Linear Mixed Models in Practice: A SAS-Oriented Approach

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.