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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…mehr
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- Produktdetails
- Verlag: John Wiley & Sons
- Seitenzahl: 504
- Erscheinungstermin: 8. August 2014
- Englisch
- ISBN-13: 9781118861813
- Artikelnr.: 41355905
- Verlag: John Wiley & Sons
- Seitenzahl: 504
- Erscheinungstermin: 8. August 2014
- Englisch
- ISBN-13: 9781118861813
- Artikelnr.: 41355905
regression line 1 1.1 A data set with several observations of variable Y at
each value of variable X 1 1.2 Simple regression analysis: Use of the
software GenStat to perform the analysis 2 1.3 Regression analysis on the
group means 9 1.4 A regression model with a term for the groups 10 1.5
Construction of the appropriate F test for the significance of the
explanatory variable when groups are present 13 1.6 The decision to specify
a model term as random: A mixed model 14 1.7 Comparison of the tests in a
mixed model with a test of lack of fit 16 1.8 The use of REsidual Maximum
Likelihood (REML) to fit the mixed model 17 1.9 Equivalence of the
different analyses when the number of observations per group is constant 21
1.10 Testing the assumptions of the analyses: Inspection of the residual
values 26 1.11 Use of the software R to perform the analyses 28 1.12 Use of
the software SAS to perform the analyses 33 1.13 Fitting a mixed model
using GenStat's Graphical User Interface (GUI) 40 1.14 Summary 46 1.15
Exercises 47 References 51 2 The need for more than one random-effect term
in a designed experiment 52 2.1 The split plot design: A design with more
than one random-effect term 52 2.2 The analysis of variance of the split
plot design: A random-effect term for the main plots 54 2.3 Consequences of
failure to recognize the main plots when analysing the split plot design 62
2.4 The use of mixed modelling to analyse the split plot design 64 2.5 A
more conservative alternative to the F and Wald statistics 66 2.6
Justification for regarding block effects as random 67 2.7 Testing the
assumptions of the analyses: Inspection of the residual values 68 2.8 Use
of R to perform the analyses 71 2.9 Use of SAS to perform the analyses 77
2.10 Summary 81 2.11 Exercises 82 References 86 3 Estimation of the
variances of random-effect terms 87 3.1 The need to estimate variance
components 87 3.2 A hierarchical random-effects model for a three-stage
assay process 87 3.3 The relationship between variance components and
stratum mean squares 91 3.4 Estimation of the variance components in the
hierarchical random-effects model 93 3.5 Design of an optimum strategy for
future sampling 95 3.6 Use of R to analyse the hierarchical three-stage
assay process 98 3.7 Use of SAS to analyse the hierarchical three-stage
assay process 100 3.8 Genetic variation: A crop field trial with an
unbalanced design 102 3.9 Production of a balanced experimental design by
'padding' with missing values 106 3.10 Specification of a treatment term as
a random-effect term: The use of mixed-model analysis to analyse an
unbalanced data set 110 3.11 Comparison of a variance component estimate
with its standard error 112 3.12 An alternative significance test for
variance components 113 3.13 Comparison among significance tests for
variance components 116 3.14 Inspection of the residual values 117 3.15
Heritability: The prediction of genetic advance under selection 117 3.16
Use of R to analyse the unbalanced field trial 122 3.17 Use of SAS to
analyse the unbalanced field trial 125 3.18 Estimation of variance
components in the regression analysis on grouped data 128 3.19 Estimation
of variance components for block effects in the split-plot experimental
design 130 3.20 Summary 132 3.21 Exercises 133 References 136 4 Interval
estimates for fixed-effect terms in mixed models 137 4.1 The concept of an
interval estimate 137 4.2 Standard errors for regression coefficients in a
mixed-model analysis 138 4.3 Standard errors for differences between
treatment means in the split-plot design 142 4.4 A significance test for
the difference between treatment means 144 4.5 The least significant
difference (LSD) between treatment means 147 4.6 Standard errors for
treatment means in designed experiments: A difference in approach between
analysis of variance and mixed-model analysis 151 4.7 Use of R to obtain
SEs of means in a designed experiment 157 4.8 Use of SAS to obtain SEs of
means in a designed experiment 159 4.9 Summary 161 4.10 Exercises 163
References 164 5 Estimation of random effects in mixed models: Best Linear
Unbiased Predictors (BLUPs) 165 5.1 The difference between the estimates of
fixed and random effects 165 5.2 The method for estimation of random
effects: The best linear unbiased predictor (BLUP) or 'shrunk estimate' 168
5.3 The relationship between the shrinkage of BLUPs and regression towards
the mean 170 5.4 Use of R for the estimation of fixed and random effects
176 5.5 Use of SAS for the estimation of random effects 178 5.6 The
Bayesian interpretation of BLUPs: Justification of a random-effect term
without invoking an underlying infinite population 182 5.7 Summary 187 5.8
Exercises 188 References 191 6 More advanced mixed models for more
elaborate data sets 192 6.1 Features of the models introduced so far: A
review 192 6.2 Further combinations of model features 192 6.3 The choice of
model terms to be specified as random 195 6.4 Disagreement concerning the
appropriate significance test when fixed and random-effect terms interact:
'The great mixed-model muddle' 197 6.5 Arguments for specifying block
effects as random 204 6.6 Examples of the choice of fixed- and
random-effect specification of terms 209 6.7 Summary 213 6.8 Exercises 215
References 216 7 Three case studies 217 7.1 Further development of mixed
modelling concepts through the analysis of specific data sets 217 7.2 A
fixed-effects model with several variates and factors 218 7.3 Use of R to
fit the fixed-effects model with several variates and factors 233 7.4 Use
of SAS to fit the fixed-effects model with several variates and factors 237
7.5 A random coefficient regression model 242 7.6 Use of R to fit the
random coefficients model 246 7.7 Use of SAS to fit the random coefficients
model 247 7.8 A random-effects model with several factors 249 7.9 Use of R
to fit the random-effects model with several factors 266 7.10 Use of SAS to
fit the random-effects model with several factors 274 7.11 Summary 282 7.12
Exercises 282 References 294 8 Meta-analysis and the multiple testing
problem 295 8.1 Meta-analysis: Combined analysis of a set of studies 295
8.2 Fixed-effect meta-analysis with estimation only of the main effect of
treatment 296 8.3 Random-effects meta-analysis with estimation of study ×
treatment interaction effects 301 8.4 A random-effect interaction between
two fixed-effect terms 303 8.5 Meta-analysis of individual-subject data
using R 307 8.6 Meta-analysis of individual-subject data using SAS 312 8.7
Meta-analysis when only summary data are available 318 8.8 The multiple
testing problem: Shrinkage of BLUPs as a defence against the Winner's Curse
326 8.9 Fitting of multiple models using R 338 8.10 Fitting of multiple
models using SAS 340 8.11 Summary 342 8.12 Exercises 343 References 348 9
The use of mixed models for the analysis of unbalanced experimental designs
350 9.1 A balanced incomplete block design 350 9.2 Imbalance due to a
missing block: Mixed-model analysis of the incomplete block design 354 9.3
Use of R to analyse the incomplete block design 358 9.4 Use of SAS to
analyse the incomplete block design 360 9.5 Relaxation of the requirement
for balance: Alpha designs 362 9.6 Approximate balance in two directions:
The alphalpha design 368 9.7 Use of R to analyse the alphalpha design 373
9.8 Use of SAS to analyse the alphalpha design 374 9.9 Summary 376 9.10
Exercises 377 References 378 10 Beyond mixed modelling 379 10.1 Review of
the uses of mixed models 379 10.2 The generalized linear mixed model
(GLMM): Fitting a logistic (sigmoidal) curve to proportions of observations
380 10.3 Use of R to fit the logistic curve 388 10.4 Use of SAS to fit the
logistic curve 390 10.5 Fitting a GLMM to a contingency table:
Trouble-shooting when the mixed modelling process fails 392 10.6 The
hierarchical generalized linear model (HGLM) 403 10.7 Use of R to fit a
GLMM and a HGLM to a contingency table 410 10.8 Use of SAS to fit a GLMM to
a contingency table 415 10.9 The role of the covariance matrix in the
specification of a mixed model 418 10.10 A more general pattern in the
covariance matrix: Analysis of pedigrees and genetic data 421 10.11
Estimation of parameters in the covariance matrix: Analysis of temporal and
spatial variation 431 10.12 Use of R to model spatial variation 441 10.13
Use of SAS to model spatial variation 444 10.14 Summary 447 10.15 Exercises
447 References 452 11 Why is the criterion for fitting mixed models called
REsidual Maximum Likelihood? 454 11.1 Maximum likelihood and residual
maximum likelihood 454 11.2 Estimation of the variance 2 from a single
observation using the maximum-likelihood criterion 455 11.3 Estimation of 2
from more than one observation 455 11.4 The -effect axis as a dimension
within the sample space 457 11.5 Simultaneous estimation of and 2 using the
maximum-likelihood criterion 460 11.6 An alternative estimate of 2 using
the REML criterion 462 11.7 Bayesian justification of the REML criterion
465 11.8 Extension to the general linear model: The fixed-effect axes as a
sub-space of the sample space 465 11.9 Application of the REML criterion to
the general linear model 470 11.10 Extension to models with more than one
random-effect term 472 11.11 Summary 473 11.12 Exercises 474 References 476
Index 477
regression line 1 1.1 A data set with several observations of variable Y at
each value of variable X 1 1.2 Simple regression analysis: Use of the
software GenStat to perform the analysis 2 1.3 Regression analysis on the
group means 9 1.4 A regression model with a term for the groups 10 1.5
Construction of the appropriate F test for the significance of the
explanatory variable when groups are present 13 1.6 The decision to specify
a model term as random: A mixed model 14 1.7 Comparison of the tests in a
mixed model with a test of lack of fit 16 1.8 The use of REsidual Maximum
Likelihood (REML) to fit the mixed model 17 1.9 Equivalence of the
different analyses when the number of observations per group is constant 21
1.10 Testing the assumptions of the analyses: Inspection of the residual
values 26 1.11 Use of the software R to perform the analyses 28 1.12 Use of
the software SAS to perform the analyses 33 1.13 Fitting a mixed model
using GenStat's Graphical User Interface (GUI) 40 1.14 Summary 46 1.15
Exercises 47 References 51 2 The need for more than one random-effect term
in a designed experiment 52 2.1 The split plot design: A design with more
than one random-effect term 52 2.2 The analysis of variance of the split
plot design: A random-effect term for the main plots 54 2.3 Consequences of
failure to recognize the main plots when analysing the split plot design 62
2.4 The use of mixed modelling to analyse the split plot design 64 2.5 A
more conservative alternative to the F and Wald statistics 66 2.6
Justification for regarding block effects as random 67 2.7 Testing the
assumptions of the analyses: Inspection of the residual values 68 2.8 Use
of R to perform the analyses 71 2.9 Use of SAS to perform the analyses 77
2.10 Summary 81 2.11 Exercises 82 References 86 3 Estimation of the
variances of random-effect terms 87 3.1 The need to estimate variance
components 87 3.2 A hierarchical random-effects model for a three-stage
assay process 87 3.3 The relationship between variance components and
stratum mean squares 91 3.4 Estimation of the variance components in the
hierarchical random-effects model 93 3.5 Design of an optimum strategy for
future sampling 95 3.6 Use of R to analyse the hierarchical three-stage
assay process 98 3.7 Use of SAS to analyse the hierarchical three-stage
assay process 100 3.8 Genetic variation: A crop field trial with an
unbalanced design 102 3.9 Production of a balanced experimental design by
'padding' with missing values 106 3.10 Specification of a treatment term as
a random-effect term: The use of mixed-model analysis to analyse an
unbalanced data set 110 3.11 Comparison of a variance component estimate
with its standard error 112 3.12 An alternative significance test for
variance components 113 3.13 Comparison among significance tests for
variance components 116 3.14 Inspection of the residual values 117 3.15
Heritability: The prediction of genetic advance under selection 117 3.16
Use of R to analyse the unbalanced field trial 122 3.17 Use of SAS to
analyse the unbalanced field trial 125 3.18 Estimation of variance
components in the regression analysis on grouped data 128 3.19 Estimation
of variance components for block effects in the split-plot experimental
design 130 3.20 Summary 132 3.21 Exercises 133 References 136 4 Interval
estimates for fixed-effect terms in mixed models 137 4.1 The concept of an
interval estimate 137 4.2 Standard errors for regression coefficients in a
mixed-model analysis 138 4.3 Standard errors for differences between
treatment means in the split-plot design 142 4.4 A significance test for
the difference between treatment means 144 4.5 The least significant
difference (LSD) between treatment means 147 4.6 Standard errors for
treatment means in designed experiments: A difference in approach between
analysis of variance and mixed-model analysis 151 4.7 Use of R to obtain
SEs of means in a designed experiment 157 4.8 Use of SAS to obtain SEs of
means in a designed experiment 159 4.9 Summary 161 4.10 Exercises 163
References 164 5 Estimation of random effects in mixed models: Best Linear
Unbiased Predictors (BLUPs) 165 5.1 The difference between the estimates of
fixed and random effects 165 5.2 The method for estimation of random
effects: The best linear unbiased predictor (BLUP) or 'shrunk estimate' 168
5.3 The relationship between the shrinkage of BLUPs and regression towards
the mean 170 5.4 Use of R for the estimation of fixed and random effects
176 5.5 Use of SAS for the estimation of random effects 178 5.6 The
Bayesian interpretation of BLUPs: Justification of a random-effect term
without invoking an underlying infinite population 182 5.7 Summary 187 5.8
Exercises 188 References 191 6 More advanced mixed models for more
elaborate data sets 192 6.1 Features of the models introduced so far: A
review 192 6.2 Further combinations of model features 192 6.3 The choice of
model terms to be specified as random 195 6.4 Disagreement concerning the
appropriate significance test when fixed and random-effect terms interact:
'The great mixed-model muddle' 197 6.5 Arguments for specifying block
effects as random 204 6.6 Examples of the choice of fixed- and
random-effect specification of terms 209 6.7 Summary 213 6.8 Exercises 215
References 216 7 Three case studies 217 7.1 Further development of mixed
modelling concepts through the analysis of specific data sets 217 7.2 A
fixed-effects model with several variates and factors 218 7.3 Use of R to
fit the fixed-effects model with several variates and factors 233 7.4 Use
of SAS to fit the fixed-effects model with several variates and factors 237
7.5 A random coefficient regression model 242 7.6 Use of R to fit the
random coefficients model 246 7.7 Use of SAS to fit the random coefficients
model 247 7.8 A random-effects model with several factors 249 7.9 Use of R
to fit the random-effects model with several factors 266 7.10 Use of SAS to
fit the random-effects model with several factors 274 7.11 Summary 282 7.12
Exercises 282 References 294 8 Meta-analysis and the multiple testing
problem 295 8.1 Meta-analysis: Combined analysis of a set of studies 295
8.2 Fixed-effect meta-analysis with estimation only of the main effect of
treatment 296 8.3 Random-effects meta-analysis with estimation of study ×
treatment interaction effects 301 8.4 A random-effect interaction between
two fixed-effect terms 303 8.5 Meta-analysis of individual-subject data
using R 307 8.6 Meta-analysis of individual-subject data using SAS 312 8.7
Meta-analysis when only summary data are available 318 8.8 The multiple
testing problem: Shrinkage of BLUPs as a defence against the Winner's Curse
326 8.9 Fitting of multiple models using R 338 8.10 Fitting of multiple
models using SAS 340 8.11 Summary 342 8.12 Exercises 343 References 348 9
The use of mixed models for the analysis of unbalanced experimental designs
350 9.1 A balanced incomplete block design 350 9.2 Imbalance due to a
missing block: Mixed-model analysis of the incomplete block design 354 9.3
Use of R to analyse the incomplete block design 358 9.4 Use of SAS to
analyse the incomplete block design 360 9.5 Relaxation of the requirement
for balance: Alpha designs 362 9.6 Approximate balance in two directions:
The alphalpha design 368 9.7 Use of R to analyse the alphalpha design 373
9.8 Use of SAS to analyse the alphalpha design 374 9.9 Summary 376 9.10
Exercises 377 References 378 10 Beyond mixed modelling 379 10.1 Review of
the uses of mixed models 379 10.2 The generalized linear mixed model
(GLMM): Fitting a logistic (sigmoidal) curve to proportions of observations
380 10.3 Use of R to fit the logistic curve 388 10.4 Use of SAS to fit the
logistic curve 390 10.5 Fitting a GLMM to a contingency table:
Trouble-shooting when the mixed modelling process fails 392 10.6 The
hierarchical generalized linear model (HGLM) 403 10.7 Use of R to fit a
GLMM and a HGLM to a contingency table 410 10.8 Use of SAS to fit a GLMM to
a contingency table 415 10.9 The role of the covariance matrix in the
specification of a mixed model 418 10.10 A more general pattern in the
covariance matrix: Analysis of pedigrees and genetic data 421 10.11
Estimation of parameters in the covariance matrix: Analysis of temporal and
spatial variation 431 10.12 Use of R to model spatial variation 441 10.13
Use of SAS to model spatial variation 444 10.14 Summary 447 10.15 Exercises
447 References 452 11 Why is the criterion for fitting mixed models called
REsidual Maximum Likelihood? 454 11.1 Maximum likelihood and residual
maximum likelihood 454 11.2 Estimation of the variance 2 from a single
observation using the maximum-likelihood criterion 455 11.3 Estimation of 2
from more than one observation 455 11.4 The -effect axis as a dimension
within the sample space 457 11.5 Simultaneous estimation of and 2 using the
maximum-likelihood criterion 460 11.6 An alternative estimate of 2 using
the REML criterion 462 11.7 Bayesian justification of the REML criterion
465 11.8 Extension to the general linear model: The fixed-effect axes as a
sub-space of the sample space 465 11.9 Application of the REML criterion to
the general linear model 470 11.10 Extension to models with more than one
random-effect term 472 11.11 Summary 473 11.12 Exercises 474 References 476
Index 477