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Generalized Linear Models for Bounded and Limited Quantitative Variables

Generalized Linear Models for Bounded and Limited Quantitative Variables

November 2019 | 128 pages | SAGE Publications, Inc

This book introduces researchers and students to the concepts and generalized linear models for analyzing quantitative random variables that have one or more bounds. Examples of bounded variables include the percentage of a population eligible to vote (bounded from 0 to 100), or reaction time in milliseconds (bounded below by 0). The human sciences deal in many variables that are bounded. Ignoring bounds can result in misestimation and improper statistical inference. Michael Smithson and Yiyun Shou's book brings together material on the analysis of limited and bounded variables that is scattered across the literature in several disciplines, and presents it in a style that is both more accessible and up-to-date. The authors provide worked examples in each chapter using real datasets from a variety of disciplines. The software used for the examples include R, SAS, and Stata. The data, software code, and detailed explanations of the example models are available on an accompanying website at

1. Introduction and Overview
Overview of this Book  
The Nature of Bounds on Variables  
The Generalized Linear Model  
2. Models for Singly-Bounded Variables
GLMs for singly-bounded variables  
Model Diagnostics  
Treatment of Boundary Cases  
3. Models for Doubly-Bounded Variables
Doubly-Bounded Variables and \Natural" Heteroskedasticity  
The Beta Distribution: Definition and Properties  
Modeling Location and Dispersion  
Estimation and Model Diagnostics  
Treatment of Cases at the Boundaries  
4. Quantile Models for Bounded Variables
Quantile regression  
Distributions for Doubly-Bounded Variables with Explicit Quantile Functions  
The CDF-Quantile GLM  
5. Censored and Truncated Variables
Types of censoring and truncation  
Tobit models  
Tobit Model Example  
Heteroskedastic and Non-Gaussian Tobit Models  
6. Extensions and Conclusions
Extensions and a General Framework  
Absolute Bounds and Censoring  
Multi-Level and Multivariate Models  
Bayesian Estimation and Modeling  
Roads Less Traveled and the State of the Art  

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.

Jeff Harden
University of Notre Dame

The authors are leaders in the world-wide effort to extend and tailor the generalized linear model to variables that are bounded and not normally distributed. The discussion of models for data recorded as proportions is worth the price of admission.

Paul Johnson
University of Kansas

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ISBN: 9781544334530