Abstract

Clustered binary data occur frequently in biostatistical work. Several approaches have been proposed for the analysis of clustered binary data. In Rosner (1984, Biometrics 40, 1025-1035), a polychotomous logistic regression model was proposed that is a generalization of the beta-binomial distribution and allows for unit- and subunit-specific covariates, while controlling for clustering effects. One assumption of this model is that all pairs of subunits within a cluster are equally correlated. This is appropriate for ophthalmologic work where clusters are generally of size 2, but may be inappropriate for larger cluster sizes. A beta-binomial mixture model is introduced to allow for multiple subclasses within a cluster and to estimate odds ratios relating outcomes for pairs of subunits within a subclass as well as in different subclasses. To include covariates, an extension of the polychotomous logistic regression model is proposed, which allows one to estimate effects of unit-, class-, and subunit-specific covariates, while controlling for clustering using the beta-binomial mixture model. This model is applied to the analysis of respiratory symptom data in children collected over a 14-year period in East Boston, Massachusetts, in relation to maternal and child smoking, where the unit is the child and symptom history is divided into early-adolescent and late-adolescent symptom experience.