Unit-Lindley mixed-effect model for proportion data

Recently, unit-Lindley distribution and its associated regression models have been developed as an alternative to Beta regression model for which continuous outcome in the unit interval (0, 1). Proportion data usually occur in clinical trials, economics and social studies with hierarchical structures. In this study, unit-Lindley mixed-effect model is proposed and the appropriate likelihood analysis methods for parameter estimation are investigated. In the case of clustered or longitudinal proportion data in mixed-effect models, the full-likelihood function does not have a closed form. Parameter estimations of unit-Lindley mixed-effect model are obtained with Laplace and adaptive Gaussian quadrature approximation methods in this study. We analyzed a dataset on the proportion of households with insufficient water supply and sewage with some sociodemographic variables in the cities of Brazil by using unit-Lindley mixed-effect model including a random intercept as federative states of Brazil. Analysis results indicate that the proposed unit-Lindley mixed-effect model provides better fit than unit-Lindley regression model and beta mixed model. Also, in the simulation study the accuracy of the estimates of approximation methods are evaluated and compared via Monte Carlo simulation study in terms of bias and mean square error.