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.

Models for Doubly Bounded Variables

Doubly Bounded Variables and “Natural” Heteroscedasticity

In this chapter and the next chapter, we will introduce models for doubly bounded variables that are capable of modeling both location (e.g., the mean) and dispersion (e.g., variance). We begin by reviewing the challenges in modeling doubly bounded variables. We then introduce the beta distribution, currently the most popular for modeling doubly bounded variables, and describe regression models using this distribution. After that, we discuss the treatment of cases on the boundaries, via hurdle and mixture models (e.g., zero-inflation in proportional data).

As we indicated early in Chapter 1, there are two types of doubly bounded variables: those with true bounds and those with bounds due to censoring or truncation. Examples of the ...

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