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.

最近，单位-Lindley 分布及其相关回归模型已被开发为 Beta 回归模型的替代方案，该模型在单位区间内获得连续结果$(0,1)$. 比例数据通常出现在具有层次结构的临床试验、经济学和社会研究中。在这项研究中，提出了unit-Lindley混合效应模型，并研究了参数估计的适当似然分析方法。在混合效应模型中的聚类或纵向比例数据的情况下，完全似然函数没有封闭形式。本研究采用拉普拉斯和自适应高斯正交逼近方法获得了unit-Lindley混合效应模型的参数估计值。我们使用单元-林德利混合效应模型（包括作为巴西联邦州的随机截距）分析了巴西城市供水和污水不足的家庭比例数据集以及一些社会人口变量。分析结果表明，所提出的单位-林德利混合效应模型比单位-林德利回归模型和β混合模型提供了更好的拟合。此外，在模拟研究中，通过蒙特卡罗模拟研究在偏差和均方误差方面评估和比较近似方法估计的准确性。