Linear Mixed Models in Practice

1 Introduction.- 2 An Example-Based Tour in Linear Mixed Models.- 2.1 Fixed Effects and Random Effects in Mixed Models.- 2.2 General Linear Mixed Models.- 2.3 Variance Components Estimation and Best Linear Unbiased Prediction.- 2.3.1 Variance Components Estimation.- 2.3.2 Best Linear Unbiased Prediction (BLUP).- 2.3.3 Examples and the SAS Procedure MIXED.- 2.4 Fixed Effects: Estimation and Hypotheses Testing.- 2.4.1 General Considerations.- 2.4.2 Examples and the SAS Procedure MIXED.- 2.5 Case Studies.- 2.5.1 Cell Proliferation.- 2.5.2 A Cross-Over Experiment.- 2.5.3 A Multicenter Trial.- 3 Linear Mixed Models for Longitudinal Data.- 3.1 Introduction.- 3.2 The Study of Natural History of Prostate Disease.- 3.3 A Two-Stage Analysis.- 3.4 The General Linear Mixed-Effects Model.- 3.4.1 The Model.- 3.4.2 Maximum Likelihood Estimation.- 3.4.3 Restricted Maximum Likelihood Estimation.- 3.4.4 Comparison between ML and REML Estimation.- 3.4.5 Model-Fitting Procedures.- 3.5 Example.- 3.5.1 The SAS Program.- 3.5.2 The SAS Output.- 3.5.3 Estimation Problems due to Small Variance Components.- 3.6 The RANDOM and REPEATED Statements.- 3.7 Testing and Estimating Contrasts of Fixed Effects.- 3.7.1 The CONTRAST Statement.- 3.7.2 Model Reduction.- 3.7.3 The ESTIMATE Statement.- 3.8 PROC MIXED versus PROC GLM.- 3.9 Tests for the Need of Random Effects.- 3.9.1 The Likelihood Ratio Test.- 3.9.2 Applied to the Prostate Data.- 3.10 Comparing Non-Nested Covariance Structures.- 3.11 Estimating the Random Effects.- 3.12 General Guidelines for Model Construction.- 3.12.1 Selection of a Preliminary Mean Structure.- 3.12.2 Selection of Random-Effects.- 3.12.3 Selection of Residual Covariance Structure.- 3.12.4 Model Reduction.- 3.13 Model Checks and Diagnostic Tools ?.- 3.13.1 Normality Assumption for the Random Effects ?.- 3.13.2 The Detection of Influential Subjects ?.- 3.13.3 Checking the Covariance Structure ?.- 4 Case Studies.- 4.1 Example 1: Variceal Pressures.- 4.2 Example 2: Growth Curves.- 4.3 Example 3: Blood Pressures.- 4.4 Example 4: Growth Data.- 4.4.1 Model 1.- 4.4.2 Model 2.- 4.4.3 Model 3.- 4.4.4 Graphical Exploration.- 4.4.5 Model 4.- 4.4.6 Model 5.- 4.4.7 Model 6.- 4.4.8 Model 7.- 4.4.9 Model 8.- 5 Linear Mixed Models and Missing Data.- 5.1 Introduction.- 5.2 Missing Data.- 5.2.1 Missing Data Patterns.- 5.2.2 Missing Data Mechanisms.- 5.2.3 Ignorability.- 5.3 Approaches to Incomplete Data.- 5.4 Complete Case Analysis.- 5.4.1 Growth Data.- 5.5 Simple Forms of Imputation.- 5.5.1 Last Observation Carried Forward.- 5.5.2 Imputing Unconditional Means.- 5.5.3 Buck’s Method: Conditional Mean Imputation.- 5.5.4 Discussion of Imputation Techniques.- 5.6 Available Case Methods.- 5.6.1 Growth Data.- 5.7 Likelihood-Based Ignorable Analysis and PROC MIXED.- 5.7.1 Growth Data.- 5.7.2 Summary.- 5.8 How Ignorable Is Missing At Random ? ?.- 5.8.1 Information and Sampling Distributions ?.- 5.8.2 Illustration ?.- 5.8.3 Example ?.- 5.8.4 Implications for PROC MIXED.- 5.9 The Expectation-Maximization Algorithm ?.- 5.10 Multiple Imputation ?.- 5.10.1 General Theory ?.- 5.10.2 Illustration: Growth Data ?.- 5.11 Exploring the Missing Data Process.- 5.11.1 Growth Data.- 5.11.2 Informative Non-Response.- 5.11.3 OSWALD for Informative Non-Response.- A Inference for Fixed Effects.- A.1 Estimation.- A.2 Hypothesis Testing.- A.3 Determination of Degrees of Freedom.- A.4 Satterthwaite’s Procedure.- B Variance Components and Standard Errors.- C Details on Table 2.10: Expected Mean Squares.- D Example 2.8: Cell Proliferation.- References.