Linear mixed models for longitudinal data:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
2009
|
| Edition: | Repr. |
| Series: | Springer series in statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. [523]-553) and index |
| Physical Description: | XXII, 568 S. graph. Darst. 25 cm |
| ISBN: | 9781441902993 |
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| adam_text | Titel: Linear mixed models for longitudinal data
Autor: Verbeke, Geert
Jahr: 2009
Contents
Preface vii
Acknowledgments ix
1 Introduction 1
2 Examples 7
2.1 The Rat Data.......................... 7
2.2 The Toenail Data (TDO)................... 9
2.3 The Baltimore Longitudinal Study of Aging (BLSA) .... 10
2.3.1 The Prostate Data................... 11
2.3.2 The Hearing Data................... 14
2.4 The Vorozole Study ...................... 15
2.5 Heights of Schoolgirls ..................... 16
2.6 Growth Data.......................... 16
xii Contents
2.7 Mastitis in Dairy Cattle.................... 18
3 A Model for Longitudinal Data 19
3.1 Introduction........................... 19
3.2 A Two-Stage Analysis..................... 20
3.2.1 Stage 1 ......................... 20
3.2.2 Stage 2......................... 20
3.2.3 Example: The Rat Data................ 21
3.2.4 Example: The Prostate Data............. 21
3.2.5 Two-Stage Analysis.................. 22
3.3 The General Linear Mixed-Effects Model .......... 23
3.3.1 The Model....................... 23
3.3.2 Example: The Rat Data................ 25
3.3.3 Example: The Prostate Data............. 26
3.3.4 A Model for the Residuai Covariance Structure ... 26
4 Exploratory Data Analysis 31
4.1 Introduction........................... 3*
4.2 Exploring the Marginai Distribution............. 31
4.2.1 The Average Evolution ................ 31
4.2.2 The Variance Structure................ 33
4.2.3 The Correlation Structure............... 34
4.3 Exploring Subject-Specific Profiles.............. 35
oc
4.3.1 Measuring the Overall Goodness-of-Fit........
4.3.2 Testing for the Need of a Model Extension ..... 37
4.3.3 Example: The Rat Data................ 38
4.3.4 Example: The Prostate Data............. 39
Contents xiii
5 Estimation of the Marginai Model 41
5.1 Introduction........................... 41
5.2 Maximum Likelihood Estimation............... 42
5.3 Restricted Maximum Likelihood Estimation ........ 43
5.3.1 Variance Estimation in Normal Populations..... 43
5.3.2 Estimation of Residuai Variance in Linear Regression 43
5.3.3 REML Estimation for the Linear Mixed Model ... 44
5.3.4 Justification of REML Estimation.......... 46
5.3.5 Comparison Between ML and REML Estimation 46
5.4 Model-Fitting Procedures................... 47
5.5 Example: The Prostate Data................. 48
5.6 Estimation Problems...................... 50
5.6.1 Small Variance Componente.............. 50
5.6.2 Model Misspecifications................ 52
6 Inference for the Marginai Model 55
6.1 Introduction........................... 55
6.2 Inference for the Fixed Effects................. 55
6.2.1 Approximate Wald Tests ............... 56
6.2.2 Approximate t-Tests and F-Tests........... 56
6.2.3 Example: The Prostate Data............. 57
6.2.4 Robust Inference.................... 61
6.2.5 Likelihood Ratio Tests................. 62
6.3 Inference for the Variance Components............ 64
6.3.1 Approximate Wald Tests ............... 64
6.3.2 Likelihood Ratio Tests................. 65
6.3.3 Example: The Rat Data................ 66
xiv Contents
6.3.4 Marginai Testing for the Need of Random Effects . . 69
6.3.5 Example: The Prostate Data............. 72
6.4 Information Criteria...................... 74
7 Inference for the Random Effects 77
7.1 Introduction........................... 77
7.2 Empirical Bayes Inference................... 78
7.3 Henderson s Mixed-Model Equations............. 79
7.4 Best Linear Unbiased Prediction (BLUP).......... 80
7.5 Shrinkage............................ 80
7.6 Example: The Random-Intercepts Model........... 81
7.7 Example: The Prostate Data................. 82
7.8 The Normality Assumption for Random Effects....... 83
7.8.1 Introduction ...................... 83
7.8.2 Impact on EB Estimates ............... 85
7.8.3 Impact on the Estimation of the Marginai Model . . 87
7.8.4 Checking the Normality Assumption......... 89
8 Fitting Linear Mixed Models with SAS 93
8.1 Introduction........................... 93
8.2 The SAS Program....................... 94
8.2.1 The PROC MIXED Statement............ 95
8.2.2 The CLASS Statement................. 96
8.2.3 The MODEL Statement................ 96
8.2.4 The ID Statement................... 97
8.2.5 The RANDOM Statement............... 97
8.2.6 The REPEATED Statement ............. 98
Contents xv
8.2.7 The CONTRAST Statement............. 101
8.2.8 The ESTIMATE Statement.............. 101
8.2.9 The MARE Statement................. 102
8.2.10 Some Additional Statements and Options...... 102
8.3 The SAS Output........................ 104
8.3.1 Information on the Iteration Procedure....... 104
8.3.2 Information on the Model Fit............. 105
8.3.3 Information Criteria.................. 107
8.3.4 Inference for the Variance Components........ 107
8.3.5 Inference for the Fixed Effects ............ Ili
8.3.6 Inference for the Random Effects........... 113
8.4 Note on the Mean Parameterization............. 114
8.5 The RANDOM and REPEATED Statements........ 117
8.6 PROC MIXED versus PROC GLM ............. 119
9 General Guidelines for Model Building 121
9.1 Introduction........................... 121
9.2 Selection of a Preliminary Mean Structure ......... 123
9.3 Selection of a Preliminary Random-Effects Structure .... 125
9.4 Selection of a Residuai Covariance Structure ........ 128
9.5 Model Reduction........................ 132
10 Exploring Serial Correlation 135
10.1 Introduction........................... 135
10.2 An Informai Check for Serial Correlation........... 136
10.3 Flexible Models for Serial Correlation............ 137
10.3.1 Introduction ...................... 137
xvi Contents
10.3.2 Fractional Polynomials.................I37
10.3.3 Example: The Prostate Data.............138
10.4 The Semi-Variogram......................141
10.4.1 Introduction ...................... 141
10.4.2 The Semi-Variogram for Random-Intercepts Models 142
10.4.3 Example: The Vorozole Study............. I44
10.4.4 The Semi-Variogram for Random-Effects Models . . I44
10.4.5 Example: The Prostate Data............. I47
10.5 SomeRemarks......................... l48
11 Locai Influence for the Linear Mixed Model 151
11.1 Introduction ..........................151
11.2 Locai Influence.........................153
11.3 The Detection of Influential Subjects.............158
11.4 Example: The Prostate Data.................162
11.5 Locai Influence Under REML Estimation..........l67
12 The Heterogeneity Model I69
12.1 Introduction........................... 169
12.2 The Heterogeneity Model................... 171
12.3 Estimation of the Heterogeneity Model............ I73
12.4 Classification of Longitudinal Profiles............. 177
12.5 Goodness-of-Fit Checks.................... 178
12.6 Example: The Prostate Data................. 180
12.7 Example: The Heights of Schoolgirls............. I83
13 Conditional Linear Mixed Models 189
13.1 Introduction ..........................189
Contents xvii
13.2 A Linear Mixed Model for the Hearing Data......... 190
13.3 Conditional Linear Mixed Models............... 194
13.4 Applied to the Hearing Data ................. 197
13.5 Relation with Fixed-Effects Models.............. 198
14 Exploring Incomplete Data 201
15 Joint Modeling of Measurements and Missingness 209
15.1 Introduction........................... 209
15.2 The Impact of Incompleteness................. 210
15.3 Simple ad hoc Methods.................... 211
15.4 Modeling Incompleteness ................... 212
15.5 Terminology........................... 214
15.6 Missing Data Patterns..................... 215
15.7 Missing Data Mechanisms................... 215
15.8 Ignorability........................... 217
15.9 A Special Case: Dropout.................... 218
16 Simple Missing Data Methods 221
16.1 Introduction........................... 221
16.2 Complete Case Analysis.................... 223
16.3 Simple Forms of Imputation.................. 223
16.3.1 Last Observation Carried Forward.......... 224
16.3.2 Imputing Unconditional Means............ 225
16.3.3 Buck s Method: Conditional Mean Imputation . . . 225
16.3.4 Discussion of Imputation Techniques......... 226
16.4 Available Case Methods.................... 227
16.5 MCAR Analysis of Toenail Data............... 227
xviii Contents
17 Selection Models 231
17.1 Introduction...........................231
17.2 A Selection Model for the Toenail Data...........233
17.2.1 MAR Analysis.....................233
17.2.2 MNAR analysis.....................234
17.3 Scope of Ignorability......................239
17.4 Growth Data..........................240
17.4.1 Analysis of Complete Growth Data..........240
17.4.2 Frequentist Analysis of Incomplete Growth Data . . 256
17.4.3 Likelihood Analysis of Incomplete Growth Data . . . 257
17.4.4 Missingness Process for the Growth Data......267
17.5 A Selection Model for Nonrandom Dropout.........269
17.6 A Selection Model for the Vorozole Study..........270
18 Pattern-Mixture Models 275
18.1 Introduction........................... 275
18.1.1 A Simple Illustration.................. 275
18.1.2 A Paradox ....................... 278
18.2 Pattern-Mixture Models.................... 280
18.3 Pattern-Mixture Model for the Toenail Data......... 281
18.4 A Pattern-Mixture Model for the Vorozole Study...... 287
18.5 Some Reflections........................ 291
19 Sensitivity Analysis for Selection Models 295
19.1 Introduction...........................295
19.2 A Modified Selection Model for Nonrandom Dropout .... 297
19.3 Locai Influence.........................298
Contents xix
19.3.1 Review of the Theory................. 299
19.3.2 Applied to the Model of Diggle and Kenward .... 300
19.3.3 Special Case: Compound Symmetry......... 302
19.3.4 Serial Correlation.................... 306
19.4 Analysis of the Rat Data ................... 307
19.5 Mastitis in Dairy Cattle.................... 312
19.5.1 Informai Sensitivity Analysis............. 312
19.5.2 Locai Influence Approach............... 319
19.6 Alternative Locai Influence Approaches ........... 326
19.7 Random-coefficient-based Models............... 328
19.8 Concluding Remarks...................... 330
20 Sensitivity Analysis for Pattern-Mixture Models 331
20.1 Introduction........................... 331
20.2 Pattern-Mixture Models and MAR.............. 332
20.2.1 MAR and ACMV ................... 333
20.2.2 Nonmonotone Patterns: A Counterexample..... 335
20.3 Multiple Imputation...................... 336
20.3.1 Parameter and Precision Estimation......... 338
20.3.2 Hypothesis Testing................... 338
20.4 Pattern-Mixture Models and Sensitivity Analysis...... 339
20.5 Identifying Restrictions Strategies .............. 343
20.5.1 Strategy Outline.................... 343
20.5.2 Identifying Restrictions................ 344
20.5.3 ACMV Restrictions.................. 347
20.5.4 Drawing from the Conditional Densities....... 350
20.6 Analysis of the Vorozole Study................ 352
xx Contents
20.6.1 Fitting a Model..................... 352
20.6.2 Hypothesis Testing................... 366
20.6.3 Model Reduction.................... 371
20.7 Thoughts............................ 373
21 How Ignorable Is Missing At Random ? 375
21.1 Introduction...........................375
21.2 Information and Sampling Distributions...........377
21.3 Illustration...........................379
21.4 Example.............................383
21.5 Implications for PROC MIXED................385
22 The Expectation-Maximization Algorithm 387
23 Design Considerations 391
23.1 Introduction........................... 391
23.2 Power Calculations Under Linear Mixed Models....... 392
23.3 Example: The Rat Data.................... 393
23.4 Power Calculations When Dropout Is to Be Expected ... 394
23.5 Example: The Rat Data.................... 397
23.5.1 Constant Pj,k k, Varying rij ............. 399
23.5.2 Constant pjtk kì Constant n, ............ 401
23.5.3 Increasing pJ lk| fc over Time, Constant rij...... 402
24 Case Studies 405
24.1 Blood Pressures ........................405
24.2 The Heat Shock Study.....................411
24.2.1 Introduction ......................411
Contents xxi
24.2.2 Analysis of Heat Shock Data............. 415
24.3 The Validation of Surrogate Endpoints from Multiple Trials 420
24.3.1 Introduction ...................... 420
24.3.2 Validation Criteria................... 421
24.3.3 Notation and Motivating Examples.......... 424
24.3.4 A Meta-Analytic Approach.............. 429
24.3.5 Data Analysis...................... 434
24.3.6 Computational Issues................. 439
24.3.7 Extensions ....................... 442
24.3.8 Reflections on Surrogacy................ 443
24.3.9 Prediction Intervals .................. 444
24.3.10 SAS Code for Random-Effects Model........ 445
24.4 The Milk Protein Content Trial................ 446
24.4.1 Introduction ...................... 446
24.4.2 Informai Sensitivity Analysis............. 448
24.4.3 Formai Sensitivity Analysis.............. 457
24.5 Hepatitis B Vaccination.................... 470
24.5.1 Time Evolution of Antibodies............. 472
24.5.2 Prediction at Year 12 ................. 481
24.5.3 SAS Code for Vaccination Models.......... 482
Appendix
A Software 485
A.l The SAS System........................ 485
A.l.l Standard Applications................. 485
A. 1.2 New Features in SAS Version 7.0........... 485
xxii Contents
A.2 Fitting Mixed Models Using MLwiN............. 489
A.3 Fitting Mixed Models Using SPlus.............. 493
A.3.1 Standard SPlus Functions............... 494
A. 3.2 OS WALD for Nonrandom Nonresponse....... 497
B Technical Details for Sensitivity Analysis 515
B.l Locai Influence: Derivation of Components of Ai...... 515
B.2 Proof of Theorem 20.1..................... 518
References 523
Index 554
|
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| id | DE-604.BV039936980 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:14:32Z |
| institution | BVB |
| isbn | 9781441902993 |
| language | English |
| lccn | 2010483807 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024795118 |
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| owner_facet | DE-19 DE-BY-UBM DE-83 DE-20 |
| physical | XXII, 568 S. graph. Darst. 25 cm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spelling | Verbeke, Geert Verfasser aut Linear mixed models for longitudinal data Geert Verbeke ; Geert Molenberghs Repr. New York, NY [u.a.] Springer 2009 XXII, 568 S. graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Includes bibliographical references (p. [523]-553) and index Linear models (Statistics) Longitudinal method Gemischtes Modell (DE-588)4156565-4 gnd rswk-swf Statistischer Test (DE-588)4077852-6 gnd rswk-swf Lineares Modell (DE-588)4134827-8 gnd rswk-swf Lineares Modell (DE-588)4134827-8 s Gemischtes Modell (DE-588)4156565-4 s Statistischer Test (DE-588)4077852-6 s DE-604 Molenberghs, Geert Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024795118&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Verbeke, Geert Molenberghs, Geert Linear mixed models for longitudinal data Linear models (Statistics) Longitudinal method Gemischtes Modell (DE-588)4156565-4 gnd Statistischer Test (DE-588)4077852-6 gnd Lineares Modell (DE-588)4134827-8 gnd |
| subject_GND | (DE-588)4156565-4 (DE-588)4077852-6 (DE-588)4134827-8 |
| title | Linear mixed models for longitudinal data |
| title_auth | Linear mixed models for longitudinal data |
| title_exact_search | Linear mixed models for longitudinal data |
| title_full | Linear mixed models for longitudinal data Geert Verbeke ; Geert Molenberghs |
| title_fullStr | Linear mixed models for longitudinal data Geert Verbeke ; Geert Molenberghs |
| title_full_unstemmed | Linear mixed models for longitudinal data Geert Verbeke ; Geert Molenberghs |
| title_short | Linear mixed models for longitudinal data |
| title_sort | linear mixed models for longitudinal data |
| topic | Linear models (Statistics) Longitudinal method Gemischtes Modell (DE-588)4156565-4 gnd Statistischer Test (DE-588)4077852-6 gnd Lineares Modell (DE-588)4134827-8 gnd |
| topic_facet | Linear models (Statistics) Longitudinal method Gemischtes Modell Statistischer Test Lineares Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024795118&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT verbekegeert linearmixedmodelsforlongitudinaldata AT molenberghsgeert linearmixedmodelsforlongitudinaldata |