Iyer, Vishwanathan R.
College of Educational and Behavioral Sciences, Department of Applied Statistics and Research Methods
University of Northern Colorado
Type of Resources
Place of Publication
University of Northern Colorado
In the applied filed of many areas such as Business and Economics, Physical, Biological, Medical, Environmental, and Social Sciences, Psychology, and Educations outcome measurements like rates, proportions, and fractions are common. Researchers have proven that using normal linear regression to analyze the relationship between outcome measurements such as rates and proportions and a set of independent variables can violates key assumptions of the method. Transformation techniques are usually applied to correct the assumptions, but this practice can violate the probability property of bounded nature of outcome variables. Even though beta regression was one of the popular methods to analyze bounded data situations, the latest development of unit-Lindley distribution and its associate regression was found to be superior. Unit-Lindley regression requires uncorrelated response variables; but in real applied fields, researchers may encounter clustered or correlated response variables. Thus, mixed models that are capable to handle correlated and clustered response variables are required. Therefore, unit-Lindley mixed model, that is capable to analyze correlated and bounded response variable and a set of independent variables is one of the suitable model choices. It is widely accepted and proven by researchers that the Bayesian parameter estimation approach is more advantageous over a classical approach in the case of mixed models in several ways. Despite the several advantages of the Bayesian approach, it has not been applied to the unit-Lindley mixed model. Therefore, this dissertation aims to develop a Bayesian approach to unit-Lindley mixed model. Additionally, the assumption of normality for random effects may not be appropriate when dealing with skewed data. Thus, this dissertation also aims to apply three distributional assumptions for random effects: normal, skew-normal, and skew-t. To achieve the research aims outlined in this dissertation, a Bayesian unit-Lindley mixed model with distributional assumptions of normal, skew-normal, and skew-t for random effects was developed. To implement them, a STAN program code was also developed and presented in this dissertation. A variety of simulated data situations were used to test all models in R. Leave-One-Out Cross-Validation Information Criterion (LOOIC) and Watanabe-Akaike Information Criterion (WAIC) were used to compare the modes. In addition, bias and RMSE of intercept and its variance were also used to support the results of LOOIC and WAIC. The results confirmed that a Bayesian unit-Lindley model with normal assumption of random effects performed better as compered to the model with two other assumptions when the data situation was approximately normal with normal random effects. However, when the response variable is skewed with a skewed random effects, model with the normal assumption was less robust than the model with skew-normal and skew-t random effects. While comparing the model with skew-normal and skew-t random effects, the model with skew-normal produced slightly less biased parameter estimation and RMSE and smaller LOOIC and WAIC. Finally, all the models were applied to analyze the child mortality rates across the countries of South Asia, and their performance was compared. After testing the models by using the simulation method and applying to a real data situation, it was confirmed that a Bayesian unit-Lindley model is ready to apply in any fields of the applied areas where measurement of outcome variable is clustered and bounded within unit interval.
Copyright is held by the author.