Bootstrap ICC estimators in analysis of small clustered binary data

Survey data are often obtained through a multilevel structure and, as such, require hierarchical modeling. While large sample approximation provides a mechanism to construct confidence intervals for the intraclass correlation coefficients (ICCs) in large datasets, challenges arise when we are faced with small-size clusters and binary outcomes. In this paper, we examine two bootstrapping methods, cluster bootstrapping and split bootstrapping. We use these methods to construct the confidence intervals for the ICCs (based on a latent variable approach) for small binary data obtained through a three-level or higher hierarchical data structure. We use 26 scenarios in our simulation study with the two bootstrapping methods. We find that the latent variable method performs well in terms of coverage. The split bootstrapping method provides confidence intervals close to the nominal coverage when the ratio of the ICC for the primary cluster to the ICC for the secondary cluster is small. While the cluster bootstrapping is preferred when the cluster size is larger and the ratio of the ICCs is larger. A numerical example based on teacher effectiveness is assessed.