A model for bimodal rates and proportions.

The beta model is the most important distribution for fitting data with the unit interval. However, the beta distribution is not suitable to model bimodal unit interval data. In this paper, we propose a bimodal beta distribution constructed by using an approach based on the alpha-skew-normal model. We discuss several properties of this distribution, such as bimodality, real moments, entropies and identifiability. Furthermore, we propose a new regression model based on the proposed model and discuss residuals. Estimation is performed by maximum likelihood. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the results. An application is provided to show the modelling competence of the proposed distribution when the data sets show bimodality.

Can Bayesian models play a role in dental caries epidemiology? Evidence from an application to the BELCAP data set.

OBJECTIVES: The aim of this study was to show the potential of Bayesian analysis in statistical modelling of dental caries data. Because of the bounded nature of the dmft (DMFT) index, zero-inflated binomial (ZIB) and beta-binomial (ZIBB) models were considered. The effects of incorporating prior information available about the parameters of models were also shown. METHODS: The data set used in this study was the Belo Horizonte Caries Prevention (BELCAP) study (Böhning et al. (1999)), consisting of five variables collected among 797 Brazilian school children designed to evaluate four programmes for reducing caries. Only the eight primary molar teeth were considered in the data set. A data augmentation algorithm was used for estimation. Firstly, noninformative priors were used to express our lack of knowledge about the regression parameters. Secondly, prior information about the probability of being a structural zero dmft and the probability of being caries affected in the subpopulation of susceptible children was incorporated. RESULTS: With noninformative priors, the best fitting model was the ZIBB. Education (OR = 0.76, 95% CrI: 0.59, 0.99), all interventions (OR = 0.46, 95% CrI: 0.35, 0.62), rinsing (OR = 0.61, 95% CrI: 0.47, 0.80) and hygiene (OR = 0.65, 95% CrI: 0.49, 0.86) were demonstrated to be factors protecting children from being caries affected. Being male increased the probability of being caries diseased (OR = 1.19, 95% CrI: 1.01, 1.42). However, after incorporating informative priors, ZIB models' estimates were not influenced, while ZIBB models reduced deviance and confirmed the association with all interventions and rinsing only. DISCUSSION: In our application, Bayesian estimates showed a similar accuracy and precision than likelihood-based estimates, although they offered many computational advantages and the possibility of expressing all forms of uncertainty in terms of probability. The overdispersion parameter could expound why the introduction of prior information had significant effects on the parameters of the ZIBB model, while ZIB estimates remained unchanged. Finally, the best performance of ZIBB compared to the ZIB model was shown to catch overdispersion in data.