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Generalized estimating equations (GEEs) provide a useful framework for estimating marginal regression parameters based on data from cluster randomized trials (CRTs), but they can result in inaccurate parameter estimates when some outcomes are informatively missing. Existing techniques to handle missing outcomes in CRTs rely on correct specification of a propensity score model, a covariate-conditional mean outcome model, or require at least one of these two models to be correct, which can be challenging in practice. In this article, we develop new weighted GEEs to simultaneously estimate the marginal mean, scale, and correlation parameters in CRTs with missing outcomes, allowing for multiple propensity score models and multiple covariate-conditional mean models to be specified. The resulting estimators are consistent provided that any one of these models is correct. An iterative algorithm is provided for implementing this more robust estimator and practical considerations for specifying multiple models are discussed. We evaluate the performance of the proposed method through Monte Carlo simulations and apply the proposed multiply robust estimator to analyze the Botswana Combination Prevention Project, a large HIV prevention CRT designed to evaluate whether a combination of HIV-prevention measures can reduce HIV incidence.

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BACKGROUND: Using four case studies, we aim to provide practical guidance and recommendations for the analysis of cluster randomised controlled trials. METHODS: Four modelling approaches (Generalized Linear Mixed Models with parameters estimated by maximum likelihood/restricted maximum likelihood; Generalized Linear Models with parameters estimated by Generalized Estimating Equations (1st order or second order) and Quadratic Inference Function, for analysing correlated individual participant level outcomes in cluster randomised controlled trials were identified after we reviewed the literature. We systematically searched the online bibliography databases of MEDLINE, EMBASE, PsycINFO (via OVID), CINAHL (via EBSCO), and SCOPUS. We identified the above-mentioned four statistical analytical approaches and applied them to four case studies of cluster randomised controlled trials with the number of clusters ranging from 10 to 100, and individual participants ranging from 748 to 9,207. Results were obtained for both continuous and binary outcomes using R and SAS statistical packages. RESULTS: The intracluster correlation coefficient (ICC) estimates for the case studies were less than 0.05 and are consistent with the observed ICC values commonly reported in primary care and community-based cluster randomised controlled trials. In most cases, the four methods produced similar results. However, in a few analyses, quadratic inference function produced different results compared to the generalized linear mixed model, first-order generalized estimating equations, and second-order generalized estimating equations, especially in trials with small to moderate numbers of clusters. CONCLUSION: This paper demonstrates the analysis of cluster randomised controlled trials with four modelling approaches. The results obtained were similar in most cases, however, for trials with few clusters we do recommend that the quadratic inference function should be used with caution, and where possible a small sample correction should be used. The generalisability of our results is limited to studies with similar features to our case studies, for example, studies with a similar-sized ICC. It is important to conduct simulation studies to comprehensively evaluate the performance of the four modelling approaches.

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Cluster randomized trials (CRTs) refer to a popular class of experiments in which randomization is carried out at the group level. While methods have been developed for planning CRTs to study the average treatment effect, and more recently, to study the heterogeneous treatment effect, the development for the latter objective has currently been limited to a continuous outcome. Despite the prevalence of binary outcomes in CRTs, determining the necessary sample size and statistical power for detecting differential treatment effects in CRTs with a binary outcome remain unclear. To address this methodological gap, we develop sample size procedures for testing treatment effect heterogeneity in two-level CRTs under a generalized linear mixed model. Closed-form sample size expressions are derived for a binary effect modifier, and in addition, a computationally efficient Monte Carlo approach is developed for a continuous effect modifier. Extensions to multiple effect modifiers are also discussed. We conduct simulations to examine the accuracy of the proposed sample size methods. We present several numerical illustrations to elucidate features of the proposed formulas and to compare our method to the approximate sample size calculation under a linear mixed model. Finally, we use data from the Strategies and Opportunities to Stop Colon Cancer in Priority Populations (STOP CRC) CRT to illustrate the proposed sample size procedure for testing treatment effect heterogeneity.

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OBJECTIVES: The design, analysis, and interpretation of cluster randomized clinical trials (RCTs) require accounting for potential correlation of observations on individuals within the same cluster. Reporting of observed intracluster correlation coefficients (ICCs) in cluster RCTs, as recommended by Consolidated Standards of Reporting Trials (CONSORT), facilitates sample size calculation of future cluster RCTs and understanding of the trial statistical power. Our objective was to summarize observed ICCs in osteoarthritis (OA) cluster RCTs. DESIGN: Systematic review of knee/hip OA cluster RCTs. We searched Cochrane Central Register of Controlled Trials for trials published from 2012, when CONSORT cluster RCTs extension was published, to September 2022. We calculated the proportion of cluster RCTs that reported observed ICCs. Of those that did, we extracted observed ICCs. PROSPERO: CRD42022365660. RESULTS: We screened 1121 references and included 20 cluster RCTs. Only 5 trials (25%) reported the observed ICC for at least one outcome variable. ICC values for pain outcomes were: 0, 0.01, 0.18; for physical function outcomes were: 0, 0.06, 0.13 (knee)/0.27 (hip); Western Ontario and McMaster Universities Arthritis Index (WOMAC) total: 0.02, 0.02; symptoms of anxiety/depression: 0.22; disability: 0; and global change: 0. One out of four (25%) trials reported an ICC that was larger than the ICC used for sample size calculation and thus was underpowered. CONCLUSIONS: Despite CONSORT statement recommendations for reporting cluster RCTs, few OA trials reported the observed ICC. Given the importance of the ICC to interpretation of trial results and future trial design, this reporting gap warrants attention.

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An important consideration in the design and analysis of randomized trials is the need to account for outcome observations being positively correlated within groups or clusters. Two notable types of designs with this consideration are individually randomized group treatment trials and cluster randomized trials. While sample size methods for testing the average treatment effect are available for both types of designs, methods for detecting treatment effect modification are relatively limited. In this article, we present new sample size formulas for testing treatment effect modification based on either a univariate or multivariate effect modifier in both individually randomized group treatment and cluster randomized trials with a continuous outcome but any types of effect modifier, while accounting for differences across study arms in the outcome variance, outcome intracluster correlation coefficient (ICC) and the cluster size. We consider cases where the effect modifier can be measured at either the individual level or cluster level, and with a univariate effect modifier, our closed-form sample size expressions provide insights into the optimal allocation of groups or clusters to maximize design efficiency. Overall, our results show that the required sample size for testing treatment effect heterogeneity with an individual-level effect modifier can be affected by unequal ICCs and variances between arms, and accounting for such between-arm heterogeneity can lead to more accurate sample size determination. We use simulations to validate our sample size formulas and illustrate their application in the context of two real trials: an individually randomized group treatment trial (the AWARE study) and a cluster randomized trial (the K-DPP study).

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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.

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BACKGROUND: The intracluster correlation coefficient is a key input parameter for sample size determination in cluster-randomised trials. Sample size is very sensitive to small differences in the intracluster correlation coefficient, so it is vital to have a robust intracluster correlation coefficient estimate. This is often problematic because either a relevant intracluster correlation coefficient estimate is not available or the available estimate is imprecise due to being based on small-scale studies with low numbers of clusters. Misspecification may lead to an underpowered or inefficiently large and potentially unethical trial. METHODS: We apply a Bayesian approach to produce an intracluster correlation coefficient estimate and hence propose sample size for a planned cluster-randomised trial of the effectiveness of a systematic voiding programme for post-stroke incontinence. A Bayesian hierarchical model is used to combine intracluster correlation coefficient estimates from other relevant trials making use of the wealth of intracluster correlation coefficient information available in published research. We employ knowledge elicitation process to assess the relevance of each intracluster correlation coefficient estimate to the planned trial setting. The team of expert reviewers assigned relevance weights to each study, and each outcome within the study, hence informing parameters of Bayesian modelling. To measure the performance of experts, agreement and reliability methods were applied. RESULTS: The 34 intracluster correlation coefficient estimates extracted from 16 previously published trials were combined in the Bayesian hierarchical model using aggregated relevance weights elicited from the experts. The intracluster correlation coefficients available from external sources were used to construct a posterior distribution of the targeted intracluster correlation coefficient which was summarised as a posterior median with a 95% credible interval informing researchers about the range of plausible sample size values. The estimated intracluster correlation coefficient determined a sample size of between 450 (25 clusters) and 480 (20 clusters), compared to 500-600 from a classical approach. The use of quantiles, and other parameters, from the estimated posterior distribution is illustrated and the impact on sample size described. CONCLUSION: Accounting for uncertainty in an unknown intracluster correlation coefficient, trials can be designed with a more robust sample size. The approach presented provides the possibility of incorporating intracluster correlation coefficients from various cluster-randomised trial settings which can differ from the planned study, with the difference being accounted for in the modelling. By using expert knowledge to elicit relevance weights and synthesising the externally available intracluster correlation coefficient estimates, information is used more efficiently than in a classical approach, where the intracluster correlation coefficient estimates tend to be less robust and overly conservative. The intracluster correlation coefficient estimate constructed is likely to produce a smaller sample size on average than the conventional strategy of choosing a conservative intracluster correlation coefficient estimate. This may therefore result in substantial time and resources savings.

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BACKGROUND: Detecting treatment effect heterogeneity is an important objective in cluster randomized trials and implementation research. While sample size procedures for testing the average treatment effect accounting for participant attrition assuming missing completely at random or missing at random have been previously developed, the impact of attrition on the power for detecting heterogeneous treatment effects in cluster randomized trials remains unknown. METHODS: We provide a sample size formula for testing for a heterogeneous treatment effect assuming the outcome is missing completely at random. We also propose an efficient Monte Carlo sample size procedure for assessing heterogeneous treatment effect assuming covariate-dependent outcome missingness (missing at random). We compare our sample size methods with the direct inflation method that divides the estimated sample size by the mean follow-up rate. We also evaluate our methods through simulation studies and illustrate them with a real-world example. RESULTS: Simulation results show that our proposed sample size methods under both missing completely at random and missing at random provide sufficient power for assessing heterogeneous treatment effect. The proposed sample size methods lead to more accurate sample size estimates than the direct inflation method when the missingness rate is high (e.g., ≥ 30%). Moreover, sample size estimation under both missing completely at random and missing at random is sensitive to the missingness rate, but not sensitive to the intracluster correlation coefficient among the missingness indicators. CONCLUSION: Our new sample size methods can assist in planning cluster randomized trials that plan to assess a heterogeneous treatment effect and participant attrition is expected to occur.

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BACKGROUND AND OBJECTIVES: To summarize intracluster correlation coefficient (ICC) estimates for pupil health outcomes from school-based cluster randomized trials (CRTs) across world regions and describe their relationship with study design characteristics and context. METHODS: School-based CRTs reporting ICCs for pupil health outcomes were identified through a literature search of MEDLINE (via Ovid). ICC estimates were summarized both overall and for different categories of study characteristics. RESULTS: Two hundred and forty-six articles reporting ICC estimates were identified. The median (interquartile range) ICC was 0.031 (0.011 to 0.08) at the school level (N = 210) and 0.063 (0.024 to 0.1) at the class level (N = 46). The distribution of ICCs at the school level was well described by the beta and exponential distributions. Besides larger ICCs in definitive trials than feasibility studies, there were no clear associations between study characteristics and ICC estimates. CONCLUSION: The distribution of school-level ICCs worldwide was similar to previous summaries from studies in the United States. The description of the distribution of ICCs will help to inform sample size calculations and assess their sensitivity when designing future school-based CRTs of health interventions.

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Objective: To determine normal variation in walking metrics in a population of lower limb amputees who use lower limb prostheses over a 6-month period and to provide a means to interpret clinically meaningful change in those community walking metrics. Design: Prospective cohort study monitoring walking behavior and subjective and objective measures of activity. Setting: Veterans Administration and university amputee clinics. Participants: 86 individuals with lower limb amputation who use protheses. Interventions: StepWatch activity monitor tracked subjects' walking for 24 weeks; Global Mobility Change Rating collected weekly. Main Outcome Measures: Association between change in Global Mobility Change Rating and change in any of the walking metrics. Results: Walking metrics including step count, cadence, cadence variability, and walking distance in a population of lower limb prosthesis users were obtained. There was a high correlation in the walking metrics indicating higher function with higher functional classification level (K-levels) but also substantial overlap in all metrics and a very weak correlation between subject-reported activity level and objective measures of walking performance. Conclusion: The overlap in walking metrics with all K-levels demonstrates that no single metric measured by StepWatch can determine K-level with 100% accuracy. As previously demonstrated in other populations, subjects' interpretations of their general activity level was inaccurate, regardless of their age or activity level. Objective measures of walking appear to provide a more accurate representation of patients' activity levels in the community than self-report. Therefore, objective measures of walking are useful in supporting K-level determinations. However, clinicians cannot rely on a single metric to determine K-level.

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The relationship between social cohesion and health has been studied for decades. Yet, due to the contextual nature of this concept, measuring social cohesion remains challenging. Using a meta-analytical framework, this review's goal was to study the ecometric measurement properties of social cohesion in order to describe dissimilarities in its measurement as well as bring a new perspective on the empirical usefulness of the concept itself. To this end, we analysed if, and to what extent, contextual-level reliability and intersubjective agreement of 78 social cohesion measurements varied under different measurement conditions like measurement instrument, spatial unit, ecometric model specification, or region. We found consistent evidence for the contextual nature of social cohesion, however, most variation existed between individuals, not contexts. While contextual dependence in response behaviour was fairly insensitive to item choices, population size within chosen spatial units of social cohesion measurements mattered. Somewhat counterintuitively, using spatial units with, on average, fewer residents did not yield systematically superior ecometric properties. Instead, our results underline that precise theory about the relevant contextual units of causal relationships between social cohesion and health is vital and cannot be replaced by empirical analysis. Although adjustment for respondent's characteristics had only small effects on ecometric properties, potential pitfalls of this analytic strategy are discussed in this paper. Finally, acknowledging the sensitivity of measuring social cohesion, we derived recommendations for future studies investigating the effects of contextual-level social characteristics on health.

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BACKGROUND: In cluster randomised controlled trials (cRCTs), groups of individuals (rather than individuals) are randomised to minimise the risk of contamination and/or efficiently use limited resources or solve logistic and administrative problems. A major concern in the primary analysis of cRCT is the use of appropriate statistical methods to account for correlation among outcomes from a particular group/cluster. This review aimed to investigate the statistical methods used in practice for analysing the primary outcomes in publicly funded cluster randomised controlled trials, adherence to the CONSORT (Consolidated Standards of Reporting Trials) reporting guidelines for cRCTs and the recruitment abilities of the cluster trials design. METHODS: We manually searched the United Kingdom's National Institute for Health Research (NIHR) online Journals Library, from 1 January 1997 to 15 July 2021 chronologically for reports of cRCTs. Information on the statistical methods used in the primary analyses was extracted. One reviewer conducted the search and extraction while the two other independent reviewers supervised and validated 25% of the total trials reviewed. RESULTS: A total of 1942 reports, published online in the NIHR Journals Library were screened for eligibility, 118 reports of cRCTs met the initial inclusion criteria, of these 79 reports containing the results of 86 trials with 100 primary outcomes analysed were finally included. Two primary outcomes were analysed at the cluster-level using a generalized linear model. At the individual-level, the generalized linear mixed model was the most used statistical method (80%, 80/100), followed by regression with robust standard errors (7%) then generalized estimating equations (6%). Ninety-five percent (95/100) of the primary outcomes in the trials were analysed with appropriate statistical methods that accounted for clustering while 5% were not. The mean observed intracluster correlation coefficient (ICC) was 0.06 (SD, 0.12; range, - 0.02 to 0.63), and the median value was 0.02 (IQR, 0.001-0.060), although 42% of the observed ICCs for the analysed primary outcomes were not reported. CONCLUSIONS: In practice, most of the publicly funded cluster trials adjusted for clustering using appropriate statistical method(s), with most of the primary analyses done at the individual level using generalized linear mixed models. However, the inadequate analysis and poor reporting of cluster trials published in the UK is still happening in recent times, despite the availability of the CONSORT reporting guidelines for cluster trials published over a decade ago.

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OBJECTIVE: To examine determinants, trends, and costs associated with 30-day all-cause readmission (R) for suicidal ideation (SI) in early-aged patients. METHODS: This was a retrospective cohort study using the 2010-2014 Nationwide Readmissions Database. Discharge records for those aged 5-24 with an SI diagnosis were analyzed. Hierarchical models (HMs) were used to assess factors of R, length of stay (LOS), and total costs of Rs. RESULTS: There were 197,603 SI index admissions (IAs). Of these, 2% had a R. The annualized trend of R rates for all age groups remained constant. Those aged 13-18 had the highest rate of Rs, while IA and R mean total costs were highest for those aged 5-12 (IA, $4,546-$5,822; R, $5,361-$7,113). The strongest risk factors for increasing R included nonelective admission and private hospital ownership. The strongest risk factors for increasing LOS and cost were major/extreme severity of illness and 30-day all-cause R. The intracluster correlation coefficient for the HMs were 0.06, 0.33, and 0.55 for the R, LOS, and cost model, respectively. CONCLUSIONS: The R rate was highest for those in the 13-18 age group, while the costs were highest for those aged 5-12.

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Each cluster consists of multiple subunits from which outcome data are collected. In a subunit randomization trial, subunits are randomized into different intervention arms. Observations from subunits within each cluster tend to be positively correlated due to the shared common frailties, so that the outcome data from a subunit randomization trial have dependency between arms as well as within each arm. For subunit randomization trials with a survival endpoint, few methods have been proposed for sample size calculation showing the clear relationship between the joint survival distribution between subunits and the sample size, especially when the number of subunits from each cluster is variable. In this paper, we propose a closed form sample size formula for weighted rank test to compare the marginal survival distributions between intervention arms under subunit randomization, possibly with variable number of subunits among clusters. We conduct extensive simulations to evaluate the performance of our formula under various design settings, and demonstrate our sample size calculation method with some real clinical trials.

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We propose group sequential methods for cluster randomized trials (CRTs) with time-to-event endpoint. The alpha spending function approach is used for sequential data monitoring. The key to this approach is determining the joint distribution of test statistics and the information fraction at the time of interim analysis. We prove that the sequentially computed log-rank statistics in CRTs do not have independent increment property. We also propose an information fraction for group sequential trials with clustered survival data and a corresponding sample size determination approach. Extensive simulation studies are conducted to evaluate the performance of our proposed testing procedure using some existing alpha spending functions in terms of expected sample size and maximal sample size. Real study examples are taken to demonstrate our method.

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BACKGROUND/AIMS: Generalized estimating equations are commonly used to fit logistic regression models to clustered binary data from cluster randomized trials. A commonly used correlation structure assumes that the intracluster correlation coefficient does not vary by treatment arm or other covariates, but the consequences of this assumption are understudied. We aim to evaluate the effect of allowing variation of the intracluster correlation coefficient by treatment or other covariates on the efficiency of analysis and show how to account for such variation in sample size calculations. METHODS: We develop formulae for the asymptotic variance of the estimated difference in outcome between treatment arms obtained when the true exchangeable correlation structure depends on the treatment arm and the working correlation structure used in the generalized estimating equations analysis is: (i) correctly specified, (ii) independent, or (iii) exchangeable with no dependence on treatment arm. These formulae require a known distribution of cluster sizes; we also develop simplifications for the case when cluster sizes do not vary and approximations that can be used when the first two moments of the cluster size distribution are known. We then extend the results to settings with adjustment for a second binary cluster-level covariate. We provide formulae to calculate the required sample size for cluster randomized trials using these variances. RESULTS: We show that the asymptotic variance of the estimated difference in outcome between treatment arms using these three working correlation structures is the same if all clusters have the same size, and this asymptotic variance is approximately the same when intracluster correlation coefficient values are small. We illustrate these results using data from a recent cluster randomized trial for infectious disease prevention in which the clusters are groups of households and modest in size (mean 9.6 individuals), with intracluster correlation coefficient values of 0.078 in the control arm and 0.057 in an intervention arm. In this application, we found a negligible difference between the variances calculated using structures (i) and (iii) and only a small increase (typically <5%) for the independent correlation structure (ii), and hence minimal effect on power or sample size requirements. The impact may be larger in other applications if there is greater variation in the ICC between treatment arms or with an additional covariate. CONCLUSION: The common approach of fitting generalized estimating equations with an exchangeable working correlation structure with a common intracluster correlation coefficient across arms likely does not substantially reduce the power or efficiency of the analysis in the setting of a large number of small or modest-sized clusters, even if the intracluster correlation coefficient varies by treatment arm. Our formulae, however, allow formal evaluation of this and may identify situations in which variation in intracluster correlation coefficient by treatment arm or another binary covariate may have a more substantial impact on power and hence sample size requirements.

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BACKGROUND: Incorporating cluster randomized trials (CRTs) into meta-analyses is challenging because appropriate standard errors of study estimates accounting for clustering are not always reported. Systematic reviews of CRTs often use a single constant external estimate of the intraclass correlation coefficient (ICC) to adjust study estimate standard errors and facilitate meta-analyses; an approach that fails to account for possible variation of ICCs among studies and the imprecision with which they are estimated. Using a large systematic review of the effects of diabetes quality improvement interventions, we investigated whether we could better account for ICC variation and uncertainty in meta-analyzed effect estimates by imputing missing ICCs from a posterior predictive distribution constructed from a database of relevant ICCs. METHODS: We constructed a dataset of ICC estimates from applicable studies. For outcomes with two or more available ICC estimates, we constructed posterior predictive ICC distributions in a Bayesian framework. For a selected continuous outcome, glycosylated hemoglobin (HbA1c), we compared the impact of incorporating a single constant ICC versus imputing ICCs drawn from the posterior predictive distribution when estimating the effect of intervention components on post treatment mean in a case study of diabetes quality improvement trials. RESULTS: Using internal and external ICC estimates, we were able to construct a database of 59 ICCs for 12 of the 13 review outcomes (range 1-10 per outcome) and estimate the posterior predictive ICC distribution for 11 review outcomes. Synthesized results were not markedly changed by our approach for HbA1c. CONCLUSION: Building posterior predictive distributions to impute missing ICCs is a feasible approach to facilitate principled meta-analyses of cluster randomized trials using prior data. Further work is needed to establish whether the application of these methods leads to improved review inferences for different reviews based on different factors (e.g., proportion of CRTs and CRTs with missing ICCs, different outcomes, variation and precision of ICCs).

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OBJECTIVES: To estimate the prevalence of time-to-event (TTE) outcomes in cluster randomized trials (CRTs) and to examine their statistical management. STUDY DESIGN AND SETTING: We searched PubMed to identify primary reports of CRTs published in six major general medical journals (2013-2018). Nature of outcomes and, for TTE outcomes, statistical methods for sample size, analysis, and measures of intracluster correlation were extracted. RESULTS: A TTE analysis was used in 17% of the CRTs (32/184) either as a primary or secondary outcome analysis, or in a sensitivity analysis. Among the five CRTs with a TTE primary outcome, two accounted for both intracluster correlation and the TTE nature of the outcome in sample size calculation; one reported a measure of intracluster correlation in the analysis. Among the 32 CRTs with a least one TTE analysis, 44% (14/32) accounted for clustering in all TTE analyses. We identified 12 additional CRTs in which there was at least one outcome not analyzed as TTE for which a TTE analysis might have been preferred. CONCLUSION: TTE outcomes are not uncommon in CRTs but appropriate statistical methods are infrequently used. Our results suggest that further methodological development and explicit recommendations for TTE outcomes in CRTs are needed.

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To calculate sample sizes in cluster randomized trials (CRTs), the cluster sizes are usually assumed to be identical across all clusters for simplicity. However, equal cluster sizes are not guaranteed in practice, especially when the number of clusters is limited. Therefore, it is important to understand the relative efficiency (RE) of equal versus unequal cluster sizes when designing CRTs with a limited number of clusters. In this paper, we are interested in the RE of two bias-corrected sandwich estimators of the treatment effect in the Generalized Estimating Equation (GEE) models for CRTs with a small number of clusters. Specifically, we derive the RE of two bias-corrected sandwich estimators for binary, continuous, or count data in CRTs under the assumption of an exchangeable working correlation structure. We consider different scenarios of cluster size distributions and investigate RE performance through simulation studies. We conclude that the number of clusters could be increased by as much as 42% to compensate for efficiency loss due to unequal cluster sizes. Finally, we propose an algorithm of increasing the number of clusters when the coefficient of variation of cluster sizes is known and unknown.

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Individual randomized trials (IRTs) and cluster randomized trials (CRTs) with binary outcomes arise in a variety of settings and are often analyzed by logistic regression (fitted using generalized estimating equations for CRTs). The effect of stratification on the required sample size is less well understood for trials with binary outcomes than for continuous outcomes. We propose easy-to-use methods for sample size estimation for stratified IRTs and CRTs and demonstrate the use of these methods for a tuberculosis prevention CRT currently being planned. For both IRTs and CRTs, we also identify the ratio of the sample size for a stratified trial vs a comparably powered unstratified trial, allowing investigators to evaluate how stratification will affect the required sample size when planning a trial. For CRTs, these can be used when the investigator has estimates of the within-stratum intracluster correlation coefficients (ICCs) or by assuming a common within-stratum ICC. Using these methods, we describe scenarios where stratification may have a practically important impact on the required sample size. We find that in the two-stratum case, for both IRTs and for CRTs with very small cluster sizes, there are unlikely to be plausible scenarios in which an important sample size reduction is achieved when the overall probability of a subject experiencing the event of interest is low. When the probability of events is not small, or when cluster sizes are large, however, there are scenarios where practically important reductions in sample size result from stratification.

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