School-level intra-cluster correlation coefficients and autocorrelations for children's accelerometer-measured physical activity in England by age and gender.

BACKGROUND: Randomised, cluster-based study designs in schools are commonly used to evaluate children's physical activity interventions. Sample size estimation relies on accurate estimation of the intra-cluster correlation coefficient (ICC), but published estimates, especially using accelerometry-measured physical activity, are few and vary depending on physical activity outcome and participant age. Less commonly-used cluster-based designs, such as stepped wedge designs, also need to account for correlations over time, e.g. cluster autocorrelation (CAC) and individual autocorrelation (IAC), but no estimates are currently available. This paper estimates the school-level ICC, CAC and IAC for England children's accelerometer-measured physical activity outcomes by age group and gender, to inform the design of future school-based cluster trials. METHODS: Data were pooled from seven large English datasets of accelerometer-measured physical activity data between 2002-18 (> 13,500 pupils, 540 primary and secondary schools). Linear mixed effect models estimated ICCs for weekday and whole week for minutes spent in moderate-to-vigorous physical activity (MVPA) and being sedentary for different age groups, stratified by gender. The CAC (1,252 schools) and IAC (34,923 pupils) were estimated by length of follow-up from pooled longitudinal data. RESULTS: School-level ICCs for weekday MVPA were higher in primary schools (from 0.07 (95% CI: 0.05, 0.10) to 0.08 (95% CI: 0.06, 0.11)) compared to secondary (from 0.04 (95% CI: 0.03, 0.07) to (95% CI: 0.04, 0.10)). Girls' ICCs were similar for primary and secondary schools, but boys' were lower in secondary. For all ages, combined the CAC was 0.60 (95% CI: 0.44-0.72), and the IAC was 0.46 (95% CI: 0.42-0.49), irrespective of follow-up time. Estimates were higher for MVPA vs sedentary time, and for weekdays vs the whole week. CONCLUSIONS: Adequately powered studies are important to evidence effective physical activity strategies. Our estimates of the ICC, CAC and IAC may be used to plan future school-based physical activity evaluations and were fairly consistent across a range of ages and settings, suggesting that results may be applied to other high income countries with similar school physical activity provision. It is important to use estimates appropriate to the study design, and that match the intended study population as closely as possible.

Efficient designs for three-sequence stepped wedge trials with continuous recruitment.

BACKGROUND/AIMS: The standard approach to designing stepped wedge trials that recruit participants in a continuous stream is to divide time into periods of equal length. But the choice of design in such cases is infinitely more flexible: each cluster could cross from the control to the intervention at any point on the continuous time-scale. We consider the case of a stepped wedge design with clusters randomised to just three sequences (designs with small numbers of sequences may be preferred for their simplicity and practicality) and investigate the choice of design that minimises the variance of the treatment effect estimator under different assumptions about the intra-cluster correlation. METHODS: We make some simplifying assumptions in order to calculate the variance: in particular that we recruit the same number of participants, m, from each cluster over the course of the trial, and that participants present at regularly spaced intervals. We consider an intra-cluster correlation that decays exponentially with separation in time between the presentation of two individuals from the same cluster, from a value of ρ for two individuals who present at the same time, to a value of ρτ for individuals presenting at the start and end of the trial recruitment interval. We restrict attention to three-sequence designs with centrosymmetry - the property that if we reverse time and swap the intervention and control conditions then the design looks the same. We obtain an expression for the variance of the treatment effect estimator adjusted for effects of time, using methods for generalised least squares estimation, and we evaluate this expression numerically for different designs, and for different parameter values. RESULTS: There is a two-dimensional space of possible three-sequence, centrosymmetric stepped wedge designs with continuous recruitment. The variance of the treatment effect estimator for given ρ and τ can be plotted as a contour map over this space. The shape of this variance surface depends on τ and on the parameter mρ/(1-ρ), but typically indicates a broad, flat region of close-to-optimal designs. The 'standard' design with equally spaced periods and 1:1:1 allocation rarely performs well, however. CONCLUSIONS: In many different settings, a relatively simple design can be found (e.g. one based on simple fractions) that offers close-to-optimal efficiency in that setting. There may also be designs that are robustly efficient over a wide range of settings. Contour maps of the kind we illustrate can help guide this choice. If efficiency is offered as one of the justifications for using a stepped wedge design, then it is worth designing with optimal efficiency in mind.

Reconsidering stepped wedge cluster randomized trial designs with implementation periods: Fewer sequences or the parallel-group design with baseline and implementation periods are potentially more efficient.

BACKGROUND/AIMS: When designing a cluster randomized trial, advantages and disadvantages of tentative designs must be weighed. The stepped wedge design is popular for multiple reasons, including its potential to increase power via improved efficiency relative to a parallel-group design. In many realistic settings, it will take time for clusters to fully implement the intervention. When designing the HEALing (Helping to End Addiction Long-termSM) Communities Study, implementation time was a major consideration, and we examined the efficiency and practicality of three designs. Specifically, a three-sequence stepped wedge design with implementation periods, a corresponding two-sequence modified design that is created by removing the middle sequence, and a parallel-group design with baseline and implementation periods. In this article, we study the relative efficiencies of these specific designs. More generally, we study the relative efficiencies of modified designs when the stepped wedge design with implementation periods has three or more sequences. We also consider different correlation structures. METHODS: We compare efficiencies of stepped wedge designs with implementation periods consisting of three to nine sequences with a variety of corresponding designs. The three-sequence design is compared to the two-sequence modified design and to the parallel-group design with baseline and implementation periods analysed via analysis of covariance. Stepped wedge designs with implementation periods consisting of four or more sequences are compared to modified designs that remove all or a subset of 'middle' sequences. Efficiencies are based on the use of linear mixed effects models. RESULTS: In the studied settings, the modified design is more efficient than the three-sequence stepped wedge design with implementation periods. The parallel-group design with baseline and implementation periods with analysis of covariance-based analysis is often more efficient than the three-sequence design. With respect to stepped wedge designs with implementation periods that are comprised of more sequences, there are often corresponding modified designs that improve efficiency. However, use of only the first and last sequences has the potential to be either relatively efficient or inefficient. Relative efficiency is impacted by the strength of the statistical correlation among outcomes from the same cluster; for example, the relative efficiencies of modified designs tend to be greater for smaller cluster auto-correlation values. CONCLUSION: If a three-sequence stepped wedge design with implementation periods is being considered for a future cluster randomized trial, then a corresponding modified design using only the first and last sequences should be considered if sole focus is on efficiency. However, a parallel-group design with baseline and implementation periods and analysis of covariance-based analysis can be a practical, efficient alternative. For stepped wedge designs with implementation periods and a larger number of sequences, modified versions that remove 'middle' sequences should be considered. Due to the potential sensitivity of design efficiencies, statistical correlation should be carefully considered.

Models for cluster randomized designs using ranked set sampling.

Cluster randomized designs (CRD) provide a rigorous development for randomization principles for studies where treatments are allocated to cluster units rather than the individual subjects within clusters. It is known that CRDs are less efficient than completely randomized designs since the randomization of treatment allocation is applied to the cluster units. To mitigate this problem, we embed a ranked set sampling design from survey sampling studies into CRD for the selection of both cluster and subsampling units. We show that ranking groups in ranked set sampling act like a covariate, reduce the expected mean squared cluster error, and increase the precision of the sampling design. We provide an optimality result to determine the sample sizes at cluster and sub-sample level. We apply the proposed sampling design to a dental study on human tooth size, and to a longitudinal study from an education intervention program.

Marginal modeling in community randomized trials with rare events: Utilization of the negative binomial regression model.

BACKGROUND/AIMS: This work is motivated by the HEALing Communities Study, which is a post-test only cluster randomized trial in which communities are randomized to two different trial arms. The primary interest is in reducing opioid overdose fatalities, which will be collected as a count outcome at the community level. Communities range in size from thousands to over one million residents, and fatalities are expected to be rare. Traditional marginal modeling approaches in the cluster randomized trial literature include the use of generalized estimating equations with an exchangeable correlation structure when utilizing subject-level data, or analogously quasi-likelihood based on an over-dispersed binomial variance when utilizing community-level data. These approaches account for and estimate the intra-cluster correlation coefficient, which should be provided in the results from a cluster randomized trial. Alternatively, the coefficient of variation or R coefficient could be reported. In this article, we show that negative binomial regression can also be utilized when communities are large and events are rare. The objectives of this article are (1) to show that the negative binomial regression approach targets the same marginal regression parameter(s) as an over-dispersed binomial model and to explain why the estimates may differ; (2) to derive formulas relating the negative binomial overdispersion parameter k with the intra-cluster correlation coefficient, coefficient of variation, and R coefficient; and (3) analyze pre-intervention data from the HEALing Communities Study to demonstrate and contrast models and to show how to report the intra-cluster correlation coefficient, coefficient of variation, and R coefficient when utilizing negative binomial regression. METHODS: Negative binomial and over-dispersed binomial regression modeling are contrasted in terms of model setup, regression parameter estimation, and formulation of the overdispersion parameter. Three specific models are used to illustrate concepts and address the third objective. RESULTS: The negative binomial regression approach targets the same marginal regression parameter(s) as an over-dispersed binomial model, although estimates may differ. Practical differences arise in regard to how overdispersion, and hence the intra-cluster correlation coefficient is modeled. The negative binomial overdispersion parameter is approximately equal to the ratio of the intra-cluster correlation coefficient and marginal probability, the square of the coefficient of variation, and the R coefficient minus 1. As a result, estimates corresponding to all four of these different types of overdispersion parameterizations can be reported when utilizing negative binomial regression. CONCLUSION: Negative binomial regression provides a valid, practical, alternative approach to the analysis of count data, and corresponding reporting of overdispersion parameters, from community randomized trials in which communities are large and events are rare.

Use of information criteria for selecting a correlation structure for longitudinal cluster randomised trials.

BACKGROUND: When designing and analysing longitudinal cluster randomised trials, such as the stepped wedge, the similarity of outcomes from the same cluster must be accounted for through the choice of a form for the within-cluster correlation structure. Several choices for this structure are commonly considered for application within the linear mixed model paradigm. The first assumes a constant intra-cluster correlation for all pairs of outcomes from the same cluster (the exchangeable/Hussey and Hughes model); the second assumes that correlations of outcomes measured in the same period are higher than outcomes measured in different periods (the block exchangeable model) and the third is the discrete-time decay model, which allows the correlation between pairs of outcomes to decay over time. Currently, there is limited guidance on how to select the most appropriate within-cluster correlation structure. METHODS: We simulated continuous outcomes under each of the three considered within-cluster correlation structures for a range of design and parameter choices, and, using the ASReml-R package, fit each linear mixed model to each simulated dataset. We evaluated the performance of the Akaike and Bayesian information criteria for selecting the correct within-cluster correlation structure for each dataset. RESULTS: For smaller total sample sizes, neither criteria performs particularly well in selecting the correct within-cluster correlation structure, with the simpler exchangeable model being favoured. Furthermore, in general, the Bayesian information criterion favours the exchangeable model. When the cluster auto-correlation (which defines the degree of dependence between observations in adjacent time periods) is large and number of periods is small, neither criteria is able to distinguish between the block exchangeable and discrete time decay models. However, for increasing numbers of clusters, periods, and subjects per cluster period, both the Akaike and Bayesian information criteria perform increasingly well in the detection of the correct within-cluster correlation structure. CONCLUSIONS: With increasing amounts of data, be they number of clusters, periods or subjects per cluster period, both the Akaike and Bayesian information criteria are increasingly likely to select the correct correlation structure. We recommend that if there are sufficient data available when planning a trial, that the Akaike or Bayesian information criterion is used to guide the choice of within-cluster correlation structure in the absence of other compelling justifications for a specific correlation structure. We also suggest that researchers conduct supplementary analyses under alternate correlation structures to gauge sensitivity to the initial choice.

Nonparametric inference of the area under ROC curve under two-phase cluster sampling.

Nonparametric inference of the area under ROC curve (AUC) has been well developed either in the presence of verification bias or clustering. However, current nonparametric methods are not able to handle cases where both verification bias and clustering are present. Such a case arises when a two-phase study design is applied to a cohort of subjects (verification bias) where each subject might have multiple test results (clustering). In such cases, the inference of AUC must account for both verification bias and intra-cluster correlation. In the present paper, we propose an IPW AUC estimator that corrects for verification bias and derive a variance formula to account for intra-cluster correlations between disease status and test results. Results of a simulation study indicate that the method that assumes independence underestimates the true variance of the IPW AUC estimator in the presence of intra-cluster correlations. The proposed method, on the other hand, provides a consistent variance estimate for the IPW AUC estimator by appropriately accounting for correlations between true disease statuses and between test results.

Intra-cluster correlations from the CLustered OUtcome Dataset bank to inform the design of longitudinal cluster trials.

BACKGROUND: Sample size calculations for longitudinal cluster randomised trials, such as crossover and stepped-wedge trials, require estimates of the assumed correlation structure. This includes both within-period intra-cluster correlations, which importantly differ from conventional intra-cluster correlations by their dependence on period, and also cluster autocorrelation coefficients to model correlation decay. There are limited resources to inform these estimates. In this article, we provide a repository of correlation estimates from a bank of real-world clustered datasets. These are provided under several assumed correlation structures, namely exchangeable, block-exchangeable and discrete-time decay correlation structures. METHODS: Longitudinal studies with clustered outcomes were collected to form the CLustered OUtcome Dataset bank. Forty-four available continuous outcomes from 29 datasets were obtained and analysed using each correlation structure. Patterns of within-period intra-cluster correlation coefficient and cluster autocorrelation coefficients were explored by study characteristics. RESULTS: The median within-period intra-cluster correlation coefficient for the discrete-time decay model was 0.05 (interquartile range: 0.02-0.09) with a median cluster autocorrelation of 0.73 (interquartile range: 0.19-0.91). The within-period intra-cluster correlation coefficients were similar for the exchangeable, block-exchangeable and discrete-time decay correlation structures. Within-period intra-cluster correlation coefficients and cluster autocorrelations were found to vary with the number of participants per cluster-period, the period-length, type of cluster (primary care, secondary care, community or school) and country income status (high-income country or low- and middle-income country). The within-period intra-cluster correlation coefficients tended to decrease with increasing period-length and slightly decrease with increasing cluster-period sizes, while the cluster autocorrelations tended to move closer to 1 with increasing cluster-period size. Using the CLustered OUtcome Dataset bank, an RShiny app has been developed for determining plausible values of correlation coefficients for use in sample size calculations. DISCUSSION: This study provides a repository of intra-cluster correlations and cluster autocorrelations for longitudinal cluster trials. This can help inform sample size calculations for future longitudinal cluster randomised trials.

Factors associated with coverage of vitamin a supplementation among Bangladeshi children: mixed modelling approach.

BACKGROUND: Vitamin A deficiency (VAD) is a prominent and widespread public health problem in developing countries, including Bangladesh. About 2% of all deaths among under-five children are attributable to VAD. Evidence-based information is required to understand the influential factors to increase vitamin A supplementation (VAS) coverage and reduce VAD. We investigated the potential factors affecting VAS coverage and its significant predictors among Bangladeshi children aged 6 to 59 months using the VAS clustered data extracted from the latest Bangladesh Demographic and Health Survey 2014. METHODS: Data were analysed using mixed logistic regression (MLR) modelling approach in the generalised linear mixed model framework. The MLR model performs better than logistic regression for analysing the clustered data because of its minimum Akaike information criterion value. The likelihood ratio test showed that the variance component was significant. Therefore, the clustering effect among children was inevitable to use. RESULTS: VAS coverage among under-five children was 63.6%, which is not optimal and below the WHO's recommendation and the country's target of 90%. Children aged 25 to 36 months (AOR = 2.07, 95% CI: 1.711 to 2.513), who had higher educated mothers (AOR = 1.37, p = 0.033, 95% CI: 1.026-1.820) and fathers (AOR = 1.32, p = 0.027, 95% CI: 1.032-1.683), whose mothers had media exposure (AOR = 1.22, p = 0.006, 95% CI: 1.059-1.408) and NGO membership (AOR = 1.24, p = 0.002, 95% CI: 1.089-1.422) were more likely to consume VAS. CONCLUSION: The relevant authorities should create proactive awareness programs for highly vulnerable local communities, specifically targeted to educate the children's mothers about the necessity and benefits of childhood nutrition.

Sample size and power calculations for open cohort longitudinal cluster randomized trials.

When calculating sample size or power for stepped wedge or other types of longitudinal cluster randomized trials, it is critical that the planned sampling structure be accurately specified. One common assumption is that participants will provide measurements in each trial period, that is, a closed cohort, and another is that each participant provides only one measurement during the course of the trial. However some studies have an "open cohort" sampling structure, where participants may provide measurements in variable numbers of periods. To date, sample size calculations for longitudinal cluster randomized trials have not accommodated open cohorts. Feldman and McKinlay (1994) provided some guidance, stating that the participant-level autocorrelation could be varied to account for the degree of overlap in different periods of the study, but did not indicate precisely how to do so. We present sample size and power formulas that allow for open cohorts and discuss the impact of the degree of "openness" on sample size and power. We consider designs where the number of participants in each cluster will be maintained throughout the trial, but individual participants may provide differing numbers of measurements. Our results are a unification of closed cohort and repeated cross-sectional sample results of Hooper et al (2016), and indicate precisely how participant autocorrelation of Feldman and McKinlay should be varied to account for an open cohort sampling structure. We discuss different types of open cohort sampling schemes and how open cohort sampling structure impacts on power in the presence of decaying within-cluster correlations and autoregressive participant-level errors.

Clustering of continuous and binary outcomes at the general practice level in individually randomised studies in primary care - a review of 10 years of primary care trials.

BACKGROUND: In randomised controlled trials, the assumption of independence of individual observations is fundamental to the design, analysis and interpretation of studies. However, in individually randomised trials in primary care, this assumption may be violated because patients are naturally clustered within primary care practices. Ignoring clustering may lead to a loss of power or, in some cases, type I error. METHODS: Clustering can be quantified by intra-cluster correlation (ICC), a measure of the similarity between individuals within a cluster with respect to a particular outcome. We reviewed 17 trials undertaken by the Department of Primary Care at the University of Southampton over the last ten years. We calculated the ICC for the primary and secondary outcomes in each trial at the practice level and determined whether ignoring practice-level clustering still gave valid inferences. Where multiple studies collected the same outcome measure, the median ICC was calculated for that outcome. RESULTS: The median intra-cluster correlation (ICC) for all outcomes was 0.016, with interquartile range 0.00-0.03. The median ICC for symptom severity was 0.02 (interquartile range (IQR) 0.01 to 0.07) and for reconsultation with new or worsening symptoms was 0.01 (IQR 0.00, 0.07). For HADS anxiety the ICC was 0.04 (IQR 0.02, 0.05) and for HADS depression was 0.02 (IQR 0.00, 0.05). The median ICC for EQ. 5D-3 L was 0.01 (IQR 0.01, 0.04). CONCLUSIONS: There is evidence of clustering in individually randomised trials primary care. The non-zero ICC suggests that, depending on study design, clustering may not be ignorable. It is important that this is fully considered at the study design phase.

Association between Rift Valley fever virus seroprevalences in livestock and humans and their respective intra-cluster correlation coefficients, Tana River County, Kenya.

We implemented a cross-sectional study in Tana River County, Kenya, a Rift Valley fever (RVF)-endemic area, to quantify the strength of association between RVF virus (RVFv) seroprevalences in livestock and humans, and their respective intra-cluster correlation coefficients (ICCs). The study involved 1932 livestock from 152 households and 552 humans from 170 households. Serum samples were collected and screened for anti-RVFv immunoglobulin G (IgG) antibodies using inhibition IgG enzyme-linked immunosorbent assay (ELISA). Data collected were analysed using generalised linear mixed effects models, with herd/household and village being fitted as random variables. The overall RVFv seroprevalences in livestock and humans were 25.41% (95% confidence interval (CI) 23.49-27.42%) and 21.20% (17.86-24.85%), respectively. The presence of at least one seropositive animal in a household was associated with an increased odds of exposure in people of 2.23 (95% CI 1.03-4.84). The ICCs associated with RVF virus seroprevalence in livestock were 0.30 (95% CI 0.19-0.44) and 0.22 (95% CI 0.12-0.38) within and between herds, respectively. These findings suggest that there is a greater variability of RVF virus exposure between than within herds. We discuss ways of using these ICC estimates in observational surveys for RVF in endemic areas and postulate that the design of the sentinel herd surveillance should consider patterns of RVF clustering to enhance its effectiveness as an early warning system for RVF epidemics.

Choosing appropriate analysis methods for cluster randomised cross-over trials with a binary outcome.

In cluster randomised cross-over (CRXO) trials, clusters receive multiple treatments in a randomised sequence over time. In such trials, there is usual correlation between patients in the same cluster. In addition, within a cluster, patients in the same period may be more similar to each other than to patients in other periods. We demonstrate that it is necessary to account for these correlations in the analysis to obtain correct Type I error rates. We then use simulation to compare different methods of analysing a binary outcome from a two-period CRXO design. Our simulations demonstrated that hierarchical models without random effects for period-within-cluster, which do not account for any extra within-period correlation, performed poorly with greatly inflated Type I errors in many scenarios. In scenarios where extra within-period correlation was present, a hierarchical model with random effects for cluster and period-within-cluster only had correct Type I errors when there were large numbers of clusters; with small numbers of clusters, the error rate was inflated. We also found that generalised estimating equations did not give correct error rates in any scenarios considered. An unweighted cluster-level summary regression performed best overall, maintaining an error rate close to 5% for all scenarios, although it lost power when extra within-period correlation was present, especially for small numbers of clusters. Results from our simulation study show that it is important to model both levels of clustering in CRXO trials, and that any extra within-period correlation should be accounted for. Copyright © 2016 John Wiley & Sons, Ltd.

Beta-binomial model for meta-analysis of odds ratios.

In meta-analysis of odds ratios (ORs), heterogeneity between the studies is usually modelled via the additive random effects model (REM). An alternative, multiplicative REM for ORs uses overdispersion. The multiplicative factor in this overdispersion model (ODM) can be interpreted as an intra-class correlation (ICC) parameter. This model naturally arises when the probabilities of an event in one or both arms of a comparative study are themselves beta-distributed, resulting in beta-binomial distributions. We propose two new estimators of the ICC for meta-analysis in this setting. One is based on the inverted Breslow-Day test, and the other on the improved gamma approximation by Kulinskaya and Dollinger (2015, p. 26) to the distribution of Cochran's Q. The performance of these and several other estimators of ICC on bias and coverage is studied by simulation. Additionally, the Mantel-Haenszel approach to estimation of ORs is extended to the beta-binomial model, and we study performance of various ICC estimators when used in the Mantel-Haenszel or the inverse-variance method to combine ORs in meta-analysis. The results of the simulations show that the improved gamma-based estimator of ICC is superior for small sample sizes, and the Breslow-Day-based estimator is the best for n⩾100. The Mantel-Haenszel-based estimator of OR is very biased and is not recommended. The inverse-variance approach is also somewhat biased for ORs≠1, but this bias is not very large in practical settings. Developed methods and R programs, provided in the Web Appendix, make the beta-binomial model a feasible alternative to the standard REM for meta-analysis of ORs. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Statistical efficiency and optimal design for stepped cluster studies under linear mixed effects models.

In stepped cluster designs the intervention is introduced into some (or all) clusters at different times and persists until the end of the study. Instances include traditional parallel cluster designs and the more recent stepped-wedge designs. We consider the precision offered by such designs under mixed-effects models with fixed time and random subject and cluster effects (including interactions with time), and explore the optimal choice of uptake times. The results apply both to cross-sectional studies where new subjects are observed at each time-point, and longitudinal studies with repeat observations on the same subjects. The efficiency of the design is expressed in terms of a 'cluster-mean correlation' which carries information about the dependency-structure of the data, and two design coefficients which reflect the pattern of uptake-times. In cross-sectional studies the cluster-mean correlation combines information about the cluster-size and the intra-cluster correlation coefficient. A formula is given for the 'design effect' in both cross-sectional and longitudinal studies. An algorithm for optimising the choice of uptake times is described and specific results obtained for the best balanced stepped designs. In large studies we show that the best design is a hybrid mixture of parallel and stepped-wedge components, with the proportion of stepped wedge clusters equal to the cluster-mean correlation. The impact of prior uncertainty in the cluster-mean correlation is considered by simulation. Some specific hybrid designs are proposed for consideration when the cluster-mean correlation cannot be reliably estimated, using a minimax principle to ensure acceptable performance across the whole range of unknown values. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.

Cluster randomised crossover trials with binary data and unbalanced cluster sizes: application to studies of near-universal interventions in intensive care.

BACKGROUND: Cluster randomised crossover trials have been utilised in recent years in the health and social sciences. Methods for analysis have been proposed; however, for binary outcomes, these have received little assessment of their appropriateness. In addition, methods for determination of sample size are currently limited to balanced cluster sizes both between clusters and between periods within clusters. This article aims to extend this work to unbalanced situations and to evaluate the properties of a variety of methods for analysis of binary data, with a particular focus on the setting of potential trials of near-universal interventions in intensive care to reduce in-hospital mortality. METHODS: We derive a formula for sample size estimation for unbalanced cluster sizes, and apply it to the intensive care setting to demonstrate the utility of the cluster crossover design. We conduct a numerical simulation of the design in the intensive care setting and for more general configurations, and we assess the performance of three cluster summary estimators and an individual-data estimator based on binomial-identity-link regression. RESULTS: For settings similar to the intensive care scenario involving large cluster sizes and small intra-cluster correlations, the sample size formulae developed and analysis methods investigated are found to be appropriate, with the unweighted cluster summary method performing well relative to the more optimal but more complex inverse-variance weighted method. More generally, we find that the unweighted and cluster-size-weighted summary methods perform well, with the relative efficiency of each largely determined systematically from the study design parameters. Performance of individual-data regression is adequate with small cluster sizes but becomes inefficient for large, unbalanced cluster sizes. When outcome prevalences are 6% or less and the within-cluster-within-period correlation is 0.05 or larger, all methods display sub-nominal confidence interval coverage, with the less prevalent the outcome the worse the coverage. LIMITATIONS: As with all simulation studies, conclusions are limited to the configurations studied. We confined attention to detecting intervention effects on an absolute risk scale using marginal models and did not explore properties of binary random effects models. CONCLUSION: Cluster crossover designs with binary outcomes can be analysed using simple cluster summary methods, and sample size in unbalanced cluster size settings can be determined using relatively straightforward formulae. However, caution needs to be applied in situations with low prevalence outcomes and moderate to high intra-cluster correlations.

Practical considerations for sample size calculation for cluster randomized trials.

Cluster randomized trials are an essential design in public health and medical research, when individual randomization is infeasible or undesirable for scientific or logistical reasons. However, the correlation among observations within clusters leads to a decrease in statistical power compared to an individually randomised trial with the same total sample size. This correlation - often quantified using the intra-cluster correlation coefficient - must be accounted for in the sample size calculation to ensure that the trial is adequately powered. In this paper, we first describe the principles of sample size calculation for parallel-arm CRTs, and explain how these calculations can be extended to CRTs with cross-over designs, with a baseline measurement and stepped-wedge designs. We introduce tools to guide researchers with their sample size calculation and discuss methods to inform the choice of the a priori estimate of the intra-cluster correlation coefficient for the calculation. We also include additional considerations with respect to anticipated attrition, a small number of clusters, and use of covariates in the randomisation process and in the analysis.

Intra-cluster correlations for ceca Salmonella prevalence and enumeration from 40 experimental floor pen trials in broiler chickens using a seeder bird challenge model.

Floor pen trials are an efficient way to evaluate the effectiveness of potential Salmonella control interventions in broiler chickens. When treatments are allocated at the pen level, and outcomes are measured at the individual bird level, floor pen studies are considered to be cluster randomized trials. Estimating the sample size required to achieve a desired level of statistical power for a cluster randomized trial requires an estimate of the intra-cluster correlation (ICC) as an input. In this study, ICCs were estimated for the untreated challenged control group from 40 broiler chicken Salmonella pen trials performed using a seeder bird challenge model. The ICCs for ceca Salmonella prevalences ranged from 0.00 to 0.64, with a median of 0.17. The ICCs for ceca Salmonella log10(MPN/g + 1) ranged from 0.00 to 0.52, with a median of 0.14. These findings indicate that the effect of pen-level clustering is substantial in Salmonella floor pen trials, and it must be considered during both the study design and analysis. In a multivariable regression analysis, ICCs for ceca Salmonella prevalences were associated with the challenge status of sampled birds, age of birds at the time of challenge, and Salmonella serovar. ICCs were lower for studies in which a combination of direct (seeder) and indirect (horizontal) challenged birds were sampled, and for studies in which birds were challenged on the day of hatch or at one day of age. ICCs were higher for studies in which Salmonella Heidelberg was used as the challenge strain. These findings may be useful for investigators that are planning pen trials to evaluate Salmonella control interventions in broiler chickens. Choosing study design elements associated with a lower ICC may improve efficiency by leading to a larger effective sample size for the same number of experimental units.

Sample sizes for cluster-randomised trials with continuous outcomes: Accounting for uncertainty in a single intra-cluster correlation estimate.

Sample size calculations for cluster-randomised trials require inclusion of an inflation factor taking into account the intra-cluster correlation coefficient. Often, estimates of the intra-cluster correlation coefficient are taken from pilot trials, which are known to have uncertainty about their estimation. Given that the value of the intra-cluster correlation coefficient has a considerable influence on the calculated sample size for a main trial, the uncertainty in the estimate can have a large impact on the ultimate sample size and consequently, the power of a main trial. As such, it is important to account for the uncertainty in the estimate of the intra-cluster correlation coefficient. While a commonly adopted approach is to utilise the upper confidence limit in the sample size calculation, this is a largely inefficient method which can result in overpowered main trials. In this paper, we present a method of estimating the sample size for a main cluster-randomised trial with a continuous outcome, using numerical methods to account for the uncertainty in the intra-cluster correlation coefficient estimate. Despite limitations with this initial study, the findings and recommendations in this paper can help to improve sample size estimations for cluster randomised controlled trials by accounting for uncertainty in the estimate of the intra-cluster correlation coefficient. We recommend this approach be applied to all trials where there is uncertainty in the intra-cluster correlation coefficient estimate, in conjunction with additional sources of information to guide the estimation of the intra-cluster correlation coefficient.

Effect sizes and intra-cluster correlation coefficients measured from the Green Dot High School study for guiding sample size calculations when designing future violence prevention cluster randomized trials in school settings.

PURPOSE: Cluster randomized controlled trials (cRCTs) are popular in school-based research designs where schools are randomized to different trial arms. To help guide future study planning, we provide information on anticipated effect sizes and intra-cluster correlation coefficients (ICCs), as well as school sizes, for dating violence (DV) and interpersonal violence outcomes based on data from a cRCT which evaluated the bystander-based violence intervention 'Green Dot'. METHODS: We utilized data from 25 schools from the Green Dot High School study. Effect size and ICC values corresponding to dating and interpersonal violence outcomes are obtained from linear mixed effect models. We also calculated the required number of schools needed for future studies utilizing available methods that do and do not consider variation in school size. RESULTS: Observed effect sizes for DV outcomes range from 0.06 to 0.11. Observed ICC values for DV outcomes range from 0.0006 to 0.0032. The upper limit of 95% CIs for the true ICCs range from 0.0023 to 0.0070. CONCLUSION: School-based evaluations with violence outcomes are expected to have small effect sizes. Observed ICCs are less than 0.005 and upper limit of of 95% CIs for the true ICCs are less than 0.01. Designing school-based cRCTs should account for the ICC, even if its value is assumed to be negligible. Furthermore, variation in school sizes should also be accounted for to avoid having too few schools to achieve the desired power.