This paper describes the specification, estimation and comparison of simplex mixed models based on the likelihood paradigm. This class of models is suitable to deal with restricted response variables, such as rates, percentages, indexes and proportions. The estimation of simplex mixed models is challenged by the intractable integral in the likelihood function. We compare results obtained with three numerical integration methods Laplace, Gauss-Hermite and Quasi-Monte Carlo to solve such integral. The specification of simplex mixed models includes the choice of a link function for which we compare models fitted with logit, probit, complement log-log and Cauchy link functions. Furthermore, results from the simplex mixed models fitted to two datasets are compared with fits of beta, linear and non-linear mixed effects models. The first is a study concerning life quality of industry workers with data collected according to a hierarchical sampling scheme. The second corresponds to water quality measurements taken at 16 operating hydroelectric power plants in Paran ́a State, Brazil. Our results showed that the simplex mixed models provide the best fit between the approaches considered for the two datasets analyzed. None of the choices of the link function outperformed the others. Simulation studies were designed to check the properties of the maximum likelihood estimators and the computational implementation. The Laplace method provides the best balance between computational complexity and accuracy. The data sets and R code are available in the supplementary material.