INSEAD Library

Includes bibliographical references and index

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Linear Mixed Models A Practical Guide Using Statistical Software Contents Chapter 1 Introduction ................................................................................................... 1 1.1 What Are Linear Mixed Models (LMMs)? .....................................................................1 1.1.1 Models with Random Effects for Clustered Data .............................................. 2 1.1.2 Models for Longitudinal or Repeated-Measures Data ....................................... 2 1.1.3 The Purpose of this Book ................................................................................ 3 1.1.4 Outline of Book Contents ................................................................................ 4 1.2 A Brief History of LMMs .............................................................................................5 1.2.1 Key Theoretical Developments .........................................................................5 1.2.2 Key Software Developments ............................................................................ 7 Chapter 2 Linear Mixed Models: An Overview .................................................................. 9 2.1 Introduction .............................................................................................................. 9 2.1.1 Types and Structures of Data Sets .................................................................. 9 2.1.1.1 Clustered Data vs. Repeated-Measures and Longitudinal Data .......... 9 2.1.1.2 Levels of Data ................................................................................... 10 2.1.2 Types of Factors and their Related Effects in an LMM.......................................11 2.1.2.1 Fixed Factors .................................................................................... 12 2.1.2.2 Random Factors ............................................................................... 12 2.1.2.3 Fixed Factors vs. Random Factors .....................................................12 2.1.2.4 Fixed Effects vs. Random Effects .......................................................13 2.1.2.5 Nested vs. Crossed Factors and their Corresponding Effects ..............13 2.2 Specification of LMMs ................................................................................................15 2.2.1 General Specification for an Individual Observation .........................................15 2.2.2 General Matrix Specification ........................................................................... 16 2.2.2.1 Covariance Structures for the D Matrix ............................................. 19 2.2.2.2 Covariance Structures for the Ri Matrix ............................................ 20 2.2.2.3 Group-Specific Covariance Parameter Values for the D and Ri Matrices ...........................................................................................21 2.2.3 Alternative Matrix Specification for All Subjects ...............................................21 2.2.4 Hierarchical Linear Model (HLM) Specification of the LMM............................... 22 2.3 The Marginal Linear Model ........................................................................................ 22 2.3.1 Specification of the Marginal Model.................................................................. 22 2.3.2 The Marginal Model Implied by an LMM .......................................................... 23 2.4 Estimation in LMMs................................................................................................... 25 2.4.1 Maximum Likelihood (ML) Estimation.............................................................. 25 2.4.1.1 Special Case: Assume 0 is Known...................................................... 26 2.4.1.2 General Case: Assume 0 is Unknown................................................. 27 2.4.2 REML Estimation ............................................................................................28 2.4.3 REML vs. ML Estimation..................................................................................28 2.5 Computational Issues ................................................................................................30 2.5.1 Algorithms for Likelihood Function Optimization ............................................. 30 2.5.2 Computational Problems with Estimation of Covariance Parameters ................31 2.6 Tools for Model Selection ........................................................................................... 33 2.6.1 Basic Concepts in Model Selection ................................................................. 34 2.6.1.1 Nested Models.................................................................................. 34 2.6.1.2 Hypotheses: Specification and Testing ............................................. 34 2.6.2 Likelihood Ratio Tests (LRTs) ......................................................................... 34 2.6.2.1 Likelihood Ratio Tests for Fixed-Effect Parameters ........................... 35 2.6.2.2 Likelihood Ratio Tests for Covariance Parameters............................. 35 2.6.3 Alternative Tests ............................................................................................ 36 2.6.3.1 Alternative Tests for Fixed-Effect Parameters ................................... 37 2.6.3.2 Alternative Tests for Covariance Parameters .................................... 38 2.6.4 Information Criteria ....................................................................................... 38 2.7 Model-Building Strategies ........................................................................................ 39 2.7.1 The Top-Down Strategy ................................................................................. 39 2.7.2 The Step-Up Strategy .....................................................................................40 2.8 Checking Model Assumptions (Diagnostics) .............................................................. 41 2.8.1 Residual Diagnostics ..................................................................................... 41 2.8.1.1 Conditional Residuals ......................................................................41 2.8.1.2 Standardized and Studentized Residuals ......................................... 42 2.8.2 Influence Diagnostics .................................................................................... 42 2.8.3 Diagnostics for Random Effects ..................................................................... 43 2.9 Other Aspects of LMMs ............................................................................................ 43 2.9.1 Predicting Random Effects: Best Linear Unbiased Predictors .......................... 43 2.9.2 Intraclass Correlation Coefficients (ICCs) ....................................................... 45 2.9.3 Problems with Model Specification (Aliasing) .................................................. 46 2.9.4 Missing Data ................................................................................................. 48 2.9.5 Centering Covariates ..................................................................................... 49 2.10 Chapter Summary...................................................................................................49 Chapter 3 Two-Level Models for Clustered Data: The Rat Pup Example .................................................................................51 3.1 Introduction .............................................................................................................51 3.2 The Rat Pup Study ................................................................................................... 51 3.2.1 Study Description ..........................................................................................51 3.2.2 Data Summary............................................................................................... 54 3.3 Overview of the Rat Pup Data Analysis ..................................................................... 58 3.3.1 Analysis Steps ............................................................................................... 58 3.3.2 Model Specification ........................................................................................60 3.3.2.1 General Model Specification .............................................................. 60 3.3.2.2 Hierarchical Model Specification ....................................................... 62 3.3.3 Hypothesis Tests ........................................................................................... 63 3.4 Analysis Steps in the Software Procedures ............................................................... 66 3.4.1 SAS ............................................................................................................... 66 3.4.2 SPSS ............................................................................................................. 74 3.4.3 R ................................................................................................................... 77 3.4.4 Stata ............................................................................................................. 82 3.4.5 HLM .............................................................................................................. 85 3.4.5.1 Data Set Preparation ........................................................................ 85 3.4.5.2 Preparing the Multivariate Data Matrix (MDM) File ............................86 3.5 Results of Hypothesis Tests ......................................................................................90 3.5.1 Likelihood Ratio Tests for Random Effects ......................................................90 3.5.2 Likelihood Ratio Tests for Residual Variance .................................................. 91 3.5.3 Ftests and Likelihood Ratio Tests for Fixed Effects ......................................... 91 3.6 Comparing Results across the Software Procedures .................................................92 3.6.1 Comparing Model 3.1 Results ....................................................................... 92 3.6.2 Comparing Model 3.2B Results .....................................................................94 3.6.3 Comparing Model 3.3 Results ....................................................................... 95 3.7 Interpreting Parameter Estimates in the Final Model ............................................... 96 3.7.1 Fixed-Effect Parameter Estimates ................................................................. 96 3.7.2 Covariance Parameter Estimates................................................................... 97 3.8 Estimating the Intraclass Correlation Coefficients (ICCs) ......................................... 98 3.9 Calculating Predicted Values ...................................................................................100 3.9.1 Litter-Specific (Conditional) Predicted Values ............................................ 100 3.9.2 Population-Averaged (Unconditional) Predicted Values .................................. 101 3.10 Diagnostics for the Final Model ............................................................................. 102 3.10.1 Residual Diagnostics .................................................................................. 102 3.10.1.1 Conditional Residuals ................................................................. 102 3.10.1.2 Conditional Studentized Residuals .............................................. 104 3.10.2 Influence Diagnostics ................................................................................. 106 3.10.2.1 Overall and Fixed-Effects Influence Diagnostics ...........................106 3.10.2.2 Influence on Covariance Parameters ............................................107 3.11 Software Notes ...................................................................................................... 108 3.11.1 Data Structure ........................................................................................... 108 3.11.2 Syntax vs. Menus ....................................................................................... 109 3.11.3 Heterogeneous Residual Variances for Level 2 Groups ................................ 109 3.11.4 Display of the Marginal Covariance and Correlation Matrices ................................................................................. 109 3.11.5 Differences in Model Fit Criteria ................................................................. 109 3.11.6 Differences in Tests for Fixed Effects ...........................................................110 3.11.7 Post-Hoc Comparisons of LS Means (Estimated Marginal Means) ......................................................................111 3.11.8 Calculation of Studentized Residuals and Influence Statistics ................................................................................... 112 3.11.9 Calculation of EBLUPs.................................................................................112 3.11.10 Tests for Covariance Parameters................................................................ 112 3.11.11 Refeernce Categories for Fixed Factors ...................................................... 112 Chapter 4 Three-Level Models for Clustered Data: The Classroom Example............................................................................ 115 4.1 Introduction ........................................................................................................... 115 4.2 The Classroom Study ..............................................................................................117 4.2.1 Study Description .........................................................................................117 4.2.2 Data Summary ............................................................................................. 118 4.2.2.1 Data Set Preparation .....................................................................119 4.2.2.2 Preparing the Multivariate Data Matrix (MDM) File ........................ 119 4.3 Overview of the Classroom Data Analysis ................................................................ 122 4.3.1 Analysis Steps .......................................................................................... 122 4.3.2 Model Specification ...................................................................................... 125 4.3.2.1 General Model Specification .......................................................... 125 4.3.2.2 Hierarchical Model Specification ....................................................126 4.3.3 Hypothesis Tests .......................................................................................... 128 4.4 Analysis Steps in the Software Procedures .............................................................. 130 4.4.1 SAS ..............................................................................................................130 4.4.2 SPSS ............................................................................................................ 136 4.4.3 R ................................................................................................................. 141 4.4.4 Stata ........................................................................................................... 144 4.4.5 HLM ............................................................................................................ 147 4.5 Results of Hypothesis Tests ....................................................................................153 4.5.1 Likelihood Ratio Test for Random Effects ..................................................... 153 4.5.2 Likelihood Ratio Tests and t-Tests for Fixed Effects ......................................154 4.6 Comparing Results across the Software Procedures ................................................155 4.6.1 Comparing Model 4.1 Results ...................................................................... 155 4.6.2 Comparing Model 4.2 Results ...................................................................... 156 4.6.3 Comparing Model 4.3 Results ...................................................................... 157 4.6.4 Comparing Model 4.4 Results ...................................................................... 159 4.7 Interpreting Parameter Estimates in the Final Model .............................................. 159 4.7.1 Fixed-Effect Parameter Estimates ................................................................ 159 4.7.2 Covariance Parameter Estimates ................................................................. 161 4.8 Estimating the Intraclass Correlation Coefficients (ICCs) ........................................ 162 4.9 Calculating Predicted Values...................................................................................165 4.9.1 Conditional and Marginal Predicted Values .................................................. 165 4.9.2 Plotting Predicted Values Using HLM............................................................ 166 4.10 Diagnostics for the Final Model ............................................................................ 167 4.10.1 Plots of the EBLUPs ................................................................................... 167 4.10.2 Residual Diagnostics ................................................................................. 169 4.11 Software Notes ..................................................................................................... 171 4.11.1 REML vs. ML Estimation............................................................................ 171 4.11.2 Setting up Three-Level Models in HLM ....................................................... 171 4.11.3 Calculation of Degrees of Freedom for t-Tests in HLM..................................171 4.11.4 Analyzing Cases with Complete Data.......................................................... 172 4.11.5 Miscellaneous Differences ..........................................................................173 Chapter 5 Models for Repeated-Measures Data: The Rat Brain Example .......................175 5.1 Introduction ...........................................................................................................175 5.2 The Rat Brain Study .............................................................................................. 176 5.2.1 Study Description ........................................................................................176 5.2.2 Data Summary ............................................................................................ 178 5.3 Overview of the Rat Brain Data Analysis ................................................................ 180 5.3.1 Analysis Steps ............................................................................................. 180 5.3.2 Model Specification ......................................................................................182 5.3.2.1 General Model Specification ...........................................................182 5.3.2.2 Hierarchical Model Specification .................................................... 184 5.3.3 Hypothesis Tests ......................................................................................... 185 5.4 Analysis Steps in the Software Procedures ............................................................. 187 5.4.1 SAS ............................................................................................................. 187 5.4.2 SPSS ........................................................................................................... 190 5.4.3 R ................................................................................................................. 193 5.4.4 Stata ........................................................................................................... 195 5.4.5 HLM ............................................................................................................ 198 5.4.5.1 Data Set Preparation ..................................................................... 198 5.4.5.2 Preparing the MDM File..................................................................199 5.5 Results of Hypothesis Tests ....................................................................................203 5.5.1 Likelihood Ratio Tests for Random Effects ................................................... 203 5.5.2 Likelihood Ratio Tests for Residual Variance ................................................ 203 5.5.3 F-Tests for Fixed Effects .............................................................................. 204 5.6 Comparing Results across the Software Procedures .............................................. 204 5.6.1 Comparing Model 5.1 Results ......................................................................204 5.6.2 Comparing Model 5.2 Results ......................................................................206 5.7 Interpreting Parameter Estimates in the Final Model ............................................ 207 5.7.1 Fixed-Effect Parameter Estimates ................................................................ 207 5.7.2 Covariance Parameter Estimates ................................................................. 209 5.8 The Implied Marginal Variance-Covariance Matrix for the Final Model................................................................................................209 5.9 Diagnostics for the Final Model ............................................................................ 211 5.10 Software Notes .....................................................................................................214 5.10.1 Heterogeneous Residual Variances for Level 1 Groups ............................... 214 5.10.2 EBLUPs for Multiple Random Effects .........................................................214 5.11 Other Analytic Approaches .................................................................................. 214 5.11.1 Kronecker Product for More Flexible Residual Covariance Structures .............................................................................. 214 5.11.2 Fitting the Marginal Model ........................................................................ 216 5.11.3 Repeated-Measures ANOVA .......................................................................217 Chapter 6 Random Coefficient Models for Longitudinal Data: The Autism Example ............................................................................... 219 6.1 Introduction .......................................................................................................... 219 6.2 The Autism Study ..................................................................................................220 6.2.1 Study Description ....................................................................................... 220 6.2.2 Data Summary ............................................................................................221 6.3 Overview of the Autism Data Analysis .................................................................... 225 6.3.1 Analysis Steps .............................................................................................226 6.3.2 Model Specification ..................................................................................... 227 6.3.2.1 General Model Specification .......................................................... 227 6.3.2.2 Hierarchical Model Specification ................................................... 229 6.3.3 Hypothesis Tests ......................................................................................... 230 6.4 Analysis Steps in the Software Procedures ............................................................. 232 6.4.1 SAS ............................................................................................................ 232 6.4.2 SPSS ...........................................................................................................236 6.4.3 R ................................................................................................................ 240 6.4.4 Stata ...........................................................................................................243 6.4.5 HLM ........................................................................................................... 246 6.4.5.1 Data Set Preparation....................................................................... 246 6.4.5.2 Preparing the MDM File...................................................................246 6.5 Results of Hypothesis Tests ................................................................................... 251 6.5.1 Likelihood Ratio Test for Random Effects .....................................................251 6.5.2 Likelihood Ratio Tests for Fixed Effects ........................................................252 6.6 Comparing Results across the Software Procedures ............................................... 253 6.6.1 Comparing Model 6.1 Results ......................................................................253 6.6.2 Comparing Model 6.2 Results ......................................................................253 6.6.3 Comparing Model 6.3 Results ......................................................................253 6.7 Interpreting Parameter Estimates in the Final Model ..............................................254 6.7.1 Fixed-Effect Parameter Estimates ................................................................ 256 6.7.2 Covariance Parameter Estimates ................................................................. 257 6.8 Calculating Predicted Values ................................................................................. 259 6.8.1 Marginal Predicted Values ........................................................................... 259 6.8.2 Conditional Predicted Values .......................................................................261 6.9 Diagnostics for the Final Model...............................................................................263 6.9.1 Residual Diagnostics ...................................................................................263 6.9.2 Diagnostics for the Random Effects ............................................................. 265 6.9.3 Observed and Predicted Values ................................................................... 266 6.10 Software Note: Computational Problems with the D Matrix ...................................268 6.11 An Alternative Approach: Fitting the Marginal Model with an Unstructured Covariance Matrix ..............................................................268 Chapter 7 Models for Clustered Longitudinal Data: The Dental Veneer Example .....................................................................273 7.1 Introduction .......................................................................................................... 273 7.2 The Dental Veneer Study ....................................................................................... 274 7.2.1 Study Description ....................................................................................... 274 7.2.2 Data Summary ........................................................................................... 275 7.3 Overview of the Dental Veneer Data Analysis ......................................................... 277 7.3.1 Analysis Steps ............................................................................................ 278 7.3.2 Model Specification ..................................................................................... 280 7.3.2.1 General Model Specification .......................................................... 280 7.3.2.2 Hierarchical Model Specification ................................................... 284 7.3.3 Hypothesis Tests .........................................................................................285 7.4 Analysis Steps in the Software Procedures ............................................................. 287 7.4.1 SAS ............................................................................................................ 287 7.4.2 SPSS ...........................................................................................................293 7.4.3 R ................................................................................................................ 296 7.4.4 Stata ...........................................................................................................300 7.4.5 HLM ........................................................................................................... 304 7.4.5.1 Data Set Preparation......................................................................304 7.4.5.2 Preparing the Multivariate Data Matrix (MDM) File ........................304 7.5 Results of Hypothesis Tests ................................................................................... 309 7.5.1 Likelihood Ratio Tests for Random Effects ................................................... 309 7.5.2 Likelihood Ratio Tests for Residual Variance ............................................... 310 7.5.3 Likelihood Ratio Tests for Fixed Effects ....................................................... 310 7.6 Comparing Results across the Software Procedures ............................................... 310 7.6.1 Comparing Model 7.1 Results ......................................................................310 7.6.2 Comparing Software Results for Model 7.2A, Model 7.2B, and Model 7.2C ........................................................................................ 312 7.6.3 Comparing Model 7.3 Results ......................................................................314 7.7 Interpreting Parameter Estimates in the Final Model ..............................................315 7.7.1 Fixed-Effect Parameter Estimates ................................................................315 7.7.2 Covariance Parameter Estimates ................................................................. 316 7.8 The Implied Marginal Variance-Covariance Matrix for the Final Model ...............................................................................................317 7.9 Diagnostics for the Final Model ..............................................................................319 7.9.1 Residual Diagnostics ...................................................................................319 7.9.2 Diagnostics for the Random Effects ............................................................. 321 7.10 Software Notes .................................................................................................... 323 7.10.1 ML vs. REML Estimation............................................................................ 323 7.10.2 The Ability to Remove Random Effects from a Model ..................................324 7.10.3 The Ability to Fit Models with Different Residual Covariance Structures .............................................................................. 324 7.10.4 Aliasing of Covariance Parameters .............................................................324 7.10.5 Displaying the Marginal Covariance and Correlation Matrices ................... 325 7.10.6 Miscellaneous Software Notes ................................................................... 325 7.11 Other Analytic Approaches .................................................................................. 326 7.11.1 Modeling the Covariance Structure ........................................................... 326 7.11.2 The Step-Up vs. Step-Down Approach to Model Building ........................... 327 7.11.3 Alternative Uses of Baseline Values for the Dependent Variable ................. 327 Appendix A Statistical Software Resources .................................................................. 329 A.1 Descriptions/Availability of Software Packages....................................................... 329 A.1.1 SAS ............................................................................................................ 329 A.1.2 SPSS .......................................................................................................... 329 A.1.3 R ................................................................................................................ 329 A.1.4 Stata .......................................................................................................... 330 A.1.5 HLM ........................................................................................................... 330 A.2 Useful Internet Links............................................................................................. 330 Appendix B Calculation of the Marginal Variance-Covariance Matrix ........................... 333 Appendix C Acronyms/Abbreviations........................................................................... 335 References...................................................................................................................337 Index........................................................................................................................... 341