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.
Arvustused
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 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
| Series Editor's Introduction |
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ix | |
| About the Authors |
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xi | |
| Acknowledgments |
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xii | |
| Companion Website for This Book |
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xiii | |
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1 Introduction and Overview |
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1 | (18) |
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1.1 Overview of This Book |
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1 | (2) |
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1.2 The Nature of Bounds on Variables |
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3 | (2) |
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1.3 The Generalized Linear Model |
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5 | (7) |
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12 | (7) |
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2 Models for Singly Bounded Variables |
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19 | (16) |
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2.1 GLMs for Singly Bounded Variables |
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19 | (8) |
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27 | (1) |
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2.3 Treatment of Boundary Cases |
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28 | (7) |
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3 Models for Doubly Bounded Variables |
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35 | (19) |
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3.1 Doubly Bounded Variables and "Natural" Heteroscedasticity |
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35 | (1) |
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3.2 The Beta Distribution: Definition and Properties |
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35 | (2) |
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3.3 Modeling Location and Dispersion |
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37 | (7) |
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3.4 Estimation and Model Diagnostics |
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44 | (6) |
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3.5 Treatment of Cases at the Boundaries |
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50 | (4) |
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4 Quantile Models for Bounded Variables |
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54 | (18) |
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54 | (1) |
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54 | (6) |
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4.3 Distributions for Doubly Bounded Variables With Explicit Quantile Functions |
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60 | (4) |
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64 | (8) |
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5 Censored and Truncated Variables |
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72 | (12) |
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5.1 Types of Censoring and Truncation |
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72 | (2) |
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74 | (3) |
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77 | (3) |
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5.4 Heteroscedastic and Non-Gaussian Tobit Models |
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80 | (4) |
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6 Extensions and Conclusions |
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84 | (21) |
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6.1 Extensions and a General Framework |
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84 | (3) |
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6.2 Absolute Bounds and Censoring |
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87 | (5) |
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6.3 Multilevel and Multivariate Models |
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92 | (5) |
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6.4 Bayesian Estimation and Modeling |
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97 | (3) |
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6.5 Roads Less Traveled and the State of the Art |
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100 | (5) |
| References |
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105 | (5) |
| Index |
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110 | |
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.
