title: Modeling overdispersed longitudinal binary data using a combined beta and normal random-effects model.
abstract: BACKGROUND
  In medical and biomedical areas, binary and binomial outcomes are very common. Such data are often collected longitudinally from a given subject repeatedly overtime, which result in clustering of the observations within subjects, leading to correlation, on the one hand. The repeated binary outcomes from a given subject, on the other hand, constitute a binomial outcome, where the prescribed mean-variance relationship is often violated, leading to the so-called overdispersion.
  
  
  METHODS
  Two longitudinal binary data sets, collected in south western Ethiopia: the Jimma infant growth study, where the child's early growth is studied, and the Jimma longitudinal family survey of youth where the adolescent's school attendance is studied over time, are considered. A new model which combines both overdispersion, and correlation simultaneously, also known as the combined model is applied. In addition, the commonly used methods for binary and binomial data, such as the simple logistic, which accounts neither for the overdispersion nor the correlation, the beta-binomial model, and the logistic-normal model, which accommodate only for the overdispersion, and correlation, respectively, are also considered for comparison purpose. As an alternative estimation technique, a Bayesian implementation of the combined model is also presented.
  
  
  RESULTS
  The combined model results in model improvement in fit, and hence the preferred one, based on likelihood comparison, and DIC criterion. Further, the two estimation approaches result in fairly similar parameter estimates and inferences in both of our case studies. Early initiation of breastfeeding has a protective effect against the risk of overweight in late infancy (p = 0.001), while proportion of overweight seems to be invariant among males and females overtime (p = 0.66). Gender is significantly associated with school attendance, where girls have a lower rate of attendance (p = 0.001) as compared to boys.
  
  
  CONCLUSIONS
  We applied a flexible modeling framework to analyze binary and binomial longitudinal data. Instead of accounting for overdispersion, and correlation separately, both can be accommodated simultaneously, by allowing two separate sets of the beta, and the normal random effects at once.
authors.affiliation: I-BioStat, Center for Statistics, Universiteit Hasselt, Diepenbeek, B-3590, Belgium. christel.faes@uhasselt.be.
authors.forename: Wondwosen
authors.initials: W
authors.lastname: Kassahun
authors.forename: Thomas
authors.initials: T
authors.lastname: Neyens
authors.forename: Geert
authors.initials: G
authors.lastname: Molenberghs
authors.forename: Christel
authors.initials: C
authors.lastname: Faes
authors.forename: Geert
authors.initials: G
authors.lastname: Verbeke