Scalable and robust regression models for continuous proportional data

Beta regression is used routinely for continuous proportional data, but it often encounters practical issues such as a lack of robustness to misspecification of the beta distribution and sensitivity to outliers. We develop an improved class of generalized linear models starting with the continuous binomial (cobin) distribution and further extending to dispersion mixtures of cobin distributions (micobin). The proposed cobin regression and micobin regression models have attractive robustness, computation, and flexibility properties. A key innovation is the Kolmogorov-Gamma data augmentation scheme, which facilitates Gibbs sampling for Bayesian computation, including in hierarchical cases involving nested, longitudinal, or spatial data. We demonstrate robustness, ability to handle responses exactly at the boundary (0 or 1), and computational efficiency relative to beta regression in simulation experiments and through analysis of the benthic macroinvertebrate multimetric index of US lakes using lake watershed covariates.