Mixed modelling is one of the most promising and exciting areas of statistical analysis, enabling more powerful interpretation of data through the recognition of random effects. However, many perceive mixed modelling as an intimidating and specialized technique. This book introduces mixed modelling analysis in a simple and straightforward way, allowing the reader to apply the technique confidently in a wide range of situations. Introduction to Mixed Modelling shows that mixed modelling is a natural extension of the more familiar statistical methods of regression analysis and analysis of variance. In doing so, it provides the ideal introduction to this important statistical technique for those engaged in the statistical analysis of data. This essential book: Demonstrates the power of mixed modelling in a wide range of disciplines, including industrial research, social sciences, genetics, clinical research, ecology and agricultural research. Illustrates how the capabilities of regression analysis can be combined with those of ANOVA by the specification of a mixed model. Introduces the criterion of Restricted Maximum Likelihood (REML) for the fitting of a mixed model to data. Presents the application of mixed model analysis to a wide range of situations and explains how to obtain and interpret Best Linear Unbiased Predictors (BLUPs). Features a supplementary website containing solutions to exercises, further examples, and links to the computer software systems GenStat and R. This book provides a comprehensive introduction to mixed modelling, ideal for final year undergraduate students, postgraduate students and professional researchers alike. Readers will come from a wide range of scientific disciplines including statistics, biology, bioinformatics, medicine, agriculture, engineering, economics, and social sciences.

Nicholas W. Galwey, Principal Scientist, GlaxoSmithKline, Harlow, Essex. A respected consultant and researcher in the pharmaceutical industry with extensive teaching experience.

Preface

ix

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

1

(44)

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

1

(1)

Simple regression analysis. Use of the software GenStat to perform the analysis

2

(8)

Regression analysis on the group means

10

(2)

A regression model with a term for the groups

12

(3)

Construction of the appropriate F test for the significance of the explanatory variable when groups are present

15

(1)

The decision to regard a model term as random: a mixed model

16

(1)

Comparison of the tests in a mixed model with a test of lack of fit

17

(1)

The use of residual maximum likelihood to fit the mixed model

18

(3)

Equivalence of the different analyses when the number of observations per group is constant

21

(6)

Testing the assumptions of the analyses: inspection of the residual values

27

(2)

Use of the software R to perform the analyses

29

(4)

Fitting a mixed model using GenStat's GUI

33

(6)

Summary

39

(1)

Exercises

40

(5)

The need for more than one random-effect term in a designed experiment

45

(28)

The split plot design: a design with more than one random-effect term

45

(2)

The analysis of variance of the split plot design: a random-effect term for the main plots

47

(8)

Consequences of failure to recognise the main plots when analysing the split plot design

55

(2)

The use of mixed modelling to analyse the split plot design

57

(3)

A more conservative alternative to the Wald statistic

60

(1)

Justification for regarding block effects as random

61

(1)

Testing the assumptions of the analyses: inspection of the residual values

62

(1)

Use of R to perform the analyses

63

(4)

Summary

67

(1)

Exercises

68

(5)

Estimation of the variances of random-effect terms

73

(52)

The need to estimate variance components

73

(1)

A hierarchical random-effect model for a three-stage assay process

73

(5)

The relationship between variance components and stratum mean squares

78

(2)

Estimation of the variance components in the hierarchical random-effect model

80

(2)

Design of an optimum strategy for future sampling

82

(3)

Use of R to analyse the hierarchical three-stage assay process

85

(2)

Genetic variation: a crop field trial with an unbalanced design

87

(5)

Production of a balanced experimental design by `padding' with missing values

92

(4)

Regarding a treatment term as a random-effect term. The use of mixed-model analysis to analyse an unbalanced data set

96

(3)

Comparison of a variance-component estimate with its standard error

99

(2)

An alternative significance test for variance components

101

(2)

Comparison among significance tests for variance components

103

(1)

Inspection of the residual values

104

(1)

Heritability. The prediction of genetic advance under selection

105

(4)

Use of R to analyse the unbalanced field trial

109

(4)

Estimation of variance components in the regression analysis on grouped data

113

(2)

Estimation of variance components for block effects in the split plot experimental design

115

(2)

Summary

117

(1)

Exercises

118

(7)

Interval estimates for fixed-effect terms in mixed models

125

(26)

The concept of an interval estimate

125

(1)

SEs for regression coefficients in a mixed-model analysis

126

(4)

SEs for differences between treatment means in the split plot design

130

(3)

A significance test for the difference between treatment means

133

(4)

The least significant difference between treatment means

137

(4)

SEs for treatment means in designed experiments: a difference in approach between analysis of variance and mixed-model analysis

141

(6)

Use of R to obtain SEs of means in a designed experiment

147

(1)

Summary

148

(2)

Exercises

150

(1)

Estimation of random effects in mixed models: best linear unbiased predictors

151

(18)

The difference between the estimates of fixed and random effects

151

(3)

The method for estimation of random effects. The best linear unbiased predictor or `shrunk estimate'

154

(2)

The relationship between the shrinkage of BLUPs and regression towards the mean

156

(6)

Use of R for the estimation of random effects

162

(2)

Summary

164

(1)

Exercises

165

(4)

More advanced mixed models for more elaborate data sets

169

(24)

Features of the models introduced so far: a review

169

(1)

Further combinations of model features

170

(2)

The choice of model terms to be regarded as random

172

(2)

Disagreement concerning the appropriate significance test when fixed- and random-effect terms interact

174

(7)

Arguments for regarding block effects as random

181

(5)

Examples of the choice of fixed- and random-effect terms

186

(4)

Summary

190

(3)

Exercises

193

(1)

Two case studies

193

(58)

Further development of mixed-modelling concepts through the analysis of specific data sets

193

(1)

A fixed-effect model with several variates and factors

194

(15)

Use of R to fit the fixed-effect model with several variates and factors

209

(5)

A random-effect model with several factors

214

(15)

Use of R to fit the random-effect model with several factors

229

(9)

Summary

238

(1)

Exercises

238

(13)

The use of mixed models for the analysis of unbalanced experimental designs

251

(24)

A balanced incomplete block design

251

(4)

Imbalance due to a missing block. Mixed-model analysis of the incomplete block design

255

(4)

Use of R to analyse the incomplete block design

259

(2)

Relaxation of the requirement for balance: alpha designs

261

(8)

Use of R to analyse the alphalpha design

269

(2)

Summary

271

(1)

Exercises

272

(3)

Beyond mixed modelling

275

(58)

Review of the uses of mixed models

275

(1)

The generalised linear mixed model. Fitting a logistic (sigmoidal) curve to proportions of observations

276

(8)

Fitting a GLMM to a contingency table. Trouble-Shooting when the mixed-modelling process fails

284

(14)

The hierarchical generalised linear model

298

(5)

The role of the covariance matrix in the specification of a mixed model

303

(4)

A more general pattern in the covariance matrix. Analysis of pedigree data

307

(10)

Estimation of parameters in the covariance matrix. Analysis of temporal and spatial variation

317

(10)

Summary

327

(1)

Exercises

327

(6)

Why is the criterion for fitting mixed models called residual maximum likelihood?

333

(24)

Maximum likelihood and residual maximum likelihood

333

(1)

Estimation of the variance σ2 from a single observation using the maximum likelihood criterion

334

(1)

Estimation of σ2 from more than one observation

334

(4)

The μ-effects axis as a dimension within the sample space

338

(1)

Simultaneous estimation of μ and σ2 using the maximum likelihood criterion

339

(3)

An alternative estimate of σ2 using the REML criterion

342

(3)

Extension to the general linear model. The fixed-effect axes as a subspace of the sample space

345

(4)

Application of the REML criterion to the general linear model

349

(2)

Extension to models with more than one random-effect term

351

(1)

Summary

352

(1)

Exercises

353

(4)

References

357

(4)

Index

361

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Introduction to Mixed Modelling : Beyond Regression and Analysis of Variance

9780470014967

0470014962

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