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**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**

- Cover
- Title Page
- Copyright
- Preface
- 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
- 1.2 Simple regression analysis: Use of the software GenStat to perform the analysis
- 1.3 Regression analysis on the group means
- 1.4 A regression model with a term for the groups
- 1.5 Construction of the appropriate F test for the significance of the explanatory variable when groups are present
- 1.6 The decision to specify a model term as random: A mixed model
- 1.7 Comparison of the tests in a mixed model with a test of lack of fit
- 1.8 The use of REsidual Maximum Likelihood (REML) to fit the mixed model
- 1.9 Equivalence of the different analyses when the number of observations per group is constant
- 1.10 Testing the assumptions of the analyses: Inspection of the residual values
- 1.11 Use of the software R to perform the analyses
- 1.12 Use of the software SAS to perform the analyses
- 1.13 Fitting a mixed model using GenStat's Graphical User Interface (GUI)
- 1.14 Summary
- 1.15 Exercises
- References

- Chapter 2: The need for more than one random-effect term in a designed experiment
- 2.1 The split plot design: A design with more than one random-effect term
- 2.2 The analysis of variance of the split plot design: A random-effect term for the main plots
- 2.3 Consequences of failure to recognize the main plots when analysing the split plot design
- 2.4 The use of mixed modelling to analyse the split plot design
- 2.5 A more conservative alternative to the F and Wald statistics
- 2.6 Justification for regarding block effects as random
- 2.7 Testing the assumptions of the analyses: Inspection of the residual values
- 2.8 Use of R to perform the analyses
- 2.9 Use of SAS to perform the analyses
- 2.10 Summary
- 2.11 Exercises
- References

- Chapter 3: Estimation of the variances of random-effect terms
- 3.1 The need to estimate variance components
- 3.2 A hierarchical random-effects model for a three-stage assay process
- 3.3 The relationship between variance components and stratum mean squares
- 3.4 Estimation of the variance components in the hierarchical random-effects model
- 3.5 Design of an optimum strategy for future sampling
- 3.6 Use of R to analyse the hierarchical three-stage assay process
- 3.7 Use of SAS to analyse the hierarchical three-stage assay process
- 3.8 Genetic variation: A crop field trial with an unbalanced design
- 3.9 Production of a balanced experimental design by ‘padding’ with missing values
- 3.10 Specification of a treatment term as a random-effect term: The use of mixed-model analysis to analyse an unbalanced data set
- 3.11 Comparison of a variance component estimate with its standard error
- 3.12 An alternative significance test for variance components
- 3.13 Comparison among significance tests for variance components
- 3.14 Inspection of the residual values
- 3.15 Heritability: The prediction of genetic advance under selection
- 3.16 Use of R to analyse the unbalanced field trial
- 3.17 Use of SAS to analyse the unbalanced field trial
- 3.18 Estimation of variance components in the regression analysis on grouped data
- 3.19 Estimation of variance components for block effects in the split-plot experimental design
- 3.20 Summary
- 3.21 Exercises
- References

- Chapter 4: Interval estimates for fixed-effect terms in mixed models
- 4.1 The concept of an interval estimate
- 4.2 Standard errors for regression coefficients in a mixed-model analysis
- 4.3 Standard errors for differences between treatment means in the split-plot design
- 4.4 A significance test for the difference between treatment means
- 4.5 The least significant difference (LSD) between treatment means
- 4.6 Standard errors for treatment means in designed experiments: A difference in approach between analysis of variance and mixed-model analysis
- 4.7 Use of R to obtain SEs of means in a designed experiment
- 4.8 Use of SAS to obtain SEs of means in a designed experiment
- 4.9 Summary
- 4.10 Exercises
- References

- Chapter 5: Estimation of random effects in mixed models: Best Linear Unbiased Predictors (BLUPs)
- 5.1 The difference between the estimates of fixed and random effects
- 5.2 The method for estimation of random effects: The best linear unbiased predictor (BLUP) or ‘shrunk estimate’
- 5.3 The relationship between the shrinkage of BLUPs and regression towards the mean
- 5.4 Use of R for the estimation of fixed and random effects
- 5.5 Use of SAS for the estimation of random effects
- 5.6 The Bayesian interpretation of BLUPs: Justification of a random-effect term without invoking an underlying infinite population
- 5.7 Summary
- 5.8 Exercises
- References

- Chapter 6: More advanced mixed models for more elaborate data sets
- 6.1 Features of the models introduced so far: A review
- 6.2 Further combinations of model features
- 6.3 The choice of model terms to be specified as random
- 6.4 Disagreement concerning the appropriate significance test when fixed- and random-effect terms interact: ‘The great mixed-model muddle’
- 6.5 Arguments for specifying block effects as random
- 6.6 Examples of the choice of fixed- and random-effect specification of terms
- 6.7 Summary
- 6.8 Exercises
- References

- Chapter 7: Three case studies
- 7.1 Further development of mixed modelling concepts through the analysis of specific data sets
- 7.2 A fixed-effects model with several variates and factors
- 7.3 Use of R to fit the fixed-effects model with several variates and factors
- 7.4 Use of SAS to fit the fixed-effects model with several variates and factors
- 7.5 A random coefficient regression model
- 7.6 Use of R to fit the random coefficients model
- 7.7 Use of SAS to fit the random coefficients model
- 7.8 A random-effects model with several factors
- 7.9 Use of R to fit the random-effects model with several factors
- 7.10 Use of SAS to fit the random-effects model with several factors
- 7.11 Summary
- 7.12 Exercises
- References

- Chapter 8: Meta-analysis and the multiple testing problem
- 8.1 Meta-analysis: Combined analysis of a set of studies
- 8.2 Fixed-effect meta-analysis with estimation only of the main effect of treatment
- 8.3 Random-effects meta-analysis with estimation of study × treatment interaction effects
- 8.4 A random-effect interaction between two fixed-effect terms
- 8.5 Meta-analysis of individual-subject data using R
- 8.6 Meta-analysis of individual-subject data using SAS
- 8.7 Meta-analysis when only summary data are available
- 8.8 The multiple testing problem: Shrinkage of BLUPs as a defence against the Winner's Curse
- 8.9 Fitting of multiple models using R
- 8.10 Fitting of multiple models using SAS
- 8.11 Summary
- 8.12 Exercises
- References

- Chapter 9: The use of mixed models for the analysis of unbalanced experimental designs
- 9.1 A balanced incomplete block design
- 9.2 Imbalance due to a missing block: Mixed-model analysis of the incomplete block design
- 9.3 Use of R to analyse the incomplete block design
- 9.4 Use of SAS to analyse the incomplete block design
- 9.5 Relaxation of the requirement for balance: Alpha designs
- 9.6 Approximate balance in two directions: The alphalpha design
- 9.7 Use of R to analyse the alphalpha design
- 9.8 Use of SAS to analyse the alphalpha design
- 9.9 Summary
- 9.10 Exercises
- References

- Chapter 10: Beyond mixed modelling
- 10.1 Review of the uses of mixed models
- 10.2 The generalized linear mixed model (GLMM): Fitting a logistic (sigmoidal) curve to proportions of observations
- 10.3 Use of R to fit the logistic curve
- 10.4 Use of SAS to fit the logistic curve
- 10.5 Fitting a GLMM to a contingency table: Trouble-shooting when the mixed modelling process fails
- 10.6 The hierarchical generalized linear model (HGLM)
- 10.7 Use of R to fit a GLMM and a HGLM to a contingency table
- 10.8 Use of SAS to fit a GLMM to a contingency table
- 10.9 The role of the covariance matrix in the specification of a mixed model
- 10.10 A more general pattern in the covariance matrix: Analysis of pedigrees and genetic data
- 10.11 Estimation of parameters in the covariance matrix: Analysis of temporal and spatial variation
- 10.12 Use of R to model spatial variation
- 10.13 Use of SAS to model spatial variation
- 10.14 Summary
- 10.15 Exercises
- References

- Chapter 11: Why is the criterion for fitting mixed models called REsidual Maximum Likelihood?
- 11.1 Maximum likelihood and residual maximum likelihood
- 11.2 Estimation of the variance σ2 from a single observation using the maximum-likelihood criterion
- 11.3 Estimation of σ2 from more than one observation
- 11.4 The μ-effect axis as a dimension within the sample space
- 11.5 Simultaneous estimation of μ and σ2 using the maximum-likelihood criterion
- 11.6 An alternative estimate of σ2 using the REML criterion
- 11.7 Bayesian justification of the REML criterion
- 11.8 Extension to the general linear model: The fixed-effect axes as a sub-space of the sample space
- 11.9 Application of the REML criterion to the general linear model
- 11.10 Extension to models with more than one random-effect term
- 11.11 Summary
- 11.12 Exercises
- References

- Index
- End User License Agreement