Research Professor, Division of Geriatrics/Institute of Gerontology, Medical School
Research Professor, Department of Biostatistics, School of Public Health
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Linear Mixed Models: A Practical Guide Using Statistical Software
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BOOK DESCRIPTION 
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 implementation, model interpretation, and visualization of clustered and longitudinal data. This easy-to-navigate reference details the use of procedures for fitting LMMs in five popular statistical software packages: SAS, SPSS, Stata, R/S-plus, and HLM.
The authors introduce basic theoretical concepts, present a heuristic approach to fitting LMMs based on both general and hierarchical model specifications, develop the model-building process step-by-step, and demonstrate the estimation, testing, and interpretation of fixed-effect parameters and covariance parameters associated with random effects. These concepts are illustrated through examples using real-world data sets that enable comparisons of model fitting options and results across the software procedures. The book also gives an overview of important options and features available in each procedure. Making popular software procedures for fitting LMMs easy-to-use, this valuable resource shows how to perform LMM analyses and provides a clear explanation of mixed modeling techniques and theories.
REVIEWS
Stata - Data Analysis and Statistical Software
CRC Press
Amazon.com
Links to additional reviews can be found on Dr. West's web page
TABLE OF CONTENTS
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 LMMs1.1.2 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
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
2.1.1.2 Levels of Data
2.1.2 Types of Factors and Their Related Effects in an LMM2.1.1.2 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
2.1.2.2 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
2.2.2 General Matrix Specification
2.2.4 Hierarchical Linear Model (HLM) Specification of the LMM
2.3 The Marginal Linear Model2.2.2 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 Subjects2.2.2.2 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
2.3.2 The Marginal Model Implied by an LMM
2.4 Estimation in LMMs2.3.2 The Marginal Model Implied by an LMM
2.4.1 Maximum Likelihood (ML) Estimation
2.4.3 REML vs. ML Estimation
2.5 Computational Issues 2.4.1.1 Special Case: Assume theta is known
2.4.1.2 General Case: Assume theta is unknown
2.4.2 REML Estimation2.4.1.2 General Case: Assume theta is unknown
2.4.3 REML vs. ML Estimation
2.5.1 Algorithms for Likelihood Function Optimization
2.5.2 Computational Problems with Estimation of Covariance Parameters
2.6 Tools for Model Selection2.5.2 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
2.6.1.2 Hypotheses: Specification and Testing
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
2.6.2.2 Likelihood Ratio Tests for Covariance Parameters
2.6.3 Alternative Tests2.6.2.2 Likelihood Ratio Tests for Covariance Parameters
2.6.3.1 Alternative Tests for Fixed-Effect Parameters
2.6.3.2 Alternative Tests for Covariance Parameters
2.6.4 Information Criteria2.6.3.2 Alternative Tests for Covariance Parameters
2.7.1 The Top-Down Strategy
2.7.2 The Step-Up Strategy
2.8 Checking Model Assumptions (Diagnostics)2.7.2 The Step-Up Strategy
2.8.1 Residual Diagnostics
2.8.3 Diagnostics for Random Effects
2.9 Other Aspects of LMMs 2.8.1.1 Conditional Residuals
2.8.1.2 Standardized and Studentized Residuals
2.8.2 Influence Diagnostics2.8.1.2 Standardized and Studentized Residuals
2.8.3 Diagnostics for Random Effects
2.9.1 Predicting Random Effects: Best Linear Unbiased Predictors
2.9.2 Intraclass Correlation Coefficients (ICCs)
2.9.3 Problems with Model Specification (Aliasing)
2.9.4 Missing Data
2.9.5 Centering Covariates
2.10 Chapter Summary2.9.2 Intraclass Correlation Coefficients (ICCs)
2.9.3 Problems with Model Specification (Aliasing)
2.9.4 Missing Data
2.9.5 Centering Covariates
3 Two-Level Models for Clustered Data: The Rat Pup Example
3.1 Introduction
3.2 The Rat Pup Study
3.9 Calculating Predicted Values
3.2 The Rat Pup Study
3.2.1 Study Description
3.2.2 Data Summary
3.3 Overview of the Rat Pup Data Analysis3.2.2 Data Summary
3.3.1 Analysis Steps
3.3.2 Model Specification
3.4 Analysis Steps in the Software Procedures3.3.2 Model Specification
3.3.2.1 General Model Specification
3.3.2.2 Hierarchical Model Specification
3.3.3 Hypothesis Tests3.3.2.2 Hierarchical Model Specification
3.4.1 SAS
3.4.2 SPSS
3.4.3 R
3.4.4 Stata
3.4.5 HLM
3.5 Results of Hypothesis Tests3.4.2 SPSS
3.4.3 R
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
3.4.5.2 Preparing the Multivariate Data Matrix (MDM) File
3.5.1 Likelihood Ratio Tests for Random Effects
3.5.2 Likelihood Ratio Tests for Residual Variance
3.5.3 F-tests and Likelihood Ratio Tests for Fixed Effects
3.6 Comparing Results across the Software Procedures3.5.2 Likelihood Ratio Tests for Residual Variance
3.5.3 F-tests and Likelihood Ratio Tests for Fixed Effects
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 Model3.6.2 Comparing Model 3.2B Results
3.6.3 Comparing Model 3.3 Results
3.7.1 Fixed-Effect Parameter Estimates
3.7.2 Covariance 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
3.9.2 Population-Averaged (Unconditional) Predicted Values
3.10 Diagnostics for the Final Model3.9.2 Population-Averaged (Unconditional) Predicted Values
3.10.1 Residual Diagnostics
3.11 Software Notes 3.10.1.1 Conditional Residuals
3.10.1.2 Conditional Studentized Residuals
3.10.2 Influence Diagnostics3.10.1.2 Conditional Studentized Residuals
3.10.2.1 Overall and Fixed-Effects Influence Diagnostics
3.10.2.2 Influence on Covariance Parameters
3.10.2.2 Influence on Covariance Parameters
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
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
4.9 Calculating Predicted Values
4.2 The Classroom Study
4.2.1 Study Description
4.2.2 Data Summary
4.3 Overview of the Classroom Data Analysis4.2.2 Data Summary
4.2.2.1 Data Set Preparation
4.2.2.2 Preparing the Multivariate Data Matrix (MDM) File
4.2.2.2 Preparing the Multivariate Data Matrix (MDM) File
4.3.1 Analysis Steps
3.2 Models Specification
4.4 Analysis Steps in the Software Procedures3.2 Models Specification
4.3.2.1 General Model Specification
4.3.2.2 Hierarchical Model Specification
4.3.3 Hypothesis Tests4.3.2.2 Hierarchical Model Specification
4.4.1 SAS
4.4.2 SPSS
4.4.3 R
4.4.4 Stata
4.4.5 HLM
4.5 Results of Hypothesis Tests4.4.2 SPSS
4.4.3 R
4.4.4 Stata
4.4.5 HLM
4.5.1 Likelihood Ratio Test for Random Effects
4.5.2 Likelihood Ratio Tests and t-tests for Fixed Effects
4.6 Comparing Results across the Software Procedures4.5.2 Likelihood Ratio Tests and t-tests for Fixed Effects
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 Model4.6.2 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
4.7.2 Covariance 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
4.9.2 Plotting Predicted Values Using HLM
4.10 Diagnostics for the Final Model4.9.2 Plotting Predicted Values Using HLM
4.10.1 Plots of the EBLUPs
4.10.2 Residual Diagnostics
4.11 Software Notes4.10.2 Residual Diagnostics
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
4.11.2 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
5.9 Diagnostics for the Final Model
5.10 Software Notes
5.2 The Rat Brain Study
5.2.1 Study Description
5.2.2 Data Summary
5.3 Overview of the Rat Brain Data Analysis5.2.2 Data Summary
5.3.1 Analysis Steps
5.3.2 Model Specification
5.4 Analysis Steps in the Software Products5.3.2 Model Specification
5.3.2.1 General Model Specification
5.3.2.2 Hierarchical Model Specification
5.3.3 Hypothesis Tests5.3.2.2 Hierarchical Model Specification
5.4.1 SAS
5.4.2 SPSS
5.4.3 R
5.4.4 Stata
5.4.5 HLM
5.5 Results of Hypothesis Tests5.4.2 SPSS
5.4.3 R
5.4.4 Stata
5.4.5 HLM
5.4.5.1 Data Set Preparation
5.4.5.2 Preparing the MDM File
5.4.5.2 Preparing the MDM File
5.5.1 Likelihood Ratio Tests for Random Effects
5.5.2 Likelihood Ratio Tests for Residual Variance
5.5.3 F-tests for Fixed Effects
5.6 Comparing Results across the Software Procedures5.5.2 Likelihood Ratio Tests for Residual Variance
5.5.3 F-tests for Fixed Effects
5.6.1 Comparing Model 5.1 Results
5.6.2 Comparing Model 5.2 Results
5.7 Interpreting Parameter Estimates in the Final Model5.6.2 Comparing Model 5.2 Results
5.7.1 Fixed-Effect Parameter Estimates
5.7.2 Covariance Parameter Estimates
5.8 The Implied Marginal Variance-Covariance Matrix for the Final Model5.7.2 Covariance Parameter Estimates
5.9 Diagnostics for the Final Model
5.10 Software Notes
5.10.1 Heterogeneous Residual Variances for Level 1 Groups
5.10.2 EBLUPs for Multiple Random Effects
5.11 Other Analytic Approaches5.10.2 EBLUPs for Multiple Random Effects
5.11.1 Kronecker Product for More Flexible Residual Covariance Structures
5.11.2 Fitting the Marginal Model
5.11.3 Repeated-Measures ANOVA
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 Autism Study
6.11 An Alternative Approach: Fitting the Marginal Model with an Unstructured Covariance Matrix
6.2 Autism Study
6.2.1 Study Description
6.2.2 Data Summary
6.3 Overview of the Autism Data Analysis6.2.2 Data Summary
6.3.1 Analysis Steps
6.3.2 Model Specification
6.4 Analysis Steps in the Software Procedures6.3.2 Model Specification
6.3.2.1 General Model Specification
6.3.2.2 Hierarchical Model Specification
6.3.3 Hypothesis Tests6.3.2.2 Hierarchical Model Specification
6.4.1 SAS
6.4.2 SPSS
6.4.3 R
6.4.4 Stata
6.4.5 HLM
6.5 Results of Hypothesis Tests6.4.2 SPSS
6.4.3 R
6.4.4 Stata
6.4.5 HLM
6.4.5.1 Data Set Preparation
6.4.5.2 Preparing the MDM File
6.4.5.2 Preparing the MDM File
6.5.1 Likelihood Ratio Tests for Random Effects
6.5.2 Likelihood Ratio Tests for Fixed Effects
6.6 Comparing Results across the Software Procedures6.5.2 Likelihood Ratio Tests for Fixed Effects
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 Model6.6.2 Comparing Model 6.2 Results
6.6.3 Comparing Model 6.3 Results
6.7.1 Fixed-Effect Parameter Estimates
6.7.2 Covariance Parameter Estimates
6.8 Calculating Predicted Values6.7.2 Covariance Parameter Estimates
6.8.1 Marginal Predicted Values
6.8.2 Conditional Predicted Values
6.9 Diagnostics for the Final Model6.8.2 Conditional Predicted Values
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 Matrix6.9.2 Diagnostics for the Random Effects
6.9.3 Observed and Predicted Values
6.11 An Alternative Approach: Fitting the Marginal Model with an Unstructured Covariance Matrix
7 Models for Clustered Longitudinal Data: The Dental Veneer Example
7.1 Introduction
7.2 The Dental Veneer Study
7.9 Diagnostics for the Final Model
7.2 The Dental Veneer Study
7.2.1 Study Description
7.2.2 Data Summary
7.3 Overview of the Dental Veneer Data Analysis7.2.2 Data Summary
7.3.1 Analysis Steps
7.3.2 Models Specification
7.4 Analysis Steps in the Software Procedures7.3.2 Models Specification
7.3.2.1 General Model Specification
7.3.2.2 Hierarchical Model Specification
7.3.3 Hypothesis Tests7.3.2.2 Hierarchical Model Specification
7.4.1 SAS
7.4.2 SPSS
7.4.3 R
7.4.4 Stata
7.4.5 HLM
7.5 Results of Hypothesis Tests7.4.2 SPSS
7.4.3 R
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
7.4.5.2 Preparing the Multivariate Data Matrix (MDM) File
7.5.1 Likelihood Ratio Tests for Random Effects
7.5.2 Likelihood Ratio Tests for Residual Variance
7.5.3 Likelihood Ratio Tests for Fixed Effects
7.6 Comparing Results across the Software Procedures7.5.2 Likelihood Ratio Tests for Residual Variance
7.5.3 Likelihood Ratio Tests for Fixed Effects
7.6.1 Comparing Model 7.1 Results
7.6.2 Comparing Software Results for Model 7.2A, Model 7.2 B, and Model 7.2C
7.6.3 Comparing Model 7.3 Results
7.7 Interpreting Parameter Estimates in the Final Model7.6.2 Comparing Software Results for Model 7.2A, Model 7.2 B, and Model 7.2C
7.6.3 Comparing Model 7.3 Results
7.7.1 Fixed-Effect Parameter Estimates
7.7.2 Covariance Parameter Estimates
7.8 The Implied Marginal Variance-Covariance Matrix for the Final Model7.7.2 Covariance Parameter Estimates
7.9 Diagnostics for the Final Model
7.9.1 Residual Diagnostics
7.9.2 Diagnostics for the Random Effects
7.10 Software Notes7.9.2 Diagnostics for the Random Effects
7.10.1 ML vs. REML Estimation
7.10.2 The Ability to Remove Random Effects from a Model
7.10.3 The Ability to Fit Models with Different Residual 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 Approaches7.10.2 The Ability to Remove Random Effects from a Model
7.10.3 The Ability to Fit Models with Different Residual 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
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
Appendix A: Statistical Software Resources
A.1 Descriptions/Availability of Software Packages
A.1.1 SASS
A.1.2 SPSS
A.1.3 R
A.1.4 Stata
A.1.5 HLM
A.2 Useful Internet LinksA.1.2 SPSS
A.1.3 R
A.1.4 Stata
A.1.5 HLM
Appendix B: Calculation of the Marginal Variance-Covariance Matrix
Appendix C: Acronyms/Abbreviations
References
Index
WHERE to BUY
http://www.crcpress.com/ecommerce_product/product_detail.jsf?catno=C4800
http://www.stata.com/bookstore/lmm.html
http://www.amazon.com/Linear-Mixed-Models-Practical-Statistical/dp/1584884800