Optimal designs in three-level cluster randomized trials with a binary outcome

Abstract

Cluster randomized trials (CRTs) were originally proposed for use when randomization at the subject level is practically infeasible or may lead to a severe estimation bias of the treatment effect. However, recruiting an additional cluster costs more than enrolling an additional subject in an individually randomized trial. Under budget constraints, researchers have proposed the optimal sample sizes in two-level CRTs. CRTs may have a three-level structure, in which two levels of clustering should be considered. In this paper, we propose optimal designs in three-level CRTs with a binary outcome, assuming nested exchangeable correlation structure in generalized estimating equation models. We provide the variance of estimators of three commonly used measures: risk difference, risk ratio, and odds ratio. For a given sampling budget, we discuss how many clusters and how many subjects per cluster are necessary to minimize the variance of each measure estimator. For known association parameters, the locally optimal design (LOD) is proposed. When association parameters are unknown but within pre-determined ranges, the MaxiMin design (MMD) is proposed to maximize the minimum of relative efficiency over the possible ranges, that is, to minimize the risk of the worst scenario.

1. Introduction

Cluster randomized trials (CRTs) were originally proposed for use when randomization at the subject level is practically infeasible or may possibly lead to a severe estimation bias of the treatment effect. In practice, implementation strategies play important roles in dissemination and implementation (D&I) research. Compared to subject-level randomized trials, CRT designs have appealing features for implementation science in public health and clinical medicine. Further, CRTs are greatly needed for effectiveness research from science to practice.^1,2 Therefore, there has been growing interest in the design of CRTs.^3–9 The unit of randomization might be hospitals, clinics, classrooms, etc. Subjects within a cluster are exposed to common factors and tend to share similar characteristics. The degree of such similarity is commonly quantified by the intracluster correlation coefficient (ICC). Recruiting an additional cluster costs more than enrolling an additional subject in an individually randomized trial; thus, researchers have proposed the optimal sample size as a function of sampling costs and the ICC in CRTs.^10–16 “Optimal” means the maximum power and precision for a given sampling budget, or the minimum sampling cost for a given power and precision. These approaches show that the optimal sample size depends strongly on the ICC. However, the ICC is usually unknown in CRTs. To overcome this shortcoming, Van Breukelen et al. considered a range of possible ICC values and presented MaxiMin designs (MMDs) based on relative efficiency (RE) under budget constraints.¹⁷ Wu et al. proposed the optimal group allocations for three measures (RD, RR, OR) in two-level CRTs with binary outcomes through the variances of the maximum likelihood estimators. ¹⁸

CRTs may have three-level structures. For example, subjects in a two-level CRT are measured at different time points. Measurements across the different time points are correlated within a subject, while subjects are correlated within a cluster. Another example is that interventions are randomly assigned to medical centers (“practices”), and health care professionals (“providers”) within the same practice are trained with the assigned intervention to provide care to participants. Participants could be correlated within a provider, while providers could be correlated within a practice. Hereafter we use a CRT with practice, provider, and participant levels as the three-level example. For simplicity, we consider the same provider size (number of participants from each provider) and equal practice sizes (number of providers per practice).

Generalized estimating equations (GEEs) proposed by Liang and Zeger¹⁹ have been commonly applied to analyze the correlated data in CRTs.^20–24 Liang and Zeger showed that the GEE approach still gives consistent estimates of the regression coefficients- provided that the marginal model is correctly specified- even if the working correlation matrix is incorrectly assumed.¹⁹ In this paper, we aim to propose an optimal design (OD) in three-level CRTs, in which “optimal” refers to the minimization of the variance of each measure estimator for a given sampling budget. We assume the nested exchangeable correlation structure²⁵ throughout and utilize the GEE models in a three-level CRT with a binary outcome. The correlation structure includes correlation among participants within the same provider in the same practice, r, and correlation among participants with different providers in the same practice, ρ. Both r and ρ are assumed to be constant across all practices. Three different link functions in GEE models are considered, e.g. identity, log, and logit, where the corresponding regression coefficients are related to risk difference (RD), risk ratio (RR), and odds ratio (OR), respectively.

For known association parameters (r,ρ), we discuss how many practices m need to be enrolled and how many providers per practice n are sufficient to minimize the variance of each measure estimator under the budget constraints when the provider size K is a pre-determined value and K is not a fixed value but within a range (K_min,K_max), respectively. This is a locally optimal design (LOD) with corresponding numbers n_LOD and m_LOD. When the association parameters (r,ρ) are unknown, but we assume that ranges for r and ρ can be obtained from other literature and similar studies, we propose MMDs in the framework of relative efficiency (RE) to minimize the risk of the worst scenario. RE is defined as the ratio of the variance of each measure estimator for practice size n_LOD to n,¹⁷ which is a function of n,r, and ρ. Our goal is to maximize the minimum of RE over the possible ranges of r and ρ.

The organization of this article is as follows. In Section 2, we briefly summarize the GEE method developed by Liang and Zeger in three-level CRTs,¹⁹ introduce the “nested exchangeable” correlation structure,²⁵ and derive the variance of the estimator of the treatment for a binary outcome in a two-group comparison. Section 3 presents the LOD for known parameter values under the assumption of a “nested exchangeable” correlation structure. In Section 4, we define the RE and propose MMDs for unknown parameter values of r and ρ. We provide guidance on applying the methods and illustrate using a real CRT, followed by a discussion about the limitations of the proposed approach and directions for future research.

2. Statistical GEE models in three-level CRTs

Let Y_ijk be a response from participant k = 1,⋯,K, for provider j = 1,⋯,n_i in practice i = 1,⋯,m. Let X_ijk = (X_ijk1,⋯,X_ijkp)′ be a covariate vector and μ_ijk = E(Y_ijk|X_ijk) be a marginal mean response given X_ijk. The marginal model is

$$g\left({\mu }_{ijk}\right)={\mathit{X}}_{ijk}^{\mathbf{\text{'}}}\mathit{\beta }.$$

Let Y_ij = (Y_ij1,⋯,Y_ijK), μ_ij = (μ_ij1, ⋯, μ_ijK), and X_ij = (X_ij1,⋯,X_ijK) be the 1 × K response vector, 1 × K marginal mean response vector, p × K covariate matrix of provider j in practice i, respectively. Let ${\mathit{Y}}_i=({\mathit{Y}}_{i1},\cdots ,{\mathit{Y}}_{i{\mathrm{n}}_i}{)}^{\mathrm{\text{'}}},{\mathit{\mu }}_i=({\mathit{\mu }}_{i1},\cdots ,{\mathit{\mu }}_{i{\mathrm{n}}_i}{)}^{\mathrm{\text{'}}},$ and ${\mathit{X}}_i=\left({\mathit{X}}_{i1},\cdots ,{\mathit{X}}_{i{n}_i}\right)$ be the matrices of responses, marginal mean responses and covariate of the providers in practice i, respectively. The mean of Y_i is denoted by μ_i = E(Y_i) and the variance of Y_i is $\mathrm{v}\mathrm{a}\mathrm{r}\left({\mathit{Y}}_i\right|{\mathit{X}}_i)=\theta {\mathit{A}}_i^{1/2}{\mathit{R}}_{i0}\left({\mathit{\omega }}_0\right){\mathit{A}}_i^{1/2},$ where ${\mathit{A}}_i=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{\gamma \left({\mu }_{i11}\right),\cdots ,\gamma \left({\mu }_{i1K}\right),\cdots ,\gamma \left({\mu }_{i{n}_{i}1}\right),\cdots ,\gamma \left({\mu }_{i{n}_{i}K}\right)\right\},$ and a Kn_i × Kn_i correlation matrix R_i0(ω₀) describes the correlation of measures within the ith practice with a vector of association parameters denoted by ω₀. Both γ and θ are dependent on the distribution of responses. If Y_ijk is binary, γ(μ_ijk) = μ_ijk(1 − μ_ijk) and θ = 1. Liang and Zeger¹⁹ showed that $\sqrt{m}(\widehat{\mathit{\beta }}-\mathit{\beta })$ is asymptotically multivariate normal with a covariance matrix ${\mathit{V}}_{\mathit{R}}=\underset{m\to \mathrm{\infty }}{\lim }m\left({\mathbf{\Sigma }}_1^{-1}{\mathbf{\Sigma }}_0{\mathbf{\Sigma }}_1^{-1}\right),$ where ${\mathbf{\Sigma }}_1=\sum _{i=1}^m{\mathit{D}}_i^{\mathrm{\text{'}}}{\mathbf{V}}_i^{-1}{\mathit{D}}_i,$ ${\mathbf{\Sigma }}_0=\sum _{i=1}^m{\mathit{D}}_i^{\mathrm{\text{'}}}{\mathbf{V}}_i^{-1}\mathrm{c}\mathrm{o}\mathrm{v}\left({\mathit{Y}}_{\mathit{i}}|{\mathit{X}}_i\right){\mathbf{V}}_i^{-1}{\mathit{D}}_i$, D_i = ∂μ_i/∂β′, and V_i is a working covariance matrix of Y_i. Let R_iw(ω) be a Kn_i × Kn_i working correlation matrix with a vector of association parameters ω. The working covariance matrix is expressed as ${\mathit{V}}_i=\theta {\mathit{A}}_i^{1/2}{\mathit{R}}_{iw}\left(\mathit{\omega }\right){\mathit{A}}_i^{1/2}$ and is unequal to var (Y_i|X_i) unless R_iw(ω) = R_i0(ω₀).

For three-level data, Teerenstra et al. proposed a “nested exchangeable” correlation structure ²⁵:

1. Correlation among participants within the same provider in the same practice, is constant, $\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\left(Y_{ij{k}_1},Y_{ij{k}_2}\right)=r$ for k₁ ≠ k₂;

2. Correlation among participants with different providers in the same practice, is constant, $\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\left(Y_{i{j}_1{k}_1},Y_{i{j}_2{k}_2}\right)=\rho$ for j₁ ≠ j₂, and any k₁, k₂;

This three-level exchangeable working correlation structure was defined as

$${\mathit{R}}_{iw}\left(r,\mathrm{\rho }\right)=\rho 1_{{Kn}_i\times {Kn}_i}+\left(r-\rho \right)\mathrm{B}\mathrm{d}\mathrm{i}\mathrm{a}{\mathrm{g}}_{n_i}\left(1_{K\times K}\right)+(1-r){\mathit{I}}_{{Kn}_i\times {Kn}_i},$$

where 1_i×i is a i × i matrix of 1’s, Bdiag_i(A) is a block diagonal matrix with matrix element A replicated i times, and I_i×i is the i × i identity matrix. Here, R_iw(r,ρ) must be positive definite (PD). Given a value of K and n_i, PD can be determined if the constraints holds, min(λ₁,λ₂,λ_3i) > 0, where λ₁ = 1 − r, λ₂ = 1 + (K − 1)r − Kρ, λ_3i = 1 + (K − 1)r + K(n_i − 1)ρ are the distinct eigenvalues of R_iw(r,ρ). The proof was provided by Web Appendix A of Li et al.²⁶ Here, the constraints are equivalent to

$$-\frac{1}{K-1}<r<1,-\frac{1+\left(K-1\right)r}{K\left(n_i-1\right)}<\rho <\frac{1+\left(K-1\right)r}{K}.$$ (1)

We assume this “nested exchangeable” correlation structure in the following sections.

Suppose we are interested in testing the treatment effect for a two-group comparison: the treated vs. control. The treatment assignment is coded in the last column of the practice covariate matrices ${\mathit{X}}_i^{\mathit{\text{'}}}$ and the corresponding last parameter of β is β_p. Let V_β denote the (p, p)th element of V_R. Thus, $\sqrt{m}(\widehat{{\beta }_p}-{\beta }_p)$ has an asymptotically normal distribution N(0,V_β), or equivalently, $\mathrm{V}\mathrm{a}\mathrm{r}\left(\widehat{{\beta }_p}\right)=V_{\beta }/m.$ For simplicity, we take p = 2, i.e. coefficients β₁ is the intercept and β₂ is the treatment effect. The practice allocations of the treatment and control groups are, m_trt = mπ and m_cont = m(1 − π), respectively, where π is a pre-determined value, e.g, 50%. The hypotheses of interest are H₀: β₂ = 0 versus H₁: β₂ = β. For a binary outcome, let p₀ and p₁ be the success rates in the control and treated group. When the identity link function, g(μ_ijk) = μ_ijk, is specified, β₂ = p₁ − p₀ is the risk difference (RD) between two groups; when the log link function, g(μ_ijk) = ln(μ_ijk), is specified, β₂ = ln(p₁/p₀) is the difference between the natural logarithms of the proportions; and when the logit link function, $g\left({\mu }_{ijk}\right)=\mathrm{l}\mathrm{n}\left({\mu }_{ijk}/\left(1-{\mu }_{ijk}\right)\right),$ is specified, ${\beta }_2=\mathrm{l}\mathrm{n}\left(\frac{p_1/(1-p_1)}{p_0/(1-p_0)}\right)$ is the difference between the natural logarithms of the odds. When the log and logit link functions are used, taking the exponential of β₂ refers to the risk ratio (RR) and the odds ratio (OR), respectively.

Given the “nested exchangeable” correlation structure, we use identity link function and have

$$\mathrm{V}\mathrm{a}\mathrm{r}\left(\widehat{{\beta }_2}\right)=\frac{{\lambda }_3}{Knm}\left(\frac{p_1\left(1-p_1\right)}{\pi }+\frac{p_0\left(1-p_0\right)}{1-\pi }\right),$$ (2)

where n_i ≡ n and the eigenvalue λ₃ = 1 + (K − 1)r + K(n − 1)ρ. If we consider the log link function, then

$$\mathrm{V}\mathrm{a}\mathrm{r}\left(\widehat{{\beta }_2}\right)=\frac{{\lambda }_3}{Knm}\left(\frac{1-p_1}{\pi p_1}+\frac{{1-p}_0}{\left(1-\pi \right)p_0}\right).$$ (3)

Using logit link function in the GEE model for a binary outcome, we have

$$\mathrm{V}\mathrm{a}\mathrm{r}\left(\widehat{{\beta }_2}\right)=\frac{{\lambda }_3}{Knm}\left(\frac{1}{\pi p_1\left(1-p_1\right)}+\frac{1}{{(1-\pi )p}_0\left(1-p_0\right)}\right).$$ (4)

Please note that Equation (4) is the same as the formula in section 4.4²⁵ and reduces to Equation (8)²⁷ when K = 1.

From the relationship between β₂ and RD, the asymptotic variance of $\widehat{RD}$ is

$$\mathrm{V}\mathrm{a}\mathrm{r}\left(\widehat{RD}\right)=\frac{{\lambda }_3}{Knm}\left(\frac{p_1\left(1-p_1\right)}{\pi }+\frac{p_0\left(1-p_0\right)}{1-\pi }\right).$$ (5)

Applying the delta method, we obtain the asymptotic variances of $\widehat{RR}$ and $\widehat{OR}$ as

$$\mathrm{V}\mathrm{a}\mathrm{r}\left(\widehat{RR}\right)=\frac{{\lambda }_3}{Knm}\left(\frac{1-p_1}{\pi p_1}+\frac{{1-p}_0}{p_0\left(1-\pi \right)}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{2p_1}{p_0}\right),$$ (6)

and

$$\mathrm{V}\mathrm{a}\mathrm{r}\left(\widehat{OR}\right)=\frac{{\lambda }_3}{Knm}\left(\frac{1}{\pi p_1\left(1-p_1\right)}+\frac{1}{{(1-\pi )p}_0\left(1-p_0\right)}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{{2p}_1/(1-p_1)}{p_0/(1-p_0)}\right).$$ (7)

3. Local optimal design

Assume the study cost per practice is c currency units (e.g. $US), and each provider costs s currency units, e denotes each participant’s cost. The total budget B in a three-level trial is defined as

$$B=m\left(c+sn+eKn\right).$$ (8)

We aim to find the optimal design (OD) given the constraint in Equation (8). The term “optimal” refers to the variance of each measure estimator being minimized for a given sampling budget. ^10,17,28,29

First, we assume that provider size K is a pre-determined value, same as B, c, s and e, for simplicity. The goal is to find the pair of m and n that minimizes the variance of each measure, which is equivalent to maximizing

$$L=\frac{Knm}{{\lambda }_3}$$ (9)

for all three measures (RD, RR, OR). Substituting $m=\frac{B}{c+\left(s+eK\right)n}$ gives

$$L=\frac{KnB}{{\lambda }_{2}c+\left({\lambda }_{2}b+K\rho c\right)n+{Kb\rho n}^2},$$

where b = s + eK. Taking the partial derivatives with respect to n gives

$$\frac{\partial L}{\partial n}\propto {\lambda }_{2}c-K{b\rho n}^2.$$

Since R_iw(r,ρ) is PD, λ₂ is positive. It can be shown that when

$$n=\sqrt{\frac{{\lambda }_{2}c}{Kb\rho }},$$ (10)

where ρ should be positive, the derivatives equals 0 and L is maximized. The local optimal design (LOD) is reached for a known pair value (r,ρ) and n in equation (9) is denoted by n_LOD. Let $\vartheta =\frac{{\lambda }_2}{K\rho }$, the parameters in LOD are given by

$$n_{\mathrm{L}\mathrm{O}\mathrm{D}}=\sqrt{\frac{\vartheta c}{b}},m_{\mathrm{L}\mathrm{O}\mathrm{D}}=\frac{B}{\sqrt{\vartheta bc}+c}.$$ (11)

Please note $\rho <\frac{\left[1+\left(K-1\right)r\right]c}{K\left(c+s+eK\right)}$ in order to be n_LOD > 1. Thus, $0<\rho <\min \left(\frac{1+\left(K-1\right)r}{K},\frac{\left[1+\left(K-1\right)r\right]}{K}\frac{c}{\left(c+s+eK\right)}\right)=\frac{\left[1+\left(K-1\right)r\right]c}{K\left(c+s+eK\right)}$ since Equation (1) also holds. For any measures (RD, RR, OR), the local optimal design is the same even if the variance of measure estimator is different. Obviously n_LOD and m_LOD may be non-integer. In reality we need to choose an integer value for practice size with either n_up = int (n_LOD) + 1 or n_down = int (n_LOD), where “int” refers to an integer part of a number. We then calculate m_up and m_down from $m=\frac{B}{c+\left(s+eK\right)n}.$ Similarly, m_up and m_down are most likely non-integers. In order to meet the limit of budget, the integer parts for m_up and m_down are taken as the values of corresponding number of practices. Then we can calculate the corresponding L using Equation (9) and the proposed optimal practice size and number of practices is the one with the larger L.

Second, when the provider size K is not a fixed value but within a range (K_min,K_max) and K_min ≥ 2, we find n_LOD and m_LOD for each value of K within this range and calculate the corresponding L in Equation (9). The design with the maximum of L within a range (K_min,K_max) is defined as LOD. Given

$$n_{\mathrm{L}\mathrm{O}\mathrm{D}}{m}_{\mathrm{L}\mathrm{O}\mathrm{D}}=\frac{B}{\sqrt{b}\left(\sqrt{b}+\sqrt{\frac{c}{\vartheta }}\right)},$$

it is easy to show that both Kn_LODm_LOD and λ₃ are increasing functions of K but $K{n}_{\mathrm{L}\mathrm{O}\mathrm{D}}{m}_{\mathrm{L}\mathrm{O}\mathrm{D}}\propto \sqrt{K}$ and λ₃ ∝ K when K ≥ 3. That is, L decreases when K increases for K ≥ 3. Therefore, the LOD is reached at K = K_min if K_min ≥ 3 and K = 3 if K_min = 2 for a known pair value (r,ρ).

Table 1 shows an example to determine LOD for r = 0.6 and ρ = 0.03, where 3 ≤ K ≤ 10, B = 300,000, c = 10,000, s = 100 and e = 10 are assumed. For each K, n and m are calculated from Equation (11). Both integers are chosen as discussed previously and the corresponding L is calculated from Equation (9). The design with K = 3, n = 43, and m = 18 is LOD since L is maximized at K = 3. Please note L is not monotone decreasing in Table 1 since the calculations are provided for (n,m) as integers only. The power estimates for RD, RR and OR are provided for p₀ = 0.3 and p₁ = 0.45. It definitely demonstrates that the power is maximized, equivalently the variance is minimized, when LOD is reached.

Table 1.

Local optimal design for B = 300000, c = 10000, s = 100 and e = 10 with known correlations r = 0.6 and ρ = 0.03

K	Practice size n	Number of practices m	L	Flag	Power [1]	Power [2]	Power [3]
3	43	18	388.3	1	0.871	0.850	0.859
4	40	18	385.0		0.868	0.847	0.856
5	39	18	385.7		0.869	0.848	0.857
6	37	18	381.3		0.865	0.844	0.853
7	36	18	379.6		0.863	0.842	0.851
8	34	18	373.2		0.858	0.836	0.845
9	33	18	370.2		0.855	0.833	0.843
10	32	18	366.9		0.852	0.830	0.839

n and m are calculated from Equation (11).

L is calculated from Equation (9).

Flag=1 refers to LOD.

^[1]RD for p₀ = 0.3 and p₁ = 0.45.

^[2]RR for p₀ = 0.3 and p₁ = 0.45.

^[3]OR for p₀ = 0.3 and p₁ = 0.45.

4. MaxiMin optimal design

First, we still assume that the provider size K is a pre-determined value. Obviously n_LOD in the Equation (11) depends on (r,ρ). In practice, the pair value of (r,ρ) could be unknown before a study starts. If the ranges, (r_min, r_max) and (ρ_min, ρ_max), can be obtained from previous studies or other literature, then we define them as the parameter space.^30,31 The range of practice size based on the practical feasibility, (n_min, n_max), is defined as the design space.^17,32,33 The objective is to identify OD within the parameter and design spaces.

Inserting (11) in (5)–(7) gives the variance of each measure estimator for the optimal design. For example,

$$\mathrm{V}\mathrm{a}\mathrm{r}\left(\widehat{RD}\right)=g(r,\mathrm{\rho })\times \frac{1}{B}\left(\frac{p_1\left(1-p_1\right)}{\pi }+\frac{p_0\left(1-p_0\right)}{1-\pi }\right),$$ (12)

where $g\left(r,\mathrm{\rho }\right)={\left(\sqrt{\rho c}+\sqrt{\frac{1+\left(K-1\right)r-K\rho }{K}\left(s+eK\right)}\right)}^2.$

Following the same definition of RE,¹⁷ the ratio of the variance of each measure estimator for practice size n_LOD to n, we use Equations (5), (8) and (12) and then define RE for measure RD as a function of n, r, and ρ,

$$\mathrm{R}\mathrm{E}\left(n,r,\mathrm{\rho }\right)=\frac{g(r,\mathrm{\rho })}{1+\left(K-1\right)r+K\left(n-1\right)\rho }\times \frac{Kn}{c+\left(s+eK\right)n}.$$ (13)

It is easy to show that REs for both RR and OR measures are the same as Equation (13). Further, the maximal value of RE(n,r,ρ) is 1 and reached when n is n_LOD. Figure 1 shows how RE changes across practice size n for a fixed r, and Figure 2 shows the trend of RE over practice size n for a fixed ρ, where K = 3, B = 300,000, c = 10,000, s = 100 and e = 10. Both figures demonstrate that RE increases until it reaches 1 and then decreases as practice size n increases. Among four REs with the different values of ρ in Figure 1, we observe that the practice size n at which RE equals 1 is the smallest when ρ = 0.7 and is the largest when ρ = 0.1. Similarly, we notice that the practice size n at which RE equals 1 is the smallest when r = 0.1 and is the largest when r = 0.7 in Figure 2.

Figure 1.

Relative efficiencies RE(n,r,ρ) as a function of n for K = 3, B = 300000, c = 10000, s = 100 and e = 10 with r = 0.8

Figure 2.

Relative efficiencies RE(n,r,ρ) as a function of n for K = 3, B = 300000, c = 10000, s = 100 and e = 10 with ρ = 0.05

MaxiMin design (MMD) is a design that maximizes some measure of performance (or minimize the risk) in the worst case scenario. ^31–34 Here, we use RE, quantified as Equation (13), as the measure of performance. Specifically, the MMD includes three steps. Step 1 defines the parameter and design spaces; Step 2 computes LOD for each pair value of (r,ρ) in the parameter space, and then computes the RE of each design in the design space; Step 3 finds its smallest RE value within the parameter space for each design in the design space and selects the design which maximizes the minimum RE among all designs in the design space. This MMD considers the worst case scenario and thus is robust against misspecification of the values of (r,ρ).

RE of any of the three measures, shown in Equation (13), is a function of n, r, and ρ given the costs c per practice, s per provider, e per participant, and the provider size K. First, Appendix 1 proves that RE(n, r, ρ) is minimized at one of the four points: (r_min, ρ_min), (r_min, ρ_max), (r_max, ρ_min), (r_max, ρ_max), i.e., the boundary of the parameter space (r_min, r_max) and (ρ_min, ρ_max). Figure 3 presents RE(n, r_min, ρ_min), RE(n, r_min, ρ_max), RE(n, r_max, ρ_min) and RE(n, r_max, ρ_max) as functions of n for K = 3, B = 300,000, c = 10,000, s = 100 and e = 10 with parameter space (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.01, ρ_max = 0.05). Next, Appendix 2 shows that the minimum of RE(n,r,ρ) is maximized by the design satisfying RE(n, r_min, ρ_max) = RE(n, r_max, ρ_min). Let $\widehat{n}$ be a solution of RE(n, r_min, ρ_max) = RE(n, r_max, ρ_min), and expressed as

$$\frac{\left[1+\left(K-1\right)r_{\mathrm{m}\mathrm{a}\mathrm{x}}-K{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right]g\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)-\left[1+\left(K-1\right)r_{\mathrm{m}\mathrm{i}\mathrm{n}}-K{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]g(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}})}{K\left({\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}g\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)-{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}g\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)\right)}.$$ (14)

As shown in Figure 3, the black vertical straight line indicates $\widehat{n}$ and locally optimal designs for (r_min, ρ_max) and (r_max, ρ_min) are added as references. By dividing g(r_min, ρ_max) and g(r_max, ρ_min) by the study cost per practice c, we notice that MMD of practice sizes depends on (r_min, ρ_max), (r_max, ρ_min), and ratio (s + eK)/c. That is, the total budget B determines the number of practices m but not practice size n.

Figure 3.

Relative efficiencies RE(n, r_min, ρ_min), RE(n, r_min, ρ_max), RE(n, r_max, ρ_min), and RE(n, r_max, ρ_max) as a function of n and locally optimal designs LOD (r_min, ρ_max) and LOD (r_max, ρ_min) for K = 3, B = 300000, c = 10000, s = 100 and e = 10 with parameter space (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.01, ρ_max = 0.05)

Now we provide a step by step approach to find an MMD for a two-arm three-level CRT with a binary outcome when the provider size K is a pre-determined value.

Step 1: Define the parameter space (r_min, r_max), (ρ_min, ρ_max) and design space (n_min, n_max), respectively.

Step 2: Calculate $\widehat{n}$ using Equation (14).

1. If it is within the range (n_min,n_max), then set $n_{\mathrm{M}\mathrm{M}\mathrm{D}}=\widehat{n}$ and the corresponding $m_{\mathrm{M}\mathrm{M}\mathrm{D}}=\mathrm{i}\mathrm{n}\mathrm{t}\left(\frac{B}{c+\left(s+eK\right)n_{\mathrm{M}\mathrm{M}\mathrm{D}}}\right).$

2. If it is outside of (n_min,n_max), calculate RE(n, r_min, ρ_min), RE(n, r_min, ρ_max), RE(n, r_max, ρ_min), and RE(n, r_max, ρ_max) for each practice size n ∈ (n_min, n_max) and take their minimum. Choose the design of (n,m) that has the maximum of minimum RE within design space, where $m=\mathrm{i}\mathrm{n}\mathrm{t}\left(\frac{B}{c+\left(s+eK\right)n}\right).$

Again, $\widehat{n}$ may be non-integer. We use the same method in Section 3 to get the integer practice size and number of practices. Please note that Equation (13) is derived using Equation (8) as well. If the calculated m_MMD from the above approach is infeasible, then the range (n_min,n_max) needs to be revised appropriately.

Table 2 shows an example to determine MMD, where the same setting as Figure 3 is assumed. We obtain $\widehat{n}=46.6$ using Equation (13). If the design space is (11, 20), then the design of (n = 20, m = 23) is MMD under the budget constraints; on the other hand, if the design space is (41, 50), then the design of (n = 47, m = 18) is MMD under the budget constraints.

Table 2.

Maximin design for K = 3, B = 300000, c = 10000, s = 100 and e = 10 with parameter space (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.01, ρ_max = 0.05)

Design space (nmin, nmax)	Practice size n	RE(n, rmin, ρmin)	RE(n, rmin, ρmax)	RE(n, rmax, ρmin)	RE(n, rmax, ρmax)	Min RE	Number of practices m	Flag
(11, 20)	11	0.5642	0.9059	0.4090	0.7346	0.4090	26
	12	0.5966	0.9257	0.4369	0.8656	0.4369	25
	13	0.6288	0.9421	0.4636	0.7935	0.4636	25
	14	0.6550	0.9556	0.4892	0.8184	0.4892	25
	15	0.6813	0.9667	0.5136	0.8408	0.5136	25
	16	0.7059	0.9757	0.5369	0.8609	0.5369	24
	17	0.7287	0.9829	0.5592	0.8788	0.5592	24
	18	0.7501	0.9886	0.5806	0.8949	0.5806	24
	19	0.7700	0.9929	0.6010	0.9093	0.6010	24
	20	0.7886	0.9961	0.6205	0.9221	0.6205	23	1
(41, 50)	41	0.9799	0.9441	0.8809	0.9975	0.8809	19
	42	0.9831	0.9394	0.8881	0.9963	0.8881	19
	43	0.9859	0.9347	0.8950	0.9949	0.8950	19
	44	0.9884	0.9299	0.9016	0.9932	0.9016	19
	45	0.9907	0.9251	0.9079	0.9913	0.9079	18
	46	0.9926	0.9202	0.9138	0.9893	0.9138	18
	47	0.9943	0.9154	0.9195	0.9872	0.9154	18	1
	48	0.9958	0.9105	0.9249	0.9849	0.9105	18
	49	0.9970	0.9056	0.9301	0.9825	0.9056	18
	50	0.9980	0.9008	0.9350	0.9799	0.9008	18

RE is calculated from Equation (13).

Flag=1 refers to MMD.

Second, when the provider size K is not a fixed value but within a range (K_min,K_max) and K_min ≥ 2, we find n_MMD and m_MMD for each value of K within this range and calculate the corresponding RE in Equation (13). The design with the maximum of RE is defined as MMD.

Table 3 demonstrates how to find MMD with parameter space (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.01, ρ_max = 0.05) where 3 ≤ K ≤ 10, B = 300,000, c = 10,000, s = 100 and e = 10 are assumed. For each K, n_MMD and m_MMD are calculated from the step by step approach and the corresponding RE is provided. If the design space is (11, 20), then the design of (K = 10, n = 20, m = 21) is MMD under the budget constraints; on the other hand, if the design space is (41, 50), then the design of (K = 3, n = 47, m = 18) is MMD under the budget constraints. SAS macros %OD_3Level_FixedK and %OD_3Level_RangeK are developed to find LOD and MMD when the corresponding parameters are provided.

Table 3.

MaxiMin design for B = 300000, c = 10000, s = 100 and e = 10 with parameter space (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.01, ρ_max = 0.05)

Design space (nmin, nmax)	K	Practice size n	Number of practices m	RE	Flag
(11, 20)	3	20	23	0.6205
	4	20	23	0.6369
	5	20	23	0.6517
	6	20	22	0.6653
	7	20	22	0.6781
	8	20	22	0.6901
	9	20	21	0.7014
	10	20	21	0.7121	1
(41, 50)	3	47	18	0.9154	1
	4	43	18	0.9032
	5	41	18	0.8876
	6	41	18	0.8638
	7	41	17	0.8421
	8	41	17	0.8222
	9	41	16	0.8037
	10	41	16	0.7866

n and m are calculated from step by step approach.

RE is calculated from Equation (13).

Flag=1 refers to MMD.

Last, we conduct a sensitivity analysis about the parameter space. The following eight different parameter spaces are considered: (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.01, ρ_max = 0.05), (r_min = 0.1, r_max = 0.3) and (ρ_min = 0.01, ρ_max = 0.05), (r_min = 0.3, r_max = 0.6) and (ρ_min = 0.01, ρ_max = 0.05), (r_min = 0.6, r_max = 0.9) and (ρ_min = 0.01, ρ_max = 0.05), (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.01, ρ_max = 0.02), (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.02, ρ_max = 0.03), (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.02, ρ_max = 0.05), (r_min = 0.1, r_max = 0.9) and (ρ_min = 0.03, ρ_max = 0.05). We still assume 3 ≤ K ≤ 10, B = 300,000, c = 10,000, s = 100 and e = 10. Table 4 shows the MMDs for two different design space (2, 20) and (2, 50). If n_max is relatively small, e.g. < $\widehat{n},$ MMDs are the same (K = 10, n = 20, m = 21). They might be different otherwise. That is, MMDs are insensitive to the parameter space when the maximum of practice size is relatively small.

Table 4.

MaxiMin design for B = 300000, c = 10000, s = 100 and e = 10 with 3 ≤ K ≤ 10

Design space (nmin, nmax)	rmin	rmax	ρmin	ρmax	K	Practice size n	Number of practices m	RE
(2, 20)	0.1	0.9	0.01	0.05	10	20	21	0.7121
	0.1	0.3	0.01	0.05	10	20	21	0.8717
	0.3	0.6	0.01	0.05	10	20	21	0.7754
	0.6	0.9	0.01	0.05	10	20	21	0.7121
	0.1	0.9	0.01	0.02	10	20	21	0.7121
	0.1	0.9	0.02	0.03	10	20	21	0.8365
	0.1	0.9	0.02	0.05	10	20	21	0.8365
	0.1	0.9	0.03	0.05	10	20	21	0.9031
(2, 50)	0.1	0.9	0.01	0.05	3	47	18	0.9154
	0.1	0.3	0.01	0.05	3	41	19	0.9441
	0.3	0.6	0.01	0.05	3	47	18	0.9446
	0.6	0.9	0.01	0.05	4	50	17	0.9446
	0.1	0.9	0.01	0.02	5	49	17	0.9466
	0.1	0.9	0.02	0.03	5	43	19	0.9751
	0.1	0.9	0.02	0.05	3	41	19	0.9441
	0.1	0.9	0.03	0.05	3	41	19	0.9441

5. Example

Teerenstra et al. discussed the Helping Hands trial (Netherlands Organization for Health Research and Development ZonMw, grant number 80–007028-98–07101).²⁵ This study aimed to change nurse behavior through two strategies and randomized the wards to either strategy. The two strategies included the state-of-the-art strategy, which is derived from literature regarding education, reminders, feedback, and targeting adequate products and facilities; and the extended strategy, which contains all elements of the state-of-the-art strategy plus activities aimed at influencing social influence in groups and enhancing leadership. The primary endpoint was adherence to hygiene guidelines (Yes vs. No) and multiple evaluations of nurses’ guideline adherence were observed. The researchers expected to improve the adherence from 60% in the state-of-the-art strategy to 70% in the extended strategy. Teerenstra et al. considered the constant behavior of nurse r = 0.6 and intra-ward coefficient correlation ρ = 0.03.²⁵ We calculated the total number of wards m = 58 to obtain 80% power using the number of nurses per ward n = 15 and number of evaluations K = 3 under the same assumptions of (r, ρ)) using Equation (4). We assume c = 2,000, s = 50 and e = 10 in this study, then the total cost 58*(2000 + 50 * 15 + 10 * 3 * 15) = 185,600 will be needed.

We now apply LOD and MMD approaches to redesign this study with the same budget B = 185,600. We consider 3 ≤ K ≤ 6 and find that LOD is K = 3, n = 25 and m = 46. The power is 83.7% under this scenario. It is worth mentioning that our proposed method does not guarantee obtaining the desired power, e.g., 80%, but to have the highest power under the budget constraints. Researchers should increase the budget if our proposed method does not reach the desired power.

On the other hand, if the researchers have no clear pictures of these two associations, then the parameter space need to be specified. Campbell et al. showed the ICC interquartile range of implantation studies in secondary care from 0.017 to 0.221.³ As Teerenstra et al. mentioned, the behavior of an individual nurse with respect to hand hygiene is constant, the parameter space 0.5 ≤ r ≤ 0.9 is reasonably assumed. Now the parameter space lies within (r_min = 0.5, r_max = 0.9) and (ρ_min = 0.017, ρ_max = 0.221) and the design space is set as (3, 50), then number of evaluations K = 3, the number of nurses per ward n = 17 and the total number of wards m = 55 is our proposed MMD given the budget B = 185,600.

6. Discussion

In this paper, we presented optimal designs based on GEE models in three-level CRTs and proposed both LODs and MMDs under budget constraints. We employed a nested exchangeable correlation structure²⁵ and derived the variance of the treatment effect under the assumption of an equal practice size and the same provider size. We derived the locally optimal design when the correlation among participants within the same provider in the same practice, r, and correlation among participants with different providers in the same practice, ρ, are known; the optimal design aims to minimize the variance of each measure estimator for a given sampling budget. If the correlation pair (r,ρ) is unknown bulongitudinal data setting with AR(1)t lies in a known range, we proposed MMDs for three-level CRTs for a range of r and ρ. We also developed SAS macros to find the LOD and MMD for practical use.

Our method can be extended in several directions. First, our proposed approach is based on the nested exchangeable correlation structure only. It is suitable when the lowest level units are exchangeable within the middle level units (‘providers’) and the middle level units are exchangeable within the highest level units (‘practices’).²⁵ We will consider more sophisticated settings, e.g., a longitudinal data setting with AR(1) correlation among repeated measures over time in our future work. Second, we assume the same practice size and same provider size. If the practice size is different across the providers, the variances of estimator of treatment effects are more complicated than Equations (2)–(4). The derivation of these formulae warrants further research. Third, when GEE models with identity or log link are used to analyze correlated binary data, the convergence issues may occur since the predicted probability is unconstrained. Fourth, the empirical sandwich estimator of the covariance matrix obtained from GEE is biased for a small number of clusters and thus can inflate type I error rates. The proposed LOD and MMD based on the asymptotic variance might be worthy of further investigation. Finally, it merits further consideration to extend treatment groups to more than two.

Appendix

The proof consists of two steps. Appendix 1 shows that RE(n,r,ρ) is minimized at one of the four points: (r_min, ρ_min), (r_min, ρ_max), (r_max, ρ_min), (r_max, ρ_max), i.e., the boundary of the parameter space (r_min, r_max) and (ρ_min, ρ_max). Next, Appendix 2 shows that the minimum of RE(n,r,ρ) is maximized by the design satisfying RE(n, r_min, ρ_max) = RE(n, r_max, ρ_min).

Appendix 1

Proof that RE(n,r,ρ) is minimized at one of the four points: (r_min,ρ_min), (r_min, ρ_max), (r_max,ρ_min), (r_max, ρ_max) within the parameter space (r_min, r_max) and (ρ_min, ρ_max)

From Equations (5), (8) and (12), it follows that the RE for measure RD as a function of n, r, and ρ for any of the three measures.

$$\mathrm{R}\mathrm{E}\left(n,r,\mathrm{\rho }\right)=\frac{g(r,\mathrm{\rho })}{1+\left(K-1\right)r+K\left(n-1\right)\rho }\times \frac{Kn}{c+\left(s+eK\right)n}.$$

Take the partial derivative with respect to ρ gives

$$\frac{\partial \mathrm{R}\mathrm{E}\left(n,r,\mathrm{\rho }\right)}{\partial \mathrm{\rho }}\propto \left\{\left[\sqrt{\frac{c}{\rho }}-\sqrt{\frac{K\left(s+eK\right)}{1+\left(K-1\right)r-K\rho }}\right]\left[1+\left(K-1\right)r+K\left(n-1\right)\rho \right]-\left(\sqrt{\rho c}+\sqrt{\frac{1+\left(K-1\right)r-K\rho }{K}\left(s+eK\right)}\right)K\left(n-1\right)\right\}.$$

Setting the right hand to zero we obtain

$${\rho }_{\mathrm{*}}=\frac{c\left[1+\left(K-1\right)r\right]}{K\left[c+{\left(s+eK\right)n}^2\right]}.$$

Then take the partial derivative with respect to r gives

$$\frac{\partial \mathrm{R}\mathrm{E}\left(n,r,\mathrm{\rho }\right)}{\partial r}\propto \left[1+\left(K-1\right)r+K\left(n-1\right)\rho \right]\sqrt{\frac{s+eK}{K\left[1+\left(K-1\right)r-K\rho \right]}}-\left(\sqrt{\rho c}+\sqrt{\frac{1+\left(K-1\right)r-K\rho }{K}\left(s+eK\right)}\right).$$

Similarly, we set the right hand to zero and have

$$r_{\mathrm{*}}=\frac{K\rho \left[c+\left(s+eK\right)n^2\right]-c}{c\left(K-1\right)}.$$

Both are actually same as n_LOD in Equation (10). We can show that $\frac{\partial \mathrm{R}\mathrm{E}\left(n,r,\mathrm{\rho }\right)}{\partial \mathrm{\rho }}>0$ if ρ_min ≤ ρ < ρ_*, and $\frac{\partial \mathrm{R}\mathrm{E}\left(n,r,\mathrm{\rho }\right)}{\partial \mathrm{\rho }}<0$ if ρ_* < ρ ≤ ρ_max, so RE(n,r,ρ) is minimized at either ρ = ρ_min or ρ = ρ_max for a fixed r. If we assume the possible range for ρ is (0.1, 0.7) and r = 0.8, Appendix Figure 1 demonstrates 3-D RE plot as a function of n and ρ while Figure 1 shows RE plots for 4 paired values (r,ρ). As seen in Figure 1, RE(n,r,ρ) is minimized at ρ = 0.1 when n < 13 and at ρ = 0.7 when n > 13. Similarly, $\frac{\partial \mathrm{R}\mathrm{E}\left(n,r,\mathrm{\rho }\right)}{\partial r}>0$ if r_min ≤ r < r_*, and $\frac{\partial \mathrm{R}\mathrm{E}\left(n,r,\mathrm{\rho }\right)}{\partial r}<0$ if r_* < r ≤ r_max, so RE(n,r,ρ) is minimized at either r = r_min or r = r_max for a fixed ρ. Appendix Figure 2 demonstrates 3-D RE plot as a function of n and r, r ∈ (0.1, 0.7) for a fixed ρ = 0.05. Shown in Figure 2 with ρ = 0.05, RE(n,r,ρ) is minimized at r = 0.7 when n < 28 and r = 0.1 when n < 28. When combining these characteristics, we conclude that RE(n,r,ρ) is minimized at (r_min,ρ_min), or (r_min,ρ_max), or (r_max,ρ_min), or (r_max,ρ_max) within the parameters space (r_min,r_max) and (ρ_min, ρ_max).

Appendix 2

Proof that the minimum of RE(n,r,ρ) is maximized by the design satisfying RE(n, r_min, ρ_max) = RE(n, r_max, ρ_min)

Inserting (r_min,ρ_min) in Equation (12) and taking the partial derivative with respect to n gives

$$\frac{\partial \mathrm{R}\mathrm{E}\left(n,r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}{\partial n}\propto ac-K{b\rho n}^2.$$

Following the proof in Section 3, RE(n, r_min, ρ_min) is a single-peaked function and maximized at

$$n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}=\sqrt{\frac{1+\left(K-1\right)r_{\mathrm{m}\mathrm{i}\mathrm{n}}-K{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}}{K{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}}\times \frac{c}{b}}$$

with maximum of 1. Similarly, RE(n, r_min, ρ_max) is maximized at

$$n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}=\sqrt{\frac{1+\left(K-1\right)r_{\mathrm{m}\mathrm{i}\mathrm{n}}-K{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{K{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}}\times \frac{c}{b}};$$

RE(n, r_max, ρ_min) is maximized at

$$n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}=\sqrt{\frac{1+\left(K-1\right)r_{\mathrm{m}\mathrm{a}\mathrm{x}}-K{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}}{K{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}}\times \frac{c}{b};}$$

and RE(n, r_max, ρ_max) is maximized at

$$n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}=\sqrt{\frac{1+\left(K-1\right)r_{\mathrm{m}\mathrm{a}\mathrm{x}}-K{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{K{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}}\times \frac{c}{b}}.$$

Since ρ_min < ρ_max, it gives $n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}>n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}$ and $n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}>n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}.$ Further, r_min < r_max is followed by $n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}<n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}$ and $n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}<n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}.$ Thus, $n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}$ is the smallest and $n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}$ is the largest. All four REs have a maximum of 1 and the maximums are reached at $n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)},$ $n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}$, $n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)},$ and $n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}$, respectively.

Following the proof of Appendix in Breukelen et al,²⁰, min RE between any two of RE(n, r_min, ρ_min), RE(n, r_min, ρ_max), RE(n, r_max, ρ_min) and RE(n, r_max, ρ_max) is maximized by the design satisfying these two REs are equal. For example, min RE for RE(n, r_min, ρ_min), and RE(n, r_min, ρ_max) is maximized by the design satisfying RE(n, r_min, ρ_min) = (n, r_min, ρ_max). For any two pair values, (r₀,ρ₀) and (r₁,ρ₁), the intersection means

$$\mathrm{R}\mathrm{E}\left(n,r_0,{\rho }_0\right)=\mathrm{R}\mathrm{E}\left(n,r_1,{\rho }_1\right)\stackrel{yields}{\to }\frac{g(r_0,{\rho }_0)}{1+\left(K-1\right)r_0+K\left(n-1\right){\rho }_0}=\frac{g(r_1,{\rho }_1)}{1+\left(K-1\right)r_1+K\left(n-1\right){\rho }_1}.$$

Its only solution is

$$n=\frac{\left[1+\left(K-1\right)r_1-K{\rho }_1\right]g\left(r_0,{\rho }_0\right)-\left[1+\left(K-1\right)r_0-K{\rho }_0\right]g(r_1,{\rho }_1)}{K\left({\rho }_{0}g\left(r_1,{\rho }_1\right)-{\rho }_{1}g\left(r_0,{\rho }_0\right)\right)}.$$

That is, there is only one intersection between any two RE(n,r,ρ)s in the function of n. Therefore, there are total six intersections across these four REs. For example, Figure 3 demonstrates REs at the four points and all six intersections. Given all facts that these four are single-peaked functions, $n_{\left(r_{\mathrm{m}\mathrm{i}\mathrm{n}},{\rho }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}$ is the smallest and $n_{\left(r_{\mathrm{m}\mathrm{a}\mathrm{x}},{\rho }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}$ is the largest, and the only one intersection between any two REs, it is obvious that the minimum of RE(n,r,ρ) for these six intersections is reached at the intersection of RE(n, r_min, ρ_max) = RE(n, r_max, ρ_min). In other words, the minimum of RE(n,r,ρ) is maximized by the design satisfying RE(n, r_min, ρ_max) = RE(n, r_max, ρ_min).

Figure 1.

3-d relative efficiencies RE(n,r,ρ) as a function of n and ρ for K = 3, B = 300000, c = 10000, s = 100 and e = 10 with r = 0.8

Figure 2.

3-d relative efficiencies RE(n,r,ρ) as a function of n and r for K = 3, B = 300000, c = 10000, s = 100 and e = 10 with ρ = 0.05