Maximin design of cluster randomized trials with heterogeneous costs and variances

Optimal designs using generalized estimating equations in cluster randomized crossover and stepped wedge trials

Cluster randomized crossover and stepped wedge cluster randomized trials are two types of longitudinal cluster randomized trials that leverage both the within- and between-cluster comparisons to estimate the treatment effect and are increasingly used in healthcare delivery and implementation science research. While the variance expressions of estimated treatment effect have been previously developed from the method of generalized estimating equations for analyzing cluster randomized crossover trials and stepped wedge cluster randomized trials, little guidance has been provided for optimal designs to ensure maximum efficiency. Here, an optimal design refers to the combination of optimal cluster-period size and optimal number of clusters that provide the smallest variance of the treatment effect estimator or maximum efficiency under a fixed total budget. In this work, we develop optimal designs for multiple-period cluster randomized crossover trials and stepped wedge cluster randomized trials with continuous outcomes, including both closed-cohort and repeated cross-sectional sampling schemes. Local optimal design algorithms are proposed when the correlation parameters in the working correlation structure are known. MaxiMin optimal design algorithms are proposed when the exact values are unavailable, but investigators may specify a range of correlation values. The closed-form formulae of local optimal design and MaxiMin optimal design are derived for multiple-period cluster randomized crossover trials, where the cluster-period size and number of clusters are decimal. The decimal estimates from closed-form formulae can then be used to investigate the performances of integer estimates from local optimal design and MaxiMin optimal design algorithms. One unique contribution from this work, compared to the previous optimal design research, is that we adopt constrained optimization techniques to obtain integer estimates under the MaxiMin optimal design. To assist practical implementation, we also develop four SAS macros to find local optimal designs and MaxiMin optimal designs.

Maximin optimal cluster randomized designs for assessing treatment effect heterogeneity

Cluster randomized trials (CRTs) are studies where treatment is randomized at the cluster level but outcomes are typically collected at the individual level. When CRTs are employed in pragmatic settings, baseline population characteristics may moderate treatment effects, leading to what is known as heterogeneous treatment effects (HTEs). Pre-specified, hypothesis-driven HTE analyses in CRTs can enable an understanding of how interventions may impact subpopulation outcomes. While closed-form sample size formulas have recently been proposed, assuming known intracluster correlation coefficients (ICCs) for both the covariate and outcome, guidance on optimal cluster randomized designs to ensure maximum power with pre-specified HTE analyses has not yet been developed. We derive new design formulas to determine the cluster size and number of clusters to achieve the locally optimal design (LOD) that minimizes variance for estimating the HTE parameter given a budget constraint. Given the LODs are based on covariate and outcome-ICC values that are usually unknown, we further develop the maximin design for assessing HTE, identifying the combination of design resources that maximize the relative efficiency of the HTE analysis in the worst case scenario. In addition, given the analysis of the average treatment effect is often of primary interest, we also establish optimal designs to accommodate multiple objectives by combining considerations for studying both the average and heterogeneous treatment effects. We illustrate our methods using the context of the Kerala Diabetes Prevention Program CRT, and provide an R Shiny app to facilitate calculation of optimal designs under a wide range of design parameters.

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

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.

Best (but oft forgotten) practices: Efficient sample sizes for commonly used trial designs

Designing studies such that they have a high level of power to detect an effect or association of interest is an important tool to improve the quality and reproducibility of findings from such studies. Since resources (research subjects, time, and money) are scarce, it is important to obtain sufficient power with minimum use of such resources. For commonly used randomized trials of the treatment effect on a continuous outcome, designs are presented that minimize the number of subjects or the amount of research budget when aiming for a desired power level. This concerns the optimal allocation of subjects to treatments and, in case of nested designs such as cluster-randomized trials and multicenter trials, also the optimal number of centers versus the number of persons per center. Since such optimal designs require knowledge of parameters of the analysis model that are not known in the design stage, in particular outcome variances, maximin designs are presented. These designs guarantee a prespecified power level for plausible ranges of the unknown parameters and minimize research costs for the worst-case values of these parameters. The focus is on a 2-group parallel design, the AB/BA crossover design, and cluster-randomized and multicenter trials with a continuous outcome. How to calculate sample sizes for maximin designs is illustrated for examples from nutrition. Several computer programs that are helpful in calculating sample sizes for optimal and maximin designs are discussed as well as some results on optimal designs for other types of outcomes.

Maximin design of cluster randomized trials with heterogeneous costs and variances

Cluster randomized trials evaluate the effect of a treatment on persons nested within clusters, with clusters being randomly assigned to treatment. The optimal sample size at the cluster and person level depends on the study cost per cluster and per person, and the outcome variance at the cluster and the person level. The variances are unknown in the design stage and can differ between treatment arms. As a solution, this paper presents a Maximin design that maximizes the minimum relative efficiency (relative to the optimal design) over the variance parameter space, for trials with two treatment arms and a quantitative outcome. This maximin relative efficiency design (MMRED) is compared with a published Maximin design which maximizes the minimum efficiency (MMED). Both designs are also compared with the optimal designs for homogeneous costs and variances (balanced design) and heterogeneous costs and homogeneous variances (cost-conscious design), for a range of variances based upon three published trials. Whereas the MMED is balanced under high uncertainty about the treatment-to-control variance ratio, the MMRED then tends towards a balanced budget allocation between arms, leading to an unbalanced sample size allocation if costs are heterogeneous, similar to the cost-conscious design. Further, the MMRED corresponds to an optimal design for an intraclass correlation (ICC) in the lower half of the assumed ICC range (optimistic), whereas the MMED is the optimal design for the maximum ICC within the ICC range (pessimistic). Attention is given to the effect of the Welch-Satterthwaite degrees of freedom for treatment effect testing on the design efficiencies.