Mixed Models : Theory and Applications with R
[BOOK DESCRIPTION]
Praise for the First Edition "This book will serve to greatly complement the growing number of texts dealing with mixed models, and I highly recommend including it in one's personal library." -Journal of the American Statistical Association Mixed modeling is a crucial area of statistics, enabling the analysis of clustered and longitudinal data. Mixed Models: Theory and Applications with R, Second Edition fills a gap in existing literature between mathematical and applied statistical books by presenting a powerful examination of mixed model theory and application with special attention given to the implementation in R. The new edition provides in-depth mathematical coverage of mixed models' statistical properties and numerical algorithms, as well as nontraditional applications, such as regrowth curves, shapes, and images. The book features the latest topics in statistics including modeling of complex clustered or longitudinal data, modeling data with multiple sources of variation, modeling biological variety and heterogeneity, Healthy Akaike Information Criterion (HAIC), parameter multidimensionality, and statistics of image processing.Mixed Models: Theory and Applications with R, Second Edition features unique applications of mixed model methodology, as well as: * Comprehensive theoretical discussions illustrated by examples and figures * Over 300 exercises, end-of-section problems, updated data sets, and R subroutines * Problems and extended projects requiring simulations in R intended to reinforce material * Summaries of major results and general points of discussion at the end of each chapter * Open problems in mixed modeling methodology, which can be used as the basis for research or PhD dissertations Ideal for graduate-level courses in mixed statistical modeling, the book is also an excellent reference for professionals in a range of fields, including cancer research, computer science, and engineering.
[TABLE OF CONTENTS]
Preface xvii
Preface to the Second Edition xix
R Software and Functions xx
Data Sets xxii
Open Problems in Mixed Models xxiii
1 Introduction: Why Mixed Models? 1 (40)
1.1 Mixed effects for clustered data 2 (2)
1.2 ANOVA, variance components, and the 4 (2)
mixed model
1.3 Other special cases of the mixed 6 (1)
effects model
1.4 Compromise between Bayesian and 7 (2)
frequentist approaches
1.5 Penalized likelihood and mixed effects 9 (2)
1.6 Healthy Akaike information criterion 11 (2)
1.7 Penalized smoothing 13 (3)
1.8 Penalized polynomial fitting 16 (2)
1.9 Restraining parameters, or what to eat 18 (2)
1.10 Ill-posed problems, Tikhonov 20 (3)
regularization, and mixed effects
1.11 Computerized tomography and linear 23 (3)
image reconstruction
1.12 GLMM for PET 26 (3)
1.13 Maple leaf shape analysis 29 (2)
1.14 DNA Western blot analysis 31 (2)
1.15 Where does the wind blow? 33 (3)
1.16 Software and books 36 (1)
1.17 Summary points 37 (4)
2 MLE for the LME Model 41 (76)
2.1 Example: weight versus height 42 (3)
2.1.1 The first R script 43 (2)
2.2 The model and log-likelihood functions 45 (15)
2.2.1 The model 45 (3)
2.2.2 Log-likelihood functions 48 (1)
2.2.3 Dimension-reduction formulas 49 (4)
2.2.4 Profile log-likelihood functions 53 (2)
2.2.5 Dimension-reduction GLS estimate 55 (1)
2.2.6 Restricted maximum likelihood 56 (3)
2.2.7 Weight versus height (continued) 59 (1)
2.3 Balanced random-coefficient model 60 (4)
2.4 LME model with random intercepts 64 (8)
2.4.1 Balanced random-intercept model 67 (4)
2.4.2 How random effect affects the 71 (1)
variance of MLE
2.5 Criterion for MLE existence 72 (2)
2.6 Criterion for the positive 74 (3)
definiteness of matrix D
2.6.1 Example of an invalid LME model 75 (2)
2.7 Pre-estimation bounds for variance 77 (2)
parameters
2.8 Maximization algorithms 79 (2)
2.9 Derivatives of the log-likelihood 81 (1)
function
2.10 Newton-Raphson algorithm 82 (3)
2.11 Fisher scoring algorithm 85 (3)
2.11.1 Simplified FS algorithm 86 (1)
2.11.2 Empirical FS algorithm 86 (1)
2.11.3 Variance-profile FS algorithm 87 (1)
2.12 EM algorithm 88 (5)
2.12.1 Fixed-point algorithm 92 (1)
2.13 Starting point 93 (2)
2.13.1 FS starting point 93 (1)
2.13.2 FP starting point 94 (1)
2.14 Algorithms for restricted MLE 95 (1)
2.14.1 Fisher scoring algorithm 95 (1)
2.14.2 EM algorithm 96 (1)
2.15 Optimization on nonnegative definite 96 (11)
matrices
2.15.1 How often can one hit the 97 (1)
boundary?
2.15.2 Allow matrix D to be not 98 (5)
nonnegative definite
2.15.3 Force matrix D to stay 103 (1)
nonnegative definite
2.15.4 Matrix D reparameterization 104 (1)
2.15.5 Criteria for convergence 105 (2)
2.16 lmeFS and lme in R 107 (4)
2.17 Appendix: proof of the existence of 111 (3)
MLE
2.18 Summary points 114 (3)
3 Statistical Properties of the LME Model 117 (68)
3.1 Introduction 117 (1)
3.2 Identifiability of the LME model 117 (3)
3.2.1 Linear regression with random 119 (1)
coefficients
3.3 Information matrix for variance 120 (11)
parameters
3.3.1 Efficiency of variance parameters 129 (2)
for balanced data
3.4 Profile-likelihood confidence 131 (2)
intervals
3.5 Statistical testing of the presence 133 (4)
of random effects
3.6 Statistical properties of MLE 137 (8)
3.6.1 Small-sample properties 137 (3)
3.6.2 Large-sample properties 140 (4)
3.6.3 ML and RML are asymptotically 144 (1)
equivalent
3.7 Estimation of random effects 145 (6)
3.7.1 Implementation in R 148 (3)
3.8 Hypothesis and membership testing 151 (3)
3.8.1 Membership test 152 (2)
3.9 Ignoring random effects 154 (3)
3.10 MINQUE for variance parameters 157 (9)
3.10.1 Example: linear regression 158 (2)
3.10.2 MINQUE for σ 160 (2)
3.10.3 MINQUE for D* 162 (3)
3.10.4 Linear model with random 165 (1)
intercepts
3.10.5 MINQUE for the balanced model 165 (1)
3.10.6 ImevarMINQUE function 166 (1)
3.11 Method of moments 166 (5)
3.11.1 ImevarMM function 171 (1)
3.12 Variance least squares estimator 171 (5)
3.12.1 Unbiased VLS estimator 173 (1)
3.12.2 Linear model with random 174 (1)
intercepts
3.12.3 Balanced desig 174 (1)
3.12.4 VLS as the first iteration of ML 175 (1)
3.12.5 ImevarUVLS function 175 (1)
3.13 Projection on D+ space 176 (1)
3.14 Comparison of the variance parameter 176 (4)
estimation
3.14.1 Imesim function 179 (1)
3.15 Asymptotically efficient estimation 180 (1)
for β
3.16 Summary points 181 (4)
4 Growth Curve Model and Generalizations 185 (60)
4.1 Linear growth curve model 185 (16)
4.1.1 Known matrix D 187 (2)
4.1.2 Maximum likelihood estimation 189 (3)
4.1.3 Method of moments for variance 192 (4)
parameters
4.1.4 Two-stage estimation 196 (1)
4.1.5 Special growth curve models 196 (4)
4.1.6 Unbiasedness and efficient 200 (1)
estimation for β
4.2 General linear growth curve model 201 (18)
4.2.1 Example: Calcium supplementation 202 (2)
for bone gain
4.2.2 Variance parameters are known 204 (3)
4.2.3 Balanced model 207 (1)
4.2.4 Likelihood-based estimation 208 (5)
4.2.5 MM estimator for variance 213 (1)
parameters
4.2.6 Two-stage estimator and 214 (1)
asymptotic properties
4.2.7 Analysis of misspecification 215 (4)
4.3 Linear model with linear covariance 219 (14)
structure
4.3.1 Method of maximum likelihood 220 (2)
4.3.2 Variance least squares 222 (1)
4.3.3 Statistical properties 223 (1)
4.3.4 LME model for longitudinal 224 (5)
autocorrelated data
4.3.5 Multidimensional LME model 229 (4)
4.4 Robust linear mixed effects model 233 (8)
4.4.1 Robust estimation of the location 235 (3)
parameter with estimated σ and c
4.4.2 Robust linear regression with 238 (1)
estimated threshold
4.4.3 Robust LME model 239 (1)
4.4.4 Alternative robust functions 239 (1)
4.4.5 Robust random effect model 240 (1)
4.5 Appendix: derivation of the MM 241 (1)
estimator
4.6 Summary points 242 (3)
5 Meta-analysis Model 245 (46)
5.1 Simple meta-analysis model 246 (27)
5.1.1 Estimation of random effects 248 (1)
5.1.2 Maximum likelihood estimation 248 (5)
5.1.3 Quadratic unbiased estimation for 253 (7)
Σ
5.1.4 Statistical inference 260 (6)
5.1.5 Robust/median meta-analysis 266 (5)
5.1.6 Random effect coefficient of 271 (2)
determination
5.2 Meta-analysis model with covariates 273 (5)
5.2.1 Maximum likelihood estimation 274 (3)
5.2.2 Quadratic unbiased estimation for 277 (1)
Σ
5.2.3 Hypothesis testing 278 (1)
5.3 Multivariate meta-analysis model 278 (11)
5.3.1 The model 280 (3)
5.3.2 Maximum likelihood estimation 283 (2)
5.3.3 Quadratic estimation of the 285 (3)
heterogeneity matrix
5.3.4 Test for homogeneity 288 (1)
5.4 Summary points 289 (2)
6 Nonlinear Marginal Model 291 (40)
6.1 Fixed matrix of random effects 292 (13)
6.1.1 Log-likelihood function 293 (2)
6.1.2 nls function in R 295 (1)
6.1.3 Computational issues of nonlinear 296 (1)
least squares
6.1.4 Distribution-free estimation 297 (1)
6.1.5 Testing for the presence of 298 (1)
random effects
6.1.6 Asymptotic properties 298 (1)
6.1.7 Example: log-Gompertz growth curve 299 (6)
6.2 Varied matrix of random effects 305 (11)
6.2.1 Maximum likelihood estimation 305 (3)
6.2.2 Distribution-free variance 308 (1)
parameter estimation
6.2.3 GEE and iteratively reweighted 309 (1)
least squares
6.2.4 Example: logistic curve with 310 (6)
random asymptote
6.3 Three types of nonlinear marginal 316 (5)
models
6.3.1 Type I nonlinear marginal model 317 (2)
6.3.2 Type II nonlinear marginal model 319 (1)
6.3.3 Type III nonlinear marginal model 319 (1)
6.3.4 Asymptotic properties under 320 (1)
distribution misspecification
6.4 Total generalized estimating 321 (7)
equations approach
6.4.1 Robust feature of total GEE 323 (1)
6.4.2 Expected Newton-Raphson algorithm 323 (1)
for total GEE
6.4.3 Total GEE for the mixed effects 324 (1)
model
6.4.4 Total GEE for the LME model 324 (1)
6.4.5 Example (continued): log-Gompertz 325 (1)
curve
6.4.6 Photodynamic tumor therapy 326 (2)
6.5 Summary points 328 (3)
7 Generalized Linear Mixed Models 331 (102)
7.1 Regression models for binary data 332 (23)
7.1.1 Approximate relationship between 336 (2)
logit and probit
7.1.2 Computation of the 338 (12)
logistic-normal integral
7.1.3 Gauss-Hermite numerical 350 (2)
quadrature for multidimensional
integrals in R
7.1.4 Log-likelihood and its numerical 352 (1)
properties
7.1.5 Unit step algorithm 353 (2)
7.2 Binary model with subject-specific 355 (7)
intercept
7.2.1 Consequences of ignoring a random 357 (1)
effect
7.2.2 ML logistic regression with a 358 (1)
fixed subject-specific intercept
7.2.3 Conditional logistic regression 359 (3)
7.3 Logistic regression with random 362 (20)
intercept
7.3.1 Maximum likelihood 362 (6)
7.3.2 Fixed sample likelihood 368 (3)
approximation
7.3.3 Quadratic approximation 371 (1)
7.3.4 Laplace approximation to the 371 (3)
likelihood
7.3.5 VARLINK estimation 374 (2)
7.3.6 Beta-binomial model 376 (2)
7.3.7 Statistical test of homogeneity 378 (3)
7.3.8 Asymptotic properties 381 (1)
7.4 Probit model with random intercept 382 (4)
7.4.1 Laplace and PQL approximations 382 (1)
7.4.2 VARLINK estimation 383 (1)
7.4.3 Heckman method for the probit 383 (1)
model
7.4.4 Generalized estimating equations 384 (2)
approach
7.4.5 Implementation in R 386 (1)
7.5 Poisson model with random intercept 386 (15)
7.5.1 Poisson regression for count data 387 (1)
7.5.2 Clustered count data 388 (1)
7.5.3 Fixed intercepts 389 (1)
7.5.4 Conditional Poisson regression 390 (1)
7.5.5 Negative binomial regression 391 (3)
7.5.6 Normally distributed intercepts 394 (2)
7.5.7 Exact GEE for any distribution 396 (1)
7.5.8 Exact GEE for balanced count data 397 (1)
7.5.9 Heckman method for the Poisson 398 (1)
model
7.5.10 Tests for overdispersion 399 (1)
7.5.11 Implementation in R 400 (1)
7.6 Random intercept model: overview 401 (1)
7.7 Mixed models with multiple random 402 (10)
effects
7.7.1 Multivariate Laplace approximation 403 (1)
7.7.2 Logistic regression 403 (4)
7.7.3 Probit regression 407 (1)
7.7.4 Poisson regression 408 (2)
7.7.5 Homogeneity tests 410 (2)
7.8 GLMM and simulation methods 412 (4)
7.8.1 General form of GLMM via the 412 (1)
exponential family
7.8.2 Monte Carlo for ML 413 (1)
7.8.3 Fixed sample likelihood approach 413 (3)
7.9 GEE for clustered marginal GLM 416 (8)
7.9.1 Variance least squares 418 (2)
7.9.2 Limitations of the GEE approach 420 (2)
7.9.3 Marginal or conditional model? 422 (1)
7.9.4 Implementation in R 423 (1)
7.10 Criteria for MLE existence for a 424 (5)
binary model
7.11 Summary points 429 (4)
8 Nonlinear Mixed Effects Model 433 (54)
8.1 Introduction 433 (1)
8.2 The model 434 (3)
8.3 Example: height of girls and boys 437 (2)
8.4 Maximum likelihood estimation 439 (3)
8.5 Two-stage estimator 442 (6)
8.5.1 Maximum likelihood estimation 445 (1)
8.5.2 Method of moments 445 (1)
8.5.3 Disadvantage of two-stage 446 (1)
estimation
8.5.4 Further discussion 446 (1)
8.5.5 Two-stage method in the presence 447 (1)
of a common parameter
8.6 First-order approximation 448 (2)
8.6.1 GEE and MLE 448 (1)
8.6.2 Method of moments and VLS 449 (1)
8.7 Lindstrom-Bates estimator 450 (6)
8.7.1 What if matrix D is not positive 452 (1)
definite?
8.7.2 Relation to the two-stage 452 (1)
estimator
8.7.3 Computational aspects of 453 (1)
penalized least squares
8.7.4 Implementation in R: the function 454 (2)
nlme
8.8 Likelihood approximations 456 (3)
8.8.1 Linear approximation of the 456 (1)
likelihood at zero
8.8.2 Laplace and PQL approximations 457 (2)
8.9 One-parameter exponential model 459 (7)
8.9.1 Maximum likelihood estimator 459 (1)
8.9.2 First-order approximation 460 (1)
8.9.3 Two-stage estimator 461 (2)
8.9.4 Lindstrom-Bates estimator 463 (3)
8.10 Asymptotic equivalence of the TS and 466 (2)
LB estimators
8.11 Bias-corrected two-stage estimator 468 (2)
8.12 Distribution misspecification 470 (3)
8.13 Partially nonlinear marginal mixed 473 (1)
model
8.14 Fixed sample likelihood approach 474 (2)
8.14.1 Example: one-parameter 475 (1)
exponential model
8.15 Estimation of random effects and 476 (2)
hypothesis testing
8.15.1 Estimation of the random effects 476 (1)
8.15.2 Hypothesis testing for the NLME 477 (1)
model
8.16 Example (continued) 478 (2)
8.17 Practical recommendations 480 (1)
8.18 Appendix: Proof of theorem on 481 (3)
equivalence
8.19 Summary points 484 (3)
9 Diagnostics and Influence Analysis 487 (52)
9.1 Introduction 487 (1)
9.2 Influence analysis for linear 488 (3)
regression
9.3 The idea of infinitesimal influence 491 (2)
9.3.1 Data influence 491 (1)
9.3.2 Model influence 492 (1)
9.4 Linear regression model 493 (17)
9.4.1 Influence of the dependent 494 (1)
variable
9.4.2 Influence of the continuous 495 (2)
explanatory variable
9.4.3 Influence of the binary 497 (1)
explanatory variable
9.4.4 Influence on the predicted value 497 (1)
9.4.5 Case or group deletion 498 (2)
9.4.6 R code 500 (1)
9.4.7 Influence on regression 501 (2)
characteristics
9.4.8 Example 1: Women's body fat 503 (4)
9.4.9 Example 2: gypsy moth study 507 (3)
9.5 Nonlinear regression model 510 (5)
9.5.1 Influence of the dependent 510 (1)
variable on the LSE
9.5.2 Influence of the explanatory 510 (1)
variable on the LSE
9.5.3 Influence on the predicted value 511 (1)
9.5.4 Influence of case deletion 511 (1)
9.5.5 Example 3: logistic growth curve 512 (3)
model
9.6 Logistic regression for binary outcome 515 (9)
9.6.1 Influence of the covariate on the 516 (1)
MLE
9.6.2 Influence on the predicted 516 (1)
probability
9.6.3 Influence of the case deletion on 517 (1)
the MLE
9.6.4 Sensitivity to misclassification 517 (5)
9.6.5 Example: Finney data 522 (2)
9.7 Influence of correlation structure 524 (1)
9.8 Influence of measurement error 525 (3)
9.9 Influence analysis for the LME model 528 (6)
9.9.1 Example: Weight versus height 532 (2)
9.10 Appendix: MLE derivative with 534 (1)
respect to Σ
9.11 Summary points 535 (4)
10 Tumor Regrowth Curves 539 (38)
10.1 Survival curves 541 (2)
10.2 Double-exponential regrowth curve 543 (14)
10.2.1 Time to regrowth, TR 546 (1)
10.2.2 Time to reach specific tumor 547 (1)
volume, T*
10.2.3 Doubling time, TD 547 (1)
10.2.4 Statistical model for regrowth 548 (1)
10.2.5 Variance estimation for tumor 549 (1)
regrowth outcomes
10.2.6 Starting values 550 (1)
10.2.7 Example: chemotherapy treatment 551 (6)
comparison
10.3 Exponential growth with fixed 557 (6)
regrowth time
10.3.1 Statistical hypothesis testing 558 (1)
10.3.2 Synergistic or supra-additive 558 (1)
effect
10.3.3 Example: combination of 559 (4)
treatments
10.4 General regrowth curve 563 (1)
10.5 Double-exponential transient 564 (7)
regrowth curve
10.5.1 Example: treatment of cellular 570 (1)
spheroids
10.6 Gompertz transient regrowth curve 571 (3)
10.6.1 Example: tumor treated in mice 572 (2)
10.7 Summary points 574 (3)
11 Statistical Analysis of Shape 577 (30)
11.1 Introduction 577 (2)
11.2 Statistical analysis of random 579 (3)
triangles
11.3 Face recognition 582 (1)
11.4 Scale-irrelevant shape model 583 (4)
11.4.1 Random effects scale-irrelevant 585 (1)
shape model
11.4.2 Scale-irrelevant shape model on 586 (1)
the log scale
11.4.3 Fixed or random size? 587 (1)
11.5 Gorilla vertebrae analysis 587 (2)
11.6 Procrustes estimation of the mean 589 (7)
shape
11.6.1 Polygon estimation 592 (1)
11.6.2 Generalized Procrustes model 592 (1)
11.6.3 Random effects shape model 593 (1)
11.6.4 Random or fixed (Procrustes) 594 (1)
effects model?
11.6.5 Maple leaf analysis 594 (2)
11.7 Fourier descriptor analysis 596 (9)
11.7.1 Analysis of a star shape 596 (6)
11.7.2 Random Fourier descriptor 602 (2)
analysis
11.7.3 Potato project 604 (1)
11.8 Summary points 605 (2)
12 Statistical Image Analysis 607 (54)
12.1 Introduction 607 (3)
12.1.1 What is a digital image? 608 (1)
12.1.2 Image arithmetic 609 (1)
12.1.3 Ensemble and repeated 609 (1)
measurements
12.1.4 Image and spatial statistics 610 (1)
12.1.5 Structured and unstructured 610 (1)
images
12.2 Testing for uniform lighting 610 (4)
12.2.1 Estimating light direction and 612 (2)
position
12.3 Kolmogorov-Smirnov image comparison 614 (4)
12.3.1 Kolmogorov-Smirnov test for 614 (1)
image comparison
12.3.2 Example: histological analysis 615 (3)
of cancer treatment
12.4 Multinomial statistical model for 618 (3)
images
12.4.1 Multinomial image comparison 620 (1)
12.5 Image entropy 621 (4)
12.5.1 Reduction of a gray image to 623 (1)
binary
12.5.2 Entropy of a gray image and 623 (2)
histogram equalization
12.6 Ensemble of unstructured images 625 (13)
12.6.1 Fixed-shift model 626 (2)
12.6.2 Random-shift model 628 (3)
12.6.3 Mixed model for gray images 631 (2)
12.6.4 Two-stage estimation 633 (2)
12.6.5 Schizophrenia MRI analysis 635 (3)
12.7 Image alignment and registration 638 (12)
12.7.1 Affine image registration 641 (1)
12.7.2 Weighted sum of squares 642 (1)
12.7.3 Nonlinear transformations 643 (1)
12.7.4 Random registration 643 (1)
12.7.5 Linear image interpolation 644 (1)
12.7.6 Computational aspects 645 (1)
12.7.7 Derivative-free algorithm for 646 (1)
image registration
12.7.8 Example: clock alignment 647 (3)
12.8 Ensemble of structured images 650 (2)
12.8.1 Fixed affine transformations 650 (1)
12.8.2 Random affine transformations 651 (1)
12.9 Modeling spatial correlation 652 (6)
12.9.1 Toeplitz correlation structure 654 (2)
12.9.2 Simultaneous estimation of 656 (2)
variance and transform parameters
12.10 Summary points 658 (3)
13 Appendix: Useful Facts and Formulas 661 (20)
13.1 Basic facts of asymptotic theory 661 (7)
13.1.1 Central Limit Theorem 661 (1)
13.1.2 Generalized Slutsky theorem 662 (2)
13.1.3 Pseudo-maximum likelihood 664 (1)
13.1.4 Estimating equations approach 665 (2)
and the sandwich formula
13.1.5 Generalized estimating equations 667 (1)
approach
13.2 Some formulas of matrix algebra 668 (4)
13.2.1 Some matrix identities 668 (1)
13.2.2 Formulas for generalized matrix 668 (1)
inverse
13.2.3 Vec and vech functions; 669 (1)
duplication matrix
13.2.4 Matrix differentiation 670 (2)
13.3 Basic facts of optimization theory 672 (9)
13.3.1 Criteria for unimodality 673 (1)
13.3.2 Criteria for global optimum 674 (1)
13.3.3 Criteria for minimum existence 674 (1)
13.3.4 Optimization algorithms in 675 (3)
statistics
13.3.5 Necessary condition for 678 (3)
optimization and criteria for
convergence
References 681 (30)
Index 711
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