Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Fixed-Effects Models
- 3 Models with Random Effects
- 4 Prediction and Bayesian Inference
- 5 Multilevel Models
- 6 Stochastic Regressors
- 7 Modeling Issues
- 8 Dynamic Models
- 9 Binary Dependent Variables
- 10 Generalized Linear Models
- 11 Categorical Dependent Variables and Survival Models
- Appendix A Elements of Matrix Algebra
- Appendix B Normal Distribution
- Appendix C Likelihood-Based Inference
- Appendix D State Space Model and the Kalman Filter
- Appendix E Symbols and Notation
- Appendix F Selected Longitudinal and Panel Data Sets
- References
- Index
3 - Models with Random Effects
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Fixed-Effects Models
- 3 Models with Random Effects
- 4 Prediction and Bayesian Inference
- 5 Multilevel Models
- 6 Stochastic Regressors
- 7 Modeling Issues
- 8 Dynamic Models
- 9 Binary Dependent Variables
- 10 Generalized Linear Models
- 11 Categorical Dependent Variables and Survival Models
- Appendix A Elements of Matrix Algebra
- Appendix B Normal Distribution
- Appendix C Likelihood-Based Inference
- Appendix D State Space Model and the Kalman Filter
- Appendix E Symbols and Notation
- Appendix F Selected Longitudinal and Panel Data Sets
- References
- Index
Summary
Abstract. This chapter considers the Chapter 2 data structure but here the heterogeneity is modeled using random quantities in lieu of fixed parameters; these random quantities are known as random effects. By introducing random quantities, the analysis of longitudinal and panel data can now be cast in the mixed linear model framework.
Although mixed linear models are an established part of statistical methodology, their use is not as widespread as regression. Thus, the chapter introduces this modeling framework, beginning with the special case of a single random intercept known as the error-components model and then focusing on the linear mixed-effects model, which is particularly important for longitudinal data. After introducing the models, this chapter describes estimation of regression coefficients and variance components, as well as hypothesis testing for regression coefficients.
Error-Components/Random-Intercepts Model
Sampling and Inference
Suppose that you are interested in studying the behavior of individuals who are randomly selected from a population. For example, in Section 3.2 we will study the effects that an individual's economic and demographic characteristics have on the amount of income tax paid. Here, the set of subjects that we will study is randomly selected from a larger database, which is itself a random sample of the U.S. taxpayers. In contrast, the Chapter 2 Medicare example dealt with a fixed set of subjects. That is, it is difficult to think of the 54 states as a subset from some “superpopulation” of states.
- Type
- Chapter
- Information
- Longitudinal and Panel DataAnalysis and Applications in the Social Sciences, pp. 72 - 124Publisher: Cambridge University PressPrint publication year: 2004