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Linear Mixed Models: A Practical Guide Using Statistical Software, Third Edition |
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Comment from the Stata technical groupThe third edition of Linear Mixed Models: A Practical Guide Using Statistical Software provides an excellent first course in the theory and methods of linear mixed models. Topics covered include fixed versus random effects, properties of estimators, nested versus crossed factors, tests of hypotheses for fixed effects (including degrees-of-freedom calculations), tests of hypotheses for variance components including likelihood-ratio tests for nested random-effects structures, approaches for fitting mixed models to complex survey data, Bayesian inference for linear mixed models, and various model diagnostics. In addition, the text provides a thorough guide through the major software applications for linear mixed models, namely, Stata, SAS, R, SPSS, and HLM. Each chapter highlights a different software package and teaches you the basics of fitting mixed models therein. The book also includes tables that compare the packages by reviewing the results obtained from fitting identical models and explaining any differences encountered. If you wish to fit linear mixed models, whether in Stata or elsewhere, we recommend this text. |
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Table of contentsView table of contents >>
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
1.2 A Brief History of LMMs
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
1.2.1 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
2.2 Specification of LMMs
2.1.1.1 Clustered Data vs. Repeated-Measures and Longitudinal Data
2.1.2 Types of Factors and Their Related Effects in an LMM
2.1.1.2 Levels of Data
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
2.2.1 General Specification for an Individual Observation
2.3 The Marginal Linear Model
2.2.2 General Matrix Specification
2.2.2.1 Covariance Structures for the D Matrix
2.2.3 Alternative Matrix Specification for All Subjects 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 2.2.4 Hierarchical Linear Model (HLM) Specification of the LMM
2.3.1 Specification of the Marginal Model
2.4 Estimation in LMMs
2.3.2 The Marginal Model Implied by an LMM
2.4.1 Maximum Likelihood (ML) Estimation
2.5 Computational Issues
2.4.1.1 Special Case: Assume Θ Is Known
2.4.2 REML Estimation 2.4.1.2 General Case: Assume Θ Is Unknown 2.4.3 REML vs. ML Estimation
2.5.1 Algorithms for Likelihood Function Optimization
2.6 Tools for Model Selection
2.5.2 Computational Problems with Estimation of Covariance Parameters
2.6.1 Basic Concepts in Model Selection
2.7 Model-Building Strategies
2.6.1.1 Nested Models
2.6.2 Likelihood Ratio Tests (LRTs)
2.6.1.2 Hypotheses: Specification and Testing
2.6.2.1 Likelihood Ratio Tests for Fixed-Effect Parameters
2.6.3 Alternative Tests
2.6.2.2 Likelihood Ratio Tests for Covariance Parameters
2.6.3.1 Alternative Tests for Fixed-Effect Parameters
2.6.4 Information Criteria
2.6.3.2 Alternative Tests for Covariance Parameters
2.7.1 The Top-Down Strategy
2.8 Checking Model Assumptions (Diagnostics)
2.7.2 The Step-Up Strategy
2.8.1 Residual Diagnostics
2.9 Other Aspects of LMMs
2.8.1.1 Raw Residuals
2.8.2 Influence Diagnostics 2.8.1.2 Standardized and Studentized Residuals 2.8.3 Diagnostics for Random Effects
2.9.1 Predicting Random Effects: Best Linear Unbiased Predictors
2.10 Chapter Summary
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
2.9.7 Bayesian Analysis of Linear Mixed Models
2.9.6.2 Hybrid Design- and Model-Based Approaches
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
3.3 Overview of the Rat Pup Data Analysis
3.2.2 Data Summary
3.3.1 Analysis Steps
3.4 Anaylsis Steps in the Software Procedures
3.3.2 Model Specification
3.3.2.1 General Model Specification
3.3.3 Hypothesis Tests
3.3.2.2 Hierarchical Model Specification
3.4.1 SAS
3.5 Results of Hypothesis Tests
3.4.2 SPSS 3.4.3 R
3.4.3.1 Analysis Using the lme() Function
3.4.4 Stata 3.4.3.2 Analysis Using the lmer() Function 3.4.5 HLM
3.4.5.1 Data Set Preparation
3.4.5.2 Preparing the Multivariate Data Matrix (MDM) File
3.5.1 Likelihood Ratio Tests for Random Effects
3.6 Comparing Results across the Software Procedures
3.5.2 Likelihood Ratio Tests for Residual Error Variance 3.5.3 F-tests and Likelihood Ratio Tests for Fixed Effects
3.6.1 Comparing Model 3.1 Results
3.7 Interpreting Parameter Estimates in the Final Model
3.6.2 Comparing Model 3.2B Results 3.6.3 Comparing Model 3.3 Results
3.7.1 Fixed-Effect Parameter Estimates
3.8 Estimating the Intraclass Correlation Coefficients (ICCs) 3.7.2 Covariance Parameter Estimates 3.9 Calculating Predicted Values
3.9.1 Litter-Specific (Conditional) Predicted Values
3.10 Diagnostics for the Final Model
3.9.2 Population-Averaged (Unconditional) Predicted Values
3.10.1 Residual Diagnostics
3.11 Software Notes and Recommendations
3.10.1.1 Conditional Residuals
3.10.2 Distribution of BLUPS
3.10.1.2 Conditional Studentized Residuals
3.10.2.1 Influence on Covariance Parameters
3.10.2.2 Influence on Fixed Effects
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
4.3 Overview of the Classroom Data Analysis
4.2.2 Data Summary
4.2.2.1 Data Set Preparation
4.2.2.2 Preparing the Multivariate Data Matrix (MDM) File
4.3.1 Analysis Steps
4.4 Analysis Steps in the Software Procedures
4.3.2 Model Specification
4.3.2.1 General Model Specification
4.3.3 Hypothesis Tests
4.3.2.2 Hierarchical Model Specification
4.4.1 SAS
4.5 Results of Hypothesis Tests
4.4.2 SPSS 4.4.3 R
4.4.3.1 Analysis Using the lme() Function
4.4.4 Stata 4.4.3.2 Analysis Using the lmer() Function 4.4.5 HLM
4.5.1 Likelihood Ratio Tests for Random Effects
4.6 Comparing Results across the Software Procedures
4.5.2 Likelihood Ratio Tests and t-Tests for Fixed Effects
4.6.1 Comparing Model 4.1 Results
4.7 Interpreting Parameter Estimates in the Final Model
4.6.2 Comparing Model 4.2 Results 4.6.3 Comparing Model 4.3 Results 4.6.4 Comparing Model 4.4 Results
4.7.1 Fixed-Effect Parameter Estimates
4.8 Estimating the Intraclass Correlation Coefficients (ICCs) 4.7.2 Covariance Parameter Estimates 4.9 Calculating Predicted Values
4.9.1 Conditional and Marginal Predicted Values
4.10 Diagnostics for the Final Model
4.9.2 Plotting Predicted Values Using HLM
4.10.1 Plots of the EBLUPs
4.11 Software Notes
4.10.2 Residual Diagnostics
4.11.1 REML vs. ML Estimation
4.12 Recommendations
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
5 Models for Repeated-Measures Data: The Rat Brain Example
5.1 Introduction
5.2 The Rat Brain Study
5.2.1 Study Description
5.3 Overview of the Rat Brain Data Analysis
5.2.2 Data Summary
5.3.1 Analysis Steps
5.4 Analysis Steps in the Sofware Procedures
5.3.2 Model Specification
5.3.2.1 General Model Specification
5.3.3 Hypothesis Tests
5.3.2.2 Hierarchical Model Specification
5.4.1 SAS
5.5 Results of Hypothesis Tests
5.4.2 SPSS 5.4.3 R
5.4.3.1 Analysis Using the lme() Function
5.4.4 Stata 5.4.3.2 Analysis Using the lmer() Function 5.4.5 HLM
5.4.5.1 Data Set Preparation
5.4.5.2 Preparing the MDM File
5.5.1 Likelihood Ratio Tests for Random Effects
5.6 Comparing Results across the Software Procedures
5.5.2 Likelihood Ratio Tests for Residual Error Variance 5.5.3 F-Tests for Fixed Effects
5.6.1 Comparing Model 5.1 Results
5.7 Interpreting Parameter Estimates in the Final Model
5.6.2 Comparing Model 5.2 Results
5.7.1 Fixed-Effect Parameter Estimates
5.8 The Implied Marginal Covariance Matrix for the Final Model 5.7.2 Covariance Parameter Estimates 5.9 Diagnostics for the Final Model 5.10 Software Notes
5.10.1 Heterogeneous Residual Error Variances for Level 1 Groups
5.11 Other Analytic Approaches
5.10.2 EBLUPs for Multiple Random Effects
5.11.1 Kronecker Product for More Flexible Residual Error Covariance Structures
5.12 Recommendations
5.11.2 Fitting the Marginal Model 5.11.3 Repeated-Measures ANOVA
6 Random Coefficient Models for Longitudinal Data: The Autism Example
6.1 Introduction
6.2 The Autism Study
6.2.1 Study Description
6.3 Overview of the Autism Data Analysis
6.2.2 Data Summary
6.3.1 Analysis Steps
6.4 Analysis Steps in the Software Procedures
6.3.2 Model Specification
6.3.2.1 General Model Specification
6.3.3 Hypothesis Tests
6.3.2.2 Hierarchical Model Specification
6.4.1 SAS
6.5 Results of Hypothesis Tests
6.4.2 SPSS 6.4.3 R
6.4.3.1 Analysis Using the lme() Function
6.4.4 Stata 6.4.3.2 Analysis Using the lmer() Function 6.4.5 HLM
6.4.5.1 Data Set Preparation
6.4.5.2 Preparing the MDM File
6.5.1 Likelihood Ratio Test for Random Effects
6.6 Comparing Results across the Software Procedures
6.5.2 Likelihood Ratio Test for Fixed Effects
6.6.1 Comparing Model 6.1 Results
6.7 Interpreting Parameter Estimates in the Final Model
6.6.2 Comparing Model 6.2 Results 6.6.3 Comparing Model 6.3 Results
6.7.1 Fixed-Effect Parameter Estimates
6.8 Calculating Predicted Values
6.7.2 Covariance Parameter Estimates
6.8.1 Marginal Predicted Values
6.9 Diagnostics for the Final Model
6.8.2 Conditional Predicted Values
6.9.1 Residual Diagnostics
6.10 Software Note: Computational Problems with the D Matrix
6.9.2 Diagnostics for the Random Effects 6.9.3 Observed and Predicted Values
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
7.3 Overview of the Dental Veneer Data Analysis
7.2.2 Data Summary
7.3.1 General Model Specification
7.4 Analysis Steps in the Software Procedures
7.3.2 Hierarchical Model Specification 7.3.3 Hypothesis Tests
7.4.1 SAS
7.5 Results of Hypothesis Tests
7.4.2 SPSS 7.4.3 R
7.4.3.1 Analysis Using the lme() Function
7.4.4 Stata 7.4.3.2 Analysis Using the lmer() Function 7.4.5 HLM
7.4.5.1 Data Set Preparation
7.4.5.2 Preparing the Multivariate Data Matrix (MDM) File
7.5.1 Likelihood Ratio Tests for Random Effects
7.6 Comparing Results across Software Procedures
7.5.2 Likelihood Ratio Tests for Residual Error Variance 7.5.3 Likelihood Ratio Tests for Fixed Effects
7.6.1 Comparing Model 7.1 Results
7.7 Interpreting Parameter Estimates in the Final Model
7.6.2 Comparing Results for Models 7.2A, 7.2B, and 7.2C 7.6.3 Comparing Model 7.3 Results
7.7.1 Fixed-Effect Parameter Estimates
7.8 The Implied Marginal Covariance Matrix for the Final Model 7.7.2 Covariance Parameter Estimates 7.9 Diagnostics for the Final Model
7.9.1 Residual Diagnostics
7.10 Software Notes and Recommendations
7.9.2 Diagnostics for the Random Effects
7.10.1 ML vs. REML Estimation
7.11 Other Analytic Approaches
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
7.11.1 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
8.3 Overview of the SAT Score Data Analysis
8.2.2 Data Summary
8.3.1 Model Specification
8.4 Analysis Steps in the Software Procedures
8.3.1.1 General Model Specification
8.3.2 Hypothesis Tests
8.3.1.2 Hierarchical Model Specification
8.4.1 SAS
8.5 Results of Hypothesis Tests
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
8.5.1 Likelihood Ratio Tests for Random Effects
8.6 Comparing Results across the Software Procedures 8.5.2 Testing the Fixed Year Effect 8.7 Interpreting Parameter Estimates in the Final Model
8.7.1 Fixed-Effect Parameter Estimates
8.8 The Implied Marginal Covariance Matrix for the Final Model 8.7.2 Covariance Parameter Estimates 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
9.3 Examining Power via Simulation 9.2.2 Examples of Direct Power Computations
9.3.1 Examples of Simulation-Based Approaches
A Statistical Software Resources
A.1 Descriptions/Availability of Software Packages
A.1.1 SAS
A.2 Useful Internet Links
A.1.2 IBM SPSS Statistics A.1.3 R A.1.4 Stata A.1.5 HLM
B Calculation of the Marginal Covariance Matrix
C Acronyms/Abbreviations Bibliography Index |
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