Preface to the Third Edition |
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xv | |
Preface to the Second Edition |
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xvii | |
Preface |
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xix | |
The Authors |
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xxi | |
Acknowledgments |
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xxiii | |
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xxv | |
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xxvii | |
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1 | (8) |
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1.1 What are Linear Mixed Models (LMMs)? |
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1 | (4) |
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1.1.1 Models with Random Effects for Clustered Data |
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2 | (1) |
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1.1.2 Models for Longitudinal or Repeated-Measures Data |
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2 | (1) |
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1.1.3 The Purpose of This Book |
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3 | (1) |
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1.1.4 Outline of Book Contents |
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4 | (1) |
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1.2 A Brief History of LMMs |
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5 | (4) |
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1.2.1 Key Theoretical Developments |
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5 | (2) |
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1.2.2 Key Software Developments |
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7 | (2) |
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2 Linear Mixed Models: An Overview |
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9 | (50) |
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9 | (6) |
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2.1.1 Types and Structures of Data Sets |
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9 | (1) |
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2.1.1.1 Clustered Data vs. Repeated-Measures and Longitudinal Data |
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9 | (1) |
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10 | (2) |
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2.1.2 Types of Factors and Their Related Effects in an LMM |
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12 | (1) |
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12 | (1) |
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12 | (1) |
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2.1.2.3 Fixed Factors vs. Random Factors |
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13 | (1) |
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2.1.2.4 Fixed Effects vs. Random Effects |
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13 | (1) |
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2.1.2.5 Nested vs. Crossed Factors and Their Corresponding Effects |
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14 | (1) |
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2.2 Specification of LMMs |
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15 | (7) |
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2.2.1 General Specification for an Individual Observation |
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16 | (1) |
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2.2.2 General Matrix Specification |
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17 | (2) |
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2.2.2.1 Covariance Structures for the D Matrix |
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19 | (1) |
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2.2.2.2 Covariance Structures for the Ri Matrix |
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20 | (1) |
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2.2.2.3 Group-Specific Covariance Parameter Values for the D and Ri Matrices |
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21 | (1) |
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2.2.3 Alternative Matrix Specification for All Subjects |
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22 | (1) |
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2.2.4 Hierarchical Linear Model (HLM) Specification of the LMM |
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22 | (1) |
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2.3 The Marginal Linear Model |
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22 | (3) |
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2.3.1 Specification of the Marginal Model |
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23 | (1) |
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2.3.2 The Marginal Model Implied by an LMM |
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23 | (2) |
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25 | (4) |
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2.4.1 Maximum Likelihood (ML) Estimation |
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25 | (1) |
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2.4.1.1 Special Case: Assume θ Is Known |
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26 | (1) |
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2.4.1.2 General Case: Assume θ Is Unknown |
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27 | (1) |
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28 | (1) |
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2.4.3 REML vs. ML Estimation |
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28 | (1) |
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29 | (5) |
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2.5.1 Algorithms for Likelihood Function Optimization |
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29 | (2) |
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2.5.2 Computational Problems with Estimation of Covariance Parameters |
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31 | (3) |
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2.6 Tools for Model Selection |
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34 | (5) |
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2.6.1 Basic Concepts in Model Selection |
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34 | (1) |
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34 | (1) |
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2.6.1.2 Hypotheses: Specification and Testing |
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34 | (1) |
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2.6.2 Likelihood Ratio Tests (LRTs) |
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35 | (1) |
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2.6.2.1 Likelihood Ratio Tests for Fixed-Effect Parameters |
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35 | (1) |
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2.6.2.2 Likelihood Ratio Tests for Covariance Parameters |
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35 | (2) |
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37 | (1) |
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2.6.3.1 Alternative Tests for Fixed-Effect Parameters |
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37 | (1) |
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2.6.3.2 Alternative Tests for Covariance Parameters |
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38 | (1) |
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2.6.4 Information Criteria |
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38 | (1) |
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2.7 Model-Building Strategies |
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39 | (2) |
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2.7.1 The Top-Down Strategy |
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39 | (1) |
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2.7.2 The Step-Up Strategy |
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40 | (1) |
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2.8 Checking Model Assumptions (Diagnostics) |
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41 | (5) |
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2.8.1 Residual Diagnostics |
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41 | (1) |
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41 | (1) |
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2.8.1.2 Standardized and Studentized Residuals |
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42 | (1) |
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2.8.2 Influence Diagnostics |
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42 | (1) |
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2.8.3 Diagnostics for Random Effects |
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43 | (3) |
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2.9 Other Aspects of LMMs |
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46 | (10) |
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2.9.1 Predicting Random Effects: Best Linear Unbiased Predictors |
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46 | (1) |
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2.9.2 Intraclass Correlation Coefficients (ICCs) |
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47 | (1) |
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2.9.3 Problems with Model Specification (Aliasing) |
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47 | (2) |
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49 | (1) |
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2.9.5 Centering Covariates |
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50 | (1) |
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2.9.6 Fitting Linear Mixed Models to Complex Sample Survey Data |
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50 | (1) |
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2.9.6.1 Purely Model-Based Approaches |
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51 | (2) |
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2.9.6.2 Hybrid Design- and Model-Based Approaches |
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53 | (2) |
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2.9.7 Bayesian Analysis of Linear Mixed Models |
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55 | (1) |
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56 | (3) |
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3 Two-Level Models for Clustered Data: The Rat Pup Example |
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59 | (82) |
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59 | (1) |
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59 | (6) |
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59 | (3) |
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62 | (3) |
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3.3 Overview of the Rat Pup Data Analysis |
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65 | (8) |
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65 | (2) |
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3.3.2 Model Specification |
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67 | (1) |
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3.3.2.1 General Model Specification |
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67 | (3) |
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3.3.2.2 Hierarchical Model Specification |
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70 | (1) |
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70 | (3) |
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3.4 Analysis Steps in the Software Procedures |
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73 | (36) |
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73 | (12) |
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85 | (6) |
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91 | (1) |
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3.4.3.1 Analysis Using the lme() Function |
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92 | (4) |
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3.4.3.2 Analysis Using the lmer() Function |
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96 | (3) |
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99 | (5) |
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104 | (1) |
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3.4.5.1 Data Set Preparation |
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104 | (1) |
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3.4.5.2 Preparing the Multivariate Data Matrix (MDM) File |
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105 | (4) |
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3.5 Results of Hypothesis Tests |
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109 | (3) |
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3.5.1 Likelihood Ratio Tests for Random Effects |
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109 | (1) |
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3.5.2 Likelihood Ratio Tests for Residual Error Variance |
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110 | (1) |
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3.5.3 F-Tests and Likelihood Ratio Tests for Fixed Effects |
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111 | (1) |
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3.6 Comparing Results across the Software Procedures |
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112 | (3) |
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3.6.1 Comparing Model 3.1 Results |
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112 | (3) |
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3.6.2 Comparing Model 3.2B Results |
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115 | (1) |
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3.6.3 Comparing Model 3.3 Results |
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115 | (1) |
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3.7 Interpreting Parameter Estimates in the Final Model |
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115 | (6) |
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3.7.1 Fixed-Effect Parameter Estimates |
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115 | (5) |
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3.7.2 Covariance Parameter Estimates |
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120 | (1) |
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3.8 Estimating the Intraclass Correlation Coefficients (ICCs) |
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121 | (3) |
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3.9 Calculating Predicted Values |
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124 | (1) |
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3.9.1 Litter-Specific (Conditional) Predicted Values |
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124 | (1) |
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3.9.2 Population-Averaged (Unconditional) Predicted Values |
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125 | (1) |
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3.10 Diagnostics for the Final Model |
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125 | (9) |
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3.10.1 Residual Diagnostics |
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125 | (1) |
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3.10.1.1 Conditional Residuals |
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125 | (2) |
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3.10.1.2 Conditional Studentized Residuals |
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127 | (2) |
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3.10.2 Distribution of BLUPs |
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129 | (1) |
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3.10.3 Influence Diagnostics |
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130 | (3) |
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3.10.3.1 Influence on Covariance Parameters |
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133 | (1) |
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3.10.3.2 Influence on Fixed Effects |
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134 | (1) |
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3.11 Software Notes and Recommendations |
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134 | (7) |
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134 | (1) |
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135 | (1) |
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3.11.3 Heterogeneous Residual Error Variances for Level 2 Groups |
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135 | (1) |
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3.11.4 Display of the Marginal Covariance and Correlation Matrices |
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135 | (1) |
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3.11.5 Differences in Model Fit Criteria |
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135 | (1) |
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3.11.6 Differences in Tests for Fixed Effects |
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136 | (2) |
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3.11.7 Post-Hoc Comparisons of LS Means (Estimated Marginal Means) |
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138 | (1) |
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3.11.8 Calculation of Studentized Residuals and Influence Statistics |
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138 | (1) |
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3.11.9 Calculation of EBLUPs |
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138 | (1) |
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3.11.10 Tests for Covariance Parameters |
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138 | (1) |
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3.11.11 Reference Categories for Fixed Factors |
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139 | (2) |
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4 Three-Level Models for Clustered Data: The Classroom Example |
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141 | (68) |
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141 | (2) |
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143 | (6) |
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143 | (2) |
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145 | (1) |
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4.2.2.1 Data Set Preparation |
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145 | (1) |
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4.2.2.2 Preparing the Multivariate Data Matrix (MDM) File |
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145 | (4) |
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4.3 Overview of the Classroom Data Analysis |
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149 | (8) |
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149 | (2) |
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4.3.2 Model Specification |
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151 | (1) |
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4.3.2.1 General Model Specification |
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151 | (1) |
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4.3.2.2 Hierarchical Model Specification |
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152 | (2) |
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154 | (3) |
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4.4 Analysis Steps in the Software Procedures |
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157 | (29) |
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157 | (7) |
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164 | (5) |
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169 | (1) |
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4.4.3.1 Analysis Using the lme() Function |
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169 | (3) |
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4.4.3.2 Analysis Using the lmer() Function |
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172 | (5) |
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177 | (3) |
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180 | (6) |
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4.5 Results of Hypothesis Tests |
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186 | (2) |
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4.5.1 Likelihood Ratio Tests for Random Effects |
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186 | (1) |
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4.5.2 Likelihood Ratio Tests and t-Tests for Fixed Effects |
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187 | (1) |
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4.6 Comparing Results Across the Software Procedures |
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188 | (7) |
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4.6.1 Comparing Model 4.1 Results |
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188 | (2) |
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4.6.2 Comparing Model 4.2 Results |
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190 | (1) |
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4.6.3 Comparing Model 4.3 Results |
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190 | (1) |
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4.6.4 Comparing Model 4.4 Results |
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190 | (5) |
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4.7 Interpreting Parameter Estimates in the Final Model |
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195 | (2) |
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4.7.1 Fixed-Effect Parameter Estimates |
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195 | (1) |
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4.7.2 Covariance Parameter Estimates |
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196 | (1) |
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4.8 Estimating the Intraclass Correlation Coefficients (ICCs) |
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197 | (2) |
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4.9 Calculating Predicted Values |
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199 | (2) |
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4.9.1 Conditional and Marginal Predicted Values |
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199 | (1) |
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4.9.2 Plotting Predicted Values Using HLM |
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200 | (1) |
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4.10 Diagnostics for the Final Model |
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201 | (4) |
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4.10.1 Plots of the EBLUPs |
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201 | (1) |
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4.10.2 Residual Diagnostics |
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202 | (3) |
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205 | (2) |
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4.11.1 REML vs. ML Estimation |
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205 | (1) |
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4.11.2 Setting up Three-Level Models in HLM |
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205 | (1) |
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4.11.3 Calculation of Degrees of Freedom for i-Tests in HLM |
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205 | (1) |
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4.11.4 Analyzing Cases with Complete Data |
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206 | (1) |
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4.11.5 Miscellaneous Differences |
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207 | (1) |
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207 | (2) |
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5 Models for Repeated-Measures Data: The Rat Brain Example |
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209 | (54) |
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209 | (1) |
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209 | (5) |
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209 | (3) |
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212 | (2) |
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5.3 Overview of the Rat Brain Data Analysis |
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214 | (8) |
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214 | (1) |
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5.3.2 Model Specification |
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215 | (1) |
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5.3.2.1 General Model Specification |
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215 | (4) |
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5.3.2.2 Hierarchical Model Specification |
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219 | (1) |
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220 | (2) |
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5.4 Analysis Steps in the Software Procedures |
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222 | (22) |
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222 | (5) |
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227 | (2) |
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229 | (1) |
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5.4.3.1 Analysis Using the lme() Function |
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230 | (2) |
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5.4.3.2 Analysis Using the lmer() Function |
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232 | (4) |
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236 | (3) |
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239 | (1) |
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5.4.5.1 Data Set Preparation |
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239 | (1) |
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5.4.5.2 Preparing the MDM File |
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240 | (4) |
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5.5 Results of Hypothesis Tests |
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244 | (1) |
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5.5.1 Likelihood Ratio Tests for Random Effects |
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244 | (1) |
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5.5.2 Likelihood Ratio Tests for Residual Error Variance |
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244 | (1) |
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5.5.3 F-Tests for Fixed Effects |
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245 | (1) |
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5.6 Comparing Results across the Software Procedures |
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245 | (1) |
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5.6.1 Comparing Model 5.1 Results |
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246 | (1) |
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5.6.2 Comparing Model 5.2 Results |
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246 | (1) |
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5.7 Interpreting Parameter Estimates in the Final Model |
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246 | (7) |
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5.7.1 Fixed-Effect Parameter Estimates |
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246 | (6) |
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5.7.2 Covariance Parameter Estimates |
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252 | (1) |
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5.8 The Implied Marginal Covariance Matrix for the Final Model |
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253 | (1) |
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5.9 Diagnostics for the Final Model |
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254 | (4) |
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258 | (1) |
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5.10.1 Heterogeneous Residual Error Variances for Level 1 Groups |
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258 | (1) |
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5.10.2 EBLUPs for Multiple Random Effects |
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258 | (1) |
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5.11 Other Analytic Approaches |
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258 | (4) |
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5.11.1 Kronecker Product for More Flexible Residual Error Covariance Structures |
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258 | (2) |
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5.11.2 Fitting the Marginal Model |
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260 | (1) |
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5.11.3 Repeated-Measures ANOVA |
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261 | (1) |
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262 | (1) |
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6 Random Coefficient Models for Longitudinal Data: The Autism Example |
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263 | (60) |
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263 | (1) |
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263 | (6) |
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263 | (2) |
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265 | (4) |
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6.3 Overview of the Autism Data Analysis |
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269 | (7) |
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269 | (2) |
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6.3.2 Model Specification |
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271 | (1) |
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6.3.2.1 General Model Specification |
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271 | (3) |
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6.3.2.2 Hierarchical Model Specification |
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274 | (1) |
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275 | (1) |
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6.4 Analysis Steps in the Software Procedures |
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276 | (24) |
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276 | (6) |
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282 | (3) |
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285 | (1) |
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6.4.3.1 Analysis Using the lme() Function |
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286 | (2) |
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6.4.3.2 Analysis Using the liner() Function |
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288 | (4) |
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292 | (3) |
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295 | (1) |
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6.4.5.1 Data Set Preparation |
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295 | (1) |
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6.4.5.2 Preparing the MDM File |
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296 | (4) |
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6.5 Results of Hypothesis Tests |
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300 | (2) |
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6.5.1 Likelihood Ratio Test for Random Effects |
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300 | (1) |
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6.5.2 Likelihood Ratio Tests for Fixed Effects |
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301 | (1) |
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6.6 Comparing Results across the Software Procedures |
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302 | (4) |
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6.6.1 Comparing Model 6.1 Results |
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302 | (1) |
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6.6.2 Comparing Model 6.2 Results |
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302 | (4) |
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6.6.3 Comparing Model 6.3 Results |
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306 | (1) |
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6.7 Interpreting Parameter Estimates in the Final Model |
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306 | (3) |
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6.7.1 Fixed-Effect Parameter Estimates |
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306 | (2) |
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6.7.2 Covariance Parameter Estimates |
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308 | (1) |
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6.8 Calculating Predicted Values |
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309 | (4) |
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6.8.1 Marginal Predicted Values |
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309 | (2) |
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6.8.2 Conditional Predicted Values |
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311 | (2) |
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6.9 Diagnostics for the Final Model |
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313 | (5) |
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6.9.1 Residual Diagnostics |
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313 | (3) |
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6.9.2 Diagnostics for the Random Effects |
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316 | (1) |
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6.9.3 Observed and Predicted Values |
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317 | (1) |
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6.10 Software Note: Computational Problems with the D Matrix |
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318 | (1) |
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319 | (1) |
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6.11 An Alternative Approach: Fitting the Marginal Model with an Unstructured Covariance Matrix |
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319 | (4) |
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322 | (1) |
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7 Models for Clustered Longitudinal Data: The Dental Veneer Example |
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323 | (66) |
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323 | (2) |
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7.2 The Dental Veneer Study |
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325 | (3) |
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325 | (1) |
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326 | (2) |
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7.3 Overview of the Dental Veneer Data Analysis |
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328 | (10) |
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328 | (3) |
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7.3.2 Model Specification |
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331 | (1) |
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7.3.2.1 General Model Specification |
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331 | (2) |
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7.3.2.2 Hierarchical Model Specification |
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333 | (3) |
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336 | (2) |
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7.4 Analysis Steps in the Software Procedures |
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338 | (27) |
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338 | (7) |
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345 | (4) |
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349 | (1) |
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7.4.3.1 Analysis Using the lme() Function |
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349 | (4) |
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7.4.3.2 Analysis Using the lmer() Function |
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353 | (3) |
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356 | (4) |
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360 | (1) |
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7.4.5.1 Data Set Preparation |
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360 | (1) |
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7.4.5.2 Preparing the Multivariate Data Matrix (MDM) File |
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361 | (4) |
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7.5 Results of Hypothesis Tests |
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365 | (1) |
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7.5.1 Likelihood Ratio Tests for Random Effects |
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365 | (1) |
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7.5.2 Likelihood Ratio Tests for Residual Error Variance |
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366 | (1) |
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7.5.3 Likelihood Ratio Tests for Fixed Effects |
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366 | (1) |
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7.6 Comparing Results across the Software Procedures |
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366 | (6) |
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7.6.1 Comparing Model 7.1 Results |
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366 | (3) |
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7.6.2 Comparing Results for Models 7.2A, 7.2B, and 7.2C |
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369 | (3) |
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7.6.3 Comparing Model 7.3 Results |
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372 | (1) |
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7.7 Interpreting Parameter Estimates in the Final Model |
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372 | (3) |
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7.7.1 Fixed-Effect Parameter Estimates |
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372 | (2) |
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7.7.2 Covariance Parameter Estimates |
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374 | (1) |
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7.8 The Implied Marginal Covariance Matrix for the Final Model |
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375 | (2) |
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7.9 Diagnostics for the Final Model |
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377 | (5) |
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7.9.1 Residual Diagnostics |
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378 | (1) |
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7.9.2 Diagnostics for the Random Effects |
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379 | (3) |
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7.10 Software Notes and Recommendations |
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382 | (3) |
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7.10.1 ML vs. REML Estimation |
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382 | (1) |
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7.10.2 The Ability to Remove Random Effects from a Model |
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382 | (1) |
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7.10.3 Considering Alternative Residual Error Covariance Structures |
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382 | (1) |
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7.10.4 Aliasing of Covariance Parameters |
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383 | (1) |
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7.10.5 Displaying the Marginal Covariance and Correlation Matrices |
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384 | (1) |
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7.10.6 Miscellaneous Software Notes |
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384 | (1) |
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7.11 Other Analytic Approaches |
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385 | (4) |
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7.11.1 Modeling the Covariance Structure |
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385 | (1) |
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7.11.2 The Step-Up vs. Step-Down Approach to Model Building |
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386 | (1) |
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7.11.3 Alternative Uses of Baseline Values for the Dependent Variable |
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386 | (3) |
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8 Models for Data with Crossed Random Factors: The SAT Score Example |
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389 | (30) |
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389 | (1) |
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389 | (5) |
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389 | (2) |
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391 | (3) |
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8.3 Overview of the SAT Score Data Analysis |
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394 | (2) |
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8.3.1 Model Specification |
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394 | (1) |
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8.3.1.1 General Model Specification |
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394 | (1) |
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8.3.1.2 Hierarchical Model Specification |
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395 | (1) |
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395 | (1) |
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8.4 Analysis Steps in the Software Procedures |
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396 | (14) |
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396 | (5) |
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401 | (2) |
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403 | (3) |
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406 | (1) |
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407 | (1) |
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8.4.5.1 Data Set Preparation |
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407 | (1) |
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8.4.5.2 Preparing the MDM File |
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408 | (1) |
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409 | (1) |
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8.5 Results of Hypothesis Tests |
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410 | (1) |
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8.5.1 Likelihood Ratio Tests for Random Effects |
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410 | (1) |
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8.5.2 Testing the Fixed Year Effect |
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411 | (1) |
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8.6 Comparing Results across the Software Procedures |
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411 | (1) |
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8.7 Interpreting Parameter Estimates in the Final Model |
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411 | (4) |
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8.7.1 Fixed-Effect Parameter Estimates |
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412 | (1) |
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8.7.2 Covariance Parameter Estimates |
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412 | (3) |
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8.8 The Implied Marginal Covariance Matrix for the Final Model |
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415 | (1) |
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8.9 Recommended Diagnostics for the Final Model |
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416 | (1) |
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8.10 Software Notes and Additional Recommendations |
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417 | (2) |
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9 Power Analysis and Sample Size Calculations for Linear Mixed Models |
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419 | (18) |
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419 | (1) |
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9.2 Direct Power Computations |
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419 | (10) |
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9.2.1 Software for Direct Power Computations |
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420 | (1) |
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9.2.2 Examples of Direct Power Computations |
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420 | (9) |
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9.3 Examining Power via Simulation |
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429 | (8) |
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9.3.1 Examples of Simulation-Based Approaches |
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430 | (7) |
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A Statistical Software Resources |
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437 | (8) |
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A.1 Descriptions/Availability of Software Packages |
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437 | (4) |
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437 | (1) |
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A.1.2 IBM SPSS Statistics |
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437 | (1) |
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437 | (1) |
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438 | (1) |
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438 | (1) |
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A.2 Useful Internet Links |
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438 | (3) |
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B Calculation of the Marginal Covariance Matrix |
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441 | (2) |
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C Acronyms / Abbreviations |
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443 | (2) |
Bibliography |
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445 | (8) |
Index |
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453 | |