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Introduction 1
What Are Linear Mixed Models (LMMs)? 1
Models with Random Effects for Clustered Data 2
Models for Longitudinal or Repeated-Measures Data 2
The Purpose of this Book 3
Outline of Book Contents 4
A Brief History of LMMs 5
Key Theoretical Developments 5
Key Software Developments 7
Linear Mixed Models: An Overview 9
Introduction 9
Types and Structures of Data Sets 9
Clustered Data vs. Repeated-Measures and Longitudinal Data 9
Levels of Data 10
Types of Factors and their Related Effects in an LMM 11
Fixed Factors 12
Random Factors 12
Fixed Factors vs. Random Factors 12
Fixed Effects vs. Random Effects 13
Nested vs. Crossed Factors and their Corresponding Effects 13
Specification of LMMs 15
General Specification for an Individual Observation 15
General Matrix Specification 16
Covariance Structures for the D Matrix 19
Covariance Structures for the R[subscript i] Matrix 20
Group-SpecificCovariance Parameter Values for the D and R[subscript i] Matrices 21
Alternative Matrix Specification for All Subjects 21
Hierarchical Linear Model (HLM) Specification of the LMM 22
The Marginal Linear Model 22
Specification of the Marginal Model 22
The Marginal Model Implied by an LMM 23
Estimation in LMMs 25
Maximum Likelihood (ML) Estimation 25
Special Case: Assume [theta] is Known 26
General Case: Assume [theta] is Unknown 27
REML Estimation 28
REML vs. ML Estimation 28
Computational Issues 30
Algorithms for Likelihood Function Optimization 30
Computational Problems with Estimation of Covariance Parameters 31
Tools for Model Selection 33
Basic Concepts in Model Selection 34
Nested Models 34
Hypotheses: Specification and Testing 34
Likelihood Ratio Tests (LRTs) 34
Likelihood Ratio Tests for Fixed-Effect Parameters 35
Likelihood Ratio Tests for Covariance Parameters 35
Alternative Tests 36
Alternative Tests for Fixed-Effect Parameters 37
Alternative Tests for Covariance Parameters 38
Information Criteria 38
Model-Building Strategies 39
The Top-Down Strategy 39
The Step-Up Strategy 40
Checking Model Assumptions (Diagnostics) 41
Residual Diagnostics 41
Conditional Residuals 41
Standardized and Studentized Residuals 42
Influence Diagnostics 42
Diagnostics for Random Effects 43
Other Aspects of LMMs 43
Predicting Random Effects: Best Linear Unbiased Predictors 43
Intraclass Correlation Coefficients (ICCs) 45
Problems with Model Specification (Aliasing) 46
Missing Data 48
Centering Covariates 49
Chapter Summary 49
Two-Level Models for Clustered Data: The Rat Pup Example 51
Introduction 51
The Rat Pup Study 51
Study Description 51
Data Summary 54
Overview of the Rat Pup Data Analysis 58
Analysis Steps 58
Model Specification 60
General Model Specification 60
Hierarchical Model Specification 63
Hypothesis Tests 63
Analysis Steps in the Software Procedures 66
SAS 66
SPSS 74
R 77
Stata 82
HLM 85
Data Set Preparation 85
Preparing the Multivariate Data Matrix (MDM) File 86
Results of Hypothesis Tests 90
Likelihood Ratio Tests for Random Effects 90
Likelihood Ratio Tests for Residual Variance 91
F-tests and Likelihood Ratio Tests for Fixed Effects 91
Comparing Results across the Software Procedures 92
Comparing Model 3.1 Results 92
Comparing Model 3.2B Results 94
Comparing Model 3.3 Results 95
Interpreting Parameter Estimates in the Final Model 96
Fixed-Effect Parameter Estimates 96
Covariance Parameter Estimates 97
Estimating the Intraclass Correlation Coefficients (ICCs) 98
Calculating Predicted Values 100
Litter-Specific (Conditional) Predicted Values 100
Population-Averaged (Unconditional) Predicted Values 101
Diagnostics for the Final Model 102
Residual Diagnostics 102
Conditional Residuals 102
Conditional Studentized Residuals 104
Influence Diagnostics 106
Overall and Fixed-Effects Influence Diagnostics 106
Influence on Covariance Parameters 107
Software Notes 108
Data Structure 108
Syntax vs. Menus 109
Heterogeneous Residual Variances for Level 2 Groups 109
Display of the Marginal Covariance and Correlation Matrices 109
Differences in Model Fit Criteria 109
Differences in Tests for Fixed Effects 110
Post-Hoc Comparisons of LS Means (Estimated Marginal Means) 111
Calculation of Studentized Residuals and Influence Statistics 112
Calculation of EBLUPs 112
Tests for Covariance Parameters 112
Refeernce Categories for Fixed Factors 112
Three-Level Models for Clustered Data: The Classroom Example 115
Introduction 115
The Classroom Study 117
Study Description 117
Data Summary 118
Data Set Preparation 119
Preparing the Multivariate Data Matrix (MDM) File 119
Overview of the Classroom Data Analysis 122
Analysis Steps 122
Model Specification 125
General Model Specification 125
Hierarchical Model Specification 126
Hypothesis Tests 128
Analysis Steps in the Software Procedures 130
SAS 130
SPSS 136
R 141
Stata 144
HLM 147
Results of Hypothesis Tests 153
Likelihood Ratio Test for Random Effects 153
Likelihood Ratio Tests and t-Tests for Fixed Effects 154
Comparing Results across the Software Procedures 155
Comparing Model 4.1 Results 155
Comparing Model 4.2 Results 156
Comparing Model 4.3 Results 157
Comparing Model 4.4 Results 159
Interpreting Parameter Estimates in the Final Model 159
Fixed-Effect Parameter Estimates 159
Covariance Parameter Estimates 161
Estimating the Intraclass Correlation Coefficients (ICCs) 162
Calculating Predicted Values 165
Conditional and Marginal Predicted Values 165
Plotting Predicted Values Using HLM 166
Diagnostics for the Final Model 167
Plots of the EBLUPs 167
Residual Diagnostics 169
Software Notes 171
REML vs. ML Estimation 171
Setting up Three-Level Models in HLM 171
Calculation of Degrees of Freedom for t-Tests in HLM 171
Analyzing Cases with Complete Data 172
Miscellaneous Differences 173
Models for Repeated-Measures Data: The Rat Brain Example 175
Introduction 175
The Rat Brain Study 176
Study Description 176
Data Summary 178
Overview of the Rat Brain Data Analysis 180
Analysis Steps 180
Model Specification 182
General Model Specification 182
Hierarchical Model Specification 184
Hypothesis Tests 185
Analysis Steps in the Software Procedures 187
SAS 187
SPSS 190
R 193
Stata 195
HLM 198
Data Set Preparation 198
Preparing the MDM File 199
Results of Hypothesis Tests 203
Likelihood Ratio Tests for Random Effects 203
Likelihood Ratio Tests for Residual Variance 203
F-Tests for Fixed Effects 204
Comparing Results across the Software Procedures 204
Comparing Model 5.1 Results 204
Comparing Model 5.2 Results 206
Interpreting Parameter Estimates in the Final Model 207
Fixed-Effect Parameter Estimates 207
Covariance Parameter Estimates 209
The Implied Marginal Variance-Covariance Matrix for the Final Model 209
Diagnostics for the Final Model 211
Software Notes 214
Heterogeneous Residual Variances for Level 1 Groups 214
EBLUPs for Multiple Random Effects 214
Other Analytic Approaches 214
Kronecker Product for More Flexible Residual Covariance Structures 214
Fitting the Marginal Model 216
Repeated-Measures ANOVA 217
Random Coefficient Models for Longitudinal Data: The Autism Example 219
Introduction 219
The Autism Study 220
Study Description 220
Data Summary 221
Overview of the Autism Data Analysis 225
Analysis Steps 226
Model Specification 227
General Model Specification 227
Hierarchical Model Specification 229
Hypothesis Tests 230
Analysis Steps in the Software Procedures 232
SAS 232
SPSS 236
R 240
Stata 243
HLM 246
Data Set Preparation 246
Preparing the MDM File 246
Results of Hypothesis Tests 251
Likelihood Ratio Test for Random Effects 251
Likelihood Ratio Tests for Fixed Effects 252
Comparing Results across the Software Procedures 253
Comparing Model 6.1 Results 253
Comparing Model 6.2 Results 253
Comparing Model 6.3 Results 253
Interpreting Parameter Estimates in the Final Model 254
Fixed-Effect Parameter Estimates 256
Covariance Parameter Estimates 257
Calculating Predicted Values 259
Marginal Predicted Values 259
Conditional Predicted Values 261
Diagnostics for the Final Model 263
Residual Diagnostics 263
Diagnostics for the Random Effects 265
Observed and Predicted Values 266
Software Note: Computational Problems with the D Matrix 268
An Alternative Approach: Fitting the Marginal Model with an Unstructured Covariance Matrix 268
Models for Clustered Longitudinal Data: The Dental Veneer Example 273
Introduction 273
The Dental Veneer Study 274
Study Description 274
Data Summary 275
Overview of the Dental Veneer Data Analysis 277
Analysis Steps 278
Model Specification 280
General Model Specification 280
Hierarchical Model Specification 284
Hypothesis Tests 285
Analysis Steps in the Software Procedures 287
SAS 287
SPSS 293
R 296
Stata 300
HLM 304
Data Set Preparation 304
Preparing the Multivariate Data Matrix (MDM) File 304
Results of Hypothesis Tests 309
Likelihood Ratio Tests for Random Effects 309
Likelihood Ratio Tests for Residual Variance 310
Likelihood Ratio Tests for Fixed Effects 310
Comparing Results across the Software Procedures 310
Comparing Model 7.1 Results 310
Comparing Software Results for Model 7.2A, Model 7.2B, and Model 7.2C 312
Comparing Model 7.3 Results 314
Interpreting Parameter Estimates in the Final Model 315
Fixed-Effect Parameter Estimates 315
Covariance Parameter Estimates 316
The Implied Marginal Variance-Covariance Matrix for the Final Model 317
Diagnostics for the Final Model 319
Residual Diagnostics 319
Diagnostics for the Random Effects 321
Software Notes 323
ML vs. REML Estimation 323
The Ability to Remove Random Effects from a Model 324
The Ability to Fit Models with Different Residual Covariance Structures 324
Aliasing of Covariance Parameters 324
Displaying the Marginal Covariance and Correlation Matrices 325
Miscellaneous Software Notes 325
Other Analytic Approaches 326
Modeling the Covariance Structure 326
The Step-Up vs. Step-Down Approach to Model Building 327
Alternative Uses of Baseline Values for the Dependent Variable 327
Statistical Software Resources 329
Descriptions/Availability of Software Packages 329
SAS 329
SPSS 329
R 329
Stata 330
HLM 330
Useful Internet Links 330
Calculation of the Marginal Variance-Covariance Matrix 333
Acronyms/Abbreviations 335
References 337
Index 341
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Add Linear Mixed Models: A Practical Guide Using Statistical Software, Simplifying the often confusing array of software programs for fitting linear mixed models (LMMs), Linear Mixed Models: A Practical Guide Using Statistical Software provides a basic introduction to primary concepts, notation, software implementatio, Linear Mixed Models: A Practical Guide Using Statistical Software to the inventory that you are selling on WonderClubX
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Add Linear Mixed Models: A Practical Guide Using Statistical Software, Simplifying the often confusing array of software programs for fitting linear mixed models (LMMs), Linear Mixed Models: A Practical Guide Using Statistical Software provides a basic introduction to primary concepts, notation, software implementatio, Linear Mixed Models: A Practical Guide Using Statistical Software to your collection on WonderClub |