Linear Mixed Models: A Practical Guide Using Statistical Software, Second Edition

434 Pages·2014·5.16 MB·English

by  Brady T. West

SECOND
Statistics
EDITION
Highly recommended by JASA, Technometrics, and other journals, the first L
edition of this bestseller showed how to easily perform complex linear mixed I
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model (LMM) analyses via a variety of software programs. Linear Mixed Models:
E
A Practical Guide Using Statistical Software, Second Edition continues to
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lead you step by step through the process of fitting LMMs. This second edition
covers additional topics on the application of LMMs that are valuable for data R
analysts in all fields. It also updates the case studies using the latest versions
M
of the software procedures and provides up-to-date information on the options
and features of the software procedures available for fitting LMMs in SAS, SPSS, I
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Stata, R/S-plus, and HLM.
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New to the Second Edition
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• A new chapter on models with crossed random effects that uses a case

study to illustrate software procedures capable of fitting these models M
• Power analysis methods for longitudinal and clustered study designs,
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including software options for power analyses and suggested approaches
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to writing simulations
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• Use of the lmer() function in the lme4 R package
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• New sections on fitting LMMs to complex sample survey data and Bayesian
S
approaches to making inferences based on LMMs
• Updated graphical procedures in the software packages
• Substantially revised index to enable more efficient reading and easier
location of material on selected topics or software options
• More practical recommendations on using the software for analysis W
e
• A new R package (WWGbook) that contains all of the data sets used in the s
t
examples ,
W
e
Ideal for anyone who uses software for statistical modeling, this book eliminates l
c
the need to read multiple software-specific texts by covering the most popular h
,

software programs for fitting LMMs in one handy guide. The authors illustrate the a
n
models and methods through real-world examples that enable comparisons of d

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model-fitting options and results across the software procedures.
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K15924_cover.indd 1 6/13/14 3:30 PM
LINEAR MIXED MODELS
A Practical Guide Using Statistical Software
SECOND EDITION
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
LINEAR MIXED MODELS
A Practical Guide Using Statistical Software
SECOND EDITION
Brady T. West
Kathleen B. Welch
Andrzej T. Gałecki
University of Michigan
Ann Arbor, USA
with contributions from Brenda W. Gillespie
First edition published in 2006.
CRC Press
Taylor & Francis Group
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© 2015 by Taylor & Francis Group, LLC
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To Laura and Carter
To all of my mentors, advisors, and teachers, especially my parents and grandparents
—B.T.W.
To Jim, my children, and grandchildren
To the memory of Fremont and June
—K.B.W.
To Viola, my children, and grandchildren
To my teachers and mentors
In memory of my parents
—A.T.G.
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
List of Tables xv
List of Figures xvii
Preface to the Second Edition xix
Preface to the First Edition xxi
The Authors xxiii
Acknowledgments xxv
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
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 ClusteredDatavs.Repeated-MeasuresandLongitudinal
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1.2 Levels of Data . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Types of Factors and Their Related Effects in an LMM . . . . . 12
2.1.2.1 Fixed Factors . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2.2 Random Factors . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2.3 Fixed Factors vs. Random Factors . . . . . . . . . . . . 13
2.1.2.4 Fixed Effects vs. Random Effects . . . . . . . . . . . . 13
2.1.2.5 Nestedvs.CrossedFactorsandTheirCorrespondingEf-
fects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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
vii
viii Contents
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 . . . . . . . . . . . . . . . . 23
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 θ Is Known . . . . . . . . . . . . 26
2.4.1.2 General Case: Assume θ Is Unknown . . . . . . . . . . 26
2.4.2 REML Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.3 REML vs. ML Estimation . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Computational Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Algorithms for Likelihood Function Optimization . . . . . . . . . 29
2.5.2 Computational Problems with Estimation of CovarianceParame-
ters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . 36
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 Raw 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.9.1 Predicting Random Effects: Best Linear Unbiased Predictors . . 46
2.9.2 Intraclass Correlation Coefficients (ICCs) . . . . . . . . . . . . . 47
2.9.3 Problems with Model Specification (Aliasing) . . . . . . . . . . . 47
2.9.4 Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.9.5 Centering Covariates . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.9.6 Fitting Linear Mixed Models to Complex Sample Survey Data . 50
2.9.6.1 Purely Model-Based Approaches . . . . . . . . . . . . . 51
2.9.6.2 Hybrid Design- and Model-Based Approaches . . . . . 52
2.9.7 BayesianAnalysis of Linear Mixed Models . . . . . . . . . . . . 55
2.10 Power Analysis for Linear Mixed Models . . . . . . . . . . . . . . . . . . 56
2.10.1 Direct Power Computations . . . . . . . . . . . . . . . . . . . . 56
2.10.2 Examining Power via Simulation . . . . . . . . . . . . . . . . . . 57
2.11 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Contents ix
3 Two-Level Models for Clustered Data: The Rat Pup Example 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 The Rat Pup Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Study Description . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.2 Data Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Overview of the Rat Pup Data Analysis . . . . . . . . . . . . . . . . . . 65
3.3.1 Analysis Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3.2 Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.2.1 General Model Specification . . . . . . . . . . . . . . . 68
3.3.2.2 Hierarchical Model Specification . . . . . . . . . . . . . 69
3.3.3 Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4 Analysis Steps in the Software Procedures . . . . . . . . . . . . . . . . . 75
3.4.1 SAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.2 SPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4.3 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4.3.1 Analysis Using the lme() Function . . . . . . . . . . . 91
3.4.3.2 Analysis Using the lmer() Function . . . . . . . . . . 96
3.4.4 Stata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4.5 HLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.4.5.1 Data Set Preparation . . . . . . . . . . . . . . . . . . . 102
3.4.5.2 Preparing the Multivariate Data Matrix (MDM) File . 103
3.5 Results of Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.5.1 Likelihood Ratio Tests for Random Effects . . . . . . . . . . . . 107
3.5.2 Likelihood Ratio Tests for Residual Variance . . . . . . . . . . . 107
3.5.3 F-tests and Likelihood Ratio Tests for Fixed Effects . . . . . . . 108
3.6 Comparing Results across the Software Procedures . . . . . . . . . . . . 109
3.6.1 Comparing Model 3.1 Results . . . . . . . . . . . . . . . . . . . 109
3.6.2 Comparing Model 3.2B Results . . . . . . . . . . . . . . . . . . 114
3.6.3 Comparing Model 3.3 Results . . . . . . . . . . . . . . . . . . . 114
3.7 Interpreting Parameter Estimates in the Final Model . . . . . . . . . . . 115
3.7.1 Fixed-Effect Parameter Estimates . . . . . . . . . . . . . . . . . 115
3.7.2 Covariance Parameter Estimates . . . . . . . . . . . . . . . . . . 118
3.8 Estimating the Intraclass Correlation Coefficients (ICCs) . . . . . . . . . 118
3.9 Calculating Predicted Values . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.9.1 Litter-Specific (Conditional) Predicted Values . . . . . . . . . . 120
3.9.2 Population-Averaged(Unconditional) Predicted Values . . . . . 121
3.10 Diagnostics for the Final Model . . . . . . . . . . . . . . . . . . . . . . . 122
3.10.1 Residual Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . 122
3.10.1.1 Conditional Residuals . . . . . . . . . . . . . . . . . . . 122
3.10.1.2 Conditional Studentized Residuals . . . . . . . . . . . . 124
3.10.2 Influence Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . 126
3.10.2.1 Overall Influence Diagnostics . . . . . . . . . . . . . . 126
3.10.2.2 Influence on Covariance Parameters . . . . . . . . . . . 128
3.10.2.3 Influence on Fixed Effects . . . . . . . . . . . . . . . . 129
3.11 Software Notes and Recommendations . . . . . . . . . . . . . . . . . . . 130
3.11.1 Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.11.2 Syntax vs. Menus . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.11.3 Heterogeneous Residual Variances for Level 2 Groups . . . . . . 130
3.11.4 Display of the Marginal Covariance and Correlation Matrices . . 130
3.11.5 Differences in Model Fit Criteria . . . . . . . . . . . . . . . . . . 131
3.11.6 Differences in Tests for Fixed Effects . . . . . . . . . . . . . . . 131