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Linear Mixed Models: A Practical Guide Using Statistical Software

Authors:
Brady T. West, Kathleen B. Welch, and Andrzej T. Galecki
Publisher: Chapman & Hall/CRC
Copyright: 2007
ISBN-13: 978-1-58488-480-4
Pages: 353; hardcover
Price: $74.50

Comment from the Stata technical group

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 discussions of 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, and various model diagnostics.

The text also provides a thorough and up-to-date guide through the major software applications for linear mixed models, namely, Stata, SAS, R, SPSS, and HLM. Each of five middle chapters highlights a different software package and teaches you the basics of fitting mixed models therein. Tables comparing each package show the results obtained from fitting identical models, as well as explaining any differences encountered.

If you wish to fit linear mixed models, whether in Stata or elsewhere, we recommend this text.


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 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 theta is known
2.4.1.2 General Case: Assume theta 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 Conditional 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
2.10 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 Analysis Steps in the Software Procedures
3.4.1 SAS
3.4.2 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
3.5 Results of Hypothesis Tests
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 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 Influence Diagnostics
3.10.2.1 Overall and Fixed-Effects Influence Diagnostics
3.10.2.2 Influence on Covariance Parameters
3.11 Software Notes
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
3.2 Models 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
4.4.4 Stata
4.4.5 HLM
4.5 Results of Hypothesis Tests
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 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
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 Software Products
5.4.1 SAS
5.4.2 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
5.5 Results of Hypothesis Tests
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 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 Variance-Covariance Matrix for the Final Model
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 Approaches
5.11.1 Kronecker Product for More Flexible Residual Covariance Structures
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
6.2.1 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
6.4.4 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 Tests for Random Effects
6.5.2 Likelihood Ratio Tests 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
6.11 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
7.2.1 Study Description
7.2.2 Data Summary
7.3 Overview of the Dental Veneer Data Analysis
7.3.1 Analysis Steps
7.3.2 Models Specification
7.3.2.1 General Model Specification
7.3.2.2 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
7.4.4 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 Variance
7.5.3 Likelihood Ratio Tests for Fixed Effects
7.6 Comparing Results across the Software Procedures
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 Model
7.7.1 Fixed-Effect Parameter Estimates
7.7.2 Covariance Parameter Estimates
7.8 The Implied Marginal Variance-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
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 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
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 Links
Appendix B: Calculation of the Marginal Variance-Covariance Matrix
Appendix C: Acronyms/Abbreviations
References
Index
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