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

Abstract

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.

Keywords: Cluster randomized trial; generalized estimating equations; intracluster correlation coefficient; logistic regression; sample size.

Figure 1.

Asymptotic standard error of estimated ${\widehat{\beta }}_1$ (SE, panels A and B) and asymptotic relative efficiency (ARE, C and D) vs. ratio of intracluster correlation coefficient (ICC) among treated compared to control clusters (${\rho }_1/{\rho }_0$, A and C) and ratio of outcome prevalence among treated compared to control clusters (${\pi }_1/{\pi }_0$, B and D) by working correlation structure, for fixed ρ₀, π₀, π₁ (A and C), ρ₁ (B and D), cluster size distribution, and K. The vertical line indicates the parameters observed in the trial by Lin et al.

Figure 2.

Asymptotic standard error of estimated ${\widehat{\beta }}_1$ (SE, panels A and B) and asymptotic relative efficiency (ARE, C and D) vs. mean cluster size (A and C) and coefficient of variation of the cluster size distribution (CV, B and D) by working correlation structure, for fixed ρ₀, ρ₁, π₀, π₁, CV (A and C), mean cluster size (B and D), and K. The vertical line indicates the parameters observed in the trial by Lin et al.

Figure 3.

Power to detect effect vs. ratio of ICC among treated compared to control clusters (${\rho }_1/{\rho }_0$, panel A), ratio of outcome prevalence among treated compared to control clusters (${\pi }_1/{\pi }_0$, B), mean cluster size (C), and coefficient of variation of the cluster size distribution (D) by combination of working correlation structure, assumed underlying correlation structure, and a priori estimate of the common ICC used to calculate sample size (SS), for fixed ρ₀, ρ₁ (B,C,D), π₀, π₁ (A,C,D), CV of cluster size distribution (A,B,C), mean cluster size (A,B,D), significance level of 0.05, desired 80% power, and null and alternative hypothesis values (A,C,D). The vertical line indicates the parameters observed in the trial by Lin et al.