Mixed models: theory and applications
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley-Interscience
2004
|
| Series: | Wiley series in probability and statistics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. [665]-695) and index |
| Physical Description: | XVIII, 704 S. Ill., graph. Darst. 25 cm |
| ISBN: | 0471601616 |
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| 100 | 1 | |a Demidenko, Eugene |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Mixed models |b theory and applications |c Eugene Demidenko |
| 264 | 1 | |a Hoboken, NJ |b Wiley-Interscience |c 2004 | |
| 300 | |a XVIII, 704 S. |b Ill., graph. Darst. |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Wiley series in probability and statistics | |
| 500 | |a Includes bibliographical references (p. [665]-695) and index | ||
| 650 | 4 | |a Analyse de variance | |
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| 650 | 4 | |a aAnalysis of variance | |
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| _version_ | 1804133251042770944 |
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| adam_text | Contents
Preface
xvii
1
Introduction:
Why Mixed Models?
1
1.1
Mixed effects for clustered data
................ 2
1.2
ANOVA, variance components, and the mixed model
.... 5
1.3
Other special cases of the mixed effects model
........ 7
1.4
A compromise between Bayesian and frequentist approaches
8
1.5
Penalized likelihood and mixed effects
............ 11
1.6
Healthy
Akaiké
information criterion
............. 13
1.7
Penalized smoothing
...................... 15
1.8
Penalized polynomial fitting
.................. 18
1.9
Restraining parameters, or what to eat
............ 19
1.10
IU-posed problems, Tikhonov regularization, and mixed effects
22
1.11
Computerized tomography and linear image reconstruction
26
1.12
GLMM for PET
........................ 30
1.13
Maple shape leaf analysis
................... 33
1.14 DNA
Western blot analysis
.................. 35
1.15
Where does the wind blow?
.................. 38
1.16
Commercial software and books
................ 40
1.17
Summary points
........................ 42
2
MLE for LME Model
45
2.1
Example: Height versus weight
................ 46
2.2
The model and log-likelihood functions
............ 48
viii Contents
2.2.1
The model
....................... 48
2.2.2
Log-likelihood functions
................ 51
2.2.3
Dimension-reduction formulas
............. 52
2.2.4
Profile log-likelihood functions
............ 56
2.2.5
Restricted maximum likelihood
............ 58
2.3
Balanced random-coefficient model
.............. 61
2.4
LME model with random intercepts
............. 66
2.4.1
Balanced random-intercept model
.......... 69
2.4.2
How random effect affects the variance of MLE
... 72
2.5
Criterion for the MLE existence
................ 74
2.6
Criterion for positive definiteness of matrix
D
........ 76
2.6.1
Example of an invalid LME model
.......... 77
2.7
Preestimation bounds for variance parameters
........ 78
2.8
Maximization algorithms
.................... 80
2.9
Derivatives of the log-likelihood function
........... 82
2.10
Newton-Raphson algorithm
.................. 83
2.11
Fisher scoring algorithm
.................... 86
2.11.1
Simplified FS algorithm
................ 87
2.11.2
Empirical FS algorithm
................ 88
2.11.3
Variance-profile FS algorithm
............. 88
2.11.4
FUl-profile FS algorithm
................ 89
2.12
EM algorithm
.......................... 90
2.12.1
Fixed-point algorithm
................. 94
2.13
Starting point
.......................... 96
2.13.1
FS starting point
.................... 96
2.13.2
FP starting points
................... 97
2.14
Algorithms for restricted MLE
................ 98
2.14.1
Fisher scoring algorithm
................ 98
2.14.2
EM algorithm
...................... 98
2.15
Optimization on
nonnegative
definite matrices
........ 99
2.15.1
How often can one hit the boundary?
........ 100
2.15.2
Allow matrix
D
to be negative definite
........ 101
2.15.3
Force matrix
D
to stay
nonnegative
definite
..... 106
2.15.4
Matrix
D
reparameterization
............. 108
2.15.5
Criteria for convergence
................ 108
2.16
Appendix: Proof of the MLE existence
............
Ill
2.17
Summary points
........................ 114
3
Statistical Properties of the LME Model
117
3.1
Introduction
........................... 117
3.2
Identifiability of the LME model
............... 117
3.2.1
Linear regression with random coefficients
...... 119
3.3
Information matrix for variance parameters
......... 121
3.3.1
Efficiency of variance parameters for balanced data
. 129
3.4
Profile-likelihood confidence intervals
............. 131
Contents ix
3.5
Statistical testing of the presence of random effects
..... 134
3.6
Statistical properties of MLE
................. 139
3.6.1
Small-sample properties
................ 139
3.6.2
Large-sample properties
................ 141
3.6.3
ML and RML are asymptotically equivalent
..... 146
3.7
Estimation of random effects
................. 147
3.8
Hypothesis and membership testing
............. 150
3.8.1
Membership test
.................... 151
3.9
Ignoring random effects
.................... 154
3.10
MINQUE for variance parameters
............... 157
3.10.1
Linear regression
.................... 157
3.10.2
MINQUE for
σ2
.................... 159
3.10.3
MINQUE for D,
.................... 162
3.10.4
Linear model with random intercepts
......... 165
3.10.5
MINQUE for the balanced model
........... 166
3.11
Method of moments
...................... 166
3.12
Variance least squares estimator
............... 171
3.12.1
Unbiased VLS estimator
................ 173
3.12.2
Linear model with random intercepts
......... 174
3.12.3
Balanced design
.................... 174
3.12.4
VLS as the first iteration of ML
........... 176
3.13
Projection on
Ђ+
space
.................... 176
3.14
Comparison of the variance parameter estimation
...... 176
3.15
Asymptotically efficient estimation for
β
........... 178
3.16
Summary points
........................ 180
Growth Curve Model and Generalizations
183
4.1
Linear growth curve model
.................. 183
4.1.1
Known matrix
D
.................... 185
4.1.2
Maximum likelihood estimation
............ 187
4.1.3
Method of moments for variance parameters
..... 191
4.1.4
Two-stage estimation
................. 196
4.1.5
Special growth curve models
............. 196
4.1.6
Unbiasedness and efficient estimation for
β
..... 200
4.2
General linear growth curve model
.............. 201
4.2.1
Example: Calcium supplementation for bone gain
. . 202
4.2.2
Variance parameters are known
............ 205
4.2.3
Balanced model
..................... 207
4.2.4
Likelihood-based estimation
.............. 208
4.2.5
MM estimator for variance parameters
........ 214
4.2.6
Two-stage estimator and asymptotic properties
. . . 215
4.2.7
Analysis of misspecification
.............. 215
4.3
Linear model with linear covariance structure
........ 220
4.3.1
Method of maximum likelihood
............ 221
4.3.2
Variance least squares
................. 223
Contents
4.3.3
Statistical properties
.................. 224
4.3.4
LME model for longitudinal autocorrelated data
. . 225
4.3.5
Multidimensional LME model
............. 230
4.4
Robust linear mixed effects model
.............. 235
4.4.1
Robust estimation of the location parameter with es¬
timated
σ
and
с
.................... 237
4.4.2
Robust linear regression with estimated threshold
. . 241
4.4.3
Robust LME model
.................. 241
4.4.4
Alternative robust functions
.............. 242
4.4.5
Robust random effect model
.............. 242
4.5
Appendix: Derivation of the MM estimator
......... 243
4.6
Summary points
........................ 244
Meta-analysis Model
247
5.1
Simple meta-analysis model
.................. 248
5.1.1
Maximum likelihood estimation
............ 250
5.1.2
Quadratic unbiased estimation for
σ2
........ 255
5.1.3
Statistical inference
.................. 262
5.1.4
Robust/median meta-analysis
............. 268
5.2
Meta-analysis model with covariates
............. 272
5.2.1
Maximum likelihood estimation
............ 273
5.2.2
Quadratic unbiased estimation for
σ2
........ 276
5.2.3
Hypothesis testing
................... 277
5.3
Multivariate meta-analysis model
............... 278
5.3.1
The model
....................... 281
5.3.2
Maximum likelihood estimation
............ 282
5.3.3
Quadratic estimation of the heterogeneity matrix
. . 285
5.3.4
Test for homogeneity
.................. 289
5.4
Summary points
........................ 289
Nonlinear Marginal Model
291
6.1
Fixed matrix of random effects
................ 292
6.1.1
Log-likelihood function
................. 293
6.1.2
Computational issues of nonlinear least squares
. . . 295
6.1.3
Distribution-free estimation
.............. 296
6.1.4
Testing for the presence of random effects
...... 297
6.1.5
Asymptotic properties
................. 298
6.1.6
Example: Log-Gompertz growth curve
........ 298
6.2
Varied matrix of random effects
................ 304
6.2.1
Maximum likelihood estimation
............ 304
6.2.2
Distribution-free variance parameter estimation
. . . 307
6.2.3
GEE and iteratively reweighted least squares
.... 307
6.2.4
Example: Logistic curve with random asymptote
. . 309
6.3
Three types of nonlinear marginal models
.......... 315
6.3.1
Type I nonlinear marginal model
........... 317
Contents xi
6.3.2
Type II nonlinear marginal model
.......... 318
6.3.3
Type HI nonlinear marginal model
.......... 319
6.3.4
Asymptotic properties under distribution misspecifi-
cation
.......................... 320
6.4
Total generalized estimating equations approach
...... 320
6.4.1
A robust feature of total GEE
............ 323
6.4.2
Expected Newton-Raphson algorithm for total GEE
323
6.4.3
Total GEE for mixed effects model
.......... 324
6.4.4
Total GEE for the LME model
............ 324
6.4.5
Example (continued): Log-Gompertz curve
..... 326
6.5
Summary points
........................ 327
Generalized Linear Mixed Models
329
7.1
Regression models for binary data
.............. 330
7.1.1
Approximate relationship between logit and
probit
. 334
7.1.2
Computation of the logistic-normal integral
..... 337
7.1.3
Log-likelihood and its numerical properties
..... 349
7.1.4
Unit step algorithm
.................. 350
7.2
Binary model with subject-specific intercept
......... 351
7.2.1
Consequences of ignoring a random effect
...... 354
7.2.2
ML logistic regression with a fixed subject-specific
intercept
........................ 355
7.2.3
Conditional logistic regression
............. 356
7.3
Logistic regression with random intercept
.......... 358
7.3.1
Maximum likelihood
.................. 359
7.3.2
Fixed sample likelihood approximation
........ 365
7.3.3
Quadratic approximation
............... 367
7.3.4
Laplace approximation to the likelihood
....... 368
7.3.5
VARLINK estimation
................. 371
7.3.6
Beta-binomial model
.................. 372
7.3.7
Statistical test of homogeneity
............ 374
7.3.8
Asymptotic properties
................. 378
7.4
Probit
model with random intercept
............. 378
7.4.1
Laplace and PQL approximations
........... 379
7.4.2
VARLINK estimation
................. 380
7.4.3
Heckman method for
probit
model
.......... 380
7.4.4
Generalized estimating equations approach
..... 380
7.5
Poisson
model with random intercept
............. 382
7.5.1
Poisson
regression for count data
........... 383
7.5.2
Clustered count data
.................. 384
7.5.3
Fixed intercepts
.................... 385
7.5.4
Conditional
Poisson
regression
............ 386
7.5.5
Negative binomial regression
............. 387
7.5.6
Normally distributed intercepts
............ 392
7.5.7
Exact GEE for any distribution
............ 393
xii Contents
7.5.8
Exact GEE for balanced count data
......... 394
7.5.9
Heekman
method for the
Poisson
model
....... 396
7.5.10
Tests for overdispersion
................ 397
7.6
Random intercept model: overview
.............. 398
7.7
Mixed models with multiple random effects
......... 399
7.7.1
Multivariate Laplace approximation
......... 400
7.7.2
Logistic regression
................... 400
7.7.3
Probit
regression
.................... 405
7.7.4
Poisson
regression
................... 406
7.8
Homogeneity tests
....................... 408
7.9
GLMM and simulation methods
................ 409
7.9.1
General form of GLMM via the exponential family
. 409
7.9.2
Monte Carlo for ML
.................. 410
7.9.3
Fixed sample likelihood approach
........... 411
7.10
GEE for clustered marginal GLM
............... 414
7.10.1
Variance least squares
................. 416
7.10.2
Limitations of the GEE approach
........... 418
7.10.3
Marginal or conditional model?
............ 420
7.11
Criteria for MLE existence for binary model
......... 422
7.12
Summary points
........................ 427
8
Nonlinear Mixed Effects Model
431
8.1
Introduction
........................... 431
8.2
The model
............................ 432
8.3
Example: Height of girls and boys
.............. 435
8.4
Maximum likelihood estimation
................ 438
8.5
Two-stage estimator
...................... 441
8.5.1
Maximum likelihood estimation
............ 442
8.5.2
Method of moments
.................. 443
8.5.3
Drawback of two-stage estimation
.......... 444
8.5.4
Further discussion
................... 444
8.5.5
Two-stage method in the presence of a common pa¬
rameter
......................... 445
8.6
First-order approximation
................... 445
8.6.1
GEE and MLE
..................... 446
8.6.2
Method of moments and VLS
............. 446
8.7
Lindstrom-Bates estimator
.................. 447
8.7.1
What if matrix
D
is not positive definite?
...... 449
8.7.2
Relation to the two-stage estimator
.......... 450
8.7.3
Computational aspects of penalized least squares
. . 451
8.8
Likelihood approximations
................... 452
8.8.1
Likelihood approximation at zero
........... 452
8.8.2
Laplace and PQL approximations
........... 453
8.9
One-parameter exponential model
.............. 455
8.9.1
Maximum likelihood estimator
............455
Contents xiii
8.9.2
First-order approximation
............... 456
8.9.3
Two-stage estimator
.................. 458
8.9.4
Lindstrom-Bates estimator
.............. 460
8.10
Asymptotic equivalence of the TS and LB estimators
.... 462
8.11
Bias-corrected two-stage estimator
.............. 463
8.12
Distribution misspecification
.................. 465
8.13
Partially nonlinear marginal mixed model
.......... 468
8.14
Fixed sample likelihood approach
............... 470
8.15
Estimation of random effects and hypothesis testing
.... 471
8.15.1
Estimation of the random effects
........... 471
8.15.2
Hypothesis testing for NLME model
......... 472
8.16
Example (continued)
...................... 473
8.17
Practical recommendations
.................. 474
8.18
Appendix: Proof of theorem on equivalence
......... 475
8.19
Summary points
........................ 479
Diagnostics and Influence Analysis
481
9.1
Introduction
........................... 481
9.2
Influence analysis for linear regression
............ 483
9.3
The idea of infinitesimal influence
............... 485
9.3.1
Data influence
..................... 486
9.3.2
Model influence
..................... 486
9.4
Linear regression model
.................... 487
9.4.1
Influence of the dependent variable
.......... 489
9.4.2
Influence of the continuous explanatory variable
. . 489
9.4.3
Influence of the binary explanatory variable
..... 491
9.4.4
Influence on the predicted value
............ 492
9.4.5
Case or group deletion
................. 493
9.4.6
Influence on regression characteristics
........ 495
9.4.7
Example
1:
Women s body fat
............ 496
9.4.8
Example
2:
Gypsy moth study
............ 501
9.5
Nonlinear regression model
.................. 503
9.5.1
Influence of the dependent variable on the
LSE
. . . 504
9.5.2
Influence of the explanatory variable on the
LSE
. . 504
9.5.3
Influence on the predicted value
............ 504
9.5.4
Influence of case deletion
............... 505
9.5.5
Example
3:
Logistic growth curve model
....... 505
9.6
Logistic regression
....................... 508
9.6.1
Influence of the covariate on the MLE
........ 509
9.6.2
Influence on the predicted probability
........ 509
9.6.3
Influence of the case deletion on the MLE
...... 510
9.6.4
Sensitivity to misclassification
............. 511
9.6.5
Example: Finney data
................. 516
9.7
Influence of correlation structure
............... 518
9.8
Influence of measurement error
................ 518
xiv Contents
9.9
Influence
analysis for the LME model
............ 522
9.9.1
Example: Weight versus height
............ 526
9.10
Appendix: MLE derivative with respect to
σ2
........ 527
9.11
Summary points
........................ 529
10
Tumor Regrowth Curves
531
10.1
Survival curves
......................... 534
10.2
Double-exponential regrowth curve
.............. 536
10.2.1
Time to regrowth, TR
................. 539
10.2.2
Time to reach specific tumor volume, T»
....... 539
10.2.3
Doubling time, TD
................... 539
10.2.4
Statistical model for regrowth
............. 540
10.2.5
Variance estimation for tumor regrowth outcomes
. 542
10.2.6
Starting values
..................... 542
10.2.7
Example: Chemotherapy treatment comparison
. . . 543
10.3
Exponential growth with fixed regrowth time
........ 547
10.3.1
Statistical hypothesis testing
............. 548
10.3.2
Synergistic or supra-additive effect
.......... 548
10.3.3
Example: Combination of treatments
......... 549
10.4
General regrowth curve
.................... 553
10.5
Double-exponential transient regrowth curve
........ 554
10.5.1
Example: Treatment of cellular spheroids
...... 561
10.6
Gompertz transient regrowth curve
.............. 562
10.6.1
Example: Tumor treated in mice
........... 563
10.7
Summary points
........................ 565
11
Statistical Analysis of Shape
567
11.1
Introduction
........................... 567
11.2
Statistical analysis of random triangles
............ 569
11.3
Face recognition
........................ 572
11.4
Scale-irrelevant shape model
.................. 574
11.4.1
Random effects scale-irrelevant shape model
..... 575
11.4.2
Scale-irrelevant shape model on the log scale
.... 576
11.4.3
Fixed or random size?
................. 577
11.5
Gorilla vertebrae analysis
................... 578
11.6
Procrustes estimation of the mean shape
........... 579
11.6.1
Polygon estimation
................... 582
11.6.2
Generalized Procrustes model
............. 583
11.6.3
Random effects shape model
............. 583
11.6.4
Random or fixed (Procrustes) effects model?
.... 585
11.7
Fourier descriptor analysis
................... 585
11.7.1
Analysis of star shape
................. 586
11.7.2
Random Fourier descriptor analysis
......... 592
11.8
Summary points
........................ 593
Contents xv
12
Statistical
Image
Analysis
595
12.1
Introduction
........................... 595
12.1.1
What is a digital image?
................ 596
12.1.2
Image arithmetic
.................... 597
12.1.3
Ensemble and repeated measurements
........ 598
12.1.4
Image and spatial statistics
.............. 598
12.1.5
Structured and unstructured images
......... 598
12.2
Testing for uniform lighting
.................. 599
12.2.1
Estimating light direction and position
........ 600
12.3
Kolmogorov-Smirnov image comparison
........... 601
12.3.1
Kolmogorov-Smirnov test for image comparison
. . 602
12.3.2
Example: Histological analysis of cancer treatment
. 602
12.4
Multinomial statistical model for images
........... 603
12.4.1
Multinomial image comparison
............ 605
12.5
Image entropy
......................... 606
12.5.1
Reduction of a gray image to binary
......... 607
12.5.2
Entropy of a gray image and histogram equalization
609
12.6
Ensemble of unstructured images
............... 610
12.6.1
Fixed-shift model
.................... 612
12.6.2
Random-shift model
.................. 613
12.6.3
Mixed model for gray images
............. 616
12.6.4
Two-stage estimation
................. 618
12.6.5
Schizophrenia
MRI
analysis
.............. 620
12.7
Image alignment and registration
............... 622
12.7.1 Affine
image registration
................ 624
12.7.2
Weighted sum of squares
................ 625
12.7.3
Nonlinear transformations
............... 625
12.7.4
Random registration
.................. 626
12.7.5
Linear image interpolation
............... 627
12.7.6
Computational aspects
................. 628
12.7.7
Derivative-free algorithm for image registration
. . . 629
12.7.8
Example: Clock alignment
............... 631
12.8
Ensemble of structured images
................ 631
12.8.1
Fixed
affine
transformations
.............. 631
12.8.2
Random
affine
transformations
............ 632
12.9
Modeling spatial correlation
.................. 633
12.9.1
Toeplitz correlation structure
............. 635
12.9.2
Simultaneous estimation of variance and transform
parameters
....................... 639
12.10
Summary points
........................ 639
xvi Contents
13 Appendix:
Useful Facts and Formulas
643
13.1
Basic facts of asymptotic theory
............... 643
13.1.1
Central Limit Theorem
................ 643
13.1.2
Generalized Slutsky theorem
............. 644
13.1.3
Pseudo-
maximum likelihood
.............. 646
13.1.4
Estimating equations approach and sandwich formula
647
13.1.5
Generalized estimating equations approach
..... 650
13.2
Some formulas of matrix algebra
............... 651
13.2.1
Some matrix identities
................. 651
13.2.2
Formulas for generalized matrix inverse
....... 651
13.2.3
Vec
and vech functions. Duplication matrix
..... 652
13.2.4
Matrix differentiation
................. 654
13.3
Basic facts of optimization theory
............... 655
13.3.1
Criteria for unimodality
................ 657
13.3.2
Criteria for global optimum
.............. 657
13.3.3
Criteria for minimum existence
............ 658
13.3.4
Optimization algorithms in statistics
......... 659
13.3.5
A necessary condition of optimum and criteria for
convergence
...................... 663
References
665
Index
697
|
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| dewey-full | 519.5/38 |
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| dewey-raw | 519.5/38 |
| dewey-search | 519.5/38 |
| dewey-sort | 3519.5 238 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T20:05:44Z |
| institution | BVB |
| isbn | 0471601616 |
| language | English |
| lccn | 2004045643 |
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| physical | XVIII, 704 S. Ill., graph. Darst. 25 cm |
| publishDate | 2004 |
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| publisher | Wiley-Interscience |
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| series2 | Wiley series in probability and statistics |
| spelling | Demidenko, Eugene Verfasser aut Mixed models theory and applications Eugene Demidenko Hoboken, NJ Wiley-Interscience 2004 XVIII, 704 S. Ill., graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Wiley series in probability and statistics Includes bibliographical references (p. [665]-695) and index Analyse de variance Variantieanalyse gtt aAnalysis of variance Gemischtes Modell (DE-588)4156565-4 gnd rswk-swf Varianzanalyse (DE-588)4187413-4 gnd rswk-swf R Programm (DE-588)4705956-4 gnd rswk-swf Gemischtes Modell (DE-588)4156565-4 s Varianzanalyse (DE-588)4187413-4 s DE-604 R Programm (DE-588)4705956-4 s 1\p DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013097084&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Demidenko, Eugene Mixed models theory and applications Analyse de variance Variantieanalyse gtt aAnalysis of variance Gemischtes Modell (DE-588)4156565-4 gnd Varianzanalyse (DE-588)4187413-4 gnd R Programm (DE-588)4705956-4 gnd |
| subject_GND | (DE-588)4156565-4 (DE-588)4187413-4 (DE-588)4705956-4 |
| title | Mixed models theory and applications |
| title_auth | Mixed models theory and applications |
| title_exact_search | Mixed models theory and applications |
| title_full | Mixed models theory and applications Eugene Demidenko |
| title_fullStr | Mixed models theory and applications Eugene Demidenko |
| title_full_unstemmed | Mixed models theory and applications Eugene Demidenko |
| title_short | Mixed models |
| title_sort | mixed models theory and applications |
| title_sub | theory and applications |
| topic | Analyse de variance Variantieanalyse gtt aAnalysis of variance Gemischtes Modell (DE-588)4156565-4 gnd Varianzanalyse (DE-588)4187413-4 gnd R Programm (DE-588)4705956-4 gnd |
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