$, that satisfies (i) the two error constraints (i.e., Equation (3)) with 0 < r₁ < n₁ for Simon’s designs or (ii) the three error constraints (i.e., Equation (6)) with 0 < s₁ ≤ r₁ < n₁ for adaptive designs, using a grid search. Different types of spatial designs can then be found by selecting a good choice $\theta \in \tilde{\Theta }$ using one of the following optimality criteria. These criteria minimize the various sample sizes or expected sample sizes in different ways and thus provide the practitioner more options to choose a design criterion most appropriate for his or her problem. Depending on the number of variables the investigator want to consider, we recommend the following optimality criteria, where the notations are developed sequentially from one criterion to another:

1. One-dimensional spatial designs

   • L1: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left\{l_{\theta }|l_{\theta }=\mathit{\text{max\hspace{0.17em}}}\left(n_{\theta },m_{\theta }\right)\right\}$; select the one with the smallest n₁,_θ, if ties are present
   • L2: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left\{n_{1,\theta }\right\}$; select the one with the smallest max (m_θ, n_θ) if ties are present
   • L3: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}E\left(N|p_0,\theta \right)$; select the one with the smallest n₁,_θ, if ties are present
2. Two-dimensional spatial designs

   • M1: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left|\left(\widehat{l},{\widehat{n}}_1\right)-\left(l_{\theta },n_{1,\theta }\right)\right|$, where l_θ = max (n_θ, m_θ), $\widehat{l}=\mathit{\text{min}}\left\{l_{\theta }|l_{\theta }=\mathit{\text{max}}\left(n_{\theta },m_{\theta }\right),\theta \in \tilde{\Theta }\right\}$ and ${\widehat{n}}_1=\mathit{\text{min}}\left\{n_{1,\theta }|\theta \in \tilde{\Theta }\right\}$
   • M2: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left|\left(l_{\theta },n_{1,\theta }\right)\right|$
   • M3: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left|\left(\widehat{l},{\widehat{EN}}_0\right)-\left(l_{\theta },E{N}_{0,\theta }\right)\right|$, where EN_0,θ = E(N|p₀, θ) and ${\widehat{EN}}_0=\mathit{\text{min}}\left\{E{N}_{0,\theta }|\theta \in \tilde{\Theta }\right\}$
   • M4: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left|\left(l_{\theta },E{N}_{0,\theta }\right)\right|$
   • M5: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left|\left({\widehat{n}}_1,{\widehat{EN}}_0\right)-\left(n_{1,\theta },E{N}_{0,\theta }\right)\right|$
   • M6: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left|\left(n_{1,\theta },E{N}_{0,\theta }\right)\right|$
3. Three-dimensional spatial designs

   • H1: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left|\left(\widehat{l},{\widehat{n}}_1,{\widehat{EN}}_0\right)-\left(l_{\theta },n_{1,\theta },E{N}_{0,\theta }\right)\right|$
   • H2: ${\mathit{\text{argmin}}}_{\theta \in \tilde{\Theta }}\left|\left(l_{\theta },n_{1,\theta },E{N}_{0,\theta }\right)\right|$

The notations n_1,θ, m_θ, and n_θ are the total sample sizes required for stage 1, for testing the alternative hypotheses involving p₁ and p₂, respectively, corresponding to a feasible solution $\theta \in \tilde{\Theta }$ and | · | is an Euclidean distance. The letters L, M, and H in the criteria stand for low (one)-, middle (two)-, and high (three)-dimensional spatial designs, O for Simon’s design for optimal designs, and C for Lin and Shih’s designs borrowed from their original notations. As mentioned before, spatial designs utilize the sample size from stage 1 (n₁), the total sample size (n or max (m, n)), and the expected sample size under the null hypothesis (E(n|p₀)). Based on these three numbers, the spatial designs use one-, two-, and three-dimensional Euclidean distance measures to search for a good design. That is, the spatial designs found under criteria L1-L3 are selected based on one-dimensional Euclidean distance, those with M1-M6 are based on two-dimensional Euclidean distance, and those with H1-H2 are based on three-dimensional Euclidean distance.

When there is only one target response, the design found under L1 has the smallest total sample size, which is the same as the Simon’s minimax design. When the number of target response rate is one, the spatial design found under criterion L3 is the same as Simon’s optimal design, which rarely has ties. When there are no ties and there are two target response rates, the criterion L3 generates the same design as the adaptive design found under C1. The criterion L2 generates the design with the smallest sample size for stage 1.

The designs found under criteria M1, M3 and M5 are based on a two-dimensional domain using the two-dimensional Euclidian distance from coordinates composed of (i) the minimum total sample size and sample size for stage 1, (ii) the minimum total and expected null sample sizes, and (iii) the minimum sample size for stage 1 and expected null sample sizes, respectively, to a desired design. The spatial designs found under criteria M2, M4, and M6 are the same designs as those found under M1, M3, and M5 except that the distances are measured from the origin (0,0). In other words, designs found under M1, M3, and M5 are the normalized versions of those obtained under M2, M4, and M6 using the minimum sample size for stage 1, total sample size, and expected null sample sizes. More generally, if the minimum total sample size for stage 1 and expected null sample sizes are on the vector line from the origin to the desired design, the two designs by the distance from the minimum and the origin will generate exactly the same design.

Similarly, the spatial design found under criterion H2 is a design based on a three-dimensional Euclidean distance from the origin to the desired design, while the design found under H1 is the normalized version of the design found under H2 by the minimum stage 1’s, total, and expected null sample sizes. Likewise for the two-dimensional spatial designs, if those three minimum values are on the vector from the origin to the design found under H1, both designs under H1 and H2 will coincide.

Except for M3 and M6, these two-dimensional and three-dimensional criteria aim to find the desired design by controlling both the sample size for stage 1 and the total sample size.

5. Comparisons of operating characteristics

We use operating characteristics to evaluate and compare performances of the spatial designs with Simon’s two-stage designs, including admissible designs from Jung et al.⁹, and adaptive designs. We recall that Simon’s designs are appropriate when the number of target response is one and adaptive designs are appropriate when the number of target response is two.

For Simon’s designs, scenarios of operating characteristics covered two sets of α and β with (α, β) ∈ {(0.05,0.2), (0.1,0.2)}. For each set of α and β, six sets of combinations for null and target response rates were empirically selected to ascertain the general performances of each method. These combination sets are (p₀, p₁) ∈ {(0.10,0.25), (0.15,0.30), (0.35,0.50), (0.55,0.70), (0.65,0.80), (0.75,0.90)}. Then Simon’s optimal and minimax designs were found by the two criteria O1 and O2, respectively, as described in Section 2. The spatial designs corresponding to Simon’s designs were then found based on the 11 spatial criteria L1-L3, M1-M6, H1-H2 using the set of feasible solutions, $\tilde{\Theta }$, that satisfies the two error constraints for Simon’s design (i.e., Equation (3)).

Similarly, we used two sets of α and (β₁, β₂) with (α, β₁, β₂) ∈ {(0.05,0.20,0.10), (0.10,0.20,0.10)} to analyze the operating characteristics of the adaptive designs. For each set of (α, β₁, β₂), six sets of null and target response rates were selected; (p₀, p₁, p₂) ∈ {(0.10,0.25,0.30), (0.15,0.30,0.35), (0.35,0.50,0.55), (0.55,0.70,0.75), (0.65,0.80,0.85), (0.75,0.90,0.95)}. The four adaptive designs were obtained based on the four criteria C1-C4 in Section 3. Their corresponding spatial designs were selected according to the 11 criteria L1-L2, M1-M6, H1-H2 using the set of feasible solutions, $\tilde{\Theta }$, that satisfies the three error constraints for the adaptive design (i.e., Equation (6)).

5.1. One target response

Tables 1 and 2 compare Simon’s designs with their corresponding spatial designs when the number of target response is one. The designs found under O1 and L3 have the smallest expected null sample sizes, but they have the largest total sample sizes among all criteria with few exceptions. These exceptions include several cases for criteria L2, M5, and M6; see, for example, when p₀ = 0.15, 0.35, 0.55 in Table 1 and when p₀ = 0.10, 0.15, 0.55 in Table 2. The criteria O2 and L1 generate the designs with the largest stage 1’s sample size and the largest expected null sample sizes, although their total sample sizes are the smallest. Interestingly, Table 2 shows when p₀ = 0.75 and α = 10%, all spatial designs, except for those found under L3, generate the same designs as the Simon’s minimax designs.

Table 1.

Spatial designs and Simon’s designs (O1 and O2) for α = 0. 05 and β = 0. 20.

p0	p1	r1/n1	r/n	E(n|p0)	E(n|p1)	Criteria	Note
0.10	0.25	2/22	7/40	28.84	22.94	O2,L1	Minimax
		1/15	7/41	26.72	15.92	M1,M2,M4	Admissible
		1/14	7/42	25.63	14.90	L2,M3,M5,M6,H1,H2	Admissible
		2/18	7/43	24.66	18.86	O1,L3	Optimal
0.15	0.30	3/23	11/48	34.51	23.95	O2,L1,M2	Minimax
		3/21	11/49	31.88	21.91	M1,M3,M4,H1,H2	Admissible
		3/19	12/55	30.37	19.87	O1,L3,M5,M6	Optimal
		3/18	14/67	31.71	18.84	L2
0.35	0.50	22/55	29/66	56.96	55.91	O2,L1	Minimax
		12/33	31/71	46.63	33.92	M2,M3,M4,H2	Admissible
		10/29	32/73	48.29	29.93	M1,H1
		12/32	32/74	45.03	69.48		Admissible
		10/27	33/77	43.51	27.88	O1,L3	Optimal
		8/22	36/85	44.26	22.86	L2,M5,M6
0.55	0.70	20/35	43/67	45.80	35.93	O2,L1,M4	Minimax
		17/30	44/69	44.01	30.92	M1,M2,M3,H1,H2	Admissible
		15/26	48/76	42.02	26.87	O1,L3	Optimal
		12/21	52/83	42.16	21.85	M5,M6
		11/19	67/109	47.52	19.82	L2
0.65	0.80	20/31	41/55	41.92	31.97	O2,L1,M2	Minimax
		19/28	43/58	37.27	28.91	M3,M4,H2	Admissible
		16/25	44/59	40.87	25.95	M1
		17/25	45/61	36.02	25.89	H1	Admissible
		12/18	49/67	35.39	18.87	O1,L2,L3,M5,M6	Optimal
0.75	0.90	17/22	33/39	27.50	22.94	O2,L1,M2,M3,M4,H2	Minimax
		16/20	36/43	25.18	39.94		Admissible
		15/19	37/44	25.58	19.89	H1
		12/16	39/46	28.15	16.93	M1
		10/13	40/48	24.64	13.87	O1,L2,L3,M5,M6	Optimal

Table 2.

Spatial designs and Simon’s designs (O1 and O2) for α = 0. 10 and β = 0.20.

p0	p1	r1/n1	r/n	E(n|p0)	E(n|p1)	Criteria	Note
0.10	0.25	1/16	5/31	23.28	16.94	O2,L1,M2	Minimax
		1/14	5/32	21.48	14.90	M1,M3,M4,H1,H2	Admissible
		1/13	5/34	20.95	13.87	O1,L3,M6	Optimal
		1/12	6/41	21.89	12.84	L2,M5
0.15	0.30	2/18	8/37	27.89	18.94	O2,L1,M2	Minimax
		2/16	8/38	25.65	16.90	M1,M3,M4,M6,H1,H2	Admissible
		3/19	8/39	25.32	19.87	O1,L3	Optimal
		2/15	9/44	26.48	15.87	M5
		2/14	10/51	27.03	14.84	L2
0.35	0.50	10/31	21/49	40.81	31.96	O2,L1,M4	Minimax
		9/26	22/52	37.10	26.92	M3	Admissible
		5/16	23/55	35.89	16.89	L2,M1,M2,M5,M6,H1,H2	Admissible
		7/20	24/58	35.16	20.87	O1,L3	Optimal
0.55	0.70	26/42	30/48	42.87	42.84	O2,L1	Minimax
		13/25	31/49	38.02	25.96	M2	Admissible
		12/22	32/51	34.61	22.91	M3,M4,H2	Admissible
		11/20	33/53	33.67	20.89	O1,L3,M1,M6,H1	Optimal
		10/18	39/63	35.62	18.86	M5
		8/15	41/66	38.06	15.87	L2
0.65	0.80	12/19	30/41	29.59	19.93	O2,L1,M1,M2,M3,M4,H1,H2	Minimax
		14/21	32/44	29.20	41.50		Admissible
		9/14	36/50	29.22	14.87	L2,M5,M6
		13/19	37/52	28.79	19.84	O1,L3	Optimal
0.75	0.90	7/10	25/30	20.51	10.93	O2,L1,L2,M1,M2,M3,M4,M5,M6,H1,H2	Minimax
		10/13	28/34	19.98	13.87	O1,L3	Optimal

Overall, except for the design criteria M5 and M6, all two- and three-dimensional spatial designs generate designs with smaller sample sizes for stage 1 than those of Simon’s minimax design and also smaller total sample sizes than those from Simon’s optimal design, indicating that spatial designs can require both smaller total sample size and smaller sample size for stage 1. For example, Table 1 shows when p₀ = 0.35 with α = 5%, the Simon’s optimal design (i.e., O1 and L3) has (r₁/n₁, r/n) = (10/27,33/77) and the Simon’s minimax design (i.e., O2 and L1) has (r₁/n₁, r/n) = (22/55, 29/66). This implies that the minimax design requires 11 (= 77–66) fewer for the whole trial than the optimal design but it itself requires 28 (= 55–27) more in stage 1 alone. As the study sample size increases, the accrual duration tends to increase and so does the whole study duration. Thus, choosing O1 requires a longer overall study duration, and choosing O2 requires a longer duration for stage 1 in the trial. On the other hand, the spatial designs under criteria M1 and H1 have (r₁/n₁, r/n) = (10/29,32/73) and those under M2, M3, M4, and H2 have (r₁/n₁, r/n) = (12/33,31/71). The former group of designs (i.e., H1 and M1) requires 26 (= 55–29) fewer and the latter group of designs (i.e., M2, M3, M4 and H2) requires 22 (= 55–33) fewer subjects in stage 1 compared to the numbers required for the minimax design (O2). Similarly, in terms of total sample sizes, the optimal design (O1) requires 4 (= 77–73) more than that for the spatial designs found under H1 and M1, and 6 (= 77–71) more than that for spatial designs found under M2, M3, M4 and H2. Therefore, spatial designs have advantages over Simon’s designs.

The search for an appropriate spatial design can be facilitated by a spatial design plot, which is a scatter plot obtained by plotting the total sample sizes versus the sample sizes required for stage 1. In a spatial design plot, the gray and green dotted curves indicate the traces of distances from the origin, (0,0), and the minimum coordinate of the domain, ($\widehat{n}$, $\widehat{n_1}$), respectively. The designs on the lower left usually have a smaller stage 1’s sample size and a smaller total sample size. As an illustration, consider the case when p₀ = 0.35 in Tables 1 and 2. Figure 1 is the spatial design plot and, as expected, the spatial designs that have smaller sample sizes for stage 1 and smaller total sample sizes are located near the bottom left corner of the spatial design plot. For two or three-dimensional spatial designs, they are located at the nearer side from the bottom left corner than Simon’s designs. Additional spatial design plots for all cases in Tables 1 and 2 are shown in Figures S1 and S2 in the supplementary material

Figure 1.

Spatial design plots for Simon’s two-stage designs when α = 0.05 (upper) and α = 0.10 (bottom) with p₀ = 0.35, p₁ = 0.50, and β = 0.20.

Tables 1 and 2 also compare admissible two-stage designs⁹ with the proposed spatial designs. Admissible designs aim to search for a design that has the smallest expected loss defined by a weighted average between the total sample size and the expected sample size corresponding to the pre-specified weight. The spatial designs found under criteria L1, L3, M3, M4, H1, and H2 are indirectly associated with the admissible designs in the sense that those criteria use the total sample size and the expected sample size to find a design. In particular, the spatial design found under criterion M4 is the same as the admissible design when the specified weight is 0.5, which can be confirmed by Tables 1 and 2. The difference between their designs and the spatial designs is that the latter compromises on the sample size for stage 1 (n₁) and the total sample size (n), not directly on Simon’s minimax and optimal designs. Our experience is that given a setup, there are usually more spatial designs than admissible designs as can be seen in Tables 1 and 2. There are also admissible designs are not spatial designs. For instance, when p₀ = 0.75 with α = 5% or p₀ = 0.65 with α = 10%, there are admissible designs that are not spatial designs. Thus, spatial designs can provide as competitive alternatives to admissible designs. Note that the admissible designs were generated using the web-based JAVA program at http://cancer.unc.edu/biostatistics/program/ivanova/SimonsTwoStageDesign.aspx.

5.2. Two target responses

Tables 3 and 4 compares adaptive designs, which have two target responses, with the corresponding spatial designs. The adaptive designs found under criteria C1 and C2 are extended versions of Simon’s optimal and minimax designs, respectively when there are two target responses. They have the smallest expected sample size under the null hypothesis (i.e., E(n|p₀)) and the smallest total sample size with p₂ (i.e., n), respectively. The spatial design found under the criterion L3 has the same designs as the adaptive design found under C1 except for the case with p₀ = 0.55 in Table 1. This is because both criteria (C1 and L3) find a design with the smallest expected sample size under p₀, but, when ties are present, the design under C1 selects the one with the smallest total sample size, while that found under L3 chooses the one with the smallest sample size for stage 1. Not surprisingly, we observe that the design found under L2 has the smallest sample size for stage 1even though this sample size appears to be frequently under-estimated and unrealistically small for some cases compared to the sample sizes required by other designs, see for example, when p₀ = 0.55 in Table 3.

Table 4.

Spatial designs and adaptive designs for α = 0. 10, β₁ = 0. 20 and β₂ = 0.10.

p0	p1	p2	s1/r1/n1	s/m	r/n	E(n|p0)	E(n|p1)	E(n|p2)	Criteria
0.10	0.25	0.30	1/3/17	5/32	4/22	23.95	24.78	23.73	C2
			1/3/20	5/28	4/27	24.73	27.03	27.05	C3,C4,L1
			1/3/17	4/27	5/29	22.35	27.79	28.40	M3,M4
			1/2/15	4/29	5/30	21.50	28.64	29.38	M1,M2,H1,H2
			1/2/14	4/26	5/34	20.25	30.54	32.14	C1,L3
			1/2/12	5/38	6/39	20.98	34.49	36.54	L2
			1/2/12	5/36	6/41	20.74	35.25	37.70	M5,M6
0.15	0.30	0.35	3/5/22	9/41	6/26	28.57	29.41	27.98	C2
			2/5/26	8/34	7/33	31.98	33.11	33.05	C3,C4,L1
			1/3/13	8/36	7/36	26.84	34.54	35.32	M1,M2,H1,H2
			2/4/18	7/34	8/36	26.57	34.38	35.24	M3,M4
			1/2/12	7/34	8/38	25.30	35.12	36.46	M6
			1/2/11	9/42	8/40	26.30	37.12	38.52	M5
			2/3/15	7/34	9/45	24.47	39.32	41.93	C1,L3
			1/2/10	10/49	9/46	27.23	41.33	43.43	L2
0.35	0.50	0.55	11/14/30	25/58	15/34	38.10	41.46	38.61	C2
			9/15/29	21/49	16/34	40.54	43.05	40.31	C4
			5/11/16	23/55	16/34	35.86	50.10	51.31	M6
			10/15/30	21/49	17/39	39.05	43.78	42.29	C3,M4
			1/2/9	20/48	21/49	43.94	48.15	48.60	L1,M1,M2
			7/9/22	22/51	21/50	37.06	48.32	49.43	M3
			2/3/10	22/52	22/52	41.01	49.70	50.85	H1
			5/6/17	21/50	22/52	36.91	49.30	50.84	H2
			2/3/9	24/57	23/55	40.03	51.20	52.94	M5
			7/8/20	22/54	25/60	34.99	54.02	57.24	C1,L3
			1/2/6	25/61	26/62	43.80	55.64	57.94	L2
0.55	0.70	0.75	17/20/30	35/55	21/32	37.38	39.35	35.98	C2
			9/13/22	30/47	29/46	43.40	46.08	46.06	C4,L1
			13/18/27	29/46	30/47	40.39	46.31	46.74	C3
			12/14/23	30/47	30/48	35.89	46.46	47.55	M4
			1/2/5	31/50	31/49	43.50	47.78	48.40	M1
			2/3/7	29/47	31/49	42.10	47.60	48.34	M2
			11/13/21	30/48	31/49	35.01	46.90	48.31	M3
			5/6/12	27/44	31/49	38.29	47.18	48.27	H2
			3/4/8	32/51	32/51	39.80	48.51	49.83	H1
			1/2/4	35/56	33/53	42.27	49.69	51.14	L2
			7/8/14	30/48	33/53	34.25	48.73	51.14	M6
			3/5/8	28/45	36/57	38.01	49.48	52.13	M5
			11/12/20	26/42	36/58	33.15	51.86	55.47	C1,L3
0.65	0.80	0.85	12/15/18	34/47	18/26	27.80	37.45	35.71	C1,L3
			15/18/23	33/45	20/26	31.05	33.91	30.53	C2
			12/16/19	30/41	20/27	29.35	36.20	34.46	C3,M3,M4
			20/23/29	30/41	24/32	31.90	35.53	34.09	C4
			1/3/5	28/38	30/41	37.50	39.99	40.43	L1,M2
			3/4/7	27/37	30/41	33.13	39.41	40.34	H1
			6/10/11	30/41	28/41	31.05	39.49	40.52	H2
			1/2/4	30/42	31/42	37.19	40.97	41.54	M1
			8/10/13	31/43	33/45	28.24	41.03	43.36	M6
			1/2/3	36/50	34/47	35.93	43.58	45.30	L2
			2/3/5	25/34	34/47	32.75	41.90	44.09	M5
0.75	0.90	0.95	12/14/16	27/32	15/18	21.59	23.70	20.54	C2
			10/12/14	24/29	23/27	21.62	27.17	27.24	C4
			13/15/17	24/29	24/28	21.19	27.53	28.10	C3
			4/5/7	24/28	24/29	23.33	28.31	28.88	L1,M2
			8/9/11	25/30	24/29	19.45	27.60	28.81	M3,M4,H2
			2/3/4	24/29	25/30	22.77	28.35	29.46	M1,H1
			8/10/11	24/29	26/31	19.28	28.02	29.86	C1,L3,M6
			2/3/4	23/28	26/31	22.67	28.71	30.11	M5
			1/2/3	22/27	27/32	25.36	29.97	31.11	L2

In summary, our cumulative comparison results suggest that spatial designs behave as expected and require smaller stage 1’s sample sizes and smaller total sample sizes than the sample sizes required in the corresponding Simon’s designs and adaptive designs, and that spatial designs are comparable to admissible designs but generate more designs. Figures 1 and 2 display the spatial design plots which for various spatial designs corresponding to Simon’s designs and adaptive designs. As expected, the balanced spatial designs are located at the bottom left corner of the plots. Figure 2 shows that all adaptive designs found under criteria C1-C4 are located at the farther side from the bottom left corner compared to two or three-dimensional spatial designs. Spatial design plots for the designs in Tables 3 and 4 are in Supplementary Figures S1 and S2. Additional cases when |p₀ − p₁| = 0.10 and 0.20 are displayed in the Supplementary Tables S2 to S5 and S6 to S9, respectively, showing a trend consistent with when |p₀ − p₁| = 0.15.

Figure 2.

Spatial design plots for adaptive designs when α = 0.05 (upper) and α = 0.10 (bottom) with p₀ = 0.35, p₁ = 0.50, p₂ = 0.55, β₁ = 0.20, and β₂ = 0.10.

6. Applications

We now construct spatial designs for two real problems: a vinorelbine, bleomycin, and gemcitabine (VBG) study that aims to investigate the efficacy of the combination therapy of VBG on patients with recurrent or refractory Hodgkin disease⁵ and a study to investigate the effect of head and neck cancer on the incidence of obstructive sleep apnea (OSA).⁷ Our goal is to study relative practical benefits of implementing spatial designs over Simon’s designs¹, admissible designs⁹, and/or adaptive designs.⁵

6.1. The vinorelbine, bleomycin, and gemcitabine (VBG) study

The VBG study initially planned to employ Simon’s two stage designs with p₀ = 40% and p₁ = 60% to achieve 90% power (i.e., β = 10%) at a 5% significance level (i.e., α = 5%). To deal with the uncertainty of the target response rate, one more target response (p₂) was then introduced into the study and the sample sizes with two response rates were determined using adaptive designs with p₀ = 40%, p₁ = 55%, p₂ = 60%, β₁ = 20%, and β₂ = 10% at a 5% significance level. Table 5 and Figure 3 display the Simon’s, adaptive, and spatial designs.

Table 5. Spatial designs for VGA and OSA studies.

The corresponding spatial design plots are available in Figures 3 and 4.

VGA
Simon’s two-stage designs
(p0 = 0.40, p1 = 0.50, α = 0.05, and β = 0.10)
r1/n1	r/n		E(n|p0)	E(n|p1)	Criteria	Note
12/29	27/54		38.06	29.97	O2,L1,M3,M4	Minimax
9/23	28/56		37.64	23.97	M1,M2,H1,H2
8/20	30/61		36.58	20.94	L2,M5,M6	Admissible
11/25	32/66		35.98	25.92	O1,L3	Optimal
Adaptive designs
(p0 = 0.40, p1 = 0.55, p2 = 0.60, α = 0.05, β1 = 0.20, and β2 = 0.10)
s1/r1/n1	s/m	r/n	E(n|p0)	E(n|p1)	E(n|p2)	Criteria
17/21/40	40/81	23/45	51.36	57.51	51.75	C2
15/22/39	34/69	23/45	53.95	59.29	53.98	C4
14/19/36	34/69	32/66	51.77	66.11	66.43	C3
1/3/9	32/66	34/69	63.53	67.98	68.49	L1,M1,M2
16/18/38	34/69	34/70	48.38	67.43	69.32	M4
13/17/32	34/70	35/71	47.10	67.75	69.94	M3
7/8/20	32/65	35/72	49.12	68.47	70.66	H2
4/5/13	35/73	36/74	52.24	69.63	71.98	H1
1/3/6	37/76	36/75	59.49	70.71	72.59	L2
9/10/22	45/93	38/80	45.71	73.77	77.66	M6
4/5/12	42/89	39/81	52.58	74.48	77.85	M5
11/12/26	38/79	39/82	43.89	74.13	78.93	C1,L3
OSA
Simon’s two-stage designs
(p0 = 0.165, p1 = 0.390, α = 0.05, and β = 0.10)
r1/n1	r/n		E(n|p0)	E(n|p1)	Criteria	Note
3/19	9/34		24.78	19.97	O2,L1	Minimax
2/14	9/35		22.69	14.95	M1,M2,M3,M4,H1,H2	Admissible
2/13	10/40		22.87	13.93	M5,M6
3/16	10/41		22.62	16.92	O1,L3	Optimal
2/12	13/56		25.93	12.91	L2
Adaptive designs
(p0 = 0.165, p1 = 0.317, p2 = 0.390, α = 0.05, β1 = 0.15, and β2 = 0.10)
s1/r1/n1	s/m	r/n	E(n|p0)	E(n|p1)	E(n|p2)	Criteria
1/6/17	15/61	7/28	51.60	51.35	43.78	L2
2/7/20	14/58	8/28	44.92	48.59	41.39	M1
3/7/20	15/64	8/29	38.34	50.58	43.96	M5,M6
3/8/22	14/59	9/29	40.44	50.05	43.42	H1
4/8/28	15/62	9/33	44.12	44.96	37.92	C2
1/7/21	14/57	8/35	52.47	49.45	43.46	L1,M2
4/9/31	14/57	10/41	46.15	47.97	43.69	C4
5/9/27	15/65	11/42	37.46	53.39	49.23	C1,L3
4/8/25	14/59	12/50	38.38	53.26	52.36	M3,H2
4/8/26	14/58	11/50	39.68	52.78	51.78	M4
7/10/38	14/57	12/52	43.20	52.52	52.26	C3

Figure 3.

Spatial design plots for Simon’s designs (upper) and adaptive two-stage designs (bottom) for the VGA study.

When there is one target response, Simon’s minimax (O2) and optimal (O1) designs have (r₁/n₁, r/n) = (12/29,27/54) and (r₁/n₁, r/n) = (11/25,32/66), respectively. This means that the total sample size required for the minimax design is 12 (= 66–54) patients fewer than that required for the optimal design. Further, the sample size required for the minimax design in stage 1 is 4 (= 29–25) patients more than that required for the optimal design. On the other hand, the sample sizes of all stage 1 in spatial designs are smaller than that required for Simon’s designs. For example, the spatial designs under criteria M1, M2, H1, and H2 (i.e., the case 2 in Figure 3) have (r₁/n₁, r/n) = (9/23, 28/56). Their sample sizes for stage 1 require 2 (= 25–23) and 6 (= 29–23) patients fewer than Simon’s optimal and minimax designs, respectively, and their total sample sizes are also 10 (= 66–56) patients fewer than that required for Simon’s optimal design and comparable to that required for Simon’s minimax design. The take home message is that spatial designs can reduce the study duration for stage 1 and, at the same time, the entire study duration is similar to that required for Simon’s minimax designs.

When there are two target response rates, Figure 3 shows an apparent advantage of spatial designs over adaptive designs (C1-C4). Based on the spatial design plot, the possible choices to balance the sample size for stage 1 and the total sample size are one of the designs under L1, L2, M1, M2, H1, and H2 (i.e., the cases 4, 7, 8, and 9 in Figure 3). These spatial designs have smaller sample sizes for stage 1 than all adaptive designs, and their total sample sizes are between those required for the adaptive designs under C3 and C4 and under C1 and C2. As mentioned before, the spatial designs under L1, L2, M1, and M2 have much smaller sample sizes for stage 1 and may be underestimated. Therefore, the spatial designs under either H1 or H2 are likely the best choice. For example, the spatial design under H2 requires 6 (= 26–20) patients fewer for stage 1 and 10 (= 82–72) to 14 (= 79–65) patients fewer for the total sample size than the corresponding numbers for the adaptive design under C1.

6.2. The obstructive sleep apnea (OSA) study

The OSA study was originally designed with three target response rates due to the wide range (i.e., 24.38% to 39.00%) of empirical target response rates for the incidence rate of OSA.⁷ However, to fix ideas, we formulate the OSA study when there is one or two target response rates similar to the VBG study. Based on the historical data, the null incidence rate of snoring and sleep apnea on healthy patients is set to 16.5% (p₀ = 0.165) and the target response rates are set to 31.69% (p₁ = 0.3169) and 39.00% (p₂ = 0.39). Table 5 and Figure 4 display the required samples sizes for Simon’s designs, adaptive designs and spatial designs.

Figure 4.

Spatial design plots for Simon’s designs (upper) and adaptive two-stage designs (bottom) for the OSA study.

For the design with one target response rate, although Simon’s minimax and optimal designs have the smallest total and stage 1’s sample sizes, the spatial designs under M1-M4 and H1-H2 (i.e., the case 4 in Figure 4) have much balanced stage 1’s and total sample sizes, i.e., (r₁/n₁, r/n) = (2/14, 9/35) with the comparable expected null sample size of 22.69. That is, the stage 1’s sample size is 2 (= 16–14) and 5 (= 19–14) patients less than those of optimal (O1) and minimax (O2) designs, respectively, and the total sample size is 6 (= 56–35) patients less and 1 (= 35–34) patient more than those of optimal (O1) and minimax (O2) designs, respectively. Certainly, these spatial designs are located at the bottom-left-most corner in the spatial design plot of Figure 4.

When there are two target response rates, the spatial designs under criteria L1, M1, M2, and H1 (i.e., the cases 2, 4, and 6 in Figure 4) show the advantage over adaptive designs, locating at the bottom-left-most corner as depicted in Figure 4. In particular, the spatial design under H1 (i.e., the case 4 in Figure 4) would be a possible choice because all of its stage 1’s and total sample sizes are less than those of the adaptive design under C1 as well as its expected null sample size is comparable to that of the design under C1. Interestingly, all spatial designs have the smaller stage 1’s sample sizes than those of the adaptive designs.

6.3. Comparisons with admissible designs

While general performance merits of admissible designs relative to spatial designs seem elusive, they can be easily compared in specific settings. For example, for the design parameters in the VGA and OSA studies, the spatial design for the VGA study is (r₁/n₁, r/n) = (9/23, 28/56), and the two spatial designs for the OSA study are (r₁/n₁, r/n) = (2/13, 10/40) and (2/12, 13/56). Table 5 shows that there are more designs that are not generated from the admissible designs for VGA and OSA studies, respectively. In particular, the spatial design with (r₁/n₁, r/n) = (9/23, 28/56), which is not an admissible design, is the best choice to compromise between n₁ and n for the VGA study.

7. Discussion

We proposed spatial designs to balance the sample size requirements for stage 1 and the entire trial with the goal of having a relatively small sample size for stage 1 among all designs with comparable total sample size. These designs are therefore especially useful to clinical trials with a slow accrual rate or for studying a disease with a low incidence rate. Our comparison studies and applications to real examples demonstrated that spatial designs possess the characteristics motivated them and have desirable properties over current designs for phase II trials. Consequently, the proposed spatial designs will provide investigators with more options to minimize cost of the trials by requiring fewer patients and shorter expected study duration. To facilitate implementation of the proposed spatial two-stage designs, we created an R/shiny package ‘spatial2stage’ and its R code is freely available at http://cansur.wayne.edu.

The spatial designs require generally two steps to find desirable designs regardless of the number of target responses: the first step is to find a set of feasible solutions and the second stop is to find spatial designs by applying spatial criteria over a set of feasible solutions. The second step is straightforward and not computationally expensive because the size of selected feasible sets is small enough to manage and it requires just simple calculations. However, the first step requires a grid search with relatively expensive computation because the solution domains are not differential and there is no analytical solution. In particular, the spatial designs with one target response will take just from several seconds to several minutes to find designs according to design configuration. However, for the case when there are two target responses, because of many design parameters and so high-dimensional domain, a grid search requires much larger computation time from several hours to one day according to design configuration.

We can extend the proposed spatial designs to the case when there are three target responses using the constraints introduced by Kim and Wong.⁷ The key component of this extension is to obtain a set of feasible solution for the case when there are three target responses. However, as shown by Kim and Wong,⁷ due to no analytical solution, a grid search should be used to find a set of feasible solution, which is computationally too expensive to achieve because of high-dimensional search domain. For this reason, to circumvent the high computational expense, Kim and Wong⁷ proposed a population-based stochastic method to find a design without finding a set of feasible solution. Thus, while the extension will be theoretically possible, it will be difficult numerically. One computationally feasible method will be to use a population-based stochastic method similar to Kim and Wong.⁷ By doing so, we could find a pseudo-set of feasible solution that is composed of a subset of a set of entire feasible solutions.

We conclude with a brief discussion on relationships between spatial designs and the graphical approach proposed by Jung et al.⁴ that searches for a suboptimal Simon’s two-stage design that requires a sample size that is a compromise among the total sample sizes required by the optimal design, the minimax design and the expected null sample sizes. Specifically, the total sample size of this suboptimal design is between those required for the minimax and optimal designs but its expected null sample size is comparable to that of the optimal design. To find the suboptimal design, they proposed a graphical approach using a scatter plot between n and N₀. However, there was no analytical criterion and the resulting solution can be subjective. The optimal design can be either Simon’s designs or a locally optimal design depending on the maximum total sample size allowed by the user. Interestingly, the suboptimal solutions of Examples 2 and 3 in Jung et al. can be derived from the spatial designs. Table S1 and Figure S3 in the Supplementary material show that the optimal solution of Example 2 in Jung et al. is exactly the same as the spatial designs found under M1-M4 and H1-H2, and the design found for Example 3 in Jung et al. is identical to the spatial designs found under L2, M1-M5 and H1-H2. This demonstrates that, when the pre-specified maximum sample size is larger or equal to the true maximum sample size, Jung et al.’s proposed designs can be found as spatial designs under our framework.

The proposed spatial designs are multifaceted and highly depends on the design configuration. Thus, we recommend a practitioner to investigate all possible spatial designs before making a decision.