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Case Studies in Bayesian Statistical Modelling and Analysis by Anthony N. Pettitt, Kerrie L. Mengersen, Clair L. Alston

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8

Bayesian Mixed Effects Models

Clair L. Alston1, Christopher M. Strickland1, Kerrie L. Mengersen1 and Graham E. Gardner2

1Queensland University of Technology, Brisbane, Australia

2Murdoch University, Perth, Australia

8.1 Introduction

In this chapter we introduce a Bayesian approach to linear mixed models. This model is an extension to the linear regression model in Chapter 4, when data are complex and hierarchical.

Mixed models, often described as a mixture of fixed and random terms, are used in cases were the data are clustered due to subpopulations, such as sires in genetics trials, years in trials that are conducted annually, assessors in experiments where the person obtaining the measurements may be subjective, individual in experiments that contain measurements in time on traits such as growth (longitudinal studies) and where higher level terms are considered random. For mixed models, the different sources of error in data are captured by treating the variation as within and between clusters.

The terms fixed and random are used somewhat loosely, throughout this chapter, in reference to their usage under the classical paradigm. In the Bayesian framework, as all parameters are treated as random, the real difference between these terms is their prior specification. This differentiation will become apparent in Section 8.3.

Demidenko (2004), Wu (2010) and Verbeke and Molenberghs (2009) provide substantial references for these models in the frequentist framework, and provide much ...

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