Generalized Linear Models for Bounded and Limited Quantitative Variables

Michael Smithson is a Professor in the Research School of Psychology at The Australian National University in Canberra, and received his PhD from the University of Oregon. He is the author of Confidence Intervals (2003), Statistics with Confidence (2000), Ignorance and Uncertainty (1989), and Fuzzy Set Analysis for the Behavioral and Social Sciences (1987), co-author of Fuzzy Set Theory: Applications in the Social Sciences (2006) and Generalized Linear Models for Categorical and Limited Dependent Variables (2014), and co-editor of Uncertainty and Risk: Multidisciplinary Perspectives (2008) and Resolving Social Dilemmas: Dynamic, Structural, and Intergroup Aspects (1999). His other publications include more than 170 refereed journal articles and book chapters. His primary research interests are in judgment and decision making under ignorance and uncertainty, statistical methods for the social sciences, and applications of fuzzy set theory to the social sciences.Dr Yiyun Shou is a research fellow in the Research School of Psychology at The Australian National University. She received her PhD degree in psychology in 2015, and was recently awarded an Australian Research Council Discovery Early Career Award (2018 - 2021). She is active in research in the areas of understanding measurement issues in psychology and developing new quantitative methods. She also conducts extensive research in judgment and decision making under uncertainty, and cross-cultural psychological assessments. She has publications in a number of respected international outlets for measurement and quantitative psychology such as Journal of Statistical Software, British Journal of Mathematical and Statistical Psychology, Psychometrika and Psychological Assessment.

This book provides a thorough and accessible look at an important class of statistical models. It communicates intuition well and shows through numerous examples that understanding how to analyze bounded outcome variables is useful for applied researchers.

• 1. Introduction and OverviewOverview of this BookThe Nature of Bounds on VariablesThe Generalized Linear ModelExamples2. Models for Singly-Bounded VariablesGLMs for singly-bounded variablesModel DiagnosticsTreatment of Boundary Cases3. Models for Doubly-Bounded VariablesDoubly-Bounded Variables and \Natural" HeteroskedasticityThe Beta Distribution: Definition and PropertiesModeling Location and DispersionEstimation and Model DiagnosticsTreatment of Cases at the Boundaries4. Quantile Models for Bounded VariablesIntroductionQuantile regressionDistributions for Doubly-Bounded Variables with Explicit Quantile FunctionsThe CDF-Quantile GLM5. Censored and Truncated VariablesTypes of censoring and truncationTobit modelsTobit Model ExampleHeteroskedastic and Non-Gaussian Tobit Models6. Extensions and ConclusionsExtensions and a General FrameworkAbsolute Bounds and CensoringMulti-Level and Multivariate ModelsBayesian Estimation and ModelingRoads Less Traveled and the State of the ArtReferences