Introduction to Mixed Modelling

Beyond Regression and Analysis of Variance

N. W. Galwey

• Publisher: Wiley
• Year: 2014
• Print ISBN:
  9781119945499
• Edition: 2
• eBook ISBN:
  9781118861820
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Mixed modelling is very useful, and easier than you think!

Mixed modelling is now well established as a powerful approach to statistical data analysis. It is based on the recognition of random-effect terms in statistical models, leading to inferences and estimates that have much wider applicability and are more realistic than those otherwise obtained.

Introduction to Mixed Modelling leads the reader into mixed modelling as a natural extension of two more familiar methods, regression analysis and analysis of variance. It provides practical guidance combined with a clear explanation of the underlying concepts.

Like the first edition, this new edition shows diverse applications of mixed models, provides guidance on the identification of random-effect terms, and explains how to obtain and interpret best linear unbiased predictors (BLUPs).   It also introduces several important new topics, including the following:

• Use of the software SAS, in addition to GenStat and R.
• Meta-analysis and the multiple testing problem.
• The Bayesian interpretation of mixed models.

Including numerous practical exercises with solutions, this book provides an ideal introduction to mixed modelling for final year undergraduate students, postgraduate students and professional researchers. It will appeal to readers from a wide range of scientific disciplines including statistics, biology, bioinformatics, medicine, agriculture, engineering, economics, archaeology and geography.

Praise for the first edition:

"One of the main strengths of the text is the bridge it provides between traditional analysis of variance and regression models and the more recently developed class of mixed models...Each chapter is well-motivated by at least one carefully chosen example...demonstrating the broad applicability of mixed models in many different disciplines...most readers will likely learn something new, and those previously unfamiliar with mixed models will obtain a solid foundation on this topic."—Kerrie Nelson University of South Carolina, in American Statistician, 2007

Chapter 1
The need for more than one random-effect term when fitting a regression line

1.1 A data set with several observations of variable Y at each value of variable X

One of the commonest, and simplest, uses of statistical analysis is the fitting of a straight line, known for historical reasons as a regression line, to describe the relationship between an explanatory variable, X and a response variable, Y. The departure of the values of Y from this line is called the residual variation, and is regarded as random. It is natural to ask whether the part of the variation in Y that is explained by the relationship with X is more than could reasonably be expected by chance: or more formally, whether it is significant relative to the residual variation. This is a simple regression analysis, and for many data sets it is all that is required. However, in some cases, several observations of Y are taken at each value of X. The data then form natural groups, and it may no longer be appropriate to analyse them as though every observation were independent: observations of Y at the same value of X may lie at a similar distance from the line. We may then be able to recognize two sources of random variation, namely

• variation among groups
• variation among observations within each group.

This is one of the simplest situations in which it is necessary to consider the possibility that there may be more than a single stratum of random variation—or, in the language of mixed modelling, that a model with more than one random-effect term may be required. In this chapter, we will examine a data set of this type and explore how the usual regression analysis is modified by the fact that the data form natural groups.

We will explore this question in a data set that relates the prices of houses in England to their latitude. There is no doubt that houses cost more in the south of England than in the north: these data will not lead to any new conclusions, but they will illustrate this trend, and the methods used to explore it. The data are displayed in a spreadsh...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Chapter 1: The need for more than one random-effect term when fitting a regression line
• Chapter 2: The need for more than one random-effect term in a designed experiment
• Chapter 3: Estimation of the variances of random-effect terms
• Chapter 4: Interval estimates for fixed-effect terms in mixed models
• Chapter 5: Estimation of random effects in mixed models: Best Linear Unbiased Predictors (BLUPs)
• Chapter 6: More advanced mixed models for more elaborate data sets
• Chapter 7: Three case studies
• Chapter 8: Meta-analysis and the multiple testing problem
• Chapter 9: The use of mixed models for the analysis of unbalanced experimental designs
• Chapter 10: Beyond mixed modelling
• Chapter 11: Why is the criterion for fitting mixed models called REsidual Maximum Likelihood?
• Index
• End User License Agreement