Preface to the Third Edition
Preface to the Second Edition
Preface
The Authors
Acknowledgments
List of Tables
List of Figures
1 Introduction
1.1 What Are Linear Mixed Models (LMMs)?
1.1.1 Models with Random Effects for Clustered Data
1.1.2 Models for Longitudinal or Repeated-Measures Data
1.1.3 The Purpose of This Book
1.1.4 Outline of Book Contents
1.2 A Brief History of LMMs
1.2.1 Key Theoretical Developments
1.2.2 Key Software Developments
2 Linear Mixed Models: An Overview
2.1 Introduction
2.1.1 Types and Structures of Data Sets
2.1.1.1 Clustered Data vs. Repeated-Measures and Longitudinal Data
2.1.1.2 Levels of Data
2.1.2 Types of Factors and Their Related Effects in an LMM
2.1.2.1 Fixed Factors
2.1.2.2 Random Factors
2.1.2.3 Fixed Factors vs. Random Factors
2.1.2.4 Fixed Effects vs. Random Effects
2.1.2.5 Nested vs. Crossed Factors and Their Corresponding Effects
2.2 Specification of LMMs
2.2.1 General Specification for an Individual Observation
2.2.2 General Matrix Specification
2.2.2.1 Covariance Structures for the D Matrix
2.2.2.2 Covariance Structures for the Ri Matrix
2.2.2.3 Group-Specific Covariance Parameter Values for the D and Ri Matrices
2.2.3 Alternative Matrix Specification for All Subjects
2.2.4 Hierarchical Linear Model (HLM) Specification of the LMM
2.3 The Marginal Linear Model
2.3.1 Specification of the Marginal Model
2.3.2 The Marginal Model Implied by an LMM
2.4 Estimation in LMMs
2.4.1 Maximum Likelihood (ML) Estimation
2.4.1.1 Special Case: Assume Θ Is Known
2.4.1.2 General Case: Assume Θ Is Unknown
2.4.2 REML Estimation
2.4.3 REML vs. ML Estimation
2.5 Computational Issues
2.5.1 Algorithms for Likelihood Function Optimization
2.5.2 Computational Problems with Estimation of Covariance Parameters
2.6 Tools for Model Selection
2.6.1 Basic Concepts in Model Selection
2.6.1.1 Nested Models
2.6.1.2 Hypotheses: Specification and Testing
2.6.2 Likelihood Ratio Tests (LRTs)
2.6.2.1 Likelihood Ratio Tests for Fixed-Effect Parameters
2.6.2.2 Likelihood Ratio Tests for Covariance Parameters
2.6.3 Alternative Tests
2.6.3.1 Alternative Tests for Fixed-Effect Parameters
2.6.3.2 Alternative Tests for Covariance Parameters
2.6.4 Information Criteria
2.7 Model-Building Strategies
2.7.1 The Top-Down Strategy
2.7.2 The Step-Up Strategy
2.8 Checking Model Assumptions (Diagnostics)
2.8.1 Residual Diagnostics
2.8.1.1 Raw Residuals
2.8.1.2 Standardized and Studentized Residuals
2.8.2 Influence Diagnostics
2.8.3 Diagnostics for Random Effects
2.9 Other Aspects of LMMs
2.9.1 Predicting Random Effects: Best Linear Unbiased Predictors
2.9.2 Intraclass Correlation Coefficients (ICCs)
2.9.3 Problems with Model Specification (Aliasing)
2.9.4 Missing Data
2.9.5 Centering Covariates
2.9.6 Fitting Linear Mixed Models to Complex Sample Survey Data
2.9.6.1 Purely Model-Based Approaches
2.9.6.2 Hybrid Design- and Model-Based Approaches
2.9.7 Bayesian Analysis of Linear Mixed Models
2.10 Chapter Summary
3 Two-Level Models for Clustered Data: The Rat Pup Example
3.1 Introduction
3.2 The Rat Pup Study
3.2.1 Study Description
3.2.2 Data Summary
3.3 Overview of the Rat Pup Data Analysis
3.3.1 Analysis Steps
3.3.2 Model Specification
3.3.2.1 General Model Specification
3.3.2.2 Hierarchical Model Specification
3.3.3 Hypothesis Tests
3.4 Anaylsis Steps in the Software Procedures
3.4.1 SAS
3.4.2 SPSS
3.4.3 R
3.4.3.1 Analysis Using the lme() Function
3.4.3.2 Analysis Using the lmer() Function
3.4.4 Stata
3.4.5 HLM
3.4.5.1 Data Set Preparation
3.4.5.2 Preparing the Multivariate Data Matrix (MDM) File
3.5 Results of Hypothesis Tests
3.5.1 Likelihood Ratio Tests for Random Effects
3.5.2 Likelihood Ratio Tests for Residual Error Variance
3.5.3 F-tests and Likelihood Ratio Tests for Fixed Effects
3.6 Comparing Results across the Software Procedures
3.6.1 Comparing Model 3.1 Results
3.6.2 Comparing Model 3.2B Results
3.6.3 Comparing Model 3.3 Results
3.7 Interpreting Parameter Estimates in the Final Model
3.7.1 Fixed-Effect Parameter Estimates
3.7.2 Covariance Parameter Estimates
3.8 Estimating the Intraclass Correlation Coefficients (ICCs)
3.9 Calculating Predicted Values
3.9.1 Litter-Specific (Conditional) Predicted Values
3.9.2 Population-Averaged (Unconditional) Predicted Values
3.10 Diagnostics for the Final Model
3.10.1 Residual Diagnostics
3.10.1.1 Conditional Residuals
3.10.1.2 Conditional Studentized Residuals
3.10.2 Distribution of BLUPS
3.10.3 Influence Diagnostics
3.10.2.1 Influence on Covariance Parameters
3.10.2.2 Influence on Fixed Effects
3.11 Software Notes and Recommendations
3.11.1 Data Structure
3.11.2 Syntax vs. Menus
3.11.3 Heterogeneous Residual Variances for Level 2 Groups
3.11.4 Display of the Marginal Covariance and Correlation Matrices
3.11.5 Differences in Model Fit Criteria
3.11.6 Differences in Tests for Fixed Effects
3.11.7 Post-Hoc Comparisons of LS Means (Estimated Marginal Means)
3.11.8 Calculation of Studentized Residuals and Influence Statistics
3.11.9 Calculation of EBLUPs
3.11.10 Tests for Covariance Parameters
3.11.11 Reference Categories for Fixed Factors
4 Three-Level Models for Clustered Data: The Classroom Example
4.1 Introduction
4.2 The Classroom Study
4.2.1 Study Description
4.2.2 Data Summary
4.2.2.1 Data Set Preparation
4.2.2.2 Preparing the Multivariate Data Matrix (MDM) File
4.3 Overview of the Classroom Data Analysis
4.3.1 Analysis Steps
4.3.2 Model Specification
4.3.2.1 General Model Specification
4.3.2.2 Hierarchical Model Specification
4.3.3 Hypothesis Tests
4.4 Analysis Steps in the Software Procedures
4.4.1 SAS
4.4.2 SPSS
4.4.3 R
4.4.3.1 Analysis Using the lme() Function
4.4.3.2 Analysis Using the lmer() Function
4.4.4 Stata
4.4.5 HLM
4.5 Results of Hypothesis Tests
4.5.1 Likelihood Ratio Tests for Random Effects
4.5.2 Likelihood Ratio Tests and t-Tests for Fixed Effects
4.6 Comparing Results across the Software Procedures
4.6.1 Comparing Model 4.1 Results
4.6.2 Comparing Model 4.2 Results
4.6.3 Comparing Model 4.3 Results
4.6.4 Comparing Model 4.4 Results
4.7 Interpreting Parameter Estimates in the Final Model
4.7.1 Fixed-Effect Parameter Estimates
4.7.2 Covariance Parameter Estimates
4.8 Estimating the Intraclass Correlation Coefficients (ICCs)
4.9 Calculating Predicted Values
4.9.1 Conditional and Marginal Predicted Values
4.9.2 Plotting Predicted Values Using HLM
4.10 Diagnostics for the Final Model
4.10.1 Plots of the EBLUPs
4.10.2 Residual Diagnostics
4.11 Software Notes
4.11.1 REML vs. ML Estimation
4.11.2 Setting up Three-Level Models in HLM
4.11.3 Calculation of Degrees of Freedom for t-Tests in HLM
4.11.4 Analyzing Cases with Complete Data
4.11.5 Miscellaneous Differences
4.12 Recommendations
5 Models for Repeated-Measures Data: The Rat Brain Example
5.1 Introduction
5.2 The Rat Brain Study
5.2.1 Study Description
5.2.2 Data Summary
5.3 Overview of the Rat Brain Data Analysis
5.3.1 Analysis Steps
5.3.2 Model Specification
5.3.2.1 General Model Specification
5.3.2.2 Hierarchical Model Specification
5.3.3 Hypothesis Tests
5.4 Analysis Steps in the Sofware Procedures
5.4.1 SAS
5.4.2 SPSS
5.4.3 R
5.4.3.1 Analysis Using the lme() Function
5.4.3.2 Analysis Using the lmer() Function
5.4.4 Stata
5.4.5 HLM
5.4.5.1 Data Set Preparation
5.4.5.2 Preparing the MDM File
5.5 Results of Hypothesis Tests
5.5.1 Likelihood Ratio Tests for Random Effects
5.5.2 Likelihood Ratio Tests for Residual Error Variance
5.5.3 F-Tests for Fixed Effects
5.6 Comparing Results across the Software Procedures
5.6.1 Comparing Model 5.1 Results
5.6.2 Comparing Model 5.2 Results
5.7 Interpreting Parameter Estimates in the Final Model
5.7.1 Fixed-Effect Parameter Estimates
5.7.2 Covariance Parameter Estimates
5.8 The Implied Marginal Covariance Matrix for the Final Model
5.9 Diagnostics for the Final Model
5.10 Software Notes
5.10.1 Heterogeneous Residual Error Variances for Level 1 Groups
5.10.2 EBLUPs for Multiple Random Effects
5.11 Other Analytic Approaches
5.11.1 Kronecker Product for More Flexible Residual Error Covariance Structures
5.11.2 Fitting the Marginal Model
5.11.3 Repeated-Measures ANOVA
5.12 Recommendations
6 Random Coefficient Models for Longitudinal Data: The Autism Example
6.1 Introduction
6.2 The Autism Study
6.2.1 Study Description
6.2.2 Data Summary
6.3 Overview of the Autism Data Analysis
6.3.1 Analysis Steps
6.3.2 Model Specification
6.3.2.1 General Model Specification
6.3.2.2 Hierarchical Model Specification
6.3.3 Hypothesis Tests
6.4 Analysis Steps in the Software Procedures
6.4.1 SAS
6.4.2 SPSS
6.4.3 R
6.4.3.1 Analysis Using the lme() Function
6.4.3.2 Analysis Using the lmer() Function
6.4.4 Stata
6.4.5 HLM
6.4.5.1 Data Set Preparation
6.4.5.2 Preparing the MDM File
6.5 Results of Hypothesis Tests
6.5.1 Likelihood Ratio Test for Random Effects
6.5.2 Likelihood Ratio Test for Fixed Effects
6.6 Comparing Results across the Software Procedures
6.6.1 Comparing Model 6.1 Results
6.6.2 Comparing Model 6.2 Results
6.6.3 Comparing Model 6.3 Results
6.7 Interpreting Parameter Estimates in the Final Model
6.7.1 Fixed-Effect Parameter Estimates
6.7.2 Covariance Parameter Estimates
6.8 Calculating Predicted Values
6.8.1 Marginal Predicted Values
6.8.2 Conditional Predicted Values
6.9 Diagnostics for the Final Model
6.9.1 Residual Diagnostics
6.9.2 Diagnostics for the Random Effects
6.9.3 Observed and Predicted Values
6.10 Software Note: Computational Problems with the
D Matrix
6.10.1 Recommendations
6.11 An Alternative Approach: Fitting the Marginal Model with an Unstructured Covariance Matrix
6.11.1 Recommendations
7 Models for Clustered Longitudinal Data: The Dental Veneer Example
7.1 Introduction
7.2 The Dental Veneer Study
7.2.1 Study Description
7.2.2 Data Summary
7.3 Overview of the Dental Veneer Data Analysis
7.3.1 General Model Specification
7.3.2 Hierarchical Model Specification
7.3.3 Hypothesis Tests
7.4 Analysis Steps in the Software Procedures
7.4.1 SAS
7.4.2 SPSS
7.4.3 R
7.4.3.1 Analysis Using the lme() Function
7.4.3.2 Analysis Using the lmer() Function
7.4.4 Stata
7.4.5 HLM
7.4.5.1 Data Set Preparation
7.4.5.2 Preparing the Multivariate Data Matrix (MDM) File
7.5 Results of Hypothesis Tests
7.5.1 Likelihood Ratio Tests for Random Effects
7.5.2 Likelihood Ratio Tests for Residual Error Variance
7.5.3 Likelihood Ratio Tests for Fixed Effects
7.6 Comparing Results across Software Procedures
7.6.1 Comparing Model 7.1 Results
7.6.2 Comparing Results for Models 7.2A, 7.2B, and 7.2C
7.6.3 Comparing Model 7.3 Results
7.7 Interpreting Parameter Estimates in the Final Model
7.7.1 Fixed-Effect Parameter Estimates
7.7.2 Covariance Parameter Estimates
7.8 The Implied Marginal Covariance Matrix for the Final Model
7.9 Diagnostics for the Final Model
7.9.1 Residual Diagnostics
7.9.2 Diagnostics for the Random Effects
7.10 Software Notes and Recommendations
7.10.1 ML vs. REML Estimation
7.10.2 The Ability to Remove Random Effects from a Model
7.10.3 Considering Alternative Residual Error Covariance Structures
7.10.4 Aliasing of Covariance Parameters
7.10.5 Displaying the Marginal Covariance and Correlation Matrices
7.10.6 Miscellaneous Software Notes
7.11 Other Analytic Approaches
7.11.1 Modeling the Covariance Structure
7.11.2 The Step-Up vs. Step-Down Approach to Model Building
7.11.3 Alternative Uses of Baseline Values for the Dependent Variable
8 Models for Data with Crossed Random Factors: The SAT Score Example
8.1 Introduction
8.2 The SAT Score Study
8.2.1 Study Description
8.2.2 Data Summary
8.3 Overview of the SAT Score Data Analysis
8.3.1 Model Specification
8.3.1.1 General Model Specification
8.3.1.2 Hierarchical Model Specification
8.3.2 Hypothesis Tests
8.4 Analysis Steps in the Software Procedures
8.4.1 SAS
8.4.2 SPSS
8.4.3 R
8.4.4 Stata
8.4.5 HLM
8.4.5.1 Data Set Preparation
8.4.5.2 Preparing the MDM File
8.4.5.3 Model Fitting
8.5 Results of Hypothesis Tests
8.5.1 Likelihood Ratio Tests for Random Effects
8.5.2 Testing the Fixed Year Effect
8.6 Comparing Results across the Software Procedures
8.7 Interpreting Parameter Estimates in the Final Model
8.7.1 Fixed-Effect Parameter Estimates
8.7.2 Covariance Parameter Estimates
8.8 The Implied Marginal Covariance Matrix for the Final Model
8.9 Recommended Diagnostics for the Final Model
8.10 Software Notes and Additional Recommendations
7 Power Analysis and Sample Size Calculations for Linear Mixed Models
9.1 Introduction
9.2 Direct Power Computations
9.2.1 Software for Direct Power Computations
9.2.2 Examples of Direct Power Computations
9.3 Examining Power via Simulation
9.3.1 Examples of Simulation-Based Approaches
A Statistical Software Resources
A.1 Descriptions/Availability of Software Packages
A.1.1 SAS
A.1.2 IBM SPSS Statistics
A.1.3 R
A.1.4 Stata
A.1.5 HLM
A.2 Useful Internet Links
B Calculation of the Marginal Covariance Matrix
C Acronyms/Abbreviations
Bibliography
Index
