Efficient design of cluster randomized trials with treatment-dependent costs and treatment-dependent unknown 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.

Optimal design for inference on the threshold of a biomarker

Enrichment designs with a continuous biomarker require the estimation of a threshold to determine the subpopulation benefitting from the treatment. This article provides the optimal allocation for inference in a two-stage enrichment design for treatment comparisons when a continuous biomarker is suspected to affect patient response. Several design criteria, associated with different trial objectives, are optimized under balanced or Neyman allocation and under equality of the first two empirical biomarker's moments. Moreover, we propose a new covariate-adaptive randomization procedure that converges to the optimum with the fastest available rate. Theoretical and simulation results show that this strategy improves the efficiency of a two-stage enrichment clinical trial, especially with smaller sample sizes and under heterogeneous responses.

The Effect of School-Linked Module-Based Friendly-Health Education on Adolescents' Sexual and Reproductive Health Knowledge, Guji Zone, Ethiopia - Cluster Randomized Controlled Trial

Background

Although access to sexual and reproductive health information is the right and a critical component of health policy, it is not well addressed in pastoral communities. This study assessed the effect of School-Lined Module-based friendly health education on adolescents' sexual and reproductive health knowledge in the pastoral community of Guji Zone, Ethiopia.

Methods

A two-arm cluster Randomized control trial study with pre-post evaluation was conducted among interventions (n=375) compared with control (n=384) in Gorodola and Wadara high schools. Comparing an intervention to a control group, pre-posttests, and post-posttests were used to evaluate the effectiveness of the intervention. The data was collected using 25 Self-administered questionnaires and analyzed using paired-sample independent t-tests and linear regressions to study the relationship between the outcome and independent variables.

Results

We collected the data from 759 adolescents among 15 intervention and 15 control clusters. The results have shown that as compared to control arms, the mean sexual and reproductive health Knowledge score was significant higher in the intervention clusters (375) 73.3%, vs (384) 66.5%%, p<0.001, 95% CI, (0.05395-0.08347). Information (β: 0.038, 95% CI: 0.028-0.052), confidence (β: 0.045, 95% CI: 0.033-0.057), knowledge (β: 0.05, 95% CI: 0.035-0.066), and compassionate care (β: 0.107, 95% CI: 0.092-0.122) were significantly associated with SRH knowledge prediction. The proportion of SRH knowledge increased from 168(44%) baseline to 244(65%) end line in the intervention versus 235(60% to 238(62%) in control arms.

Conclusion

The execution of school-linked module-based friendly health education has proved to have a significant effect on mean SRH knowledge. Individual-level and behavioral-level factors significantly explain variability in enhancing SRH knowledge in the pastoral community. We recommend scaling up the School-Linked Module-based friendly health education intervention.

Trial Registration

We registered clinical trial PACTR202107905622610 on 16 July 2021.

Designing multicenter individually randomized group treatment trials

In an individually randomized group treatment (IRGT) trial, participant outcomes can be positively correlated due to, for example, shared therapists in treatment delivery. Oftentimes, because of limited treatment resources or participants at one location, an IRGT trial can be carried out across multiple centers. This design can be subject to potential correlations in the participant outcomes between arms within the same center. While the design of a single-center IRGT trial has been studied, little is known about the planning of a multicenter IRGT trial. To address this gap, this paper provides analytical sample size formulas for designing multicenter IRGT trials with a continuous endpoint under the linear mixed model framework. We found that accounting for the additional center-level correlation at the design stage can lead to sample size reduction, and the magnitude of reduction depends on the amount of between-therapist correlation. However, if the variance components of therapist-level random effects are considered as input parameters in the design stage, accounting for the additional center-level variance component has no impact on the sample size estimation. We presented our findings through numeric illustrations and performed simulation studies to validate our sample size procedures under different scenarios. Optimal design configurations under the multicenter IRGT trials have also been discussed, and two real-world trial examples are drawn to illustrate the use of our method.

Effect of a school-linked life skills intervention on adolescents' sexual and reproductive health skills in Guji zone, Ethiopia (CRT)-A generalized linear model

Background

Although appropriate life skills are recognized as a fundamental right for adolescents and a critical component of health policy, they are often overlooked and rarely researched in pastoral communities. Life skills encompass the ability to adopt positive behaviors, which enable individuals to effectively deal with the demands and challenges of everyday life. This study aimed to evaluate the effectiveness of a school-linked life skills intervention program on adolescents' sexual and reproductive health skills in the pastoral community of Guji zone.

Methods

A two-arm cluster randomized control trial with a pretest-posttest experimental design was conducted, involving the intervention group (N = 375) and the control group (N = 384). This study assessed the effect of a school-linked adolescent-friendly life skills intervention in comparison to the usual RH curriculum, used as a control arm. Pretest-posttest and posttest-posttest scores of the control group and trial groups were compared, and the data were collected using 27 self-administered questions. The collected data were analyzed using paired-sample independent t-tests and a generalized linear model to examine the relationship between the dependent and independent variables.

Results

Data were collected from 759 adolescents in 15 intervention and 15 control clusters. The findings have shown that the proportion of mean life skills score was significantly higher in the intervention clusters than controls [(375) 70.49% vs. (384) 62.25%, P < 0.001 95% CI (0.06 and 0.1)]. Adolescents who were trained in school-linked life skills (β = 1.915, 95% CI: 1.411-2.418), were confident to make safe and informed decisions (β = 1.999, 95% CI: 1.562-2.436), and had life skills to deal with SRH issues (β = 1.66, 95% CI: 1.233-2.087) were significantly correlated with predicting the relevant life skills. The proportion of adolescents with SRH life skills increased from 384 (52%) at baseline to 375 (70%) at end line in the intervention group compared to 384 (60.31%) at baseline to 384 (62.31%) in control arms, respectively.

Conclusions

The implementation of a school-linked life skills intervention program proved to have a significant effect on SRH life skills development. Furthermore, individual-level and behavioral-level variables were significant in explaining variability in life skills development within the pastoral community. Therefore, we recommend scaling up this intervention in all high schools.

Trial registration

Trial registration PACTR202107905622610, registered on 16 July 2021.

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.

Optimal allocation to treatments in a sequential multiple assignment randomized trial

One of the main questions in the design of a trial is how many subjects should be assigned to each treatment condition. Previous research has shown that equal randomization is not necessarily the best choice. We study the optimal allocation for a novel trial design, the sequential multiple assignment randomized trial, where subjects receive a sequence of treatments across various stages. A subject's randomization probabilities to treatments in the next stage depend on whether he or she responded to treatment in the current stage. We consider a prototypical sequential multiple assignment randomized trial design with two stages. Within such a design, many pairwise comparisons of treatment sequences can be made, and a multiple-objective optimal design strategy is proposed to consider all such comparisons simultaneously. The optimal design is sought under either a fixed total sample size or a fixed budget. A Shiny App is made available to find the optimal allocations and to evaluate the efficiency of competing designs. As the optimal design depends on the response rates to first-stage treatments, maximin optimal design methodology is used to find robust optimal designs. The proposed methodology is illustrated using a sequential multiple assignment randomized trial example on weight loss management.

Optimal design of cluster randomized trials allowing unequal allocation of clusters and unequal cluster size between arms

There are sometimes cost, scientific, or logistical reasons to allocate individuals unequally in an individually randomized trial. In cluster randomized trials we can allocate clusters unequally and/or allow different cluster size between trial arms. We consider parallel group designs with a continuous outcome, and optimal designs that require the smallest number of individuals to be measured given the number of clusters. Previous authors have derived the optimal allocation ratio for clusters under different variance and/or intracluster correlations (ICCs) between arms, allowing different but prespecified cluster sizes by arm. We derive closed-form expressions to identify the optimal proportions of clusters and of individuals measured for each arm, thereby defining optimal cluster sizes, when cluster size can be chosen freely. When ICCs differ between arms but the variance is equal, the optimal design allocates more than half the clusters to the arm with the higher ICC, but (typically only slightly) less than half the individuals and hence a smaller cluster size. We also describe optimal design under constraints on the number of clusters or cluster size in one or both arms. This methodology allows trialists to consider a range for the number of clusters in the trial and for each to identify the optimal design. Except if there is clear prior evidence for the ICC and variance by arm, a range of values will need to be considered. Researchers should choose a design with adequate power across the range, while also keeping enough clusters in each arm to permit the intended analysis method.

Optimal allocations for two treatment comparisons within the proportional odds cumulative logits model

This paper studies optimal treatment allocations for two treatment comparisons when the outcome is ordinal and analyzed by a proportional odds cumulative logits model. The variance of the treatment effect estimator is used as optimality criterion. The optimal design is sought so that this variance is minimal for a given total sample size or a given budget, meaning that the power for the test on treatment effect is maximal, or it is sought so that a required power level is achieved at a minimal total sample size or budget. Results are presented for three, five and seven ordered response categories, three treatment effect sizes and a skewed, bell-shaped or polarized distribution of the response probabilities. The optimal proportion subjects in the intervention condition decreases with the number of response categories and the costs for the intervention relative to those for the control. The relation between the optimal proportion and effect size depends on the distribution of the response probabilities. The widely used balanced design is not always the most efficient; its efficiency as compared to the optimal design decreases with increasing cost ratio. The optimal design is highly robust to misspecification of the response probabilities and treatment effect size. The optimal design methodology is illustrated using two pharmaceutical examples. A Shiny app is available to find the optimal treatment allocation, to evaluate the efficiency of the balanced design and to study the relation between budget or sample size and power.

Optimal two-stage sampling for mean estimation in multilevel populations when cluster size is informative

To estimate the mean of a quantitative variable in a hierarchical population, it is logistically convenient to sample in two stages (two-stage sampling), i.e. selecting first clusters, and then individuals from the sampled clusters. Allowing cluster size to vary in the population and to be related to the mean of the outcome variable of interest (informative cluster size), the following competing sampling designs are considered: sampling clusters with probability proportional to cluster size, and then the same number of individuals per cluster; drawing clusters with equal probability, and then the same percentage of individuals per cluster; and selecting clusters with equal probability, and then the same number of individuals per cluster. For each design, optimal sample sizes are derived under a budget constraint. The three optimal two-stage sampling designs are compared, in terms of efficiency, with each other and with simple random sampling of individuals. Sampling clusters with probability proportional to size is recommended. To overcome the dependency of the optimal design on unknown nuisance parameters, maximin designs are derived. The results are illustrated, assuming probability proportional to size sampling of clusters, with the planning of a hypothetical survey to compare adolescent alcohol consumption between France and Italy.

Optimal allocation of subjects in a matched pair cluster-randomized trial with fixed number of heterogeneous clusters

In cluster-randomized trials, investigators randomize clusters of individuals such as households, medical practices, schools or classrooms despite the unit of interest are the individuals. It results in the loss of efficiency in terms of the estimation of the unknown parameters as well as the power of the test for testing the treatment effects. To recoup this efficiency loss, some studies pair similar clusters and randomize treatment within pairs. However, the clusters within a treatment arm might be heterogeneous in nature. In this article, we propose a locally optimal design that accounts the clusters heterogeneity and optimally allocates the subjects within each cluster. To address the dependency of design on the unknown parameters, we also discuss Bayesian optimal designs. Performances of proposed designs are investigated numerically through some data examples.

Efficient design of cluster randomized trials with treatment-dependent costs and treatment-dependent unknown variances

Cluster randomized trials evaluate the effect of a treatment on persons nested within clusters, where treatment is randomly assigned to clusters. Current equations for the optimal sample size at the cluster and person level assume that the outcome variances and/or the study costs are known and homogeneous between treatment arms. This paper presents efficient yet robust designs for cluster randomized trials with treatment‐dependent costs and treatment‐dependent unknown variances, and compares these with 2 practical designs. First, the maximin design (MMD) is derived, which maximizes the minimum efficiency (minimizes the maximum sampling variance) of the treatment effect estimator over a range of treatment‐to‐control variance ratios. The MMD is then compared with the optimal design for homogeneous variances and costs (balanced design), and with that for homogeneous variances and treatment‐dependent costs (cost‐considered design). The results show that the balanced design is the MMD if the treatment‐to control cost ratio is the same at both design levels (cluster, person) and within the range for the treatment‐to‐control variance ratio. It still is highly efficient and better than the cost‐considered design if the cost ratio is within the range for the squared variance ratio. Outside that range, the cost‐considered design is better and highly efficient, but it is not the MMD. An example shows sample size calculation for the MMD, and the computer code (SPSS and R) is provided as supplementary material. The MMD is recommended for trial planning if the study costs are treatment‐dependent and homogeneity of variances cannot be assumed.