Augmented mixed beta regression models for periodontal proportion data

Diana M. Galvis
Dipankar Badyophadyay
Víctor H. Lachos

Continuous (clustered) proportion data often arise in various domains of medicine andpublic health where the response variable of interest is a proportion (or percentage) quantify-ing disease status for the cluster units, ranging between zero and one. However, due to thepresence of relatively disease-free as well as highly diseased subjects in any study, the proportion values can lie in the interval [0, 1]. While Beta regression can be adapted for assessing covariate effects here, it’s versatility is often challenged due to the presence/excess of zeros and ones because the Beta support lies in the interval (0, 1). To circumvent this, we augment the probabilities of zero and one with the Beta density, controlling for the clustering effect.Our approach is Bayesian with the ability to borrow information across various stages of thecomplex model hierarchy, and produces a computationally convenient framework amenable toavailable freeware. The marginal likelihood is tractable, and can be used to develop Bayesiancase-deletion influence diagnostics based on q-divergence measures. Both simulation studiesand application to a real dataset from a clinical periodontology study quantify the gain in modelfit and parameter estimation over other adhoc alternatives, and provide quantitative insight intoassessing the true covariate effects on the proportion responses.

Augmented Beta
Beta density
Kullback-Leibler divergence
Periodon- tal disease