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4 - Mixed Models

Published online by Cambridge University Press:  06 July 2010

David Ruppert
Affiliation:
Cornell University, New York
M. P. Wand
Affiliation:
University of New South Wales, Sydney
R. J. Carroll
Affiliation:
Texas A & M University
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Summary

Introduction

Mixed models are an extension of regression models that allow for the incorporation of random effects. However, they also turn out to be closely related to smoothing. In fact, we will show in Section 4.9 that the penalized spline smoother exactly corresponds to the optimal predictor in a mixed model framework. This link allows for mixed model methodology and software to be used in semiparametric regression analysis, as we will demonstrate in subsequent chapters.

This chapter begins with a brief review of mixed models. Readers with detailed knowledge of mixed models could skip these sections and proceed directly to Section 4.9.

Mixed Models

Much of the early work on mixed models – in particular, the special case of variance component models – was motivated by the analysis of data from animal breeding experiments and driven by the need to incorporate heritabilities and genetic correlations in a parsimonious fashion. They have also played an important role in establishing quality control procedures and determination of sampling designs, among other applications. Overviews of this vast topic are provided by Searle, Casella, and McCulloch (1992), Vonesh and Chinchilli (1997), Pinheiro and Bates (2000), Verbeke and Molenberghs (2000), and McCulloch and Searle (2001).

A more contemporary application of mixed models is the analysis of longitudinal data sets (see e.g. Laird and Ware 1982; Diggle et al. 2002). We will use this setting to illustrate the essence of mixed modeling.

Figure 4.1 shows two representations of data pertaining to weight measurements of 48 pigs for nine successive weeks. Figure 4.1(a) is simply a scatterplot of the weights against their corresponding week number; in Figure 4.1(b), lines are drawn connecting those measurements that belong to the same pig.

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Publisher: Cambridge University Press
Print publication year: 2003

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  • Mixed Models
  • David Ruppert, Cornell University, New York, M. P. Wand, University of New South Wales, Sydney, R. J. Carroll, Texas A & M University
  • Book: Semiparametric Regression
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755453.006
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  • Mixed Models
  • David Ruppert, Cornell University, New York, M. P. Wand, University of New South Wales, Sydney, R. J. Carroll, Texas A & M University
  • Book: Semiparametric Regression
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755453.006
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Mixed Models
  • David Ruppert, Cornell University, New York, M. P. Wand, University of New South Wales, Sydney, R. J. Carroll, Texas A & M University
  • Book: Semiparametric Regression
  • Online publication: 06 July 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511755453.006
Available formats
×