Optimal designs in open-cohort longitudinal cluster randomized trials with a continuous outcome

Abstract

Although sample size calculation for open-cohort longitudinal cluster randomized trials (LCRTs) under a fixed design framework was developed by Kasza et al. (2021), unifying the closed cohort and repeated cross-sectional sampling provided in Hooper et al. (2016) when a churn rate is constant, there has been no prior efforts in developing optimal open-cohort LCRTs that maximizes the design efficiency. This work assumes a pre-specified number of periods $T$ and a constant number of replaced individuals at each period in open-cohort LCRTs. We propose algorithms for deriving optimal sample size under a cost-efficiency framework and arrive at the local optimal design (LOD) for fixed correlation parameters and MaxiMin optimal design for addressing uncertainty in correlation parameters. When correlation parameters are known, as the number of replaced individuals increases, for open-cohort PA-LCRTs, the optimal cluster-period size generally decreases and then increases whereas the optimal number of clusters and power under LOD first increase and then decrease. In contrast, for CRXO trials and standard SW-CRTs, the optimal cluster-period size and churn rate under LOD increase whereas the optimal number of clusters and power under LOD decrease. When correlation parameters are unknown, but the parameter space is available, with a small number of replaced individuals, there is no difference in optimal designs between PA-LCRTs and CRXO trials. The number of replaced individuals also has less impact on the optimal cluster-period size than optimal number of clusters. We demonstrate our new optimal design methods using the context of two real-world LCRTs.

1. Introduction

Cluster randomized trials (CRTs) randomize each intact cluster with individual members, rather than each individual, to a particular treatment condition; a cluster in a CRT may refer to a medical center, clinic, clinician, and classroom, among others. Longitudinal CRTs (LCRTs) are a popular class of cluster randomized designs and include parallel-arm LCRTs (PA-LCRTs), cluster randomized crossover (CRXO) trials, and stepped wedge cluster randomized trials (SW-CRTs). Table 1 demonstrates a PA-LCRT with 1:1 randomization; a two-sequence CRXO trial with 1:1 randomization; and a SW-CRT with 4 treatment sequences, each with 20 clusters and 6 time periods. For these LCRTs, researchers have often considered two different sampling designs including the closed-cohort design, where the same individuals are followed across time periods and the outcome is measured repeated at every time period, and a repeated cross-sectional design, where different individuals are enrolled at each time period and the outcomes are collected once before the end of follow-up.^1,2 Statistical methods for sample size calculation and data analysis in these two sampling designs often assume that every individual completes the entire measurement schedule. However, this assumption may not hold, especially in a closed-cohort design, where some individuals may contribute only a portion of their scheduled measurements, possibly due to earlier withdrawal from the study or later entry into the study in a middle period. In other words, some LCRTs are "open-cohort" designs where each individual is allowed to contribute a different number of measurements. The churn rate (sometimes referred to as the attrition rate) is defined as the proportion of individuals in one time period who do not appear in a different time period.³ When the churn rate approaches one, the open-cohort design becomes the repeated cross-sectional design. On the other hand, as the churn rate approaches zero, the open-cohort design reduces to the closed-cohort design.⁴

Table 1.

Example schematics for three possible longitudinal cluster randomized designs with 20 clusters

(a) An example of a PA-LCRT treatment schedule
Cluster	Treatment sequence	Time Periods
		Period 1	Period 2	Period 3	Period 4	Period 5	Period 6
1, ⋯,10	1	Intervention	Intervention	Intervention	Intervention	Intervention	Intervention
11, ⋯,20	2	Control	Control	Control	Control	Control	Control
(b) An example of a two-sequence CRXO trial treatment schedule
Cluster	Treatment sequence	Time Periods
		Period 1	Period 2	Period 3	Period 4	Period 5	Period 6
1, ⋯,10	1	Intervention	Control	Intervention	Control	Intervention	Control
11, ⋯,20	2	Control	Intervention	Control	Intervention	Control	Intervention
(c) An example of a SW-CRT treatment schedule
Cluster	Treatment sequence	Time Periods
		Period 1 (Baseline)	Period 2 (Step 1)	Period 3 (Step 2)	Period 4 (Step 3)	Period 5 (Step 4)	Period 6 (Extended period)
1,⋯ ,5	1	Control	Intervention	Intervention	Intervention	Intervention	Intervention
6, ⋯ ,10	2	Control	Control	Intervention	Intervention	Intervention	Intervention
11,⋯,15	3	Control	Control	Control	Intervention	Intervention	Intervention
16, ⋯,20	4	Control	Control	Control	Control	Intervention	Intervention

Damsgaard et al. evaluated the effects of providing healthy school meals based on the New Nordic Diet (NND) on a metabolic syndrome score, and individual cardiometabolic markers and body composition in children aged 8–11.⁵ This two-sequence CRXO trial involved 823 Danish schoolchildren across 9 schools. The unit of randomization is year-group within schools. Children received freshly prepared school lunch and snacks or usual packed lunch from home each for three months. Dietary intake and physical activity were measured for seven consecutive days at baseline, months 3 and 6. Participanťs churn rates were 10% and 11.5%, due to school changes or meal preferences. As a second example, through a four-period SW-CRT, Mhurchu et al. investigated the impact of a free school breakfast program over a school year.⁶ The trial involved 424 students across 14 New Zealand schools which initially did not have the free school breakfast program. The primary outcome was school attendance and secondary outcomes included academic performance and hunger levels. This study defined a churn rate as the number of students remained after the last period compared to the enrolled number of students. A 12.5% churn rate was observed and attributed to students transferring schools. These trials demonstrated examples of open-cohort LCRTs with a range of churn rates.

For closed-cohort and repeated cross-sectional CRTs, Liu et al. proposed the optimal designs under a fixed total budget, which refer to the combination of optimal cluster-period size $n$ (number of observations per cluster per period) and optimal number of clusters $m$, based on generalized estimating equations (GEEs).^7,8 Optimal design methods, however, have not been developed to accommodate open-cohort LCRTs, where some individuals may leave the study after one or more periods but are immediately replaced by new individuals. Assuming a constant churn rate and under a linear mixed model, Kasza et al. first developed closed-form variance formulas for sample size calculation in open-cohort LCRTs,³ a unification of the closed cohort and repeated cross-sectional sampling provided in Hooper et al.⁹ Building on this sample size expression, this work develops the theory of local optimal design and MaxiMin optimal design to obtain optimal open-cohort PA-LCRTs, two-sequence multiple-period CRXO trials, as well as SW-CRTs under a cost-efficiency framework. Through this development, we are able to investigate the impact of an assumed number of replaced individuals at each period on optimal open-cohort LCRTs under a cost-efficiency framework. We will develop SAS macros for the corresponding local optimal designs and MaxiMin optimal designs to facilitate the implementation of our methods to improve the design of future open-cohort LCRTs.

The remainder of the article is organized as follows: In Sections 2, we summarize the fixed-design sample size formulae specifically for each open-cohort LCRT. Sections 3 and 4 focus on the development of local optimal designs and MaxiMin optimal designs under known and unknown correlation coefficients, respectively. We further extend the MaxiMin optimal design algorithm to account for the uncertainty in the number of replaced individuals at each period - a unique design parameter in open-cohort LCRTs. We then illustrate the proposed optimal designs through several numerical examples in Section 5 and provide insights into their operating characteristics. Section 6 concludes with an overview of our contributions and limitations, along with possible future directions. Data sharing is not applicable to this article as no new data are created or analyzed in this study.

2. Open-cohort longitudinal cluster randomized trials

2.1. Statistical models

Let $Y_{ijt}$ be a continuous outcome from individual $j=1,\cdots ,n$ in cluster $i=1,\cdots ,m$ and period $t=1,\cdots ,T$,

$$\begin{gathered} Y_{ijt}={\delta }_t+X_{it}\beta +P_i+Q_{it}+{\gamma }_{ij}+{ϵ}_{ijt}, \\ {P}_i\sim N(0,{\sigma }_P^2),Q_{it}\sim N(0,{\sigma }_Q^2),{\gamma }_{ij}\sim N(0,{\sigma }_{\gamma }^2),{ϵ}_{ijt}\sim N(0,{\sigma }_{ϵ}^2), \end{gathered}$$

where $n$ is an equal cluster-period size per period, $m$ is the number of clusters, ${\delta }_t$ is the $t$th period effect, $X_{it}$ is the treatment indicator of cluster $i$ in period $t$ (=1 if receiving intervention, and =0 otherwise), and $\beta$ is the treatment effect. Let $L$ denote the total number of treatment sequences. The cluster-level random effect $P_i\sim N(0,{\sigma }_P^2)$ and cluster-period-level random effect $Q_{it}\sim N(0,{\sigma }_Q^2)$ capture the dependence between multiple measurements on the same individual. Individual-level random effect ${\gamma }_{ij}\sim N(0,{\sigma }_{\gamma }^2)$ captures the dependence between multiple measurements on the same individual. All random effects and the individual error term ${ϵ}_{ijt}\sim N(0,{\sigma }_{ϵ}^2)$ are assumed independent of each other.

Let ${\alpha }_0$ be the constant within-period correlation (between outcomes from different individuals within the same cluster during the same period), ${\alpha }_1$ be the constant inter-period correlation (between outcomes from different individuals within the same cluster but across periods), ${\alpha }_2$ be the constant within-individual correlation (between outcomes from the same individual across periods), and $R$ be the number of replaced individuals at each period respectively. Suppose we wish to test the hypothesis ${\mathrm{H}}_0:\beta =0$ versus ${\mathrm{H}}_1:\beta =\widehat{\beta }$. Based on the above linear mixed model, Kasza et al. obtained the variance of the treatment effect estimator as³

$$\text{Var}(\widehat{\beta })=\frac{{\sigma }^2}{n}\times \frac{m[1+(n-1){\alpha }_0](1-r)[1+(T-1)r]}{m{X}_{\cdot \cdot }-{\sum }_{t=1}^T{(X_{\cdot t})}^2+[{(X_{\cdot \cdot })}^2+m(T-1)X_{\cdot \cdot }-(T-1){\sum }_{t=1}^T{(X_{\cdot t})}^2-m{\sum }_{i=1}^m{(X_{i\cdot })}^2]r},$$ (1)

where ${\sigma }^2={\sigma }_Q^2+{\sigma }_P^2+{\sigma }_{\gamma }^2+{\sigma }_{ϵ}^2$ is the total variance, $X_{\cdot \cdot }={\sum }_{t=1}^T{\sum }_{i=1}^m{X}_{it}$, $X_{\cdot t}={\sum }_{i=1}^m{X}_{it}$, $X_{i\cdot }={\sum }_{t=1}^T{X}_{it}$, are design constants determined by the randomization schedule, $r=\frac{n{\alpha }_0{\alpha }_1+(1-\omega )(1-{\alpha }_0){\alpha }_2}{1+(n-1){\alpha }_0}$ is a function of the correlation parameters as well as the constant churn rate $\omega$. Here, the churn rate $\omega =R∕n$ is defined as the proportion of individuals that will leave the study at the end of each period, or equivalently, the proportion of replaced individuals that will join the study in the next period. We also refer to $r$ as the churn-rate adjusted cluster-period mean correlation (CPMC). Two important special cases emerge from this variance expression. That is, $r=\frac{n{\alpha }_0{\alpha }_1}{1+(n-1){\alpha }_0}$ when $\omega =1$ (repeated cross-sectional LCRTs) and $r=\frac{n{\alpha }_0{\alpha }_1+(1-{\alpha }_0){\alpha }_2}{1+(n-1){\alpha }_0}$ when $\omega =0$ (closed-cohort LCRTs), in which cases the variance expression (1) becomes Equations (5) and (10) in Hooper et al.⁹ and $r$ is the cluster-period mean correlation in Hooper et al.⁹ Thus, Equations (1) unifies the closed cohort and repeated cross-sectional sampling provided in Hooper et al.⁹ In what follows, we first simplify (1) under each specific LCRT design. Throughout, we focus on a two-sided test with a nominal type I error of 5%, a target 80% power, an assumed treatment effect size of $\beta =0.2$ and a total variance of ${\sigma }^2=1$.

2.2. Open-cohort PA-LCRTs

In a PA-LCRT, all individuals within a cluster receive the same treatment condition, either I (intervention) or C (control), at all periods, and individual outcomes are measured during each period when individuals are present, and $\pi$ is the proportion of clusters receiving the I assignment. Here, the number of treatment sequence $L=2$. From the treatment matrix, with an assumption of 1:1 randomization, e.g., $\pi =0.5$, we have

$$X_{\cdot \cdot }=\frac{mT}{2},X_{\cdot t}=\frac{m}{2},$$

for $t=1,\cdots T$ and $X_{1\cdot }=\cdots =X_{\frac{m}{2}\cdot }=T$, $X_{\left(\frac{m}{2}+1\right)\cdot }=\cdots X_{m\cdot }=0$. Then the variance of treatment effect estimator, Equation (1), is simplified as

$$\text{Var}(\widehat{\beta })=\frac{4{\sigma }^2}{Tmn}\times [1+(n-1){\alpha }_0][1+(T-1)r],$$ (2)

and $r>-1∕(T-1)$ is required for a valid variance estimate. The derivation is provided in Appendix 1. When $\omega =1$, $\text{Var}(\widehat{\beta })=\frac{4{\sigma }^2}{Tmn}\times [1+(n-1){\alpha }_0+(T-1)n{\alpha }_0{\alpha }_1]$, an variance of treatment effect estimator for repeated cross-sectional PA-LCRTs; When $\omega =0,\text{Var}(\widehat{\beta })=\frac{4{\sigma }^2}{Tmn}\times \{1+(n-1){\alpha }_0+(T-1)[n{\alpha }_0{\alpha }_1+(1-{\alpha }_0){\alpha }_2]\}$, an variance for closed-cohort PA-LCRTs. The interpretation of (2) is straightforward, that is, the variance under an individually randomized trial, multiplied by the usual design effect due to cluster randomization, $[1+(n-1){\alpha }_0]$, and further multiplied the second design effect due to churn-rate adjusted longitudinal follow-up, $[1+(T-1)r]$.

Appendix Figures 1A-1E present the required number of clusters $m$ across churn rates from 0 to 1 for the different number of periods $T$ and cluster-period size $n$. To obtain at least 80% power of detecting a treatment effect at a two-sided type I error of 5%, the required number of clusters decreases as churn rate $\omega$ increases for a specific value of cluster-period size $n$. Given a fixed value of (${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$), the higher churn rate leads to a smaller churn-rate adjusted CPCM $r$ and then a lower variance of treatment effect estimator, which requires a lower number of clusters. We also notice that the churn rate has a larger impact on the required number of clusters with a smaller cluster-period size $n$, e.g. 5. The decreasing trend of the required number of clusters appears more noticeable when $T$ increases. This is because $\text{Var}(\widehat{\beta })\sim \frac{1}{T}\{1+(T-1)r\}$ decreases more dramatically when $T$ increases with a smaller $r$.

2.3. Open-cohort multiple-period two-sequence CRXO trials

For two-sequence CRXO trials with an even number of periods $T$, the treatment assignment is either IC (intervention in the first period, followed by control in the second period) or CI (control in the first period, followed by intervention in the second period) and repeat the same pattern in the subsequent periods (e.g., ICICIC and CICICI for $T=6$) and $\pi$ is the proportion of clusters receiving the IC treatment sequence. For multiple-period two-sequence CRXO trials, with an assumption of 1:1 randomization, e.g., $\pi =0.5$,

$$X_{\cdot \cdot }=\frac{mT}{2},X_{\cdot t}=\frac{m}{2},$$

for $t=1,\cdots T$ and $X_{1\cdot }=\cdots =\cdots X_{m\cdot }=\frac{m}{2}$. Then the variance of treatment effect estimator is simplified as

$$\text{Var}(\widehat{\beta })=\frac{4{\sigma }^2}{Tmn}[1+(n-1){\alpha }_0](1-r),$$ (3)

and $r<1$ is required for a valid variance estimate. The derivation is provided in Appendix 2. For repeated cross-sectional CRXO trials, $\text{Var}(\widehat{\beta })=\frac{4{\sigma }^2}{Tmn}\times [1+(n-1){\alpha }_0-n{\alpha }_0{\alpha }_1]$, whereas for closed-cohort CRXO trials, $\text{Var}(\widehat{\beta })=\frac{4{\sigma }^2}{Tmn}\times \{1+(n-1){\alpha }_0-[n{\alpha }_0{\alpha }_1+(1-{\alpha }_0){\alpha }_2]\}$. Equation (3) also has a straightforward interpretation, that is, the variance under an individually randomized trial, multiplied by the usual design effect due to cluster randomization, $[1+(n-1){\alpha }_0]$, and further multiplied by one minus the churn-rate adjusted CPMC, $1-r$. This variance expression clearly indicates that adding a crossover component may substantially reduce of the variance compared to a PA-LCRT, as the overall design effect in an open-cohort CRXO trial is no larger than the usual design effect due to cluster randomization.

Similarly, Appendix Figures 2A-2E present the required number of clusters $m$ across churn rates from 0 to 1 for the different period $T$ and cluster-period size $n$. Like PA-LCRTs, the churn rate has a larger impact on the required number of clusters with a smaller cluster-period size $n$, e.g. 5. However, unlike PA-LCRTs, the required number of clusters increases as churn rate $\omega$ increases for a specific value of cluster-period size $n$. Given a fixed value of (${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$), the higher churn rate results in a smaller $r$ and then a higher variance of treatment effect estimator, which is opposite to the patterns observed in PA-LCRTs. At a specific churn rate $\omega$, the required number of clusters can decrease substantially when the number of periods $T$ increases; and thus, the increasing trend of the required number of clusters is not more obvious across the churn rates $\omega$ for a larger value of $T$.

2.4. Open-cohort SW-CRTs

In a typical SW-CRT, 1) all clusters start from the control condition in period 1; 2) at each subsequent period, a randomly chosen subset of clusters receive the intervention condition and maintain the intervention status until the end of the study; 3) at the end of the study, all clusters receive the intervention. The range of number of treatment sequence is $2\le L\le T-1$ in SW-CRTs, where a standard SW-CRT has $L=T-1$. Following Liu and Li,⁸ we assume an equal number of clusters crossover to intervention at each period $h=2,\cdots ,L$. In other words, we consider SW-CRTs including a single baseline period, $L$ periods where the treatment is being rolled out, and $T-L$ periods where the intervention is maintained; see Table 1 for an example schematic. Then, the total number of clusters is $m=hL$, and $X_{\cdot t}={\sum }_{i=1}^m{X}_{it}=h{\sum }_{i=1}^L{X}_{it}$. We can further simplify the design constants as

$$\begin{gathered} X_{\cdot \cdot }=h\frac{L(2T-L-1)}{2}, \\ \sum _{t=1}^T{(X_{\cdot t})}^2=h^{2}L\left\{\frac{1}{6}(L-1)(2L-1)+(T-L)L\right\} \end{gathered}$$

and

$$\sum _{i=1}^m{(X_{i\cdot })}^2=\frac{h}{6}\{T(T-1)(2T-1)-(T-L-1)(T-L)[2(T-L)-1]\}.$$

The derivation is provided in Appendix 3. Under this design with two parameters of $L$ and $T$, we cannot obtain the simplified expressions of variance of treatment effect estimator for SW-CRTs as (2) and (3) for PA-LCRTs and CRXO trials, respectively.

Similarly, Appendix Figures 3A-3E show the required number of clusters $m$ across the churn rates from 0 to 1 for the different number of treatment sequence $L$ and cluster-period size $n$ at the number of periods $T=6$. Like PA-LCRTs and CRXO trials, the churn rate has more effect on the required number of clusters with a smaller cluster-period size $n$. Similar to CRXO trials, the required number of clusters has an increasing trend as churn rate $\omega$ increases for a specific value of cluster-period size $n$. In addition, the churn rate has less effect on the required number of clusters with a larger number of treatment sequences $L$. In practice, the researchers often use $L=T-1$ in SW-CRTs.^10-19 It is consistent with our finding that a standard SW-CRT has more stable performance for open-cohort SW-CRTs.

Although the general variance does not further simplify, we are able to obtain the simplified expressions of variance for standard SW-CRTs under $L=T-1$. That is, we have

$$X_{\cdot \cdot }=\frac{mT}{2},X_{\cdot t}=(t-1)h,X_{i\cdot }=T-i,$$

and the variance Equation (1) can be simplified as

$$\text{Var}(\widehat{\beta })=\frac{12{\sigma }^2}{Tmn}\times \frac{T-1}{T-2}\times \left(1-\frac{1}{2+(T-1)r}\right)\times [1+(n-1){\alpha }_0](1-r),$$ (4)

where $-1∕(T-1)<r<1$ should hold for a valid variance estimate. In general, a higher churn rate $\omega$ results in a smaller $r$ for a fixed value of (${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$) Then since the final variance depends on the CPMC through $\left(1-\frac{1}{2+(T-1)r}\right)\times (1-r)$, whose monotonicity depends on values of $r$ over [0,1]. A higher churn rate may lead to a larger or smaller variance. An exception is the standard SW-CRTs with $T=3$, where a larger churn rate always leads to a higher variance. We provide a further investigation of the monotonicity of $\text{Var}(\widehat{\beta })$ in $r$ in the Appendix 4.

3. Local optimal designs

Here, we develop a cost-efficiency framework to accommodate open-cohort longitudinal CRTs. We assume the cost per cluster recruitment as $c$ currency units, the cost per individual enrollment as $s$ currency unit and the cost per outcome measurement as $e$ currency units. Under an open-cohort design, new individuals will replace an existing individual if he/she leaves the cluster at any period during the study, in a random fashion such that there is no systematic bias due to the open-cohort enrollment mechanism. Of note, we follow the convention in the optimal design literature and assume an equal cluster-period size $n$ across clusters. In an open-cohort trial with a constant churn rate $\omega$, $\omega n$ individuals each period for $(T-1)$ periods will be added to replace the individuals leaving the study. That is, the total number of enrolled individuals would be $n+(T-1)\omega n=[1+(T-1)\omega ]n$. Thus, the total cost becomes

$$B=m\{c+s[1+(T-1)\omega ]n+eTn\}.$$

Under a repeated cross-sectional design with $\omega =1$, the cost function becomes $B=m\{c+(s+e)Tn\}$; and under a closed-cohort design with $\omega =0$, the cost function reduces to $B=m\{c+sn+eTn\}$. Both are consistent with Liu and Li,⁸ and thus the proposed cost function is a generalization to open-cohort designs. The notation of $\omega n$ represents number of the replaced individuals at each period and its values spread over [0, $n$]. We rewrite the expression of total cost as

$$B=m\{c+[s+eT]n+s(T-1)R\},$$ (5)

where $R=\omega n$ and is an integer, as $m$ and $n$ in a specific trial, thus $\omega$ is not an arbitrary decimal but a value from $0,1∕n,\cdots (n-1)∕n,1$.

We refer to the enrollment feasibility of a cluster-period size, e.g., ($n_{\text{min}}$, $n_{\max }$) and number of clusters, e.g., ($m_{\text{min}}$, $m_{\max }$), as a design space, where cluster-period size and number of clusters should be greater than or equal to number of the replaced individuals at each period $R$ and number of treatment sequence $L$, respectively. Under the consideration of a total budget $B$, the range of $n$ is detailed as

$$\{\begin{array}{c}n\ge \max (n_{\text{min}},R)\hfill \\ n\le \text{min}\left(n_{\max },\text{int}\left(\frac{\left(\frac{B}{m}-c-s(T-1)R\right)}{\{s+eT\}}\right)\right)\text{at a specific}m\hfill \end{array}\phantom{\}}.$$ (6)

We define the local optimal design as a design within the design space with the lowest variance of treatment effect estimator in the design space given a fixed budget $B$.^20-23 For any open-cohort LCRTs, we propose the following algorithm to obtain integer estimates:

Algorithm 1: Local optimal design algorithm with a fixed number of replaced individuals at each period

$$\begin{array}{cc}\hfill & \mathrm{Input}:\text{Design space}(n_{\text{min}},n_{\max })\text{and}(m_{\text{min}},m_{\max });\hfill \\ \hfill & \text{Correlation parameters}({\alpha }_0,{\alpha }_1,{\alpha }_2);\hfill \\ \hfill & \text{Number of the replaced individuals at each period}R;\hfill \\ \hfill & \text{Total budget}B;\text{unit costs}c,s,\text{and}e;\hfill \\ \hfill & \text{Total number of periods}T;\text{variance}{\sigma }^2;\hfill \\ \hfill & \mathrm{Output}:n_{\text{LOD}},m_{\text{LOD}}\hfill \\ \hfill & 1.\text{Calculate}{n}_{\text{start}}=\max (n_{\text{min}},R);\hfill \\ \hfill & 2.\text{Find a multiplier of number of treatment sequence}L\text{which is greater}\hfill \\ \hfill & \text{than or equals to}{m}_{\text{min}}\text{and set it as}{m}_{\text{start}};\hfill \\ \hfill & 3.\text{For}m=m_{\text{start}}\text{to}{m}_{\max }\text{by}L\text{do}\hfill \\ \hfill & 1)\text{Calculate}{n}_{\text{end}}=\text{min}\left(n_{\max },\text{int}\left(\frac{\left(\frac{B}{m}-c-s(T-1)R\right)}{\{s+eT\}}\right)\right);\hfill \\ \hfill & 2)\text{If}{n}_{\text{end}}\ge n_{\text{start}},\text{for}n=n_{\text{start}}\text{to}{n}_{\text{end}}\text{do}\hfill \\ \hfill & \text{Calculate}\omega =R∕n\text{and}\text{the}\text{variance},\text{e.g.},\text{Equation}(2)\text{for}\text{PA-}\hfill \\ \hfill & \text{LCRTs};\hfill \\ \hfill & 4.\text{Select the design with the smallest variance in the design space};\hfill \\ \hfill & 5.\text{Return:}{n}_{\text{LOD}},m_{\text{LOD}}\hfill \end{array}$$

Table 2 shows the local optimal designs for PA-LCRT, CRXO trials, and standard SW-CRTs with $T=6$, respectively. We assume that a treatment effect of $\beta =0.2$, ${\sigma }^2=1$, $B=400,000$, $c=3000$, $s=200$, and $e=50$. The results indicate that, as $R$ increases, for PA-LCRTs, the optimal cluster-period size generally decreases and then increases whereas the optimal number of clusters and power under local optimal design first increase and then decrease. Churn rate under local optimal design is 1 for $R\ge 2$, that is, repeated cross-sectional design PA-LCRT has the minimum of variance. In contrast, as $R$ increases, for CRXO trials and standard SW-CRTs, the optimal cluster-period size and churn rate under local optimal design increase whereas the optimal number of clusters and power under local optimal design decrease. For all scenarios, local optimal designs for CRXO trials have the highest power (0.75 - >0.999) whereas local optimal design for PA-LCRTs (0.64 - 0.84) and SW-CRTs have the opposite trend in power (0.41 – 0.98).

Table 2.

Local optimal designs of cluster-period size ($n$), number of clusters ($m$), power and churn rate under local optimal designs to detect a treatment effect size of 0.2 and variance of ${\sigma }^2=1$ for known correlation parameters $({\alpha }_0,{\alpha }_1,{\alpha }_2)=(0.07,0.01,0.9)$ assuming a total budget $B=400,000$, cost per cluster $c=3000$, cost per individual $s=200$, and cost per outcome $e=50$ in open-cohort LCRTs with $T=6$

$R$	PA-LCRTs $(L=2)$				Multiple-period CRXO trials $(L=2)$				Standard SW-CRTs $(L=5)$
	$n_{\mathrm{LOD}}$	$m_{\mathrm{LOD}}$	${\text{Power}}_{\mathrm{LOD}}$	${\omega }_{\mathrm{LOD}}$	$n_{\mathrm{LOD}}$	$m_{\mathrm{LOD}}$	${\text{Power}}_{\mathrm{LOD}}$	${\omega }_{\mathrm{LOD}}$	$n_{\mathrm{LOD}}$	$m_{\mathrm{LOD}}$	${\text{Power}}_{\mathrm{LOD}}$	${\omega }_{\mathrm{LOD}}$
0	19	32	0.658	0	3	88	>0.999	0	4	80	0.984	0
1	17	32	0.637	0.059	8	50	>0.999	0.125	8	50	0.912	0.125
2	2	66	0.774	1	10	40	0.999	0.200	10	40	0.847	0.200
3	3	52	0.814	1	13	32	0.996	0.231	14	30	0.775	0.214
4	4	44	0.836	1	11	32	0.989	0.364	12	30	0.730	0.333
5	5	38	0.842	1	17	24	0.980	0.294	16	25	0.700	0.313
6	6	32	0.826	1	15	24	0.970	0.400	14	25	0.663	0.429
7	7	28	0.814	1	20	20	0.957	0.350	20	20	0.630	0.350
8	8	26	0.818	1	18	20	0.944	0.444	18	20	0.606	0.444
9	9	24	0.814	1	20	18	0.927	0.450	16	20	0.578	0.563
10	10	22	0.804	1	24	16	0.911	0.417	14	20	0.548	0.714
11	11	20	0.787	1	22	16	0.896	0.500	25	15	0.533	0.440
12	12	18	0.763	1	27	14	0.876	0.444	23	15	0.519	0.522
13	13	16	0.730	1	25	14	0.862	0.520	21	15	0.504	0.619
14	14	16	0.745	1	23	14	0.843	0.609	19	15	0.489	0.737
15	15	14	0.702	1	30	12	0.823	0.500	17	15	0.479	0.882
16	16	14	0.714	1	28	12	0.809	0.571	42	10	0.433	0.381
17	17	14	0.726	1	26	12	0.791	0.654	40	10	0.429	0.425
18	18	12	0.669	1	38	10	0.771	0.474	38	10	0.424	0.474
19	19	12	0.679	1	36	10	0.761	0.528	36	10	0.419	0.528
20	20	12	0.688	1	34	10	0.750	0.588	34	10	0.414	0.588

treatment effect=0.2; outcome variance=1; ${\alpha }_0$: within-period correlation; ${\alpha }_1$: inter-period correlation; ${\alpha }_2$: within-individual correlation; $T$: number of periods; $L$: number of treatment sequences; treatment sequence is either I or C for PA-LCRTs and ICICIC or CICICI for multiple-period CRXO trials; $n$: number of individuals per cluster-period; $m$: total number of clusters; $n_{\max }=2000$; $m_{\max }=2000$; $c=3000$: cost per cluster; $s=200$: cost per individual; $e=200$; cost per outcome; total budget $B=400,000$; the type I error of 5%.

Figures 1A-1F presents the local optimal designs of $\{n,m\}$ as well as their power and churn rate under local optimal designs within a range of 0 to 20 replaced individuals at each period $R$ for different periods $T$. The other parameters remain the same in Table 2. At a specific $T$, PA-LCRT demonstrates that the $n_{\text{LOD}}$ decreases and then increases but the $m_{\text{LOD}}$ and power under local optimal design increases and then decreases as $R$ increases. However, a CRXO trial demonstrates that the $n_{\text{LOD}}$ and churn rate under local optimal design increase but the $m_{\text{LOD}}$ and power under local optimal design decrease as $R$ increases. SW-CRTs have the similar observations to CRXO trials as $R$ increases. When $T$ increases, $m_{\text{LOD}}$ decreases while other findings are same for all three open-cohort LCRTs. In addition, an increasing $T$ has a more obvious effect on optimal cluster-period size when $R$ is larger than 10 in standard SW-CRTs.

Figure 1.

Local optimal designs of cluster-period size ($n$), number of clusters ($m$), power and churn rate under local optimal designs to detect a treatment effect size of 0.2 and variance of ${\sigma }^2=1$ for known correlation parameter $({\alpha }_0,{\alpha }_1,{\alpha }_2)=(0.07,0.01,0.9)$ assuming a total budget $B=400,000$, cost per cluster $c=3000$, cost per individual $s=200$, and cost per outcome $e=50$ in open-cohort LCRTs

Algorithm 1 aims to obtain integer estimates of {$n_{\text{LOD}}$, $m_{\text{LOD}}$}, however, the previous optimal research also provided the decimal values of {$n_{\text{LOD}}$, $m_{\text{LOD}}$},^7,8,22-25 through the closed-form formulae of $\{n,m\}$ under a constraint of the exact budget. Although the decimal estimates are not practical in a clinical trial, they can function as true values and are used for comparing the performance of integer estimates obtained from the proposed algorithm. Therefore, we rewrite the variance of treatment effect estimator (2) under a constraint of the exact budget, Equation (5),

$$\begin{gathered} \text{Var}(\widehat{\beta })=\frac{4{\sigma }^2}{TB}\times \left\{\frac{c+s(T-1)R}{n}+[s+eT]\right\}\times \\ \left\{(1-{\alpha }_0)[1+(T-1){\alpha }_2]+n{\alpha }_0[1+(T-1){\alpha }_1]-(1-{\alpha }_0)(T-1){\alpha }_2\times \frac{R}{n}\right\}. \end{gathered}$$ (7)

However, we cannot derive the closed-form formulae of $\{n,m\}$, similar to previous optimal research, to minimize the Equation (7). Our solution is to find the minimum of Equation (7) when the decimal values of $n$ are allowed using nonlinearly constrained Quasi-Newton optimization through the SAS NLPQN Subroutine,²⁶ the same approach adopted in Liu et al.⁸ This research does not provide the formulae of decimal values of $\{n_{\text{LOD}},m_{\text{LOD}}\}$ but obtains the minimum of Equation (7) at the decimal values of $n_{\text{LOD}}$. We will apply this proposed method in the following section, when the correlation parameters (${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$) are unknown.

4. MaxiMin optimal designs

Although the local optimal design is helpful in elucidating key factors that drive an optimal open-cohort LCRT, it rests upon precise knowledge of the correlation parameter which may not be available when designing a particular study. Often times, it is more realistic to have a range of plausible values for the correlation parameters; see, for example, the empirical estimates reported by Korevaar et al.²⁷ Accordingly, methods and recommendations for obtaining correlation parameters from routinely collected data are also discussed in Ouyang et al.²⁸ Building on the assumption that a plausible range of correlation parameters, referred to be parameter space, can be easier to obtain than a precise point value in the design stage, we expect to obtain the candidates of local optimal design for any correlation parameters (${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$) within the parameter space. As the previous researchers did,^7,8,22-25 we define relative efficiency (RE) as a ratio of the optimal variance achieved by the local optimal design to the variance of treatment effect estimator, e.g., Equation (3), and then minimize RE within the parameter space. We further develop the MaxiMin optimal design that can maximize the minimized RE within the design space, thus leading to a robust design to assumptions on correlation parameters.

We first propose the MaxiMin optimal design (MMD) algorithm to obtain the integer estimates for open-cohort LCRTs using optimization techniques with a mid-value in the parameter space of each parameter as a starting point for (${\alpha }_0$, ${\alpha }_1$, ${\alpha }_2$) to find the minimum RE within the parameter space, where the approach was suggested by Lu et al.⁸ Since Algorithm 1 proposed in Section 3 only provides the integer estimates of {$n$, $m$} having the smallest variance of treatment effect estimator, we realize that the variance of treatment effect estimator could be smaller if allowing the decimal values of {$n$, $m$} and thus consider integer values as well as decimal values in the design space when calculating RE. Therefore, we implement the optimization technique twice: 1) to identify the decimal estimates $n$ which have the smallest variance of treatment effect estimator, e.g., Equation (7) for PA-LCRTs, within the constraints of a valid variance estimate and Equation (6); 2) to find the minimum value of RE across correlation coefficients within the parameter space. The detailed steps are as follows.

Algorithm 2: MaxiMin optimal design algorithm with a fixed number of replaced individuals at each period

$$\begin{array}{cc}\hfill & \mathrm{Input}:\text{Parameter space}({\alpha }_{0\text{min}},{\alpha }_{0\max }),({\alpha }_{1\text{min}},{\alpha }_{1\max }),\text{and}({\alpha }_{2\text{min}},{\alpha }_{2\max });\hfill \\ \hfill & \text{Design space}(n_{\text{min}},n_{\max })\text{and}(m_{\text{min}},m_{\max });\hfill \\ \hfill & \text{Number of the replaced individuals at each period}R;\hfill \\ \hfill & \text{Total budget}B;\text{unit costs}c,s,\text{and}e;\hfill \\ \hfill & \text{Total number of periods}T;\text{variance}{\sigma }^2;\hfill \\ \hfill & \mathrm{Output}:n_{\text{MMD}},m_{\text{MMD}}\hfill \\ \hfill & 1.{\text{RE}}_{\text{MMD}}=0(\text{Initializtion});\hfill \\ \hfill & 2.\text{Calculate}{n}_{\text{start}}=\max (n_{\text{min}},R);\hfill \\ \hfill & 3.\text{Find a multiplier of number of treatment sequence}L\text{which is}\hfill \\ \hfill & \text{greater then or equals to}{m}_{\text{min}}\text{and set it as}{m}_{\text{start}};\hfill \\ \hfill & 4.\text{For}m=m_{\text{start}}\text{to}{m}_{\max }\text{by}L\text{do}\hfill \\ \hfill & 1)\text{Calculate}{n}_{\text{end}}=\text{min}\left(n_{\max },\text{int}\left(\frac{\left(\frac{B}{m}-c-s(T-1)R\right)}{\{s+eT\}}\right)\right);\hfill \\ \hfill & 2)\text{If}{n}_{\text{end}}\ge n_{\text{start}},\text{set}n=n_{\text{end}}\text{do}\hfill \\ \hfill & \mathrm{a})\text{Set}\text{the}\text{initial}\text{value}\text{of}\text{correlation}\text{coefficients}\text{for}\hfill \\ \hfill & \text{optimization:}\hfill \\ \hfill & {\alpha }_{\mathrm{i},\text{int}}=\frac{{\alpha }_{\mathrm{i},\text{min}}+{\alpha }_{\mathrm{i},\max }}{2},\mathrm{i}=0,1,2;\hfill \\ \hfill & \mathrm{b})\text{Under}\text{the}\text{current correlation coefficients},\text{adopt Quasi-}\hfill \\ \hfill & \text{Newton algorithm to find the minimum value of variance},\hfill \\ \hfill & \text{e.g.},\text{Equation}(7)\text{for PA-LCRTs},\text{where the cluster-period}\hfill \\ \hfill & \text{size}\text{could}\text{be}\text{decimal},\text{denoted by}{n}_{\text{dec}},\text{and}{m}_{\text{dec}}=\hfill \\ \hfill & B∕(c+[s+eT]n_{\text{dec}}+s(T-1)R);\hfill \\ \hfill & \mathrm{c})\text{Calculate}\omega =R∕n\text{and RE}=\frac{{\text{Var of m}}_{\text{dec}}\text{and}{n}_{\text{dec}}}{\text{Var of m and n}},\text{where Var}\hfill \\ \hfill & \text{is calculated in Equation},\text{e.g.},\text{Equation}(2)\text{for PA-LCRTs};\hfill \\ \hfill & \mathrm{d})\text{Adopt Quasi-Newton Algorithm to find the minimum value}\hfill \\ \hfill & \text{of RE across correlation coefficients under current}m\text{and}n,\hfill \\ \hfill & {\text{denoting as RE}}_{m,n};\hfill \\ \hfill & \mathrm{e}){\text{If RE}}_{m,n}>{\text{RE}}_{\text{MMD}},\text{then}\hfill \\ \hfill & \mathrm{i}.n_{\text{MMD}}=n\hfill \\ \hfill & \text{ii}.m_{\text{MMD}}=m\hfill \\ \hfill & \text{iii}.{\text{RE}}_{\text{MMD}}={\text{RE}}_{m,n};\hfill \\ \hfill & 5.\text{return}{n}_{\text{MMD}},m_{\text{MMD}}\hfill \end{array}$$

In Step 4-2), we do not consider all possible values of cluster-period size $n$ within the design space and under a budget constraint but the maximum of these values, since the running time is much less as well as the power is more at a larger $n$. Table 3 presents the MaxiMin optimal designs for open-cohort PA-LCRTs, multiple-period two-sequence CRXO trials and standard SW-CRTs, where the parameters are the same in Table 2 except that we consider the parameter spaces $({\alpha }_{0\text{min}},{\alpha }_{0\max })=(0.05,0.2)$, $({\alpha }_{1\text{min}},{\alpha }_{1\max })=(0.02,0.2)$, and $({\alpha }_{2\text{min}},{\alpha }_{2\max })=(0.2,0.5)$ for deriving MaxiMin optimal designs. When $R$ is small, e.g., ≤ 5, there is no difference of $n_{\text{MMD}}$ and $m_{\text{MMD}}$ between PA-LCRTs and CRXO trials. As $R$ increases, the optimal cluster-period size generally increases whereas the optimal number of clusters decreases. Under MaxiMin optimal design, RE vary and there is obvious pattern churn rate across the values of $R$. For standard SW-CRTs, the optimal cluster-period size has the opposite trend of optimal number of clusters. Additionally, Figures 2A-2F, which present the MaxiMin optimal designs of $\{n,m\}$ as well as the corresponding REs and churn rates for the different number of periods $T$, visually provide more details about the trend than Table 3. Among all three open-cohort LCRTs, we notice that $R$ does have a more substantial impact on $m_{\text{MMD}}$ than $n_{\text{MMD}}$. It suggests that, under an optimal design routine, the information change due to individual replacement is usually compensated by adding or removing clusters rather than modifying the cluster-period sizes.

Table 3.

MaxiMin optimal designs of cluster-period size ($n$), number of clusters ($m$), RE and churn rate under MaxiMin optimal designs to detect a treatment effect size of 0.2 and variance of ${\sigma }^2=1$ for correlation parameters $({\alpha }_{0\min },{\alpha }_{0\max })=(0.05,0.2)$, $({\alpha }_{1\min },{\alpha }_{1\max })=(0.02,0.2)$, and $({\alpha }_{2\min },{\alpha }_{2\max })=(0.2,0.5)$ assuming a total budget $B=400,000$, cost per cluster $c=3000$, cost per individual $s=200$, and cost per outcome $e=50$ in open-cohort LCRTs with $T=6$

$R$	PA-LCRTs $(L=2)$				Multipe-period CRXO trials $(L=2)$				Standard SW-CRTs $(L=5)$
	$n_{\mathrm{MMD}}$	$m_{\mathrm{MMD}}$	${\mathrm{RE}}_{\mathrm{MMD}}$	${\omega }_{\mathrm{MMD}}$	$n_{\mathrm{MMD}}$	$m_{\mathrm{MMD}}$	${\mathrm{RE}}_{\mathrm{MMD}}$	${\omega }_{\mathrm{MMD}}$	$n_{\mathrm{MMD}}$	$m_{\mathrm{MMD}}$	${\mathrm{RE}}_{\mathrm{MMD}}$	${\omega }_{\mathrm{MMD}}$
0	2	98	0.980	0	2	98	0.980	0	2	95	0.950	0
1	2	78	0.975	0.500	2	78	0.975	0.500	2	75	0.938	0.500
2	2	66	0.990	1	2	66	0.990	1	2	65	0.975	1
3	3	52	0.975	1	3	52	0.975	1	28	20	0.724	0.107
4	4	44	0.990	1	4	44	0.990	1	26	20	0.739	0.154
5	5	38	0.998	1	5	38	0.998	1	6	35	0.958	0.833
6	7	32	0.856	0.857	26	18	0.801	0.231	22	20	0.804	0.273
7	8	28	0.855	0.875	30	16	0.767	0.233	20	20	0.842	0.350
8	8	26	0.975	1	8	26	0.975	1	18	20	0.879	0.444
9	9	24	0.990	1	9	24	0.990	1	16	20	0.914	0.563
10	10	22	0.990	1	10	22	0.990	1	14	20	0.949	0.714
11	12	20	0.915	0.917	29	14	0.833	0.379	12	20	0.985	0.917
12	14	18	0.862	0.857	27	14	0.868	0.444	23	15	0.863	0.522
13	18	16	0.798	0.722	25	14	0.902	0.520	21	15	0.898	0.619
14	16	16	0.887	0.875	32	12	0.835	0.438	19	15	0.933	0.737
15	21	14	0.797	0.714	30	12	0.867	0.500	17	15	0.968	0.883
16	19	14	0.867	0.842	28	12	0.897	0.571	42	10	0.748	0.381
17	17	14	0.998	1	17	14	0.998	1	40	10	0.778	0.425
18	24	12	0.813	0.750	24	12	0.951	0.750	38	10	0.807	0.474
19	22	12	0.879	0.864	22	12	0.973	0.864	36	10	0.837	0.528
20	20	12	0.990	1	20	12	0.990	1	34	10	0.865	0.588

treatment effect=0.2; outcome variance=1; ${\alpha }_0$: within-period correlation; ${\alpha }_1$: inter-period correlation; ${\alpha }_2$: within-individual correlation; $T$: number of periods; $L$: number of treatment sequences; treatment sequence is either I or C for PA-LCRTs and ICICIC or CICICI for Multiple-period CRXO trials; $n$: number of individuals per cluster-period; $m$: total number of clusters; NA: not applicable; $n_{\max }=2000$; $m_{\max }=2000$; $c=3000$: cost per cluster; $s=200$: cost per individual; $e=200$; cost per outcome; total budget $B=400,000$; RE is calculated as the ratio of the variance of treatment effect estimator for LOD to each design in the cluster-period size $n$ within the range $(2,n_{\max })$ and number of clusters $m$ within the range (2, $m_{\max }$); the type I error of 5%.

Figure 2.

Maximin optimal designs of cluster-period size ($n$), number of clusters ($m$), RE and churn rate under MaxiMin optimal designs to detect a treatment effect size of 0.2 and variance of ${\sigma }^2=1$ for correlation parameters $({\alpha }_{0\min },{\alpha }_{0\max })=(0.05,0.2)$, $({\alpha }_{1\min },{\alpha }_{1\max })=(0.02,0.2)$, and $({\alpha }_{2\min },{\alpha }_{2\max })=(0.2,0.5)$ assuming a total budget $B=400,000$, cost per cluster $c=3000$, cost per individual $s=200$, and cost per outcome $e=50$ in open-cohort LCRTs

5. Numerical examples

The study designs for two examples are illustrated in Figure 3. First, we consider the first example in Introduction Section. The NND trial conducted by Damsgaard et al.⁵ included two consecutive 3-month periods $(T=2)$, where children received school meals based on the NND and their usual packed lunch from home (control). During the 3-month NND period the children were served a mid-morning snack, an ad libitum hot lunch meal and an afternoon snack. The trial enrolled nine schools with two year-groups $(m=18)$, 4 classes per year-group and 22 children per class $(n=88)$ to obtain 80% power to detect an effect size of 0.32 in the sample size calculation.⁵ We plan to apply our proposed algorithms to this trial, where the total budget is calculated from the original sample size and the following unit costs. In Step 1, we use the School Nutrition and Meal Cost Study (SNMCS) as a reference: $3.81 per student for lunch and $2.72 for breakfast.²⁹ For a unit cost, the student cost per period (45 school days) is $s=(3.81+2.72)\times 45=293.85$. We assume the enrollment cost per year-group (e.g., operational, administrative, and other indirect costs) is $c=2000$ and measurement cost per period $e=50$. Given that this study reported the churn rates of 10% and 11.5%, here we use $R=3$ as an example. With 18 year-groups, 4 classes per year-group and 22 children per class, the total budget is $B=m\{c+[s+eT]n+s(T-1)R\}=18\times \{2000+393.85\times 88+293.85\times 3\}=\$675726.3$. In Step 2, we aim to propose local optimal design with the assumption of ${\alpha }_0=0.05$, ${\alpha }_1=0.01$, and ${\alpha }_2=0.5$. Using the Algorithm 1 in Section 3, we have a local optimal design $(n_{\text{LOD}},m_{\text{LOD}})=(13,84)$ for $R=3$ with at least 99.9% power and cost of $672134.4. The number of students per year group, 13, is too small to be realistic. Thus, we set $n_{\text{min}}=60$ and then obtain a local optimal design of (64,24) with a 98.5% power and cost of $674110.8. Comparing the 98.5% power under local optimal design with the 80% power proposed in the original trial, we have a much larger one with a smaller total cost ($674110.8 vs. $675726.3). In Step 3, to be conservative, we also consider a reasonably wide range of correlation parameters,^27,28 e.g., $({\alpha }_{0\text{min}},{\alpha }_{0\max })=(0.01,0.1)$, $({\alpha }_{1\text{min}},{\alpha }_{1\max })=(0.01,0.1)$, and $({\alpha }_{2\text{min}},{\alpha }_{2\max })=(0.2,0.8)$ and then obtain the MaxiMin optimal design $(n_{\text{MMD}},m_{\text{MMD}})=(29,46)$ for $R=3$ with a cost of $657947.2 and RE=0.694. Again, because of feasibility with $n_{\text{min}}=60$, we obtain MaxiMin optimal designs of (64, 24) for $R=3$ with a cost of $674110.8 and RE=0.950.

Figure 3.

Study design illustration for two examples

Solomon et al. conducted Devon Active Villages Evaluation (DAVE) trial through employing a SW-CRT with $L=4$ and $T=5$ at the randomization level of rural UK villages, aiming to improve an adulťs participation in physical activity through a community-level intervention.¹⁸ 10 households within each village from 128 rural villages ($m=128$ and $n=10$) were randomly selected to collect the outcomes at each time period through a postal survey. The study results showed that the mean (±standard deviation) per group is 2317 ± 2964 (m) and 2450 ± 3014 (m) for the outcome of “minutes spent in moderate-and-vigorous activity per week”, respectively as well as within-period correlation ${\alpha }_0=0.01$.³⁰ Our proposed algorithms are applied to this outcome as follows. In Step 1, using Equation (4), we need 628 villages to obtain 80.2% power at two-sided significance level of 5% to evaluate the effect of community-level intervention with an assumption of $({\alpha }_0,{\alpha }_1,{\alpha }_2)=(0.01,0.005,0.5)$, where the treatment effect is 130±3000 (m), cluster-period size (number of households per time period per village) is $n=10$, and $R=1$, equiavalent, $\omega =0.1$. In Step 2, we assume $c=500$, and $s=50$, $e=10$, then the total budget is $B=m\{c+[s+eT]n+s(T-1)R\}=628\times (5000+100\times 10+50\times 4\times 1)=3893,600$ for this proposed study. In Step 3, we identify the LOD under the cost consideration. With the same assumptions of the treatment effect size, standard deviation, and $({\alpha }_0,{\alpha }_1,{\alpha }_2)=(0.01,0.005,0.5)$, our algorithm returns $\{n_{\text{LOD}},m_{\text{LOD}}\}=(56,360)$ with 98.0% power and an actual cost is 3,888,000. However, the cluster-period size configuration may be too large to be realistic in the trial context. We then set the maximum cluster-period size to be 20 and our algorithm returns $\{n_{\text{LOD}},m_{\text{LOD}}\}=(20,540)$ with 93.8% power and an actual cost is 3,888,000. The proposed study under the cost consideration has an approximately 15% power increase (93.8% vs. 80.2%) with a smaller actual cost (3,888,000 vs. 3,893,600 ). In Step 4, we also consider a range for ${\alpha }_1$, $({\alpha }_{1\min },{\alpha }_{1\max })=(0.001,0.009)$, and our algorithm returns the MMD estimates at $R=1:(2,720)$ with an actual cost of 3,888,000 and RE=0.999. Liu et al. also demonstrated the proposed optimal designs based on GEEs using DAVE trial, in which repeated cross-sectional SW-CRT ($R=0$) and $s=10$ with the total budget of 4147,000 were considered.⁸ The use of a relatively smaller budget (3893,600) to illustrate methods in this work may partly explain the differences observed between the optimal designs.

6. Discussion

Previously, sample size calculation for open-cohort LCRTs with three sampling schemes, under a fixed design (rather than an optimal design) framework, were developed by Kasza et al.³ With the assumptions of a constant churn rate and the withdrawn individuals replaced by new ones during the trial, it unifies the closed cohort and repeated cross-sectional sampling provided in Hooper et al.⁹ To our knowledge, research on optimal designs under a fixed budget constraint for open-cohort LCRTs has not been developed; thus our work fills in this gap by proposing local optimal design and MaxiMin optimal design algorithms for open-cohort LCRTs under a cost-efficiency framework.

To elaborate, our main contributions to the optimal cluster randomized design literature are four-fold. First, we explore the effect of churn rate on the number of clusters when the cluster-period size is fixed for all three LCRTs. We find that the churn rate significantly effects on the required number of clusters when a cluster-period size is small, e.g. 5. As churn rate increases for a specific value of cluster-period size, the required number of clusters decreases for PA-LCRTs but increases for CRXO trials and SW-CRTs. When the number of periods increases, the decreasing trend of the required number of clusters is more apparent for PA-LCRTs and the increasing trend is not as obvious for CRXO trials across the churn rates. The required number of clusters decreases but not substantially so for SW-CRTs if the churn rate $\omega <0.3$ and when the total number of periods increases. An open-cohort CRXO trial needs a much smaller number of clusters than an open-cohort PA-LCRT for a fixed cluster-period size since the variance of treatment effect estimator

$$\frac{4{\sigma }^2}{Tmn}[1+(n-1){\alpha }_0](1-r)<\frac{4{\sigma }^2}{Tmn}\times [1+(n-1){\alpha }_0][1+(T-1)r].$$

Since the churn rate has less effect on the required number of clusters with a larger number of treatment sequences $L$, using a standard SW-CRT might suffice in many settings. Second, we investigate the effect of number of the replaced individuals at each period on the optimal cluster-period size and optimal number of clusters for all three LCRTs, where only standard SW-CRTs are considered. Similar to our optimal design research based on GEEs for two-level and three-level PA-CRTs,^{7,8,22,23,31,32} we cannot obtain the closed-form formulate of local optimal design and MaxiMin optimal design in all open-cohort LCRTs. When calculating the variance minimum in RE, we use a larger design space allowing decimal values of a cluster-period size, for which the optimization technique is implemented. Again, we implement the optimization technique to find the minimum value of RE across correlation coefficients within the parameter space. The computation time is longer but acceptable when the design space is reasonable. Third, our results reveal that 1) when correlation parameters are known, as $R$, the number of replaced individuals at each period, increases, for open-cohort PA-LCRTs, the optimal cluster-period size generally decreases and then increases whereas the optimal number of clusters and power under local optimal design first increase and then decrease. In contrast, as $R$ increases, for CRXO trials and standard SW-CRTs, the optimal cluster-period size and churn rate under local optimal design increase whereas the optimal number of clusters and power under local optimal design decrease. In addition, repeated cross-sectional design PA-LCRTs have the minimum of variance for $R\ge 2$ and an increasing $T$ has a more effect on optimal cluster-period size when $R$ is larger than 10 in standard SW-CRTs. 2) when correlation parameters are unknown, but the parameter space is available, for all three LCRTs with a fixed $R$, there is no difference of $n_{\mathrm{MMD}}$ and $m_{\mathrm{MMD}}$ between PA-LCRTs and CRXO trials for $R\le 5$. $R$ has little impact on the optimal cluster-period size, but number of clusters can be easily affected by $R$. Lastly, we develop SAS macros to find the local optimal design and MaxiMin optimal design for each type of open-cohort LCRTs to facilitate their implementation in practice.

Our proposed algorithms consider the fixed number of replaced individuals at each period rather than a churn rate. It leads to the fact that the churn rate changes during the optimization process, e.g., Step 3-2) in Algorithm 1, although we assume a constant rate within each trial. We realize that the churn rate $\omega$ is not an arbitrary decimal but a value from $0,1∕n,\cdots (n-1)∕n,1$. When using Equation (1) in the sample size calculation, we must keep in mind that $R=\omega n$ should be an integer. It is of no practical significance if we cannot guarantee that $R$ is an integer, regardless of optimal designs. As such, using the fixed number of replaced individuals at each period is preferred in optimal designs. For the same reason, we cannot directly apply the previous research using to the open-cohort study besides that $r=\frac{n{\alpha }_0{\alpha }_1+(1-\omega )(1-{\alpha }_0){\alpha }_2}{1+(n-1){\alpha }_0}$ is a function of $n$.⁸ It is worth noting that, in developing the MaxiMin optimal designs, we have followed the convention by focusing on addressing the uncertainty in the correlation parameters but fixed the value of the number of replaced individuals at each period. In principle, one could further address the uncertainty in the number of replaced individuals at each period simultaneously with the uncertainty in the correlation parameters and develop an extended MaxiMin optimal design. That is, our MaxiMin optimal design can be extended to address an uncertain number of replaced individuals at each period through assuming an available range of plausible values - $(R_{\min },R_{\max })$. To proceed with this idea, one could first identify $n_{\mathrm{MMD}}$ and $m_{\mathrm{MMD}}$ for each value of $R$ within a range of $(R_{\min },R_{\max })$ and calculate the corresponding RE in Algorithm 2. The design with the maximum of RE is referred to MaxiMin optimal design. We use Table 3 as an example, $(R_{\min }=0,R_{\max }=20)$. For PA-CRTs and CRXO trials, two designs, $\{n_{\mathrm{MMD}}=5,m_{\mathrm{MMD}}=38\}$ at $R=5$ and $\{n_{\mathrm{MMD}}=17,m_{\mathrm{MMD}}=14\}$ at $R=17$ are MaxiMin optimal designs having the maximum of RE, e.g., 0.998. For standard SW-CRTs and CRXO trials, a design of $\{n_{\mathrm{MMD}}=12,m_{\mathrm{MMD}}=20\}$ at $R=11$ has the largest RE of 0.985 and thus is MaxiMin optimal design. Interestingly, there is a possibility that multiple designs are identified as MaxiMin optimal designs. This suggests that, although the concept of MaxiMin optimal design can be extended to address uncertainty in the expected number of replaced individuals at each period, the non-unique optimal design may introduce challenges in interpretation and implementation. The operational details and properties of alternative MaxiMin optimal design that accounts for the uncertainty in the number of replaced individuals at each period, perhaps by maximizing power, are interesting topics for future research.

Our study has several limitations and inspires important directions for future research. Above all, our framework assumes a constant churn rate. However, as the NND study showed, the churn rates tend to vary and increase in the later period. In other words, they are not constant across the different period. We suggest using the highest number of replaced individuals when implementing our proposed algorithms as well as the sample size calculation as a conservative option. Second, we assume an equal cluster-period size across clusters in open-cohort LCRTs for simplicity. In practice, such scenarios cannot be guaranteed. Relative efficiency of equal versus unequal cluster-period size have been investigated in two-level and three-level PA-CRTs.^24,33-36 The effect of number of the replaced individuals in open-cohort studies with an unequal cluster-period size is worthy exploring in depth. Third, we assume 1:1 randomization in PA-LCRTs and CRXO trials and an equal number of clusters crossover to intervention at each period $h$ for SW-CRTs. That is, optimal number of clusters is equivalent to optimal number of clusters per sequence in these open-cohort LCRTs. Otherwise, Equations (2)-(4) as well as local optimal designs and MaxiMin optimal designs for PA-LCRTs and CRXO trials would be more complex and require additional study. Fourth, the proposed algorithms assume a pre-specified fixed number of periods $T$. However, our optimal design could be extended to determine the cluster-period size and number of clusters to reach the maximized power in local optimal design and the maximum of RE within the parameter space within a range of $(T_{\min },T_{\max })$ under budget requirements. Fifth, this research assumes “random” or “non-informative” drop out. Some trials may not have random withdrawals, and the theory for optimal designs to accommodate non-random withdrawals requires additional future work. Lastly, this work is based on the assumption of constant within-period, inter-period, and within-individual correlations. We replace the individuals who leave their cluster, by new individuals, in order to keep cluster-period size constant throughout the trial and avoid the missing outcome measures. Such a correlation structure and design setting represent a reasonable starting point for design derivation rather than a claim that they cover all practical design settings. Alternative specifications, such as decaying correlation models, may be more appropriate depending on the study context, but this type of correlation models generally preclude tractable variance derivations for design optimization although the optimal designs can be solved numerically. For example, Kasza et al.³ considered decaying correlation structures that allow for more general within-cluster and within-individual dependence under fixed design settings. Extending optimal design theory to such a decaying correlation structure is an important direction for future research.

Appendix 1. Variance of the treatment effect estimator under open-cohort PA-LCRTs.

For PA-LCRTs, the number of treatment sequence $L=2$. From the treatment matrix, we have

$$X_{\cdot \cdot }=\frac{mT}{2},X_{\cdot t}=\frac{m}{2},$$

for $t=1,\cdots T$ and

$$X_{1\cdot }=\cdots =X_{\frac{m}{2}\cdot }=T,X_{\left(\frac{m}{2}+1\right)\cdot }=\cdots X_{m\cdot }=0,$$

Thus,

$$\begin{gathered} \text{Var}(\widehat{\beta }) \\ =\frac{{\sigma }^2}{n}\times \frac{m[1+(n-1){\alpha }_0](1-r)[1+(T-1)r]}{m\frac{mT}{2}-T{\left(\frac{m}{2}\right)}^2+\left[{\left(\frac{mT}{2}\right)}^2+m(T-1)\frac{mT}{2}-(T-1)T{\left(\frac{m}{2}\right)}^2-m\frac{m}{2}{T}^2\right]r} \\ =\frac{{\sigma }^2}{mn}\times \frac{[1+(n-1){\alpha }_0](1-r)[1+(T-1)r]}{\frac{T}{4}(1-r)} \\ =\frac{4{\sigma }^2}{Tmn}\times [1+(n-1){\alpha }_0][1+(T-1)r]. \end{gathered}$$

Appendix 2. Variance of the treatment effect estimator under open-cohort multiple-period two-sequence CRXO trials.

For multiple-period two-sequence CRXO trials, the number of treatment sequence $L=2$. From the treatment matrix, we have

$$X_{\cdot \cdot }=\frac{mT}{2},X_{\cdot t}=\frac{m}{2},$$

for $t=1,\cdots T$ and

$$X_{1\cdot }=\cdots =\cdots X_{m\cdot }=\frac{T}{2}.$$

Thus,

$$\begin{gathered} \text{Var}(\widehat{\beta }) \\ =\frac{{\sigma }^2}{n}\times \frac{m[1+(n-1){\alpha }_0](1-r)[1+(T-1)r]}{m\frac{mT}{2}-T{\left(\frac{m}{2}\right)}^2+\left[{\left(\frac{mT}{2}\right)}^2+m(T-1)\frac{mT}{2}-(T-1)T{\left(\frac{m}{2}\right)}^2-m^2{\left(\frac{T}{2}\right)}^2\right]r} \\ =\frac{{\sigma }^2}{mn}\times \frac{[1+(n-1){\alpha }_0](1-r)[1+(T-1)r]}{\frac{T}{4}+\frac{T}{4}(T-1)r} \\ =\frac{4{\sigma }^2}{Tmn}\times \frac{[1+(n-1){\alpha }_0](1-r)[1+(T-1)r]}{1+(T-1)r}=\frac{4{\sigma }^2}{Tmn}[1+(n-1){\alpha }_0](1-r). \end{gathered}$$

Appendix 3. Simplified notations under open-cohort stepped wedge cluster randomized trials.

For $T$ periods, $L$ has the potential values from $2,\cdots ,T-1$, and we can denote $X_{it}$ as follows

$$\{\begin{array}{c}{X}_{1.}=T-1\hfill \\ X_{2.}=T-2\hfill \\ \dots \hfill \\ X_{(L-1).}=T-L+1\hfill \\ X_{L.}=T-L\hfill \end{array}\phantom{\}},\text{and}\{\begin{array}{c}{X}_{\cdot 1}=0\hfill \\ X_{\cdot 2}=h\hfill \\ \dots \hfill \\ X_{\cdot L}=(L-1)h\hfill \\ \hfill \\ X_{\cdot (L+1)}=\cdots X_{\cdot T}=Lh\hfill \end{array}\phantom{\}}.$$

Thus, $X_{\cdot \cdot }={\sum }_{t=1}^T{\sum }_{i=1}^m{X}_{it}=h{\sum }_{t=1}^T{\sum }_{i=1}^L{X}_{it}=h\frac{L(2T-L-1)}{2}$,

$$\begin{gathered} \sum _{t=1}^T{(X_{\cdot t})}^2=\sum _{t=1}^T{\left(h\sum _{i=1}^L{X}_{it}\right)}^2=h^2\sum _{t=1}^T{\left(\sum _{i=1}^L{X}_{it}\right)}^2 \\ =h^2\left\{\sum _{t=1}^T{\left(\sum _{i=1}^L{X}_{it}\right)}^2+\sum _{t=L+1}^T{\left(\sum _{i=1}^L{X}_{it}\right)}^2\right\}=h^2\left\{\sum _{i=1}^{L-1}{i}^2+(T-L)L^2\right\} \\ =h^{2}L\left\{\frac{1}{6}(L-1)(2L-1)+(T-L)L\right\}; \end{gathered}$$

and

$$\begin{gathered} \sum _{i=1}^m{(X_{i\cdot })}^2=h\sum _{i=1}^L{(X_{i\cdot })}^2=h\left(\sum _{i=1}^{T-1}{i}^2-\sum _{i=1}^{T-L-1}{i}^2\right) \\ =\frac{h}{6}\{T(T-1)(2T-1)-(T-L-1)(T-L)[2(T-L)-1]\}. \end{gathered}$$

When $L=T-1$, $X_{\cdot \cdot }=\frac{mT}{2}$, $X_{\cdot t}=(t-1)h,X_{i\cdot }=T-i$, then

$$\sum _{t=1}^T{(X_{\cdot t})}^2=\frac{h^2}{6}T(T-1)(2T-1)=\frac{m^2}{6}\frac{T(2T-1)}{T-1},$$

and

$$\sum _{i=1}^m{(X_{i\cdot })}^2=\frac{h}{6}T(T-1)(2T-1)=\frac{m}{6}T(2T-1).$$

Appendix 4. Monotonicity of $\mathbf{Var}(\widehat{\beta })$ in open-cohort SW-CRTs

Denote

$$g(r)=\left(1-\frac{1}{2+(T-1)r}\right)\times (1-r)=\frac{1+(T-1)r}{2+(T-1)r}\times (1-r).$$

We differentiate $g(r)$ and get

$$g^{\prime }(r)=\frac{(T+1)-{[2+(T-1)r]}^2}{{[2+(T-1)r]}^2}.$$

Here the numerator is negative when

$$r>\frac{\sqrt{T+1}-2}{T-1}.$$

So, then if $T=3$, then $\frac{\sqrt{T+1}-2}{T-1}=0$, and $g(r)$ is always decreasing in $r$. Otherwise, $g(r)$ is an increasing function and then decreasing function of $r$, where the maximum is reached at $\frac{\sqrt{T+1}-2}{T-1}$.