An overview of optimal designs under a given budget in cluster randomized trials with a binary outcome

Abstract

Cluster randomized trial design may raise financial concerns because the cost to recruit an additional cluster is much higher than to enroll an additional subject in subject-level randomized trials. Therefore, it is desirable to develop an optimal design. For local optimal designs, optimization means the minimum variance of the estimated treatment effect under the total budget. The local optimal design derived from the variance needs the input of an association parameter $ρ$ in terms of a “working” correlation structure $R (ρ)$ in the generalized estimating equation models. When the range of $ρ$ instead of an exact value is available, the parameter space is defined as the range of $ρ$ and the design space is defined as enrollment feasibility, for example, the number of clusters or cluster size. For any value $ρ$ within the range, the optimal design and relative efficiency for each design in the design space is obtained. Then, for each design in the design space, the minimum relative efficiency within the parameter space is calculated. MaxiMin design is the optimal design that maximizes the minimum relative efficiency among all designs in the design space. Our contributions are threefold. First, for three common measures (risk difference, risk ratio, and odds ratio), we summarize all available local optimal designs and MaxiMin designs utilizing generalized estimating equation models when the group allocation proportion is predetermined for two-level and three-level parallel cluster randomized trials. We then propose the local optimal designs and MaxiMin designs using the same models when the group allocation proportion is undecided. Second, for partially nested designs, we develop the optimal designs for three common measures under the setting of equal number of subjects per cluster and exchangeable working correlation structure in the intervention group. Third, we create three new Statistical Analysis System (SAS) macros and update two existing SAS macros for all the optimal designs. We provide two examples to illustrate our methods.

Keywords

Cluster randomized trial generalized estimating equation local optimal design MaxiMin design partially nested design two-level parallel CRTs three-level parallel CRTs

1. Introduction

Health services and primary care researchers increasingly use cluster randomized trials (CRTs) due to their logistic advantages.^1–5 In these CRTs, all subjects within a cluster receive the same treatment (known as parallel CRT [PA-CRT]), which can prevent treatment contamination; thus, they are greatly needed for effectiveness research from science to practice.^6,7 Gravenstein et al. compared the relative effectiveness of two licensed influenza vaccines in US nursing homes through PA-CRTs.⁸ On the other hand, partially nested designs (PNDs) are defined as clinical trials where the subjects randomly assigned to the intervention are clustered while the subjects receiving the control treatments are unclustered. In such studies, dependence may exist among subjects in the intervention group but not in the control group. For example, the Scleroderma Patient-centered Intervention Network COVID-19 Home-isolation Activities Together (SPIN-CHAT) Trial is a pragmatic PND in which participants randomized into an intervention group were clustered and received the 4-week SPIN-CHAT Program via videoconference.⁹ Participants randomly assigned to the waitlist control were not clustered and only completed trial measures.

The studies above have two-level structures, where subjects are nested into clusters, but PA-CRTs may also use a three-level structure. For example, the Helping Hands trial (Netherlands Organization for Health Research and Development: ZonMw) randomized hospital wards to one of two strategies for increasing hand-hygiene-guideline adherence among nurses, with nurses then nested within wards and adherence observed at multiple time points. This way, the multiple evaluations from the same nurse could be correlated. Another example is that treatments are randomly assigned to practices, and providers within the same practice are trained with the assigned treatment. Participants within a provider and providers within a practice could be correlated. The researchers have proposed the sample size formulae for three-level PA-CRTs recently.^10–15 Hereafter, we use a CRT with practice, provider, and subject levels as a three-level PA-CRT example.

In a CRT design, the cost to recruit an additional cluster is much higher than to enroll an additional subject in subject-level randomized trials. Therefore, it is desirable to develop an optimal design (OD), which has the minimum variance of the estimated treatment effect under the total budget B. The OD derived from the variance needs the input of an association parameter $ρ$ in terms of a “working” correlation structure $R (ρ)$ in the generalized estimating equation (GEE) models; thus, the OD is locally optimal only. In other words, an incorrect prespecification of $ρ$ may lead to the wrong OD proposal. However, when the range of $ρ$ is available, the MaxiMin design (MMD) is a solution. The parameter space is defined as the range of $ρ,$ and the design space is defined as the enrollment feasibility of the number of clusters or cluster size. For any value $ρ$ within the range, we first compute the OD and relative efficiency (RE) for each design in the design space. Then, for each design in the design space, we compute the minimum RE within the parameter space and select the design that maximizes the minimum RE among all designs in the design space. As such, even if the parameter value $ρ$ is wrongly assumed, the choice of maximizing the minimum RE within the range of $ρ$ guarantees the optimization in the parameter space. Therefore, ODs include local optimal design (LOD) and MMD.

This paper discusses the two-group PA-CRTs with a binary outcome and assumes equal unit costs in the two groups through utilizing GEE models throughout.¹⁶ The group allocation proportion $π$ is defined as the percentage of clusters allocated to an intervention group. It is identical to the percentage of subjects allocated to an intervention group when an equal cluster size in two-level PA-CRTs, and the equal practice and provider size in three-level PA-CRTs are considered. For two-level two-group PA-CRTs, researchers Moerbeek et al., Raudenbush, Van Breukelen et al., and Liu J et al. derived the optimal number of clusters m and cluster size n for both LOD and MMD when $π$ is prespecified.^10,17–19 With the same assumptions of an equal cluster size and the exchangeable working correlation structure in GEE models, this paper proposes the ODs of $(m, π, n)$ when $π$ is undecided for three common measures: risk difference (RD), risk ratio (RR), and odds ratio (OR). The design setting in the three-level two-group PA-CRTs includes group allocation proportion $π$ , number of practices $m$ , practice size (number of providers per practice) n, provider size (number of subjects per provider) K, and association parameters $(γ, ρ),$ where $γ$ denotes correlation among subjects within the same provider in the same practice and $ρ$ denotes correlation among subjects for different providers in the same practice. Through utilizing GEE models with the assumption of the “nested exchangeable” correlation structure,²⁰ Liu et al. derived the optimal number of practices and practice size $(m, n)$ for two scenarios of provider size K under the assumptions of a predetermined group allocation proportion $π$ , an equal practice size across all practices and an equal provider size across all providers.²¹ In this paper, when the group allocation proportion is undecided, for three common measures, we propose the optimal number of practices, group allocation proportion, and practice size $(m, π, n)$ using the same GEE models for two scenarios of provider size K under the same assumptions. For PNDs with a binary outcome, Roberts et al. provided the sample size formulae (equations (14) and (17)) for RD and OR through GEE models.²² Moerbeek et al. proposed LODs that ensure adequate power to detect a treatment effect with either minimal cost or a minimal number of subjects using the multilevel linear regression models.²³ In this paper, we provide the variance of the estimated treatment effect, equivalently the sample size formula, utilizing GEE models under the assumptions of an equal cluster size and the exchangeable working correlation structure in the intervention group, and then propose the OD for three common measures.

The outline is as follows: Section 2 summarizes the ODs previously proposed for two-level PA-CRTs when a group allocation proportion $π$ is predetermined and proposes the ODs with an optimized group allocation proportion when $π$ is undecided. Section 3 summarizes the research of Liu et al. proposed for three-level PA-CRTs with a predetermined group allocation proportion $π,$ ²¹ and extends the ODs with an optimized group allocation proportion when $π$ is undecided. Section 4 derives the ODs of the number of clusters and cluster size in the intervention group and the number of subjects in the control group in PNDs. Section 5 illustrates how to apply the proposed methods. In Section 6, we discuss the limitations of the proposed methods and directions for future research.

1.1. Notations

To test the treatment effect in cluster trials, we utilize GEE models $g (μ) = X^{'} β$ with a working correlation structure, where $g (\cdot)$ is a link function, $μ$ is the marginal expected value of the outcome, $X$ is a covariate vector, and $β$ is the corresponding coefficient. For simplicity, $β_{0}$ is the intercept, and $β_{1}$ is the marginal treatment effect. We are interested in two-group cluster trials with a binary outcome. Let $p_{0}$ and $p_{1}$ be the success rates in the control and intervention groups. The null and alternative hypothesis are $p_{0} = p_{1}$ and $p_{0} \neq p_{1}$ , respectively. When the identity link function is specified, $β_{1} = p_{1} - p_{0}$ is RD; when the log link function is specified, $β_{1} = ln (p_{1} / p_{0})$ is the difference between the natural logarithms of the proportions; and when the logit link function is specified, $β_{1} = ln (\frac{p_{1} / (1 - p_{1})}{p_{0} / (1 - p_{0})})$ is the log OR—the difference between the natural logarithms of the odds. When the log or logit link function is used, taking the exponential of $β_{1}$ refers to RR or OR, respectively.

In two-level cluster trials, the working correlation structure is denoted by $R (ρ),$ where $ρ$ is an association parameter. We define the study cost per cluster as c currency units, and the study cost per subject as s currency units. In three-level cluster trials, the working correlation structure is denoted by $R (γ, ρ),$ where both $γ$ and $ρ$ are association parameters at the top and middle cluster levels. We define the study cost per practice in the enrollment stage as c currency units, the study cost per provider as s currency units, and the study cost per subject as $e .$ An OD refers to either LOD or MMD, depending on a specific scenario.

2. Two-level PA-CRTs

For two-level PA-CRTs, we assume the same exchangeable working correlation structure $R (ρ)$ and an equal cluster size of n for both groups in GEE models. Total budget B includes m clusters and n subjects per cluster, that is,

\begin{aligned} B = m (c + s n) . \end{aligned}

(1)

We use derivations similar to the research^21,24 and obtain the marginal variances of the estimated treatment effect as follows:

\begin{aligned} Var (\hat{β_{2}}) = \frac{1 + (n - 1) ρ}{n m} \times g (π, p_{0}, p_{1}), \end{aligned}

(2)

where

\begin{aligned} g (π, p_{0}, p_{1}) = {\begin{cases} \frac{p_{1} (1 - p_{1})}{π} + \frac{p_{0} (1 - p_{0})}{1 - π}, & identity\ link\ function \to RD \frac{1 - p_{1}}{π p_{1}} + \frac{1 - p_{0}}{(1 - π) p_{0}}, & log\ link\ function \to RR . \frac{1}{π p_{1} (1 - p_{1})} + \frac{1}{(1 - π) p_{0} (1 - p_{0})}, & logit\ link\ function \to OR \end{cases} \end{aligned}

Please note that Wu et al. also presented the same formulae.²⁵ Equation (2) shows that the variances need the input from the design factors

(m, π, n)

and the parameters of

ρ

and

(p_{0}, p_{1}) .

2.1 Local optimal designs

This section assumes that $ρ$ and $(p_{0}, p_{1})$ are prespecified. For example, $ρ$ and $p_{0}$ are pulled from similar studies including the control group, and $p_{1}$ from the alternative hypothesis.

2.1.1. $π$ is a predetermined value

First, assume that the group allocation proportion $π$ is a predetermined value, for example, $π = 0.5$ for a 1:1 randomization. The optimization in LOD is to find a pair of m and n to minimize the variances of the estimated treatment effect of equation (2) given the constraint in equation (1). For all three link functions, it is equivalent to minimizing $\frac{1 + (n - 1) ρ}{n m} .$ The following formulae are easily obtained.

\begin{aligned} n_{LOD} = \sqrt{\frac{θ c}{s}}, m_{LOD} = \frac{B}{\sqrt{θ s c} + c}, \end{aligned}

(3)

where

θ = \frac{1 - ρ}{ρ} .

Equation (3) is identical to the ODs using the linear regression model.^10,17,18 The proof are detailed in Liu et al.¹⁹ Here,

n_{LOD}

does not depend on

π,

and

(p_{0}, p_{1})

. The budget

B,

θ,

and unit costs

(c, s)

determine

m_{LOD}

. Therefore, the optimal number of clusters per group and cluster size are

m_{t - LOD} = m_{LOD} \times π,

m_{c - LOD} = m_{LOD} \times (1 - π),

and

n_{LOD},

where

m_{t - LOD}

is the optimal number of clusters in the intervention group and

m_{c - LOD}

is the optimal number of clusters in the control group.

2.1.2.

π

is undecided

Second, assume that the group allocation proportion $π$ is undecided. For any $π \in (0, 1)$ , the marginal variances of the estimated treatment effect under the OD is

Var (\hat{β_{2}}) = \frac{1 + (n_{LOD} - 1) ρ}{n_{LOD} m_{LOD}} \times g (π, p_{0}, p_{1}) .

Obviously, this variance is minimized when

g (π, p_{0}, p_{1})

is minimized. It can be shown that when

\begin{aligned} π_{LOD} = \frac{1}{1 + \sqrt{φ}}, \end{aligned}

(4)

LOD is reached for all three link functions and

φ = {\begin{array}{ll} \frac{p_{0} (1 - p_{0})}{p_{1} (1 - p_{1})}, & identity\ link\ function \to RD \\ \frac{p_{1} (1 - p_{0})}{p_{0} (1 - p_{1})}, & log\ link\ function \to RR \\ \frac{p_{1} (1 - p_{1})}{p_{0} (1 - p_{0})}, & logit\ link\ function \to OR \end{array} .

Therefore, the optimal group allocation proportion, number of clusters per group, and cluster size are

π_{LOD},

m_{t - LOD} = m_{LOD} \times π_{LOD},

m_{c - LOD} = m_{LOD} \times (1 - π_{LOD}),

and

n_{LOD},

respectively. Wu et al. proposed the optimal group allocation proportions to maximize the cost efficiency for three measures including RD, RR, and OR where the unit costs and intraclass correlation coefficients (ICCs) in the two groups may be unequal.²⁵ When two ICCs and the unit costs are equal for both groups, their optimal allocations are identical to equation (4) for all three measures.

From the equation (4), we can prove that (a) $π_{LOD}$ is an increasing function of $p_{1}$ until 0.5 after which it decreases for RD; (b) $π_{LOD}$ is a decreasing function of $p_{1}$ for RR; (c) $π_{LOD}$ is a decreasing function of $p_{1}$ until 0.5 after which it increases for OR. Figure 1 shows the optimal allocations for three measures of treatment effect (RD, RR, and OR) with $p_{0} = 0.3$ and $p_{0} = 0.6$ and confirms these findings.

The proposed OD in the Section 2.1 is a design with the number of clusters of $m_{t - LOD}$ , and $m_{c - LOD}$ with an equal cluster size of $n_{LOD} .$ When the group allocation proportion is prespecified, the choice among the three measures has no effect of the optimal number of clusters and cluster size. Otherwise, the choice affects the optimal group allocation proportion.

2.2 Maximin design

When the group allocation proportion $π$ is a predetermined value, equation (3) shows that the OD depends on the total budget B, unit costs $(c, s)$ , and $ρ$ ; otherwise, equation (4) presents that the OD also depends on the $(p_{0}, p_{1})$ . Since these parameters need to be prespecified, the OD is locally optimal only. Given the fact that the budget and unit cost can be known but $ρ$ and $(p_{0}, p_{1})$ may not, the proposed designs in Section 2.1 are not robust when the different values of $ρ$ and $(p_{0}, p_{1})$ are chosen. Suppose the range of $ρ$ is available, for example, from previous studies, $(ρ_{min}, ρ_{max})$ . The parameter space is defined as the range of $ρ$ , and the ranges of $(p_{0}, p_{1})$ . The design space is defined as enrollment feasibility of the number of clusters or cluster size. For simplicity, they can be assumed to be from 2 to 200. MMD is proposed for such a scenario.^18,26 Specifically, it is detailed in the following steps for this research:

Step 1.

Define the parameter space $(ρ, \; p_{0}, p_{1})$ and the design space $(m, n) .$

Step 2.

For each value in the parameter space, compute the LOD and RE for each design in the design space.

Step 3.

For each design in the design space, compute the minimum RE within the parameter space.

Step 4.

Select the design with the maximum of smallest RE among all designs in the design space, defined as $(m_{MMD}, n_{MMD}) .$

Similar to Section 2.1, we discuss two different cases about the group allocation proportion $π$ .

2.2.1.

π

is a predetermined value

When the group allocation proportion $π$ is predetermined, inserting (3) in (2) gives the marginal variance of the estimated treatment effect for the OD

\begin{aligned} Var (\hat{β_{2}}) = \frac{g (π, p_{0}, \; p_{1})}{B} \times h (ρ), \end{aligned}

(5)

where

h (ρ) = {(\sqrt{(1 - ρ) s} + \sqrt{ρ c})}^{2}

We follow the definition of RE: the ratio of the variance of the estimated treatment effect for each design in the design space to LOD.¹⁸ The RE for measure RD using equations (2) and (5) is expressed as a function of n and $ρ,$

\begin{aligned} RE (n, ρ) = \frac{{(\sqrt{(1 - ρ) s} + \sqrt{ρ c})}^{2}}{[1 + (n - 1) ρ]} \frac{n}{c + s n} . \end{aligned}

(6)

The two other measures, RR and OR, have the same formula (6) given the expression of equation (2). In addition, it is identical to equation (A1).¹⁸ Please note this RE is unrelated to

(p_{0}, p_{1})

, so we can simplify parameter space based on

ρ

only. Therefore, the parameter space in Step 1 is simplified as

(ρ_{min}, ρ_{max})

. The MMD considers the worst-case scenario and thus is robust against misspecification of the values of

ρ .

Following the proof,¹⁸ the MMD is summarized as below:

Step 1.

Define the parameter space $(ρ_{min}, ρ_{max}) .$

Step 2.

Calculate

\begin{aligned} n_{MMD} = \frac{[1 - ρ_{min}] h (ρ_{max}) - [1 - ρ_{max}] h (ρ_{min})}{ρ_{max} h (ρ_{min}) - ρ_{min} h (ρ_{max})}, \end{aligned}

(7)

and

m_{MMD} = B / (c + s \times n_{MMD}) .

Step 3.

Obtain $m_{t - MMD} = m_{MMD} \times π$ and $m_{c - MMD} = m_{MMD} \times (1 - π)$ , where $m_{t - MMD}$ is the optimal number of clusters in the intervention group and $m_{c - MMD}$ is the optimal number of clusters in the control group.

The above equation (7) is the same as equation (8) in the research of Van Breukelen et al.¹⁸

2.2.2.

π

is undecided

Please note that the equation (6) is also unrelated to $π$ . Thus, for any $π \in (0, 1)$ , the optimal number of clusters and cluster size is the same as in Section 2.2.1. The corresponding variance of the estimated treatment effect for the OD is

\begin{aligned} Var (\hat{β_{2}}) = \frac{1 + (n_{MMD} - 1) ρ}{n_{MMD} m_{MMD}} \times g (π, p_{0}, p_{1}) . \end{aligned}

(8)

Please note that this variance is an increasing function of

ρ

and maximized at

ρ_{max}

. After inserting (4), LOD for

π

, and

m_{MMD} = B / (c + s \times n_{MMD})

in (8), we have the updated variance of the estimated treatment effect

\begin{aligned} Var (\hat{β_{2}}) = f (p_{0}, \; p_{1}) \times (c + s \times n_{MMD}) \times \frac{1 + (n_{MMD} - 1) ρ}{n_{MMD} B}, \end{aligned}

(9)

where

f (p_{0}, p_{1}) = {\begin{array}{ll} {(\sqrt{p_{1} (1 - p_{1})} + \sqrt{p_{1} (1 - p_{0})})}^{2}, & identity link function \to RD \\ {(\sqrt{\frac{1 - p_{1}}{p_{1}}} + \sqrt{\frac{1 - p_{0}}{p_{0}}})}^{2}, & log link function \to RR \\ {(\sqrt{\frac{1}{p_{1} (1 - p_{1})}} + \sqrt{\frac{1}{p_{0} (1 - p_{0})}})}^{2}, & logit link function \to OR \end{array}

The derivations are provided in Appendix 1. The factor

1 + (n - 1) ρ

is known as the inflation factor or design effect.^27,28 When

ρ = 0

Var (\hat{β_{2}})

is identical to the variance of the estimated treatment effect in subject-level randomized trials. Here, RE is defined as the ratio of the variance of the estimated treatment effect for group allocation proportion

π_{LOD}

π

. The RE using equations (8) and (9) is expressed as a function of

π,

p_{0},

and

p_{1},

\begin{aligned} RE (π, \; φ) = \frac{{(1 + \sqrt{φ})}^{2}}{\frac{1}{π} + \frac{φ}{1 - π}}, \end{aligned}

(10)

where the derivations are provided in the Appendix 2. Our goal is to maximize the minimum of RE over the possible ranges for

p_{0}

and

p_{1}

. Even if

(p_{0}, p_{1})

are misspecified at the stage of study design, MMD minimizes the risk of the worst scenario in the framework of RE. We notice that equation (10) is the same as RCE in Web Appendix A.1 when

γ = 1

.²⁵ Following the proof in Wu et al.,²⁵ the MMD for the group allocation proportion

π

\begin{aligned} π_{MMD} (\underline{φ}, \bar{φ}) = \frac{{(1 + \sqrt{\underline{φ}})}^{2} - {(1 + \sqrt{\bar{φ}})}^{2}}{{(1 + \sqrt{\bar{φ}})}^{2} (\underline{φ} - 1) - {(1 + \sqrt{\underline{φ}})}^{2} (\bar{φ} - 1)}, \end{aligned}

(11)

where

\underline{φ} = min φ

and

\bar{φ} = max φ .

Suppose

p_{0} \in [p_{a}, p_{b}]

and

p_{1} \in [p_{c}, p_{d}]

\underline{φ}

and

\bar{φ}

are detailed in the footnote of Tables 1 and 2; the proof is provided in Appendix 3.

Table 1.

Optimal designs for two-level PA-CRTs.

Association
parameter ρ
and $(p_{0}, p_{1})$	$π$	$π_{OD}$	$m_{OD}$	$n_{OD}$
Known	Predetermined	NA	$\frac{B}{\sqrt{ϑ s c} + c}$	$\sqrt{\frac{ϑ c}{s}}$
Known	Undecided	$\frac{1}{1 + \sqrt{φ}}$	$\frac{B}{\sqrt{ϑ s c} + c}$	$\sqrt{\frac{ϑ c}{s}}$
ρ range $(ρ_{min}, ρ_{max})$ $p_{0} \in [p_{a}, p_{b}]$ $p_{1} \in [p_{c}, p_{d}]$	Predetermined^[1]	NA	$\frac{B}{c + s \times n_{OD}}$	$\frac{[1 - ρ_{min}] h (ρ_{max}) - [1 - ρ_{max}] h (ρ_{min})}{ρ_{max} h (ρ_{min}) - ρ_{min} h (ρ_{max})}$
	Undecided	$\frac{{(1 + \sqrt{\underline{φ}})}^{2} - {(1 + \sqrt{\bar{φ}})}^{2}}{{(1 + \sqrt{\bar{φ}})}^{2} (\underline{φ} - 1) - {(1 + \sqrt{\underline{φ}})}^{2} (\bar{φ} - 1)}$	$\frac{B}{c + s \times n_{OD}}$	$\frac{[1 - ρ_{min}] h (ρ_{max}) - [1 - ρ_{max}] h (ρ_{min})}{ρ_{max} h (ρ_{min}) - ρ_{min} h (ρ_{max})}$

$m :$ number of clusters; $n :$ number of subjects per cluster; NA: not applicable; $π :$ % clusters (subjects) allocated to intervention group.

Total budget $B = m (c + s n)$ ; $c :$ cost per cluster; $s :$ cost per subject; $ϑ = \frac{1 - ρ}{ρ};$ $φ = {\begin{array}{lll} \frac{p_{0} (1 - p_{0})}{p_{1} (1 - p_{1})}, & for\ RD \\ \frac{p_{1} (1 - p_{0})}{p_{0} (1 - p_{1})}, & for\ RR \\ \frac{p_{1} (1 - p_{1})}{p_{0} (1 - p_{0})}, & for\ OR \end{array};$ $h (ρ) = {(\sqrt{(1 - ρ) s} + \sqrt{ρ c})}^{2};$

$\underline{φ} = {\begin{array}{lll} {\begin{array}{lll} \frac{min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}{max (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}, p_{c} \geq 0.5 or p_{d} \leq 0.5 \\ \frac{min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}{0.25}, p_{c} \leq 0.5 \leq p_{d} \end{array} for\ RD \\ \frac{p_{c} (1 - p_{b})}{p_{b} (1 - p_{c})}, for\ RR \\ {\begin{array}{lll} \frac{min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}{max (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}, p_{a} \geq 0.5 or p_{b} \leq 0.5 \\ \frac{min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}{0.25}, p_{a} \leq 0.5 \leq p_{b} \end{array} for\ OR \end{array}$ $\bar{φ} = {\begin{array}{lll} {\begin{array}{lll} \frac{max (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}{min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}, p_{a} \geq 0.5 or p_{b} \leq 0.5 \\ \frac{0.25}{min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}, p_{a} \leq 0.5 \leq p_{b} \end{array} for\ RD \\ \frac{p_{d} (1 - p_{a})}{p_{a} (1 - p_{d})}, for\ RR . \\ {\begin{array}{lll} \frac{max (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}{min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}, p_{c} \geq 0.5 or p_{d} \leq 0.5 \\ \frac{0.25}{min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}, p_{c} \leq 0.5 \leq p_{d} \end{array} for\ OR \end{array}$

^[1]

$[p_{a}, p_{b}]$ and $[p_{c}, p_{d}]$ are unnecessary.

Table 2.

Optimal design for three-level PA-CRTs.

Association
parameters
$(γ, ρ)$ and
$(p_{0}, p_{1})$	$π$	$K$	$π_{OD}$	$m_{OD}$	$n_{OD}$	$K_{OD}$
Known	Predetermined	Predetermined	NA	$\frac{B}{\sqrt{ϑ b c} + c}$	$\sqrt{\frac{ϑ c}{b}}$	NA
	Predetermined	Ranges $(K_{min}, K_{max})$	NA	$\frac{B}{\sqrt{ϑ b c} + c}$ at $K_{OD}$	$\sqrt{\frac{ϑ c}{b}}$ at $K_{OD}$	$= {\begin{array}{lll} K_{min} i f K_{min} \geq 3 \\ 3 i f K_{min} = 2 \end{array}$
	Undecided	Predetermined	$\frac{1}{1 + \sqrt{φ}}$	$\frac{B}{\sqrt{ϑ b c} + c}$	$\sqrt{\frac{ϑ c}{b}}$	NA
	Undecided	Range $(K_{min}, K_{max})$	$\frac{1}{1 + \sqrt{φ}}$	$\frac{B}{\sqrt{ϑ b c} + c}$ at $K_{OD}$	$\sqrt{\frac{ϑ c}{b}}$ at $K_{OD}$	$= {\begin{array}{lll} K_{min} i f K_{min} \geq 3 \\ 3 i f K_{min} = 2 \end{array}$
Ranges $(γ_{min}, γ_{max})$ and $(ρ_{min}, ρ_{max})$ $p_{0} \in [p_{a}, p_{b}]$ $p_{1} \in [p_{c}, p_{d}]$	Predetermined^[1]	Predetermined	NA	$\frac{B}{c + (s + e K) n_{OD}}$	$\frac{\begin{matrix} [1 + (K - 1) γ_{max} - K ρ_{min}] h (γ_{min}, \; ρ_{max}) \\ - [1 + (K - 1) γ_{min} - K ρ_{max}] h (γ_{max}, \; ρ_{min}) \end{matrix}}{K (ρ_{max} h (γ_{max}, \; ρ_{min}) - ρ_{min} h (γ_{min}, \; ρ_{max}))}$	NA
		Range $(K_{min}, K_{max})$	NA	For each value of K,
				Step 1. Calculate $n$ $\frac{[1 + (K - 1) γ_{max} - K ρ_{min}] h (γ_{min}, \; ρ_{max}) - [1 + (K - 1) γ_{min} - K ρ_{max}] h (γ_{max}, \; ρ_{min})}{K (ρ_{max} h (γ_{max}, \; ρ_{min}) - ρ_{min} h (γ_{min}, \; ρ_{max}))} .$
				Step 2. Calculate
				$RE (n, γ_{min}, \; ρ_{min}),$ $RE (n, γ_{min}, \; ρ_{max}),$ $RE (n, γ_{max}, \;\; ρ_{min}),$ and $RE (n, γ_{max}, \; ρ_{max}) .$
				Step 3. Take the minimum of 4 REs.
				Step 4. The design with the maximum of RE in Steps 1–3 within $(K_{min}, K_{max})$ is denoted as
				$(γ_{OD}, ρ_{OD}, n_{OD}, K_{OD}) .$
				Step 5. Calculate $m_{OD} = \frac{B}{c + (s + e \times K_{OD}) n_{OD}} .$
	Undecided	Predetermined	[2]	$\frac{B}{c + (s + e K) n_{OD}}$ $\frac{\begin{matrix} [1 + (K - 1) γ_{max} - K ρ_{min}] h (γ_{min}, \; ρ_{max}) \\ - [1 + (K - 1) γ_{min} - K ρ_{max}] h (γ_{max}, \; ρ_{min}) \end{matrix}}{K (ρ_{max} h (γ_{max}, \; ρ_{min}) - ρ_{min} h (γ_{min}, \; ρ_{max}))}$		NA
		Range $(K_{min}, K_{max})$	[2]	For each value of K,
				Step 1. Calculate $n$ $\frac{[1 + (K - 1) γ_{max} - K ρ_{min}] h (γ_{min}, \; ρ_{max}) - [1 + (K - 1) γ_{min} - K ρ_{max}] h (γ_{max}, \; ρ_{min})}{K (ρ_{max} h (γ_{max}, \; ρ_{min}) - ρ_{min} h (γ_{min}, \; ρ_{max}))} .$
				Step 2. Calculate
				$RE (n, γ_{min}, \; ρ_{min}),$ $RE (n, γ_{min}, \; ρ_{max}),$ $RE (n, γ_{max}, \;\; ρ_{min}),$ and $RE (n, γ_{max}, \; ρ_{max}) .$
				Step 3. Take the minimum of 4 REs.
				Step 4. The design with the maximum of RE in Steps 1–3 within $(K_{min}, K_{max})$ is denoted as
				$(γ_{OD}, ρ_{OD}, n_{OD}, K_{OD}) .$
				Step 5. Calculate $m_{OD} = \frac{B}{c + (s + e \times K_{OD}) n_{OD}} .$

$K :$ number of subjects per provider; $m :$ number of practices; $n :$ number of providers per practice; NA: not applicable; $π :$ % practice allocated to intervention group; $γ :$ correlation among subjects within the same provider in the same practice; $ρ :$ correlation among subjects with the different providers in the same practice.

Total budget $B = m (c + s n + e K n);$ $c :$ cost per practice; $s :$ cost per provider; $e :$ cost per subject;

$b = s + e K; ϑ = \frac{1 + (K - 1) γ - K ρ}{K ρ}; φ = {\begin{array}{lll} \frac{p_{0} (1 - p_{0})}{p_{1} (1 - p_{1})}, f o r R D \\ \frac{p_{1} (1 - p_{0})}{p_{0} (1 - p_{1})}, f o r R R \\ \frac{p_{1} (1 - p_{1})}{p_{0} (1 - p_{0})}, f o r O R \end{array};$ $h (r, ρ) = {(\sqrt{ρ c} + \sqrt{\frac{1 + (K - 1) γ - K ρ}{K} (s + e K)})}^{2};$ $RE (n, γ, ρ) = \frac{h (γ, ρ)}{1 + (K - 1) γ + K (n - 1) ρ} \times \frac{K n}{c + (s + e K) n};$

^[1]

$[p_{a}, p_{b}]$ and $[p_{c}, p_{d}]$ are unnecessary.

^[2]

$π_{OD} = \frac{{(1 + \sqrt{\underline{φ}})}^{2} - {(1 + \sqrt{\bar{φ}})}^{2}}{{(1 + \sqrt{\bar{φ}})}^{2} (\underline{φ} - 1) - {(1 + \sqrt{\underline{φ}})}^{2} (\bar{φ} - 1)},$ where $\underline{φ} = {\begin{cases} {\begin{array}{lll} \frac{min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}{max (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}, p_{c} \geq 0.5 or p_{d} \leq 0.5 \\ \frac{min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}{0.25}, p_{c} \leq 0.5 \leq p_{d} \end{array} for\;RD \\ \frac{p_{c} (1 - p_{b})}{p_{b} (1 - p_{c})}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;for\;RR; \\ {\begin{array}{lll} \frac{min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}{max (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}, p_{a} \geq 0.5 or p_{b} \leq 0.5 \\ \frac{min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}{0.25}, p_{a} \leq 0.5 \leq p_{b} \end{array} for\;OR \end{cases}$ $\bar{φ} = {\begin{array}{lll} {\begin{array}{lll} \frac{max (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}{min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}, p_{a} \geq 0.5 or p_{b} \leq 0.5 \\ \frac{0.25}{min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}, p_{a} \leq 0.5 \leq p_{b} \end{array} for\ RD \\ \frac{p_{d} (1 - p_{a})}{p_{a} (1 - p_{d})}, for\ RR . \\ {\begin{array}{lll} \frac{max (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))}{min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}, p_{c} \geq 0.5 or p_{d} \leq 0.5 \\ \frac{0.25}{min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))}, p_{c} \leq 0.5 \leq p_{d} \end{array} for\ OR \end{array}$

When $π$ , $ρ$ , and $(p_{0}, p_{1})$ are not prespecified, we summarize the approach to find a MMD:

Step 1.

Define the parameter space $(ρ_{min}, ρ_{max})$ , $p_{0} \in [p_{a}, p_{b}]$ , and $p_{1} \in [p_{c}, p_{d}] .$

Step 2.

Select a treatment effect measure from RD, RR, and OR.

Step 3.

Calculate $n_{MMD}$ , $m_{MMD}$ , and $π_{MMD}$ using equations (7) and (11).

Step 4.

Obtain $m_{t - MMD} = m_{MMD} \times π_{MMD}$ and $m_{c - MMD} = m_{MMD} \times (1 - π_{MMD}) .$

The proposed OD in Section 2.2 is a design with the number of clusters of $m_{t - MMD}$ , and $m_{c - MMD}$ with an equal cluster size of $n_{MMD} .$ Similar to Section 2.1.2, the choice among the three measures in MMD has no effect of the optimal number of clusters and cluster size when the group allocation proportion is prespecified. However, the choice determines the optimal group allocation proportion when the group allocation proportion is undecided. Figure 2 shows the trend of three measures when only one parameter of $[p_{a}, p_{b}]$ and $[p_{c}, p_{d}]$ varies. For example, $p_{b} = 0.6$ , $p_{c} = 0.2$ , and $p_{d} = 0.9$ . When $p_{a}$ increases from 0.1 to 0.5, $π_{MMD}$ decreases for the RD but increases for the RR and OR. The same finding is observed when $p_{b}$ increases from 0.15 to 0.6, $p_{a} = 0.1$ , $p_{c} = 0.2$ , and $p_{d} = 0.9$ . We observe the opposite trend when $p_{c}$ increases from 0.2 to 0.7. for the remaining two figures. When $p_{d}$ increases from 0.3 to 0.9, $π_{MMD}$ decreases for the RD and RR but increases for the OR.

The ODs for two-level PA-CRTs are detailed in Table 1. Statistical Analysis System (SAS) macro, originally developed by Liu et al.,¹⁹ %OD_2Level_FixedAllo is revised to find the LOD and MMD of the number of clusters per group and cluster size. SAS macros %OD_2Level_OptAllo are developed to find the LOD and MMD of the group allocation proportion, the number of clusters per group, and cluster size.

3. Three-level PA-CRTs

For three-level data, the total budget B accommodates m practices, n providers per practice, and K subjects per provider, that is,

\begin{aligned} B = m (c + s n + e K n) . \end{aligned}

(12)

Teerenstra et al. proposed a “nested exchangeable” correlation structure

R (ρ, γ)

in GEE models:²⁰

Correlation among subjects within the same provider in the same practice is constant, $Corr (Y_{i j k_{1}}, Y_{i j k_{2}}) = γ$ for $k_{1} \neq k_{2}$ .

Correlation among subjects with different providers in the same practice is constant, $Corr (Y_{i j_{1} k_{1}}, Y_{i j_{2} k_{2}}) = ρ$ for $j_{1} \neq j_{2},$ and any $k_{1}, k_{2}$ ,

where

Y_{i j k}

is a response from subject

k = 1, \dots, K

, for provider

j = 1, \dots, n

in practice

i = 1, \dots, m .

In order to obtain the marginal variance of the estimated treatment effect, this correlation structure should be positive definite (PD). With the assumptions of the equal practice size n and equal provider size

K,

PD can be determined by the following constraints:

\begin{aligned} - \frac{1}{K - 1} < γ < 1, - \frac{1 + (K - 1) γ}{K (n - 1)} < ρ < \frac{1 + (K - 1) γ}{K} . \end{aligned}

(13)

The proof was provided by Web Appendix A of Li et al.²⁹ Given the “nested exchangeable” correlation structure in the GEE model, Liu et al. derived the following marginal variance estimators:²¹

\begin{aligned} Var (\hat{β_{2}}) = \frac{λ_{3}}{K n m} \times g (π, p_{0}, p_{1}), \end{aligned}

(14)

where

λ_{3} = 1 + (K - 1) γ + K (n - 1) ρ

, and

g (π, p_{0}, p_{1})

is defined as Section 2. Equation (14) shows that the variance need the input from the design factors

(m, π, n)

and the parameters of

(γ, ρ),

and

(p_{0}, p_{1}) .

3.1 Local optimal design

This section assumes that $(γ, ρ)$ and $(p_{0}, p_{1})$ are prespecified.

3.1.1. $π$ is a predetermined value

For a predetermined group allocation proportion $π,$ Liu et al. proposed LOD.²¹ It is summarized as follows.

First, assume that the provider size K is a predetermined value,

\begin{aligned} n_{LOD} = \sqrt{\frac{ϑ c}{b}}, m_{LOD} = \frac{B}{\sqrt{ϑ b c} + c} . \end{aligned}

(15)

where

b = s + e K,

for example, cost per provider including the costs for K subjects,

λ_{2} = 1 + (K - 1) γ - K ρ, ϑ = \frac{λ_{2}}{K ρ} .

In order to be

n_{LOD} > 1

0 < ρ < \frac{[1 + (K - 1) r] c}{K (c + s + e K)}

should hold. Here, the optimal number of practices per group and practice size are

m_{t - LOD} = m_{LOD} \times π,

m_{c - LOD} = m_{LOD} \times (1 - π),

and

n_{LOD},

where

m_{t - LOD}

is the optimal number of practices in the intervention group and

m_{c - LOD}

is the optimal number of practices in the control group. The proposed OD is a design with the number of practices of

m_{t - LOD}

, and

m_{c - LOD}

with the equal practice and provider size of

n_{LOD}

and K. For a special case,

K = 1,

n_{LOD} = \sqrt{\frac{ϑ c}{s + e}}, m_{LOD} = \frac{B}{\sqrt{ϑ (s + e) c} + c}, ϑ = \frac{1 - ρ}{ρ} .

It is the same as equation (3) in two-level CRTs, except that s is replaced by

s + e

Second, when the provider size K is not a fixed value but within a range $(K_{min}, K_{max})$ and $K_{min} \geq 2$ , where the assumption of an equal provider size still holds, the LOD is reached at $K = K_{min}$ if $K_{min} \geq 3$ and $K = 3$ if $K_{min} = 2$ , denoted by $K_{LOD}$ . The proof is detailed in Liu et al.²¹ Therefore, the optimal provider size, number of practices per group, and practice size are $K_{LOD}, m_{t - LOD} = m_{LOD} \times π, m_{c - LOD} = m_{LOD} \times (1 - π),$ and $n_{LOD},$ respectively.

3.1.2.

π

is undecided

When the group allocation proportion $π$ is undecided, for any $π \in (0, 1)$ , the marginal variance of the estimated treatment effect under the OD is

Var (\hat{β_{2}}) = \frac{1 + (K_{LOD} - 1) γ + K_{LOD} (n_{LOD} - 1) ρ}{K_{LOD} n_{LOD} m_{LOD}} \times g (π, p_{0}, p_{1}),

where

K_{LOD} = K

when K is a predetermined value and

K_{LOD}

when K is within a range

(K_{min}, K_{max})

. Then,

π_{LOD}

is the same as in equation (4) since the same

g (π, p_{0}, p_{1})

need to be minimized. Therefore, the optimal group allocation proportion, number of practices per group, an equal practice size, and an equal provide size are

π_{LOD}

m_{t - LOD} = m_{LOD} \times π_{LOD}

m_{c - LOD} = m_{LOD} \times (1 - π_{LOD})

n_{LOD}

, and

K_{LOD}

The proposed OD in Section 3.1 is a design with the number of practices of $m_{t - LOD}$ , and $m_{c - LOD}$ with the equal practice and provider size of $n_{LOD}$ , and $K_{LOD}$ .

3.2 Maximin design

For three-level PA-CRTs, equation (15) shows that the OD depends on the total budget B, unit costs $(c, s, e)$ , and $(r, \; ρ)$ when the group allocation proportion $π$ is a predetermined value. MMD steps in three-level PA-CRTs are the same as those in two-level PA-CRTs except that the parameter space is $(γ, \; ρ)$ . Similar to Section 3.1, we discuss two different cases about the group allocation proportion $π$ .

3.2.1. $π$ is a predetermined value

For a predetermined group allocation proportion $π,$ the MMDs proposed by Liu et al. are simplified as follows.²¹ When the provider size K is a predetermined value,

Step 1.
Define the parameter space $(γ_{min}, γ_{max}), \; (ρ_{min}, ρ_{max})$ , and a fixed provider size $K .$
Step 2.
Calculate
$\begin{aligned} n_{MMD} = \frac{[1 + (K - 1) γ_{max} - K ρ_{min}] h (γ_{min}, \; ρ_{max}) - [1 + (K - 1) γ_{min} - K ρ_{max}] h (γ_{max}, \; ρ_{min})}{K (ρ_{max} h (γ_{max}, \; ρ_{min}) - ρ_{min} h (γ_{min}, \; ρ_{max}))}, \end{aligned}$
(16)
where $h (r, ρ) = {(\sqrt{ρ c} + \sqrt{\frac{1 + (K - 1) γ - K ρ}{K} (s + e K)})}^{2},$

and $m_{MMD} = \frac{B}{c + (s + e K) n_{MMD}} .$
Step 3.
Obtain $m_{t - MMD} = m_{MMD} \times π$ and $m_{c - MMD} = m_{MMD} \times (1 - π)$ .

The proposed OD is a design with the number of practices of $m_{t - MMD}$ and $m_{c - MMD}$ with the equal practice size of $n_{MMD}$ and provider size of K. For a special case, $K = 1,$ $n_{MMD}$ and $m_{MMD}$ are the same as equation (7) in two-level CRTs, except that s is replaced by $s + e$ .

When the provider size K is not a fixed value but within a range $(K_{min}, K_{max})$ and $K_{min} \geq 2$ ,

Step 1.
Define the parameter space $(γ_{min}, γ_{max}), \; (ρ_{min}, ρ_{max})$ and design space $(K_{min}, K_{max}) .$
Step 2.
Calculate n from equation (16) for each value of K and take the minimum of $RE (n, γ_{min}, \; ρ_{min}),$ $RE (n, γ_{min}, \; ρ_{max}),$ $RE (n, γ_{max}, \;\; ρ_{min}),$ and $RE (n, γ_{max}, \; ρ_{max})$ , where RE is defined as equation (17)
$\begin{aligned} RE (n, γ, ρ) = \frac{h (γ, ρ)}{1 + (K - 1) γ + K (n - 1) ρ} \times \frac{K n}{c + (s + e K) n} . \end{aligned}$
(17)
Step 3.
The design with the maximum of RE in Step 2 is denoted as $(γ_{MMD}, ρ_{MMD}, n_{MMD}, K_{MMD})$ and the n
$m_{MMD} = \frac{B}{c + (s + e \times K_{MMD}) n_{MMD}} .$

Step 4.
Obtain $m_{t - MMD} = m_{MMD} \times π$ and $m_{c - MMD} = m_{MMD} \times (1 - π)$ .

The proposed OD is a design with the number of practices of $m_{t - MMD}$ and $m_{c - MMD}$ with the equal practice and provider size of $(n_{MMD}, K_{MMD})$ .
3.2.2. $π$ is undecided

The equations (16) and (17) are unrelated to $π$ . Thus, for any $π \in (0, 1)$ , the optimal number of practices and practice size are the same as in Section 3.2.1. First, when the provider size K is a predetermined value, then the corresponding variance of the estimated treatment effect for the OD is

\begin{aligned} Var (\hat{β_{2}}) = \frac{1 + (K - 1) γ + K (n_{MMD} - 1) ρ}{K n_{MMD} m_{MMD}} \times g (π, p_{0}, p_{1}), \end{aligned}

(18)

Please note that this variance is an increasing function of

γ

and

ρ .

It is maximized at

γ_{max}

and

ρ_{max}

. Second, the provider size K is not a fixed value but within a range

(K_{min}, K_{max})

and

K_{min} \geq 2

\begin{aligned} Var (\hat{β_{2}}) = \frac{1 + (K_{MMD} - 1) γ + K_{MMD} (n_{MMD} - 1) ρ}{K_{MMD} n_{MMD} m_{MMD}} \times g (π, p_{0}, p_{1}), \end{aligned}

(19)

where

(r_{MMD}, ρ_{MMD}, n_{MMD}, K_{MMD},)

are obtained in Step 3 in the previous section. After inserting (4), LOD for

π

, and

m_{MMD} = \frac{B}{c + (s + e K) n_{MMD}}

in equation (18) or (19), we have the updated variance of the estimated treatment effect

Var (\hat{β_{2}}) = f (p_{0}, \; p_{1}) \times (c + s \times n_{MMD} + e K \times n_{MMD}) \times \frac{1 + (K - 1) γ + K (n_{MMD} - 1) ρ}{K n_{MMD} B},

Var (\hat{β_{2}}) = f (p_{0}, \; p_{1}) \times (c + s \times n_{MMD} + e \times K_{MMD} \times n_{MMD}) \times \frac{1 + (K_{MMD} - 1) γ + K_{MMD} (n_{MMD} - 1) ρ}{K_{MMD} \times n_{MMD} B},

Similarly, RE for group allocation proportion

π_{LOD}

π

is the same as equation (10). That is,

π_{MMD}

is also the same as equation (11).

In summary, the approaches to find a MMD are detailed as follows: (a)

when the provider size K is a predetermined value, Step 1.

Define the parameter space $(γ_{min}, γ_{max}), \; (ρ_{min}, ρ_{max})$ , $p_{0} \in [p_{a}, p_{b}]$ and $p_{1} \in [p_{c}, p_{d}] .$

Step 2.

Select a treatment effect measure from RD, RR, and OR.

Step 3.

Calculate $n_{MMD}$ , $m_{MMD}$ , and $π_{MMD}$ using equations (16) and (11).

Step 4.

Obtain $m_{t - MMD} = m_{MMD} \times π_{MMD}$ and $m_{c - MMD} = m_{MMD} \times (1 - π_{MMD})$ .

The proposed OD is a design with the number of clusters of

m_{t - MMD}

, and

m_{c - MMD}

with the equal practice size of

n_{MMD}

and provider size of K.

(b)

When the provider size K is not a fixed value but within a range $(K_{min}, K_{max})$ and $K_{min} \geq 2$ , Step 1.

Define the parameter space $(γ_{min}, γ_{max}), \; (ρ_{min}, ρ_{max})$ and design space $(K_{min}, K_{max}) .$

Step 2.

Calculate n from equation (16) for each value of K, and take the minimum of $RE (n, γ_{min}, \; ρ_{min}),$ $RE (n, γ_{min}, \; ρ_{max}),$ $RE (n, γ_{max}, \;\; ρ_{min}),$ and $RE (n, γ_{max}, \; ρ_{max})$ , where RE is defined as equation (17).

Step 3.

The design with the maximum of RE in Step 2 is denoted as $(γ_{MMD}, ρ_{MMD}, n_{MMD}, K_{MMD},)$ and then

m_{MMD} = \frac{B}{c + (s + e \times K_{MMD}) n_{MMD}} .

Step 4.

Calculate $π_{MMD}$ using equation (11).

Step 5.

Obtain $m_{t - MMD} = m_{MMD} \times π_{MMD}$ and $m_{c - MMD} = m_{MMD} \times (1 - π_{MMD})$ .

The proposed OD is a design with the number of clusters of $m_{t - MMD}$ and $m_{c - MMD}$ with the equal practice and provider size of $(n_{MMD}, K_{MMD})$ .

The ODs for three-level PA-CRTs are summarized in Table 2. We revise SAS macro, originally developed by Liu et al.,²¹ %OD-3Level-FixedAllo to find the LOD and MMD of number of practices and practice sizes. We develop SAS macros %OD-3Level-OptAllo to find the LOD and MMD of group allocation proportion, the number of practices, and practice sizes.

4. Partially nested designs

PNDs can be viewed as cluster trials with the total number of clusters, $m = m_{t} + m_{c},$ where $m_{t}$ is the number of clusters in the intervention group with an equal cluster size of n and $m_{c}$ is the number of clusters in the control group with a cluster size of 1. The percentage of subjects allocated to the intervention group is denoted by $π = \frac{m_{t} \times n}{m_{t} \times n + m_{c}} .$ Specifically, $m_{t} = \frac{π}{n - (n - 1) π} m$ and $m_{c} = \frac{n (1 - π)}{n - (n - 1) π} m .$ Assume the exchangeable working correlation structure $R (ρ)$ in the intervention group and $R (ρ) = 1$ in the control group. Through similar derivations in Section 2, we have the following variances of the estimated treatment effect:

\begin{aligned} Var (\hat{β_{2}}) = \frac{n - (n - 1) π}{n m} \times g (n, π, p_{0}, p_{1}), \end{aligned}

(20)

where

g (n, π, p_{0}, p_{1}) = {\begin{array}{l} \frac{p_{1} (1 - p_{1})}{π} (1 + (n - 1) ρ) + \frac{p_{0} (1 - p_{0})}{1 - π}, & identity link function \to RD \\ \frac{1 - p_{1}}{π p_{1}} (1 + (n - 1) ρ) + \frac{1 - p_{0}}{(1 - π) p_{0}}, & log link function \to RR \\ \frac{1 + (n - 1) ρ}{π p_{1} (1 - p_{1})} + \frac{1}{(1 - π) p_{0} (1 - p_{0})}, & logit link function \to OR \end{array}

Here, the function

g (n, π, p_{0}, p_{1})

depends on the cluster size of n in the intervention group, unlike in two-level and three-level PA-CRTs. The total budget B is

\begin{aligned} B = m_{t} (c + n s) + m_{c} s = \frac{π c + n s}{n - (n - 1) π} m . \end{aligned}

(21)

4.1 Local optimal designs

Similarly, this section assumes that $ρ$ and $(p_{0}, p_{1})$ are prespecified. In scenarios when $π$ is a predetermined value, the OD is to find a pair of m and n to minimize the variance of the estimated treatment effect in equation (20) given the constraint in equation (21). If we consider the identity link function, it is equivalent to minimizing

L = \frac{1}{B} (\frac{π c}{n} + s) ([\frac{p_{1} (1 - p_{1}) (1 - ρ)}{π} + \frac{p_{0} (1 - p_{0})}{1 - π}] + \frac{p_{1} (1 - p_{1}) ρ n}{π}) .

It can be shown that when

\begin{aligned} n_{LOD} = \sqrt{\frac{π c}{ρ s} [(1 - ρ) + φ \frac{π}{1 - π}]}, \end{aligned}

(22)

where

φ

defined in Section 2.1.2, the derivative of L equals 0 and L is minimized. The LOD is reached for a known pair value

ρ

and

π

is denoted by

n_{LOD}

, and

\begin{aligned} m_{t - LOD} = \frac{π}{π c + n_{LOD} \times s} B, m_{c - LOD} = \frac{n_{LOD} \times (1 - π)}{π c + n_{LOD} \times s} B . \end{aligned}

(23)

Through the similar procedure, we can obtain

n_{LOD}

when the log and logit link functions are considered. The formulas of

n_{LOD},

m_{t - LOD},

and

m_{c - LOD}

for all three link functions are the same. The proposed LOD is a design with the number of clusters of

m_{t - LOD}

with an equal cluster size of

n_{LOD}

and

m_{c - LOD}

subjects in the control group.

Please note that the group allocation proportion is identical to the percentage of subjects allocated to the intervention group in Sections 2 and 3 since we consider an equal cluster size in two-level PA-CRTs, and the equal practice and provider size in three-level PA-CRTs. As such, the LODs in two-level and three-level PA-CRTs, equations (3) and (15), are independent on $π$ . However, the LOD in the PNDs, equations (22) and (23), depends on $π$ heavily. This makes the OD derivations more complicated when group allocation proportion is undecided. Thus, instead of an undecided $π$ shown in two-level and three-level PA-CRTs, we suggest using a predetermined $π$ in the PNDs.

4.2 Maximin design

Assume that $π$ is a predetermined value. Inserting (22) and (23) in (20) gives the variance of each measure estimator for the OD. Unlike equations (6) and (17), RE in the PNDs contains $(p_{0}, p_{1})$ since $g (n, π, p_{0}, p_{1})$ depends on $π$ in equation (20). To simplify the case, we also assume $(p_{0}, p_{1})$ are predetermined. Under the assumptions that $π$ and $(p_{0}, p_{1})$ are predetermined, when $ρ$ increases, $n_{LOD}$ decreases and the variance estimator under the LODs is an increasing function of $ρ$ since $n_{LOD} \times ρ$ increases for $n_{LOD} > 1$ . Therefore, the variance of LODs is maximized at $ρ = ρ_{max}$ .

When $π$ and $(p_{0}, p_{1})$ are predetermined, the MMD algorithm shows as follows:

Step 1.
Define the parameter space, $(ρ_{min}, ρ_{max})$ .
Step 2.
Consider $ρ = ρ_{max}$ , calculate $n_{MMD}$ , $m_{t - MMD}$ , and $m_{c - MMD}$ using equations (22) and (23).

The proposed OD is a design with the number of clusters of $m_{t - MMD}$ with an equal cluster size of $n_{MMD}$ and $m_{c - MMD}$ subjects in the control group. The ODs for PNDs are detailed in Table 3. We develop SAS macros %OD_PND to find the LOD and MMD for number of clusters and cluster sizes in the intervention group and number of subjects in the control group when $π$ is predetermined.

Table 3.
Optimal designs for PNDs.

Association parameter ρ $(p_{0}, p_{1})$ $π$ $m_{t - OD}$ $n_{OD}$ $m_{c - OD}$

Known Known Predetermined $\frac{π}{π c + n_{OD} \times s} B$ $\sqrt{\frac{π c}{ρ s} [(1 - ρ) + φ \frac{π}{1 - π}]}$ $\frac{n_{OD} \times (1 - π)}{π c + n_{OD} \times s} B$

ρ range $(ρ_{min}, ρ_{max})$ Known Predetermined $\frac{π}{π c + n_{OD} \times s} B$ $\sqrt{\frac{π c}{ρ_{max} s} [(1 - ρ_{max}) + φ \frac{π}{1 - π}]}$ $\frac{n_{OD} \times (1 - π)}{π c + n_{OD} \times s} B$

$m :$ total number of clusters $= m_{t} + m_{c}$ ; $m_{t}$ : the number of clusters in the intervention group with an equal cluster size of n; $m_{c}$ is the number of subjects in the control group; $n :$ number of subjects per cluster in the intervention group; $π :$ % subjects allocated to intervention group.

Total budget $B = m_{t} (c + s n) + m_{c} s$ ; $c :$ cost per cluster; $s :$ cost per subject; $φ = {\begin{array}{ll} \frac{p_{0} (1 - p_{0})}{p_{1} (1 - p_{1})}, & for\ RD \\ \frac{p_{1} (1 - p_{0})}{p_{0} (1 - p_{1})}, & for\ RR \\ \frac{p_{1} (1 - p_{1})}{p_{0} (1 - p_{0})}, & for\ OR \end{array} .$

5. Examples

Association parameter ρ	$(p_{0}, p_{1})$	$π$	$m_{t - OD}$	$n_{OD}$	$m_{c - OD}$
Known	Known	Predetermined	$\frac{π}{π c + n_{OD} \times s} B$	$\sqrt{\frac{π c}{ρ s} [(1 - ρ) + φ \frac{π}{1 - π}]}$	$\frac{n_{OD} \times (1 - π)}{π c + n_{OD} \times s} B$
ρ range $(ρ_{min}, ρ_{max})$	Known	Predetermined	$\frac{π}{π c + n_{OD} \times s} B$	$\sqrt{\frac{π c}{ρ_{max} s} [(1 - ρ_{max}) + φ \frac{π}{1 - π}]}$	$\frac{n_{OD} \times (1 - π)}{π c + n_{OD} \times s} B$

5.1 Helping hands trial

The Netherlands Organization for Health Research and Development ZonMw conducted the Helping Hands trial (grant number: 80-007028-98-07101), and the objective was to improve nurse behavior of guideline adherence through two strategies. The state-of-the-art strategy was derived from the literature including education, reminders, feedback, and targeting adequate products and facilities, and the extended strategy contained 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 nurse's adherence to hygiene guidelines or not. The researchers expected to improve the adherence from 60% in the state-of-the-art strategy to 70% in the extended strategy. These OD results for RD, RR, and OR are provided in Tables 4 and 5. In the following paragraphs, we use the ODs for RD as an illustration.

Table 4.
Optimal designs for an example in two-level PA-CRTs.

RD RR OR

Association parameter ρ and $(p_{0}, p_{1})$ $π$ $π_{OD}$ $m_{OD}$ $n_{OD}$ Power /RE^[1] $π_{OD}$ $m_{OD}$ $n_{OD}$ Power/RE^[1] $π_{OD}$ $m_{OD}$ $n_{OD}$ Power/RE^[1]

$ρ = 0.03$ $p_{0} = 0.60$ $p_{1} = 0.70$ 0.50 NA 71.5 18.0 0.868 NA 71.5 18.0 0.860 NA 71.5 18.0 0.862

Undecided 0.483 71.5 18.0 0.868 0.445 71.5 18.0 0.864 0.517 71.5 18.0 0.863

ρ range $(0.017, 0.221)$ $p_{0} \in [0.1, 0.6]$ $p_{1} \in [0.2, 0.9]$ 0.5 NA 92.1 11.7 0.899 NA 92.1 11.7 0.899 NA 92.1 11.7 0.899

Undecided 0.476 92.1 11.7 0.899 0.405 92.1 11.7 0.899 0.524 92.1 11.7 0.899

		RD	RR	OR
$ρ = 0.03$ $p_{0} = 0.60$ $p_{1} = 0.70$	0.50	NA	71.5	18.0	0.868	NA	71.5	18.0	0.860	NA	71.5	18.0	0.862
Undecided	0.483	71.5	18.0	0.868	0.445	71.5	18.0	0.864	0.517	71.5	18.0	0.863
ρ range $(0.017, 0.221)$ $p_{0} \in [0.1, 0.6]$ $p_{1} \in [0.2, 0.9]$	0.5	NA	92.1	11.7	0.899	NA	92.1	11.7	0.899	NA	92.1	11.7	0.899
Undecided	0.476	92.1	11.7	0.899	0.405	92.1	11.7	0.899	0.524	92.1	11.7	0.899

$m :$ number of inpatient wards; $n :$ number of nurses per ward; $π :$ % inpatient wards allocated to the extended strategy.

Total budget $B = 400, 000$ ; $c =$ 2000; $s =$ 200.

^[1]

Power for LOD; RE for MMD.

Table 5.

Optimal designs for an example in three-level PA-CRTs.

			RD					RR					OR
Association parameters $(γ, ρ)$ and $(p_{0}, p_{1})$	$π$	$K$	$π_{OD}$	$m_{OD}$	$n_{OD}$	$K_{OD}$	Power/RE^[1]	$π_{OD}$	$m_{OD}$	$n_{OD}$	$K_{OD}$	Power/RE^[1]	$π_{OD}$	$m_{OD}$	$n_{OD}$	$K_{OD}$	Power/RE^[1]
$γ = 0.6$ $ρ = 0.03$ $p_{0} = 0.60$ $p_{1} = 0.70$	0.5	4	NA	64.2	10.6	NA	0.789	NA	64.2	10.6	NA	0.780	NA	64.2	10.6	NA	0.783
	0.5	range $(3, 6)$	NA	66.1	11.6	3	0.812	NA	66.1	11.6	3	0.803	NA	66.1	11.6	3	0.806
	Undecided	4	0.483	64.2	10.6	NA	0.790	0.445	64.2	10.6	NA	0.785	0.517	64.2	10.6	NA	0.783
	Undecided	range $(3, 6)$	0.483	66.1	11.6	3	0.812	0.445	66.1	11.6	3	0.807	0.517	66.1	11.6	3	0.806
$γ$ range $(0. 5, 0.9)$ ; ρ range $(0.017, 0.221)$ ; $p_{0} \in [0.1, 0.6]$ $p_{1} \in [0.2, 0.9]$	0.5	4	NA	85.7	6.7	NA	0.866	NA	85.7	6.7	NA	0.866	NA	85.7	6.7	NA	0.866
	0.5	range $(3, 6)$	NA	87.3	7.4	3	0.871	NA	87.3	7.4	3	0.871	NA	87.3	7.4	3	0.871
	Undecided	4	0.476	85.7	6.7	NA	0.866	0.405	85.7	6.7	NA	0.866	0.524	85.7	6.7	NA	0.866
	Undecided	range $(3, 6)$	0.476	87.3	7.4	3	0.871	0.405	87.3	7.4	3	0.871	0.524	87.3	7.4	3	0.871

$K :$ the number of evaluation per nurse ; $m :$ number of inpatient wards; $n :$ number of nurses per ward; $π :$ % inpatient wards allocated to the extended strategy.

Total budget $B = 400, 000$ ; $c =$ 2000; $s =$ 200.

^[1]

Power for LOD; RE for MMD.

Table 6.

Optimal designs for an example in PNDs.

		RD			RR			OR
Association parameter ρ and $(p_{0}, p_{1})$	$π$	$m_{OD}$	$n_{OD}$	Power /RE^[1]	$m_{OD}$	$n_{OD}$	Power/RE^[1]	$m_{OD}$	$n_{OD}$	Power/RE^[1]
$ρ = 0.05$ $p_{0} = 0.51$ $p_{1} = 0.67$	0.50	$m_{t} = 14.7$ $m_{c} = 212.4$	14.4	0.842	$m_{t} = 13.0$ $m_{c} = 221.1$	17.0	0.847	$m_{t} = 15.4$ $m_{c} = 208.9$	13.5	0.817
ρ range $(0.017, 0.221)$ $p_{0} = 0.51$ $p_{1} = 0.67$	0.50	$m_{t} = 24.7$ $m_{c} = 162.4$	6.6	0.657	$m_{t} = 22.2$ $m_{c} = 174.8$	7.9	0.682	$m_{t} = 25.7$ $m_{c} = 157.6$	6.1	0.622

$m = m_{t} + m_{c}$ ; $m_{c}$ : the number of patients in the waitlist control; $m_{t}$ : the number of training groups with an equal cluster size of n; $n :$ number of patients per training group; $π :$ % patients allocated to the intervention.

Total budget $B = 114, 400$ ; $c =$ 2000; $s =$ 200.

^[1]

Power for LOD; RE for MMD.

First, we propose a two-level PA-CRT, where inpatient wards are randomized to either strategy. The primary outcome from each nurse is collected at the end of study.

Teerenstra et al. supposed the constant intra-ward coefficient correlation $ρ = 0.03$ .²⁰ We assume the cost as follows, $B = 400, 000, c = 2000, s = 200$ . These OD results for two-level PA-CRTs are shown in Table 4. When $ρ = 0.03$ and $π = 0.50,$ 71.5 wards with 18 nurses per ward will obtain the power maximum of 86.8%. When $π$ is undecided, 71.5 wards with 18 nurses per ward and $π_{OD} = 0.483$ will obtain the maximum power of 86.8%. On the other hand, if the researchers have no clear pictures of the association, 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 0.221.¹ For the range of $ρ = (0.017, 0.221)$ and $π = 0.5$ , MMDs show that 92.1 wards with 11.7 nurses per ward will obtain the RE of 0.899. In addition, if $p_{0}$ and $p_{1}$ are assumed to be unknown but the ranges of $p_{0}$ and $p_{1}$ are, for example, $p_{0} \in [0.1, 0.6],$ $p_{1} \in [0.2, 0.9] .$ When $π$ is undecided, 92.1 wards with 11.7 nurses per ward and $π_{OD} = 0.476$ will obtain the RE of 0.899.

Second, we propose a three-level PA-CRT. This trial randomizes the inpatient wards to either strategy and collects the multiple evaluations of primary outcome from each nurse. Teerenstra et al. considered the constant behavior of nurse $γ = 0.6$ and intra-ward coefficient correlation $ρ = 0.03$ when they proposed the sample size formula using GEE models in three-level PA-CRTs.²⁰ We assume the cost as follows, for example, $B = 400, 000, c = 2000, s = 200, e = 50$ . These OD results for three-level PA-CRTs are shown in Table 5. Using an equal allocation $π = 0.5$ and $K = 4$ , the total number of wards $m = 64.2$ and $n = 10.6$ is proposed to achieve 78.9% power. When $3 \leq K \leq 6$ , LOD is reached at $m = 66.1, n = 11.6$ , and $K = 3$ with a power of 81.2%. As Teerenstra et al. mentioned, the behavior of an individual nurse with respect to hand hygiene is constant, the parameter space $0.5 \leq γ \leq 0.9$ is reasonably assumed. For the range of $ρ = (0.017, 0.221),$ $0.5 \leq γ \leq 0.9,$ $π = 0.5$ and $K = 4$ , MMDs show that 85.7 wards with 6.7 nurses per ward will obtain the RE of 0.866. If $p_{0}$ and $p_{1}$ are assumed to be unknown but the ranges of $p_{0}$ and $p_{1}$ are $p_{0} \in [0.1, 0.6],$ $p_{1} \in [0.2, 0.9]$ , then (a) when $π$ is undecided, 85.7 wards with 6.7 nurses per ward and $π_{OD} = 0.476$ will obtain the RE of 0.866; (b) when $3 \leq K \leq 6$ , MMD is reached at $m = 87.3, n = 7.4$ and $K = 3$ with the RE of 0.871.

When $π$ is undecided, the optimal group allocation proportions $π_{OD}$ are consistent with the findings shown in Figure 1. For example, $π_{OD}$ for RR (0.445) is less than $π_{OD}$ for RD (0.483) but both $π_{OD}$ 's are <0.5, while $π_{OD}$ for RR >0.5 for OR (0.517) in LODs. For practical use, we suggest choosing $int\; (m_{OD}) + 1$ , where “int” refers to an integer part of a number, as the optimal number of ward in order to obtain more power; either $int\; (n_{OD}) + 1$ or $int\; (n_{OD})$ as the optimal number of nurses per ward, depending on the budget.

5.2 SPIN-CHAT trial

The videoconference-based intervention was designed to improve symptoms of anxiety and other mental health outcomes among patients with systemic sclerosis who were at risk of poor mental health during the COVID-19 pandemic. The SPIN-CHAT Trial was conducted to evaluate its effect, where all enrolled patients were randomized to receive either the intervention or waitlist control with a 1:1 allocation ratio.⁹ For patients randomized to the intervention, sessions were delivered using the GoToMeeting® video conferencing platform and 8 patients were assigned to each training group to maximize effective interaction and participation. Patients randomly assigned to the waitlist control were not clustered and only completed trial measures. This is a PND study because of the correlation among patients within training groups in the intervention but not in the waitlist control. The primary outcome of PROMIS Anxiety 4a score immediately postintervention was a continuous outcome.^9,30 Thombs et al. also considered anxiety symptom reduction of at least 1 minimal clinically important difference (MCID) as an outcome.³⁰ Table 2 from 161 patients showed that the proportions of patients who have PROMIS Anxiety 4a score with $\geq$ 1 MCID at 6 weeks postintervention in the intervention and waitlist control groups are 67% and 51%, respectively.³⁰ Using our variance estimator (20) in Section 4, $π = 0.5,$ and $ρ = 0.05$ ,⁹ 22 training groups with eight patients per training group $(n = 8)$ and 176 patients in the waitlist control are required to obtain approximately 80.0% power at the 5% type I error $.$

We assume the cost as follows, $B = 114, 400, c = 2000, s = 200$ . The OD results for RD, RR, and OR are provided in Table 6. The design of 14.7 training groups with 14.4 patients per training group and 212.4 patients in the waitlist control obtains the power of 84.2%. For the range of $ρ = (0.017, 0.221)$ , MMDs show that 24.7 training groups with 6.6 patients per training group and 162.4 patients in the waitlist control obtain the RE of 65.7%. For practical use, we suggest choosing $int\; (m_{t}) + 1$ as the optimal number of training group; $int\; (n_{OD})$ as the optimal number of patients per training group; and $(int\; (m_{t}) + 1) \times int\; (n_{OD})$ as the optimal number of patients in the waitlist control.

6. Discussion

This paper discusses the two-group PA-CRTs with a binary outcome and assumes equal unit costs in the two groups. When using the different link functions, the coefficient $β_{1}$ of the treatment effect in the GEE model has relationships with RD, RR, and OR. We assume the exchangeable working correlation structure and an equal cluster size in two-level PA-CRTs and PNDs; and the “nested exchangeable” correlation structure, the equal practice size, and provider size in three-level PA-CRTs. We have summarized all available ODs, when the group allocation proportion is predetermined, for two-level and three-level PA-CRTs. When the group allocation proportion is undecided, we have proposed the optimal group allocation proportion, the number of clusters, and the cluster size in two-level PA-CRTs; optimal group allocation proportion, number of practices, and practice size under two scenarios of provider size in three-level PA-CRTs. For PNDs, the optimal number of clusters per group and cluster size are obtained when the percentage of subjects allocated to the intervention is predetermined. Obviously, the optimal number of clusters and cluster sizes (equations (3) and (7)), and the number of practices and practice sizes (equations (15) and (16)) may be noninteger. Given the fact that the number of clusters plays a more important role than the cluster size in the power estimation, we suggest using $int\; (m_{OD}) + 1$ as the optimal number of clusters in practice. The proposed optimal cluster size is either $int\; (n_{OD}) + 1$ or $int\; (n_{OD})$ , depending on the budget.

In addition to offering a review of the existing methods, we have contributed fourfold. First, for two-level PA-CRTs, we have summarized the ODs with a predetermined group allocation proportion $π$ and proposed the ODs with an optimized group allocation proportion when $π$ is undecided; for three-level PA-CRTs, when a group allocation proportion is predetermined, we have summarized the research of Liu et al.²¹ Then, we have extended the ODs with an optimized group allocation proportion when $π$ is undecided. For a special case of $K = 1$ , we have noted that the ODs are the same as those proposed in two-level PA-CRTs. Second, we have noticed that when the group allocation proportion is undecided, the OD of group allocation proportion $π_{LOD}$ is independent of the association parameter $ρ$ in two-level PA-CRTs and association parameters ( $γ, ρ$ ) in three-level PA-CRTs. Third, for PNDs, Moerbeek et al. considered multilevel linear regression models and proposed a LOD of RD for binary outcomes.²³ We have proposed LODs and MMDs for three measures including RD, RR, and OR. Last, we have developed SAS macros, %OD_PND, %OD_2Level-OptAllo, and %OD_3Level-OptAllo, and revised two SAS macros, %OD-2Level-FixedAllo and %OD-3Level-FixedAllo, to find LOD and MMD for practical use.

For two-level, two-group PA-CRTs with continuous outcomes, Van Breukelen et al. considered the generalized mixed models and proposed the ODs including unequal variance, homogeneous and heterogeneous costs, unequal ICCs, and cluster sizes between two groups.³¹ Van Breukelen et al. also proposed the MMD of CRTs with heterogeneous costs and variances.²⁶ The assumptions of the common variance and ICC between two groups are the limitations of the GEE models.

Our research could be extended in several directions. First, this paper assumes an equal cluster size. The extension to unequal cost is worthy of further investigation. For example, assume the study cost per practice is $c_{q}$ currency units (e.g., $US), and each provider costs $s_{q}$ currency units, $e_{q}$ denotes each subject's cost in group $q, q = 0, 1$ . The total budget B in a three-level PA-CRT is defined as $B = m_{cont} (c_{0} + s_{0} n + e_{0} K n) + m_{trt} (c_{1} + s_{1} n + e_{1} K n) .$ Second, we assume that the allocation proportion $π$ is a predetermined value in PNDs. How to optimize $π$ could be considered as well. Third, a cluster randomized crossover trial (CRCT) allows each cluster to randomly receive both intervention and control intervention during consecutive time periods. In the future, we plan to develop LODs and MMDs in the CRCTs. Fourth, a stepped wedge design (SW-CRT) is a type of three-level CRT in which all subjects within a cluster receive the same treatment at each step. All clusters receive the control at step 0; at each subsequent step, some clusters initiate the intervention and keep the intervention afterward. All clusters receive the intervention eventually. This unique characteristic of SW-CRTs addresses many ethical concerns. It also bypasses logistical barriers, for example, geographical concerns, by allowing the researchers to assign the intervention to a smaller fraction of the enrolled clusters at each step. Therefore, SW-CRTs are becoming increasingly popular.^32–39 Copas et al. defined two types of SW-CRTs: (a) in a closed cohort SW-CRT, the same subject is followed over time with outcomes measured at every step, while (b) in a repeated cross-section SW-CRT, different subjects are enrolled at each step and the outcomes are collected within the step.⁴⁰ We plan to propose the ODs for both types of SW-CRTs.

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Footnotes

Acknowledgments

We thank the Alvin J. Siteman Cancer Center at Washington University School of Medicine and Barnes-Jewish Hospital in St Louis, MO (P30 CA91842), National Institutes of Health (NIH) grants U01CA2098611 and P50 CA244431, Institute of Clinical and Translational Sciences (ICTS) grant CTRFP2019-05, for supporting this research. Lei Liu's work was supported by the Washington University ICTS grant UL1TR000448 from the National Center for Advancing Translational Sciences (NCATS) of the National Institutes of Health (NIH). Aimee James's was supported by NIH grant R01CA233848. The content is solely the responsibility of the authors and does not necessarily represent the official view of the NIH.

Declaration of conflicting interests

The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: Dr Lei Liu is a consultant to Adial LLC.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: P30 CA91842, U01CA2098611, P50 CA244431, CTRFP2019-05, UL1TR000448, R01CA233848.

ORCID iDs

Jingxia Liu

Lei Liu

Supplemental material

Supplemental material for this article is available online.

Appendix 1

Insert $π_{LOD} = \frac{1}{1 + \sqrt{φ}}$ into $g (π, p_{0}, p_{1}) .$ (1)

For identity link function, $φ = \frac{p_{0} (1 - p_{0})}{p_{1} (1 - p_{1})}$ and $g (π, p_{0}, p_{1}) = \frac{p_{1} (1 - p_{1})}{π} + \frac{p_{0} (1 - p_{0})}{1 - π}$ , then

g (π, p_{0}, p_{1}) = p_{1} (1 - p_{1}) (1 + \sqrt{φ}) + \frac{p_{0} (1 - p_{0}) (1 + \sqrt{φ})}{\sqrt{φ}}

= p_{1} (1 - p_{1}) + \sqrt{p_{0} (1 - p_{0}) p_{1} (1 - p_{1})} + p_{0} (1 - p_{0}) + \sqrt{p_{0} (1 - p_{0}) p_{1} (1 - p_{1})}

= p_{1} (1 - p_{1}) + 2 \sqrt{p_{0} (1 - p_{0}) p_{1} (1 - p_{1})} + p_{0} (1 - p_{0})

= {(\sqrt{p_{1} (1 - p_{1})} + \sqrt{p_{0} (1 - p_{0})})}^{2};

(2)

For log link function, $φ = \frac{p_{1} (1 - p_{0})}{p_{0} (1 - p_{1})}$ and $g (π, p_{0}, p_{1}) = \frac{1 - p_{1}}{π p_{1}} + \frac{1 - p_{0}}{(1 - π) p_{0}}$ , then

g (π, p_{0}, p_{1}) = \frac{1 - p_{1}}{p_{1}} (1 + \sqrt{φ}) + \frac{1 - p_{0}}{p_{0}} \frac{1 + \sqrt{φ}}{\sqrt{φ}}

= \frac{1 - p_{1}}{p_{1}} + \sqrt{\frac{(1 - p_{0}) (1 - p_{1})}{p_{0} p_{1}}} + \frac{1 - p_{0}}{p_{0}} + \frac{1 - p_{0}}{p_{0}} \sqrt{\frac{p_{0} (1 - p_{1})}{p_{1} (1 - p_{0})}}

= \frac{1 - p_{1}}{p_{1}} + 2 \sqrt{\frac{(1 - p_{0}) (1 - p_{1})}{p_{0} p_{1}}} + \frac{1 - p_{0}}{p_{0}}

= {(\sqrt{\frac{1 - p_{1}}{p_{1}}} + \sqrt{\frac{1 - p_{0}}{p_{0}}})}^{2};

(3)

For logit link function, $φ = \frac{p_{1} (1 - p_{1})}{p_{0} (1 - p_{0})}$ and $g (π, p_{0}, p_{1}) = \frac{1}{π p_{1} (1 - p_{1})} + \frac{1}{(1 - π) p_{0} (1 - p_{0})}$ , then

g (π, p_{0}, p_{1}) = \frac{(1 + \sqrt{φ})}{p_{1} (1 - p_{1})} + \frac{(1 + \sqrt{φ})}{\sqrt{φ} p_{0} (1 - p_{0})}

= \frac{1}{p_{1} (1 - p_{1})} + \sqrt{\frac{1}{p_{0} (1 - p_{0}) p_{1} (1 - p_{1})}} + \frac{1}{p_{0} (1 - p_{0})} + \sqrt{\frac{1}{p_{0} (1 - p_{0}) p_{1} (1 - p_{1})}}

= \frac{1}{p_{1} (1 - p_{1})} + 2 \sqrt{\frac{1}{p_{0} (1 - p_{0}) p_{1} (1 - p_{1})}} + \frac{1}{p_{0} (1 - p_{0})}

= {(\sqrt{\frac{1}{p_{1} (1 - p_{1})}} + \sqrt{\frac{1}{p_{0} (1 - p_{0})}})}^{2} .

Appendix 2

For any $π \in (0, 1)$ , the variance of estimated treatment effect for the OD is equation (8), where the optimal number of clusters and cluster size are provided in Section 2.2.1. RE is defined as the ratio of the variance of estimated treatment effect for group allocation proportion $π_{LOD}$ to $π,$ where $π_{LOD} = \frac{1}{1 + \sqrt{φ}}$ . The RE is calculated using equations (8)–(9) and $m_{MMD} = B / (c + s \times n_{MMD}),$

RE (π, \; φ) = \frac{f (p_{0}, \; p_{1}) \times (c + s \times n_{MMD}) \times \frac{1 + (n_{MMD} - 1) ρ}{n_{MMD} B}}{\frac{1 + (n_{MMD} - 1) ρ}{n_{MMD} m_{MMD}} \times g (π, p_{0}, p_{1})} = \frac{f (p_{0}, \; p_{1})}{g (π, p_{0}, p_{1})} .

(1)

For identity link function, $g (π, p_{0}, p_{1}) = \frac{p_{1} (1 - p_{1})}{π} + \frac{p_{0} (1 - p_{0})}{1 - π}$ and $f (p_{0}, \; p_{1}) = {(\sqrt{p_{1} (1 - p_{1})} + \sqrt{p_{0} (1 - p_{0})})}^{2},$ then $RE (π, \; φ) = \frac{{(\sqrt{p_{1} (1 - p_{1})} + \sqrt{p_{0} (1 - p_{0})})}^{2}}{\frac{p_{1} (1 - p_{1})}{π} + \frac{p_{0} (1 - p_{0})}{1 - π}} = \frac{{(1 + \sqrt{\frac{p_{0} (1 - p_{0})}{p_{1} (1 - p_{1})}})}^{2}}{\frac{1}{π} + \frac{p_{0} (1 - p_{0})}{(1 - π) p_{1} (1 - p_{1})}} = \frac{{(1 + \sqrt{φ})}^{2}}{\frac{1}{π} + \frac{φ}{1 - π}} .$

(2)

For log link function, $g (π, p_{0}, p_{1}) = \frac{1 - p_{1}}{π p_{1}} + \frac{1 - p_{0}}{(1 - π) p_{0}}$ and $f (p_{0}, \; p_{1}) = {(\sqrt{\frac{1 - p_{1}}{p_{1}}} + \sqrt{\frac{1 - p_{0}}{p_{0}}})}^{2},$ then $RE (π, \; φ) = \frac{{(\sqrt{\frac{1 - p_{1}}{p_{1}}} + \sqrt{\frac{1 - p_{0}}{p_{0}}})}^{2}}{\frac{1 - p_{1}}{π p_{1}} + \frac{1 - p_{0}}{(1 - π) p_{0}}} = \frac{{(1 + \sqrt{\frac{1 - p_{0}}{p_{0}} \times \frac{p_{1}}{1 - p_{1}}})}^{2}}{\frac{1}{π} + \frac{1}{1 - π} \times \frac{1 - p_{0}}{p_{0}} \times \frac{p_{1}}{1 - p_{1}}} = \frac{{(1 + \sqrt{φ})}^{2}}{\frac{1}{π} + \frac{φ}{1 - π}} .$

(3)

For logit link function, $g (π, p_{0}, p_{1}) = \frac{1}{π p_{1} (1 - p_{1})} + \frac{1}{(1 - π) p_{0} (1 - p_{0})}$ and $f (p_{0}, \; p_{1}) = {(\sqrt{\frac{1}{p_{1} (1 - p_{1})}} + \sqrt{\frac{1}{p_{0} (1 - p_{0})}})}^{2},$ then $RE (π, \; φ) = \frac{{(\sqrt{\frac{1}{p_{1} (1 - p_{1})}} + \sqrt{\frac{1}{p_{0} (1 - p_{0})}})}^{2}}{\frac{1}{π p_{1} (1 - p_{1})} + \frac{1}{(1 - π) p_{0} (1 - p_{0})}} = \frac{{(1 + \sqrt{\frac{p_{1} (1 - p_{1})}{p_{0} (1 - p_{0})}})}^{2}}{\frac{1}{π} + \frac{p_{1} (1 - p_{1})}{(1 - π) p_{0} (1 - p_{0})}} = \frac{{(1 + \sqrt{φ})}^{2}}{\frac{1}{π} + \frac{φ}{1 - π}} .$

Appendix 3

Suppose $p_{0} \in [p_{a}, p_{b}]$ and $p_{1} \in [p_{c}, p_{d}]$ . Let $\underline{φ} = min φ$ and $\bar{φ} = max φ .$ Note that $p (1 - p)$ is an increasing function, maximized at p = 0.5, and a decreasing function of p. Therefore, A.

If $p_{a} \geq 0.5$ , then $p_{0} (1 - p_{0})$ is maximized at $p_{a}$ and minimized at $p_{b}$ .

If $p_{b} \leq 0.5$ , then $p_{0} (1 - p_{0})$ is minimized at $p_{a}$ and maximized at $p_{b}$ .

If $p_{a} \leq 0.5 \leq p_{b}$ , then $p_{0} (1 - p_{0})$ is minimized at $p_{a}$ or $p_{b}$ and maximized at 0.5.

If $p_{c} \geq 0.5$ , then $p_{1} (1 - p_{1})$ is maximized at $p_{c}$ and minimized at $p_{d}$ .

If $p_{d} \leq 0.5$ , then $p_{1} (1 - p_{1})$ is minimized at $p_{c}$ and maximized at $p_{d}$ .

If $p_{c} \leq 0.5 \leq p_{d}$ , then $p_{1} (1 - p_{1})$ is minimized at $p_{c}$ or $p_{d}$ and maximized at 0.5.

(1)

For identity link function, $φ = \frac{p_{0} (1 - p_{0})}{p_{1} (1 - p_{1})}$ .

Obviously, $\underline{φ}$ is reached when $p_{0} (1 - p_{0})$ is minimized and $p_{1} (1 - p_{1})$ is maximized. From the scenarios of A–C, the minimum of $p_{0} (1 - p_{0})$ is $min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))$ . On the other hand, from the scenarios of D–E, the maximum of $p_{1} (1 - p_{1})$ is $max (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))$ . If $p_{c} \leq 0.5 \leq p_{d}$ , then max $p_{1} (1 - p_{1}) = 0.25$ .

$\bar{φ}$ is reached when $p_{0} (1 - p_{0})$ is maximized and $p_{1} (1 - p_{1})$ is minimized. From the scenarios of A and B, the maximum of $p_{0} (1 - p_{0})$ is $max (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))$ . If $p_{a} \leq 0.5 \leq p_{b}$ , max $p_{0} (1 - p_{0}) = 0.25$ . On the other hand, from the scenarios of D–F, the minimum of $p_{1} (1 - p_{1})$ is $min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))$ .

(2)

For log link function, $φ = \frac{p_{1} (1 - p_{0})}{p_{0} (1 - p_{1})} = \frac{1 - p_{0}}{p_{0}} \times \frac{p_{1}}{1 - p_{1}}$ .

$\frac{1 - p_{0}}{p_{0}} = \frac{1}{p_{0}} - 1$ is a decreasing function of $p_{0}$ , so it reaches the minimum of $\frac{1 - p_{b}}{p_{b}}$ at $p_{b}$ and the maximum of $\frac{1 - p_{a}}{p_{a}}$ at $p_{a}$ ; while $\frac{p}{1 - p} = - 1 + \frac{1}{1 - p}$ is an increasing function, $\frac{p_{1}}{1 - p_{1}}$ is minimized at $p_{c}$ and maximized at $p_{d}$ .

Thus, $\underline{φ} = \frac{1 - p_{b}}{p_{b}} \times \frac{p_{c}}{1 - p_{c}} = \frac{p_{c} (1 - p_{b})}{p_{b} (1 - p_{c})};$ and $\bar{φ} = \frac{1 - p_{a}}{p_{a}} \times \frac{p_{d}}{1 - p_{d}} = \frac{p_{d} (1 - p_{a})}{p_{a} (1 - p_{d})}$ .

(3)

For logit link function, $φ = \frac{p_{1} (1 - p_{1})}{p_{0} (1 - p_{0})}$ .

$\underline{φ}$ is reached when $p_{0} (1 - p_{0})$ is maximized and $p_{1} (1 - p_{1})$ is minimized. From the scenarios of A and B, the maximum of $p_{0} (1 - p_{0})$ is $max (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))$ . If $p_{a} \leq 0.5 \leq p_{b}$ , max $p_{0} (1 - p_{0}) = 0.25$ . On the other hand, from the scenarios of D–F, the minimum of $p_{1} (1 - p_{1})$ is $min (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))$ .

$\bar{φ}$ is reached when $p_{0} (1 - p_{0})$ is minimized and $p_{1} (1 - p_{1})$ is maximized. From the scenarios of A–C, the minimum of $p_{0} (1 - p_{0})$ is $min (p_{a} (1 - p_{a}), p_{b} (1 - p_{b}))$ . On the other hand, from the scenarios of D and E, the maximum of $p_{1} (1 - p_{1})$ is $max (p_{c} (1 - p_{c}), p_{d} (1 - p_{d}))$ . If $p_{c} \leq 0.5 \leq p_{d}$ , then max $p_{1} (1 - p_{1}) = 0.25$ .

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Supplementary Material

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An overview of optimal designs under a given budget in cluster randomized trials with a binary outcome

Abstract

Keywords

1. Introduction

1.1. Notations

2. Two-level PA-CRTs

2.1.1. π is a predetermined value

3.1.1. π is a predetermined value

3.2 Maximin design

3.2.1. π is a predetermined value

5.1 Helping hands trial

6. Discussion

Supplemental Material

sj-sas-3-smm-10.1177_09622802231172026 - Supplemental material for An overview of optimal designs under a given budget in cluster randomized trials with a binary outcome

Supplemental Material

sj-sas-4-smm-10.1177_09622802231172026 - Supplemental material for An overview of optimal designs under a given budget in cluster randomized trials with a binary outcome

Supplemental Material

sj-sas-5-smm-10.1177_09622802231172026 - Supplemental material for An overview of optimal designs under a given budget in cluster randomized trials with a binary outcome

Supplemental Material

sj-sas-6-smm-10.1177_09622802231172026 - Supplemental material for An overview of optimal designs under a given budget in cluster randomized trials with a binary outcome

Supplemental Material

sj-sas-7-smm-10.1177_09622802231172026 - Supplemental material for An overview of optimal designs under a given budget in cluster randomized trials with a binary outcome

Footnotes

Acknowledgments

Declaration of conflicting interests

Funding

ORCID iDs

Supplemental material

Appendix 1

Appendix 2

Appendix 3

References

Supplementary Material

2.1.1. $π$ is a predetermined value

3.1.1. $π$ is a predetermined value

3.2.1. $π$ is a predetermined value