Power and sample size calculations for cluster randomized trials with binary outcomes when intracluster correlation coefficients vary by treatment arm

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 [Formula: see text]) 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.

Estimating the Intracluster Correlation Coefficient for the Clinical Sign “Trachomatous Inflammation—Follicular” in Population-Based Trachoma Prevalence Surveys: Results From a Meta-Regression Analysis of 261 Standardized Preintervention Surveys Carried Out in Ethiopia, Mozambique, and Nigeria

Sample sizes in cluster surveys must be greater than those in surveys using simple random sampling in order to obtain similarly precise prevalence estimates, because results from subjects examined in the same cluster cannot be assumed to be independent. Therefore, a crucial aspect of cluster sampling is estimation of the intracluster correlation coefficient (ρ): the degree of relatedness of outcomes in a given cluster, defined as the proportion of total variance accounted for by between-cluster variation. In infectious disease epidemiology, this coefficient is related to transmission patterns and the natural history of infection; its value also depends on particulars of survey design. Estimation of ρ is often difficult due to the lack of comparable survey data with which to calculate summary estimates. Here we use a parametric bootstrap model to estimate ρ for the ocular clinical sign “trachomatous inflammation—follicular” (TF) among children aged 1–9 years within population-based trachoma prevalence surveys. We present results from a meta-regression analysis of data from 261 such surveys completed using standardized methods in Ethiopia, Mozambique, and Nigeria in 2012–2015. Consistent with the underlying theory, we found that ρ increased with increasing overall TF prevalence and smaller numbers of children examined per cluster. Estimates of ρ for TF were independently higher in Ethiopia than in the other countries.