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Generalized Linear Models for Bounded and Limited Quantitative Variables
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Generalized Linear Models for Bounded and Limited Quantitative Variables

Volume: 181
September 2019 | 136 pages | SAGE Publications, Inc
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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.

 
1. Introduction and Overview
Overview of this Book

 
The Nature of Bounds on Variables

 
The Generalized Linear Model

 
Examples

 
 
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
Introduction

 
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

 
 
References

Supplements

Resource Center

The data, R, SAS, and Stata software code, and detailed explanations of the example models are available at the SAGE Research Methods Resource Center site. 

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

Sample Materials & Chapters

1. Introduction and Overview

Michael Smithson

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... More About Author

Yiyun Shou

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... More About Author

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