Linear mixed-effects models using R: a step-by-step approach
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
[2013]
|
| Series: | Springer Texts in Statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxxii, 542 Seiten Illustrationen, Diagramme |
| ISBN: | 9781461438991 9781489996671 |
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MARC
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| 245 | 1 | 0 | |a Linear mixed-effects models using R |b a step-by-step approach |c Andrzej Gałecki ; Tomasz Burzykowski |
| 264 | 1 | |a New York, NY |b Springer |c [2013] | |
| 264 | 4 | |c © 2013 | |
| 300 | |a xxxii, 542 Seiten |b Illustrationen, Diagramme | ||
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| adam_text | Contents
Part I Introduction
1 Introduction........................................................... 3
1.1 The Aim of the Book.............................................. 3
1.2 Implementation of Linear Mixed-Effects Models in R............... 3
1.3 The Structure of the Book........................................ 5
1.4 Technical Notes ............................................... 8
2 Case Studies........................................................ 11
2.1 Introduction.................................................... 11
2.2 Age-Related Macular Degeneration Trial.......................... 12
2.2.1 Raw Data............................................... 13
2.2.2 Data for Analysis ..................................... 14
2.3 Progressive Resistance Training Study .......................... 20
2.3.1 Raw Data............................................... 20
2.3.2 Data for Analysis ..................................... 22
2.4 The Study of Instructional Improvement Project ................. 24
2.4.1 Raw Data .............................................. 24
2.4.2 Data for Analysis ..................................... 26
2.4.3 Data Hierarchy ........................................ 28
2.5 The Flemish Community Attainment-Targets Study.................. 31
2.5.1 Raw Data .......................................... 32
2.5.2 Data for Analysis ..................................... 34
2.6 Chapter Summary................................................. 34
3 Data Exploration...................................................... 39
3.1 Introduction.................................................... 39
3.2 ARMD Trial: Visual Acuity....................................... 39
3.2.1 Patterns of Missing Data............................... 41
3.2.2 Mean-Value Profiles ................................... 42
3.2.3 Sample Variances and Correlations of Visual
Acuity Measurements.................................... 45
xi
xii Contents
3.3 PRT Study: Muscle Fiber Specific Force .................... 48
3.4 SII Project: Gain in the Math Achievement Score ............. 53
3.4.1 School-Level Data ................................. 55
3.4.2 Class-Level Data .................................... 58
3.4.3 Pupil-Level Data .................................... 60
3.5 FCAT Study: Target Score .................................... 63
3.6 Chapter Summary ........................................... 64
Part II Linear Models for Independent Observations
4 Linear Models with Homogeneous Variance............................ 69
4.1 Introduction ............................................. 69
4.2 Model Specification ........................................ 70
4.2.1 Model Equation at the Level of the Observation....... 70
4.2.2 Model Equation for All Data ......................... 71
4.3 Offset ...................................................... 71
4.4 Estimation................................................. 72
4.4.1 Ordinary Least Squares .............................. 72
4.4.2 Maximum-Likelihood Estimation...................... 73
4.4.3 Restricted Maximum-Likelihood Estimation........... 74
4.4.4 Uncertainty in Parameter Estimates .................. 75
4.5 Model Diagnostics.......................................... 75
4.5.1 Residuals ......................................... 76
4.5.2 Residual Diagnostics............................... 78
4.5.3 Influence Diagnostics ............................... 80
4.6 Inference.................................................. 81
4.6.1 The Wald, Likelihood Ratio, and Score Tests ......... 81
4.6.2 Confidence Intervals for Parameters ................. 84
4.7 Model Reduction and Selection................................ 84
4.7.1 Model Reduction.................................... 85
4.7.2 Model Selection Criteria......................... 86
4.8 Chapter Summary ............................................. 88
5 Fitting Linear Models with Homogeneous Variance:
The ImO and gls O Functions...................................... 89
5.1 Introduction ............................................... 89
5.2 Specifying the Mean Structure Using a Model Formula....... 89
5.2.1 The Formula Syntax ................................. 90
5.2.2 Representation of R Formula: The terms Class ........ 94
5.3 From a Formula to the Design Matrix.......................... 96
5.3.1 Creating a Model Frame............................... 96
5.3.2 Creating a Design Matrix .......................... 102
5.4 Using the lm() and gls () Functions to Fit a Linear Model.... 107
5.5 Extracting Information from a Model-Fit Object.............. 108
Contents xiii
5.6 Tests of Linear Hypotheses for Fixed Effects ................ 109
5.7 Chapter Summary ............................................. 110
6 ARMD Trial: Linear Model with Homogeneous Variance............... 113
6.1 Introduction ................................................ 113
6.2 A Linear Model with Independent Residual Errors
with Homogeneous Variance.................................... 113
6.3 Fitting a Linear Model Using the lm() Function .............. 114
6.4 Fitting a Linear Model Using the gls ( ) Function............ 119
6.5 Chapter Summary ............................................. 120
7 Linear Models with Heterogeneous Variance.......................... 123
7.1 Introduction ................................................ 123
7.2 Model Specification ......................................... 124
7.2.1 Known Variance Weights .............................. 124
7.2.2 Variance Function ................................... 125
7.3 Details of the Model Specification........................... 127
7.3.1 Groups of Variance Functions ........................ 127
7.3.2 Aliasing in Variance Parameters ..................... 129
7.4 Estimation................................................... 130
7.4.1 Weighted Least Squares .............................. 130
7.4.2 Likelihood Optimization.............................. 131
7.4.3 Constrained Versus Unconstrained
Parameterization of the Variance Parameters ......... 135
7.4.4 Uncertainty in Parameter Estimation ................. 135
7.5 Model Diagnostics............................................ 136
7.5.1 Pearson Residuals ................................... 136
7.5.2 Influence Diagnostics ............................... 137
7.6 Inference ................................................... 138
7.6.1 Tests of Statistical Significance.................... 138
7.6.2 Confidence Intervals for Parameters ................. 140
7.7 Model Reduction and Selection................................ 140
7.8 Mean-Variance Models ........................................ 141
7.8.1 Estimation .......................................... 141
7.8.2 Model Diagnostics and Inference ..................... 145
7.9 Chapter Summary ............................................. 146
8 Fitting Linear Models with Heterogeneous Variance:
The glsO Function........................................ 149
8.1 Introduction ................................................ 149
8.2 Variance-Function Representation: The varFunc Class ......... 149
8.2.1 Variance-Function Constructors....................... 150
8.2.2 Initialization of Objects of Class varFunc........... 151
8.3 Inspecting and Modifying Objects of Class varFunc ........... 152
8.4 Using the glsO Function to Fit Linear Models
with Heterogeneous Variance ................................. 154
xiv
Contents
8.5 Extracting Information From a Model-fit Object
ofdassgls.................................................. 156
8.6 Chapter Summary............................................. 158
9 ARMD Trial: Linear Model with Heterogeneous Variance.............. 159
9.1 Introduction ............................................... 159
9.2 A Linear Model with Independent Residual Errors
and Heterogeneous Variance .................................. 159
9.2.1 Fitting the Model Using the gls () Function........ 160
9.3 Linear Models with the varPower(-) Variance-Function........ 162
9.3.1 Fitting the Models Using the gls () Function....... 163
9.3.2 Model-Fit Evaluation .............................. 168
9.4 Chapter Summary ............................................ 171
Part III Linear Fixed-Effects Models for Correlated Data
10 Linear Model with Fixed Effects and Correlated Errors............. 177
10.1 Introduction................................................ 177
10.2 Model Specification......................................... 178
10.3 Details of Model Specification................................ 179
10.3.1 Variance Structure ................................ 180
10.3.2 Correlation Structure................................ 181
10.3.3 Serial Correlation Structures...................... 182
10.3.4 Spatial Correlation Structures .................... 183
10.4 Estimation.................................................. 185
10.4.1 Weighted Least Squares ............................ 185
10.4.2 Likelihood-Based Estimation ....................... 186
10.4.3 Constrained Versus Unconstrained
Parameterization of the Variance-Covariance
Matrix .............................................. 188
10.4.4 Uncertainty in Parameter Estimation ................. 190
10.5 Model Diagnostics............................................ 190
10.5.1 Residual Diagnostics............................... 191
10.5.2 Influence Diagnostics ............................... 192
10.6 Inference and Model Selection............................... 192
10.7 Mean-Variance Models......................................... 194
10.8 Chapter Summary ............................................ 196
11 Fitting Linear Models with Fixed Effects and Correlated Errors:
The gls () Function............................................... 197
11.1 Introduction.................................................. 197
11.2 Correlation-Structure Representation: The corStruct Class... 197
11.2.1 Correlation-Structure Constructor Functions ......... 198
11.3 Inspecting and Modifying Objects of Class corStruct .......... 199
11.3.1 Coefficients of Correlation Structures............... 199
11.3.2 Semivariogram........................................ 200
11.3.3 The corMatrixQ Function.............................. 202
Contents
XV
11.4 Illustration of Correlation Structures....................... 202
11.4.1 Compound Symmetry: The corCompSymm
Class ................................................ 203
11.4.2 Autoregressive Structure of Order 1:
The corARl Class...................................... 204
11.4.3 Exponential Structure: The corExp Class............... 206
11.5 Using the glsO Function....................................... 209
11.6 Extracting Information from a Model-Fit Object
ofClassg/ճ·................................................... 210
11.7 Chapter Summary............................................... 211
12 ARMD Trial: Modeling Correlated Errors for Visual Acuity............ 213
12.1 Introduction ................................................. 213
12.2 The Model with Heteroscedastic, Independent
Residual Errors Revisited..................................... 213
12.2.1 Empirical Semivariogram............................... 214
12.3 A Linear Model with a Compound-Symmetry
Correlation Structure ........................................ 216
12.3.1 Model Specification................................. 216
12.3.2 Syntax and Results ................................... 217
12.4 Heteroscedastic Autoregressive Residual Errors................ 220
12.4.1 Model Specification................................... 220
12.4.2 Syntax and Results ................................... 221
12.5 General Correlation Matrix for Residual Errors ............... 223
12.5.1 Model Specification................................... 223
12.5.2 Syntax and Results ................................... 224
12.6 Model-Fit Diagnostics......................................... 227
12.6.1 Scatterplots of Raw Residuals ........................ 227
12.6.2 Scatterplots of Pearson Residuals .................... 229
12.6.3 Normalized Residuals ................................. 232
12.7 Inference About the Mean Structure............................ 234
12.7.1 Models with the General Correlation Structure
and Power Variance Function .......................... 236
12.7.2 Syntax and Results ................................... 236
12.8 Chapter Summary .............................................. 238
Part IV Linear Mixed-Effects Models
13 Linear Mixed-Effects Model ......................................... 245
13.1 Introduction ................................................. 245
13.2 The Classical Linear Mixed-Effects Model ..................... 246
13.2.1 Specification at a Level of a Grouping Factor........ 246
13.2.2 Specification for All Data............................ 248
13.3 The Extended Linear Mixed-Effects Model ...................... 249
xvi Contents
13.4 Distributions Defined by the y and b Random Variables ....... 250
13.4.1 Unconditional Distribution of Random Effects......... 250
13.4.2 Conditional Distribution of y Given the
Random Effects ...................................... 250
13.4.3 Additional Distributions Defined by y and b ......... 252
13.5 Estimation................................................... 254
13.5.1 The Marginal Model Implied by the Classical
Linear Mixed-Effects Model........................... 254
13.5.2 Maximum-Likelihood Estimation........................ 256
13.5.3 Penalized Least Squares ............................. 257
13.5.4 Constrained Versus Unconstrained
Parameterization of the Variance-Covariance
Matrix .............................................. 261
13.5.5 Uncertainty in Parameter Estimation .................. 263
13.5.6 Alternative Estimation Approaches.................... 264
13.6 Model Diagnostics............................................ 264
13.6.1 Normality of Random Effects ......................... 264
13.6.2 Residual Diagnostics................................. 265
13.6.3 Influence Diagnostics ............................ 267
13.7 Inference and Model Selection................................ 267
13.7.1 Testing Hypotheses About the Fixed Effects........... 267
13.7.2 Testing Hypotheses About the Variance-
Covariance Parameters................................. 268
13.7.3 Confidence Intervals for Parameters .................. 269
13.8 Mean-Variance Models...................................... 270
13.8.1 Single-Level Mean-Variance Linear
Mixed-Effects Models.................................. 270
13.8.2 Multilevel Hierarchies ............................... 272
13.8.3 Inference........................................... 272
13.9 Chapter Summary ............................................ 273
14 Fitting Linear Mixed-Effects Models: The İme () Function........... 275
14.1 Introduction ............................................... 275
14.2 Representation of a Positive-Definite Matrix: The pdMat Class... 276
14.2.1 Constructor Functions for the pdMat Class............. 276
14.2.2 Inspecting and Modifying Objects of Class pdMat...... 279
14.3 Random-Effects Structure Representation:
The reStruct class ........................................... 283
14.3.1 Constructor Function for the reStruct Class........... 284
14.3.2 Inspecting and Modifying Objects of Class reStruct... 286
14.4 The Random Part of the Model Representation:
The ImeStruct Class .......................................... 290
14.5 Using the Function Ime () to Specify and Fit Linear
Mixed-Effects Models.......................................... 292
Contents xvii
14.6 Extracting Information from a Model-Fit Object
of Class Ime.................................................. 293
14.7 Tests of Hypotheses About the Model Parameters....,......... 297
14.8 Chapter Summary .............................................. 300
15 Fitting Linear Mixed-Effects Models: The lmer () Function.......... 303
15.1 Introduction.................................................. 303
15.2 Specification of Models with Crossed and Nested
Random Effects ............................................... 304
15.2.1 A Hypothetical Experiment with the Effects
of Plates Nested Within Machines...................... 304
15.2.2 A Hypothetical Experiment with the Effects
of Plates Crossed with the Effects of Machines...... 305
15.2.3 General Case.......................................... 306
15.3 Using the Function lmer () to Specify and Fit Linear
Mixed-Effects Models ......................................... 308
15.3.1 The lmer () Formula .................................. 308
15.4 Extracting Information from a Model-Fit Object
of Class mer.............................................. 312
15.5 Tests of Hypotheses About the Model Parameters ............... 314
15.6 Illustration of Computations ................................. 315
15.7 Chapter Summary .............................................. 325
16 ARMD Trial: Modeling Visual Acuity............................... 327
16.1 Introduction ................................................. 327
16.2 A Model with Random Intercepts and Homogeneous
Residual Variance............................................. 327
16.2.1 Model Specification................................... 328
16.2.2 R Syntax and Results.................................. 330
16.3 A Model with Random Intercepts and the varPower(-)
Residual Variance Function.................................... 334
16.3.1 Model Specification................................... 334
16.3.2 R Syntax and Results.................................. 336
16.3.3 Diagnostic Plots ..................................... 339
16.4 Models with Random Intercepts and Slopes and the
varPower(-) Residual Variance-Function........................ 346
16.4.1 Model with a General Matrix T ....................... 346
16.4.2 Model with a Diagonal Matrix V....................... 348
16.4.3 Model with a Diagonal Matrix X
and a Constant Treatment Effect ...................... 353
16.5 An Alternative Residual Variance Function: varldent(-) ....... 356
16.6 Testing Hypotheses About Random Effects....................... 361
16.6.1 Test for Random Intercepts ........................... 362
16.6.2 Test for Random Slopes ............................... 364
xviii Contents
16.7 Analysis Using the Function ImerC) ............................ 367
16.7.1 Basic Results ........................................ 367
16.7.2 Simulation-Based p-Values:
The simulate.mer() Method.............................. 372
16.7.3 Test for Random Intercepts.............................. 376
16.7.4 Test for Random Slopes ................................. 379
16.8 Chapter Summary ............................................... 380
17 PRT Trial: Modeling Muscle Fiber Specific-Force....................... 385
17.1 Introduction .................................................. 385
17.2 A Model with Occasion-Specific Random Intercepts
for Type-1 Fibers ............................................. 385
17.2.1 Model Specification................................... 386
17.2.2 R Syntax and Results................................... 388
17.3 A Mean-Variance Model with Occasion-Specific
Random Intercepts for Type-1 Fibers............................ 397
17.3.1 R Syntax and Results.................................. 397
17.4 A Model with Heteroscedastic Fiber-Typex Occasion-
Specific Random Intercepts.................................... 400
17.4.1 Model Specification................................... 400
17.4.2 R Syntax and Results.................................. 402
17.5 A Model with Heteroscedastic Fiber-Type x Occasion-
Specific Random Intercepts (Alternative Specification).......... 411
17.5.1 Model Specification................—.................. 411
17.5.2 R Syntax and Results................................... 412
17.6 A Model with Heteroscedastic Fiber-Typex Occasion-
Specific Random Intercepts and a Structured
Matrix ......................................................... 415
17.6.1 Model Specification................................... 415
17.6.2 R Syntax and Results................................. 416
17.7 A Model with Homoscedastic Fiber-Typex Occasion-
Specific Random Intercepts and a Structured
Matrix T ...................................................... 419
17.7.1 Model Specification...................................... 419
17.7.2 R Syntax and Results................................... 420
17.8 A Joint Model for Two Dependent Variables....................... 422
17.8.1 Model Specification.................................. — 422
17.8.2 R Syntax and Results................................. 423
17.9 Chapter Summary ............................................... 429
18 SII Project: Modeling Gains in Mathematics Achievement-Scores ... 431
18.1 Introduction................................................... 431
18.2 A Model with Fixed Effects for School-
and Pupil-Specific Covariates and Random Intercepts
for Schools and Classes....................................... 431
18.2.1 Model Specification.................................... 432
18.2.2 R Syntax and Results.................................. 433
Contents
XIX
18.3 A Model with an Interaction Between School-
and Pupil-Level Covariates..................................... 438
18.3.1 Model Specification................................... 438
18.3.2 R Syntax and Results................................... 439
18.4 A Model with Fixed Effects of Pupil-Level
Covariates Only ............................................... 442
18.4.1 Model Specification.................................... 442
18.4.2 R Syntax and Results................................... 442
18.5 A Model with a Third-Degree Polynomial
of a Pupil-Level Covariate in the Mean Structure............. 444
18.5.1 Model Specification.................................... 444
18.5.2 R Syntax and Results................................... 444
18.6 A Model with a Spline of a Pupil-Level Covariate
in the Mean Structure.......................................... 448
18.6.1 Model Specification.................................... 448
18.6.2 R Syntax and Results................................... 449
18.7 The Final Model with Only Pupil-Level Variables
in the Mean Structure.......................................... 450
18.7.1 Model Specification.................................... 450
18.7.2 R Syntax and Results................................... 450
18.8 Analysis Using the Function ImerO ............................ 457
18.9 Chapter Summary .............................................. 462
19 FCAT Study: Modeling Attainment-Target Scores......................... 465
19.1 Introduction.................................................... 465
19.2 A Fixed-Effects Linear Model Fitted Using
the Function ImO .............................................. 465
19.2.1 Model Specification................................... 466
19.2.2 R Syntax and Results................................... 466
19.3 A Linear Mixed-Effects Model with Crossed Random
Effects Fitted Using the Function ImerO ....................... 468
19.3.1 Model Specification.................................... 469
19.3.2 R Syntax and Results................................... 469
19.4 A Linear Mixed-Effects Model with Crossed Random
Effects Fitted Using the Function İme ()..................... 478
19.5 A Linear Mixed-Effects Model with Crossed Random
Effects and Heteroscedastic Residual Errors Fitted
Using lme().................................................... 485
19.5.1 Model Specification.................................... 485
19.5.2 R Syntax and Results................................... 486
19.6 Chapter Summary ................................................ 489
20 Extensions of the R Tools for Linear Mixed-Effects Models........... 49 İ
20.1 Introduction ................................................... 491
20.2 The New pdMatClass: pdKronecker................................. 491
20.2.1 Creating Objects of Class pdKronecker.................. 493
XX
Contents
20.2.2 Extracting Information from Objects of Class
pdKronecker........................................ 494
20.3 Influence Diagnostics....................................... 497
20.3.1 Preparatory Steps .................................. 497
20.3.2 Influence Diagnostics .............................. 501
20.4 Simulation of the Dependent Variable ....................... 509
20.5 Power Analysis ............................................. 511
20.5.1 Post Hoc Power Calculations....................... 512
20.5.2 A Priori Power Calculations
for a Hypothetical Study............................. 515
20.5.3 Power Evaluation Using Simulations ................. 521
Acronyms............................................................. 525
References............................................................. 527
Function Index....................................................... 531
Subject Index
537
|
| any_adam_object | 1 |
| author | Gałecki, Andrzej T. Burzykowski, Tomasz |
| author_GND | (DE-588)1146895003 (DE-588)1213542766 |
| author_facet | Gałecki, Andrzej T. Burzykowski, Tomasz |
| author_role | aut aut |
| author_sort | Gałecki, Andrzej T. |
| author_variant | a t g at atg t b tb |
| building | Verbundindex |
| bvnumber | BV040772759 |
| classification_rvk | CM 4000 SK 830 SK 840 SK 850 ST 250 |
| ctrlnum | (OCoLC)846534045 (DE-599)BVBBV040772759 |
| 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 | Informatik Psychologie Mathematik |
| format | Book |
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| id | DE-604.BV040772759 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:33:36Z |
| institution | BVB |
| isbn | 9781461438991 9781489996671 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025751118 |
| oclc_num | 846534045 |
| open_access_boolean | |
| owner | DE-824 DE-20 DE-11 DE-188 DE-355 DE-BY-UBR |
| owner_facet | DE-824 DE-20 DE-11 DE-188 DE-355 DE-BY-UBR |
| physical | xxxii, 542 Seiten Illustrationen, Diagramme |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer Texts in Statistics |
| spelling | Gałecki, Andrzej T. Verfasser (DE-588)1146895003 aut Linear mixed-effects models using R a step-by-step approach Andrzej Gałecki ; Tomasz Burzykowski New York, NY Springer [2013] © 2013 xxxii, 542 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Springer Texts in Statistics Software (DE-588)4055382-6 gnd rswk-swf Lineares Modell (DE-588)4134827-8 gnd rswk-swf R Programm (DE-588)4705956-4 gnd rswk-swf Gemischtes Modell (DE-588)4156565-4 gnd rswk-swf Lineares Modell (DE-588)4134827-8 s Gemischtes Modell (DE-588)4156565-4 s Software (DE-588)4055382-6 s R Programm (DE-588)4705956-4 s DE-604 Burzykowski, Tomasz Verfasser (DE-588)1213542766 aut Erscheint auch als Online-Ausgabe 978-1-4614-3900-4 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025751118&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gałecki, Andrzej T. Burzykowski, Tomasz Linear mixed-effects models using R a step-by-step approach Software (DE-588)4055382-6 gnd Lineares Modell (DE-588)4134827-8 gnd R Programm (DE-588)4705956-4 gnd Gemischtes Modell (DE-588)4156565-4 gnd |
| subject_GND | (DE-588)4055382-6 (DE-588)4134827-8 (DE-588)4705956-4 (DE-588)4156565-4 |
| title | Linear mixed-effects models using R a step-by-step approach |
| title_auth | Linear mixed-effects models using R a step-by-step approach |
| title_exact_search | Linear mixed-effects models using R a step-by-step approach |
| title_full | Linear mixed-effects models using R a step-by-step approach Andrzej Gałecki ; Tomasz Burzykowski |
| title_fullStr | Linear mixed-effects models using R a step-by-step approach Andrzej Gałecki ; Tomasz Burzykowski |
| title_full_unstemmed | Linear mixed-effects models using R a step-by-step approach Andrzej Gałecki ; Tomasz Burzykowski |
| title_short | Linear mixed-effects models using R |
| title_sort | linear mixed effects models using r a step by step approach |
| title_sub | a step-by-step approach |
| topic | Software (DE-588)4055382-6 gnd Lineares Modell (DE-588)4134827-8 gnd R Programm (DE-588)4705956-4 gnd Gemischtes Modell (DE-588)4156565-4 gnd |
| topic_facet | Software Lineares Modell R Programm Gemischtes Modell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025751118&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gałeckiandrzejt linearmixedeffectsmodelsusingrastepbystepapproach AT burzykowskitomasz linearmixedeffectsmodelsusingrastepbystepapproach |