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by N. W. Galwey
Introduction to Mixed Modelling: Beyond Regression and Analysis of Variance, 2nd Edition
Cover
Title Page
Copyright
Preface
References
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
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