Generalized linear mixed models: modern concepts, methods and applications
Gespeichert in:
| Beteilige Person: | |
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| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Boca Raton, Fla. [u.a.]
CRC Press
2013
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| Schriftenreihe: | Texts in statistical science
A Chapman & Hall book |
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| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025781858&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XXV, 529 S. graph. Darst. 24 cm |
| ISBN: | 9781439815120 1439815127 |
Internformat
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| 100 | 1 | |a Stroup, Walter W. |e Verfasser |0 (DE-588)128490977 |4 aut | |
| 245 | 1 | 0 | |a Generalized linear mixed models |b modern concepts, methods and applications |c Walter W. Stroup |
| 264 | 1 | |a Boca Raton, Fla. [u.a.] |b CRC Press |c 2013 | |
| 300 | |a XXV, 529 S. |b graph. Darst. |c 24 cm | ||
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| 490 | 0 | |a Texts in statistical science | |
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| 653 | |a Linear models (Statistics) / Data processing | ||
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Datensatz im Suchindex
| _version_ | 1819350301863313408 |
|---|---|
| adam_text | Titel: Generalized linear mixed models
Autor: Stroup, Walter W
Jahr: 2013
Contents
Preface.............................................................................................................................................xv
Acknowledgments.....................................................................................................................xxv
Part I The Big Picture
1. Modeling Basics......................................................................................................................3
1.1 What Is a Model?...........................................................................................................3
1.2 Two Model Forms: Model Equation and Probability Distribution........................4
1.2.1 Twist Illustrating the Weakness of the Model Equation Form.................5
1.3 Types of Model Effects.................................................................................................9
1.3.1 Extension of the Linear Regression Example to Illustrate an
Important Distinction between Types of Model Effects............................9
1.4 Writing Models in Matrix Form................................................................................12
1.4.1 Fixed-Effects-Only Models...........................................................................13
1.4.2 Mixed Models: Models with Fixed and Random Effects.........................18
1.5 Summary: Essential Elements for a Complete Statement of the Model..............20
Exercises..................................................................................................................................21
2. Design Matters......................................................................................................................25
2.1 Introductory Ideas for Translating Design and Objectives into Models............25
2.1.1 Chapter Organization...................................................................................26
2.2 Describing Data Architecture to Facilitate Model Specification......................27
2.2.1 Every Data Set Has a Plot Plan .................................................................27
2.2.2 Terminology for Treatment and Design Structure...................................28
2.2.3 Nested and Cross-Classification: Alternative Ways
of Organizing Treatment and Design Structure.......................................29
2.3 From Plot Plan to Linear Predictor...........................................................................31
2.3.1 Unit of Replication Approach......................................................................31
2.3.2 What Would Fisher Do? ............................................................................32
2.3.2.1 Complication...................................................................................33
2.3.2.2 Linear Predictor for Nested Schools............................................34
2.3.3 Matching the Objective.................................................................................35
2.4 Distribution Matters...................................................................................................38
2.4.1 Model Effects: Fixed or Random.................................................................38
2.4.2 Response Variable Distribution...................................................................40
2.4.3 Fixed or Random?: Tough Calls...................................................................41
2.5 More Complex Example: Multiple Factors with Different
Units of Replication....................................................................................................44
2.5.1 Variations on the Multifactor, Multisize Unit of Replication Theme.....46
viii Contents
Exercises..................................................................................................................................49
2.A Appendix A: Common Response Variable (y|b) Distributions............................51
2.B Appendix B: Communicating Your Model to Software or How SAS®
PROC GLIMMIX Thinks ........................................................................................52
2.B.1 General Principles.......................................................................................................52
3. Setting the Stage...................................................................................................................65
3.1 Goals for Inference with Models: Overview...........................................................65
3.2 Basic Tools of Inference..............................................................................................68
3.2.1 Estimable Functions......................................................................................68
3.2.2 Linear Combinations of Fixed and Random Effects: Predictable
Functions.........................................................................................................69
3.2.3 Three Issues for Inference............................................................................69
3.2.3.1 Model Scale vs. Data Scale............................................................70
3.2.3.2 Inference Space...............................................................................70
3.2.3.3 Inference Based on Conditional and Marginal Models...........70
3.3 Issue I: Data Scale vs. Model Scale...........................................................................71
3.3.1 Model Scale Estimation.................................................................................73
3.3.2 Data Scale........................................................................................................74
3.4 Issue II: Inference Space.............................................................................................90
3.4.1 Broad Inference..............................................................................................91
3.4.2 Narrow Inference...........................................................................................95
3.4.2.1 GLIMMIX Implementation...........................................................96
3.5 Issue III: Conditional and Marginal Models...........................................................99
3.5.1 Normal Approximation................................................................................99
3.5.2 Binomial GLMM..........................................................................................101
3.5.3 Conditional and Marginal Distribution...................................................102
3.5.4 Visualizing the Marginal p.d.f...................................................................104
3.5.5 What Do the Normal Approximation and the GLMM Estimate?........106
3.5.6 Gaussian Conditional and Marginal Models..........................................107
3.5.7 Non-Gaussian Marginal vs. Conditional Model.....................................109
3.5.8 One Last Aspect of the Conditional Model: The Role of Residual
in Gaussian LMM vs. One-Parameter Non-Gaussian GLMM..............112
3.6 Summary....................................................................................................................115
Exercises................................................................................................................................116
Part II Estimation and Inference Essentials
4. Estimation............................................................................................................................121
4.1 Introduction...............................................................................................................121
4.2 Essential Background...............................................................................................121
4.2.1 Exponential Family......................................................................................122
4.2.1.1 Essential Terminology and Results...........................................123
4.2.2 Maximum Likelihood Estimation.............................................................125
4.2.3 Newton-Raphson and Fisher Scoring......................................................126
4.2.4 Quasi-Likelihood.........................................................................................127
Contents ix
4.3 Fixed Effects Only.....................................................................................................128
4.3.1 Relation to Least Squares Estimation.......................................................129
4.3.1.1 Pseudo-Likelihood for GLM.......................................................130
4.3.1.2 Gaussian Linear Models and Ordinary Least Squares..........130
4.4 Gaussian Mixed Models...........................................................................................131
4.4.1 Mixed Model Equations for p and b..........................................................131
4.4.2 Relation to Least Squares............................................................................132
4.4.3 Unknown G and R: ML and REML Variance-Covariance
Component Estimation...............................................................................134
4.4.3.1 ANOVA Estimator........................................................................135
4.4.3.2 Maximum Likelihood..................................................................135
4.4.3.3 Restricted Maximum Likelihood...............................................138
4.5 Generalized Linear Mixed Models.........................................................................140
4.5.1 Pseudo-Likelihood for GLMM...................................................................141
4.5.2 Variance-Covariance Estimation with Pseudo-Likelihood..................141
4.5.3 Integral Approximation: Laplace and Quadrature.................................142
4.6 Summary....................................................................................................................145
Exercises................................................................................................................................146
Inference, Part I: Model Effects.......................................................................................149
5.1 Introduction...............................................................................................................149
5.2 Essential Background...............................................................................................149
5.2.1 Estimable and Predictable Functions........................................................150
5.2.1.1 Estimability and GLMMs...........................................................151
5.2.2 Basics of Interval Estimates and Test Statistics.......................................151
5.2.3 Approximate Distribution of Estimable and Predictable Functions.... 152
5.2.3.1 Distribution of [3 in the LM with Known V............................152
5.2.3.2 Distribution of the Quadratic Form Defined on p for the
LM with Known V.......................................................................152
5.2.3.3 LM with Unknown V..................................................................153
5.2.3.4 LM with Unknown V: Case 1?V = a2S.....................................153
5.2.3.5 LM with Unknown V: Case 2?All Covariance
Components Must Be Estimated................................................154
5.2.3.6 GLM................................................................................................156
5.2.3.7 GLM: Case 1?No Scale Parameter to Estimate.......................156
5.2.3.8 GLM: Case 2?Estimated Scale Parameter(s)...........................156
5.2.3.9 Mixed Models...............................................................................157
5.3 Approaches to Testing..............................................................................................159
5.3.1 Likelihood Ratio and Deviance.................................................................160
5.3.2 Wald and Approximate F-statistics...........................................................161
5.3.2.1 A Special Case: The Gaussian LM with V=Ia2........................161
5.3.3 Multiple Effect Models and Order of Testing..........................................162
5.4 Inference Using Model-Based Statistics.................................................................165
5.4.1 Naive Statistics and Degrees of Freedom.................................................166
5.4.2 Satterthwaite Degrees of Freedom Approximation................................167
5.4.3 Bias Correction for Model-Based Standard Errors and Test Statistics.....168
Contents
5.5 Inference Using Empirical Standard Error...........................................................170
5.5.1 Sandwich (a.k.a Robust or Empirical) Estimator.....................................170
5.5.2 Bias Correction for Sandwich Estimators.................................................171
5.6 Summary of Main Ideas and General Guidelines for Implementation............173
Exercises................................................................................................................................174
Inference, Part II: Covariance Components..................................................................179
6.1 Introduction...............................................................................................................179
6.2 Formal Testing of Covariance Components..........................................................179
6.2.1 ANOVA-Based Tests for Variance-Component-Only LMMs................180
6.2.2 Wald Statistics for Covariance Component
Testing and Why They Should Not Be Used...........................................181
6.2.3 Likelihood Ratio Tests for Covariance Components..............................182
6.2.3.1 One-Way ANOVA: Test for Homogeneity of Variance...........183
6.2.3.2 Repeated Measures Example: Selecting a Parsimonious
Covariance Model........................................................................183
6.2.4 Consequences of PL versus Integral Approximation for GLMMs.......186
6.2.4.1 R-Side or Working Correlation Model.......................................187
6.2.4.2 What Would Fisher Do? The G-Side Approach...................188
6.2.4.3 R-Side versus G-Side: Consequences for Covariance
Model Selection............................................................................189
6.3 Fit Statistics to Compare Covariance Models.......................................................191
6.3.1 AIC and AICC................................................................................................191
6.3.2 BIC..................................................................................................................192
6.3.3 Application to Comparison of Covariance Models................................192
6.4 Interval Estimation...................................................................................................194
6.4.1 Wald Approach Based on the x2................................................................195
6.4.2 Likelihood-Based Approach......................................................................195
6.5 Summary....................................................................................................................195
Exercises................................................................................................................................196
Part HI Working with GLMMs
7. Treatment and Explanatory Variable Structure...........................................................203
7.1 Types of Treatment Structures................................................................................203
7.2 Types of Estimable Functions..................................................................................204
7.2.1 Relation to Classical ANOVA Reduction Sums of Squares...................204
7.2.2 How Do We Know What We Are Testing?..............................................205
7.2.3 How to Decide What to Test Rather than Letting It Be
Decided for Us..............................................................................................206
7.2.4 Multiplicity...................................................................................................206
7.3 Multiple Factor Models: Overview.........................................................................209
7.4 Multifactor Models with All Factors Qualitative.................................................211
7.4.1 Review of Options.......................................................................................212
7.4.2 Tools for Qualitative Factorial Inference: SLICE,
SLICEDLFF, and Other Tools...................................................................213
7.4.3 Multiplicity Adjustments............................................................................216
Contents xi
7.5 Multifactor: Some Factors Qualitative, Some Factors Quantitative...................219
7.5.1 Generic Form of the Linear Predictor.......................................................219
7.5.2 Many Uses of the Generic Linear Predictor.............................................220
7.5.2.1 Latent Growth Curve Models.....................................................220
7.5.2.2 Analysis of Covariance................................................................220
7.5.2.3 Factorial Treatment Design.........................................................226
7.6 Multifactor: All Factors Quantitative.....................................................................229
7.6.1 Second-Order Polynomial, a.k.a. Classical Response Surface
Linear Predictors..........................................................................................229
7.6.2 Other Quantitative-by-Quantitative Models...........................................231
7.6.2.1 Nonlinear Mean Models.............................................................231
7.6.2.2 Spline or Segmented Regression................................................235
7.7 Summary....................................................................................................................236
8. Multilevel Models..............................................................................................................239
8.1 Types of Design Structure: Single- and Multilevel Models Defined.................239
8.2 Types of Multilevel Models and How They Arise...............................................240
8.2.1 Units of Replication: Not Just in Designed Experiments.......................241
8.2.2 What Would Fisher Do? Revisited: Topographical and
Treatment Component................................................................................242
8.3 Role of Blocking in Multilevel Models...................................................................245
8.3.1 Block Effects Fixed vs. Block Effects Random Revisited....................246
8.3.2 Fixed Blocks, Multilevel Designs, and Spurious Nonestimability.......248
8.4 Working with Multilevel Designs..........................................................................250
8.4.1 Examples of Multilevel Structures............................................................250
8.4.2 Multifactor Treatment and Multilevel Design Structures:
How They Fit Together................................................................................259
8.5 Marginal and Conditional Multilevel Models......................................................264
8.5.1 Gaussian Data...............................................................................................265
8.5.2 Non-Gaussian Models.................................................................................267
8.6 Summary....................................................................................................................267
Exercises................................................................................................................................268
9. Best Linear Unbiased Prediction.....................................................................................271
9.1 Review of Estimable and Predictable Functions..................................................271
9.2 BLUP in Random-Effects-Only Models.................................................................272
9.2.1 One-Way Random Effects Model..............................................................273
9.2.2 Two-Way Random Effects Nested Model.................................................276
9.2.2.1 Analysis: Balanced Case..............................................................278
9.2.2.2 Unbalanced Case..........................................................................280
9.3 Gaussian Data with Fixed and Random Effects...................................................284
9.3.1 Mixed-Model Analysis with BLUP to Modify the Inference Space.....285
9.3.2 Relationship between BLUP and Fixed Effect Estimators.....................288
9.4 Advanced Applications with Complex Z Matrices..............................................292
9.5 Summary....................................................................................................................2%
10. Rates and Proportions.......................................................................................................299
10.1 Types of Rate and Proportion Data........................................................................299
xjl Contents
10.2 Discrete Proportions: Binary and Binomial Data.................................................299
10.2.1 Pseudo-Likelihood or Integral Approximation.......................................300
10.2.2 Example of Explanatory-Response Models..............................................303
10.2.3 Models for Contingency Tables.................................................................313
10.3 Alternative Link Functions for Binomial Data.....................................................317
10.3.1 Role of Residual in Binomial Models....................................................322
10.4 Continuous Proportions...........................................................................................326
10.4.1 Beta Distribution..........................................................................................326
10.4.2 Continuous Proportion Example Using the Beta Distribution.............327
10.5 Summary....................................................................................................................330
Exercises................................................................................................................................331
11. Counts...................................................................................................................................335
11.1 Introduction...............................................................................................................335
11.1.1 Count Data and the Poisson Distribution................................................335
11.1.2 Example Comparing Pre-GLM ANOVA-Based
Analysis to Poisson GLM............................................................................336
11.2 Overdispersion in Count Data................................................................................340
11.2.1 Overdispersion Denned..............................................................................340
11.2.2 Detecting Overdispersion...........................................................................342
11.2.3 Strategies.......................................................................................................346
11.2.3.1 Scale Parameter.............................................................................347
11.2.3.2 What Would Fisher Do? Revisited.........................................348
11.2.3.3 Alternative Distributions............................................................350
11.3 More on Alternative Distributions.........................................................................352
11.3.1 Negative Binomial.......................................................................................352
11.3.2 Generalized Poisson....................................................................................354
11.4 Conditional and Marginal.......................................................................................356
11.5 Too Many Zeroes.......................................................................................................361
11.5.1 Formal Description of Zero-Inflated and Hurdle Models.....................362
11.5.2 GLMM for Poisson and Negative Binomial Zero-Inflated and
Hurdle Models.............................................................................................362
11.6 Summary....................................................................................................................369
Exercises................................................................................................................................369
12. Time-to-Event Data............................................................................................................375
12.1 Introduction: Probability Concepts for Time-to-Event Data..............................375
12.2 Gamma GLMMs.......................................................................................................376
12.2.1 Hierarchical (Split-Plot) Gamma GLMM..................................................377
12.2.1.1 What Happens If We Fit This Model Using a
Gaussian LMM?............................................................................378
12.2.1.2 Gamma Generalized Linear Model...........................................379
12.2.2 Response Surface for Time-to-Event: An Example Using the
Box-Behnken Design...................................................................................381
12.2.2.1 Gaussian LMM.............................................................................382
12.2.2.2 Gamma GLMM............................................................................384
12.3 GLMMs and Survival Analysis..............................................................................386
12.3.1 Basic Concepts and Terminology..............................................................387
Contents xiii
12.3.2 Exponential Survival GLMM for Uncensored Data...............................388
12.3.3 Exponential Survival GLMM for Censored Data...................................391
12.4 Summary....................................................................................................................394
13. Multinomial Data...............................................................................................................397
13.1 Overview....................................................................................................................397
13.2 Multinomial Data with Ordered Categories.........................................................398
13.3 Nominal Categories: Generalized Logit Models..................................................404
13.4 Model Comparison...................................................................................................408
13.5 Summary....................................................................................................................410
Exercises................................................................................................................................410
14. Correlated Errors, Part I: Repeated Measures..............................................................413
14.1 Overview....................................................................................................................413
14.1.1 What Are Repeated Measures/Longitudinal Data.................................413
14.1.2 Pre-GLMM Methods...................................................................................414
14.2 Gaussian Data: Correlation and Covariance Models for LMMs........................417
14.3 Covariance Model Selection....................................................................................418
14.3.1 Why Does It Matter?....................................................................................419
14.3.2 Covariance Model Selection Methods......................................................420
14.4 Non-Gaussian Case..................................................................................................429
14.4.1 GEE-Type Models.........................................................................................429
14.4.2 GLMMs..........................................................................................................431
14.5 Issues for Non-Gaussian Repeated Measures.......................................................434
14.5.1 How Do Correlated Errors Arise? Deciding What
We Are Modeling........................................................................................434
14.5.2 Covariance Model Selection and Non-Gaussian
Repeated Measures.....................................................................................435
14.5.3 Inference Space, Standard Errors, and Test Statistics.............................435
14.6 Summary....................................................................................................................437
Exercises................................................................................................................................438
15. Correlated Errors, Part II: Spatial Variability...............................................................443
15.1 Overview....................................................................................................................443
15.1.1 Types of Spatial Variability........................................................................443
15.1.2 Pre-GLMM Methods...................................................................................447
15.1.2.1 Nearest-Neighbor Adjustment...................................................447
15.1.2.2 Blocking.........................................................................................448
15.2 Gaussian Case with Covariance Model.................................................................448
15.2.1 Covariance Model Selection.......................................................................449
15.2.2 Impact of Spatial Variability on Inference................................................452
15.3 Spatial Covariance Modeling by Smoothing Spline............................................453
15.4 Non-Gaussian Case..................................................................................................456
15.4.1 Randomized Complete Block Model........................................................457
15.4.2 Incomplete Block Model.............................................................................457
15.4.3 GLIMMIX Statements..................................................................................457
15.4.3.1 RGB.................................................................................................457
15.4.3.2 Lattice Incomplete Blocks............................................................457
xiv Contents
15.4.4 GEE-Type R-Side Spatial Correlation Model........................................458
15.4.5 G-Side Spatial Correlation Model..........................................................459
15.4.5.1 G-Side Spatial Radial Smoothing Model..................................460
15.4.5.2 Relevant Output...........................................................................460
15.5 Summary....................................................................................................................464
Exercise..................................................................................................................................465
16. Power, Sample Size, and Planning.................................................................................467
16.1 Basics of GLMM-Based Power and Precision Analysis.......................................467
16.1.1 Essential GLMM Theory for Power and Precision Analysis.................468
16.1.2 Using SAS PROC GLIMMIX to Implement a Power Analysis..............469
16.2 Gaussian Example.....................................................................................................474
16.3 Power for Binomial GLMMs....................................................................................479
16.4 GLMM-Based Power Analysis for Count Data.....................................................484
16.5 Power and Planning for Repeated Measures........................................................487
16.5.1 Straightforward Cases: Gaussian and One-Parameter
Exponential Family......................................................................................488
16.5.2 On the Frontier: The Two-Parameter Exponential Family....................490
16.6 Summary....................................................................................................................492
Exercises................................................................................................................................494
Appendices: Essential Matrix Operations and Results.....................................................499
Appendix A: Matrix Operations.............................................................................................501
Appendix B: Distribution Theory for Matrices...................................................................509
References...................................................................................................................................513
Index.............................................................................................................................................519
|
| any_adam_object | 1 |
| author | Stroup, Walter W. |
| author_GND | (DE-588)128490977 |
| author_facet | Stroup, Walter W. |
| author_role | aut |
| author_sort | Stroup, Walter W. |
| author_variant | w w s ww wws |
| building | Verbundindex |
| bvnumber | BV040801755 |
| classification_rvk | SK 840 |
| classification_tum | MAT 620f |
| ctrlnum | (OCoLC)821565132 (DE-599)BSZ373353790 |
| dewey-full | 519.535 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.535 |
| dewey-search | 519.535 |
| dewey-sort | 3519.535 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV040801755 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T16:26:00Z |
| institution | BVB |
| isbn | 9781439815120 1439815127 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025781858 |
| oclc_num | 821565132 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-83 DE-1028 DE-11 |
| owner_facet | DE-19 DE-BY-UBM DE-83 DE-1028 DE-11 |
| physical | XXV, 529 S. graph. Darst. 24 cm |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Texts in statistical science A Chapman & Hall book |
| spellingShingle | Stroup, Walter W. Generalized linear mixed models modern concepts, methods and applications Datenverarbeitung Lineares Modell (DE-588)4134827-8 gnd |
| subject_GND | (DE-588)4134827-8 |
| title | Generalized linear mixed models modern concepts, methods and applications |
| title_auth | Generalized linear mixed models modern concepts, methods and applications |
| title_exact_search | Generalized linear mixed models modern concepts, methods and applications |
| title_full | Generalized linear mixed models modern concepts, methods and applications Walter W. Stroup |
| title_fullStr | Generalized linear mixed models modern concepts, methods and applications Walter W. Stroup |
| title_full_unstemmed | Generalized linear mixed models modern concepts, methods and applications Walter W. Stroup |
| title_short | Generalized linear mixed models |
| title_sort | generalized linear mixed models modern concepts methods and applications |
| title_sub | modern concepts, methods and applications |
| topic | Datenverarbeitung Lineares Modell (DE-588)4134827-8 gnd |
| topic_facet | Datenverarbeitung Lineares Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025781858&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT stroupwalterw generalizedlinearmixedmodelsmodernconceptsmethodsandapplications |