The dissemination of the MIXED procedure in SAS has provided a whole class of statistical models for routine use. We believe that both the ideas be­ hind the techniques and their implementation in SAS are not at all straight­ forward and users from various applied backgrounds, including the phar­ maceutical industry, have experienced difficulties in using the procedure effectively. Courses and consultancy on PROC MIXED have been in great demand in recent years, illustrating the clear need for resource material to aid the user. This book is intended as a contribution to bridging this gap. We hope the book will be of value to a wide audience, including applied statisticians and many biomedical researchers, particularly in the pharma­ ceutical industry, medical and public health research organizations, con­ tract research organizations, and academic departments. This implies that our book is explanatory rather than research oriented and that it empha­ sizes practice rather than mathematical rigor. In this respect, clear guidance and advice on practical issues are the main focus of the text. Nevertheless, this does not imply that more advanced topics have been avoided. Sections containing material of a deeper level have been sign posted by means of an asterisk.

The SAS routines on mixed models have applications in many areas of statistics, especially biostatistics, but the procedures are not well- documented. Based on short courses given by the authors, this book provides practical guidance for SAS users.

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.- 2.4 Fixed Effects: Estimation and Hypotheses Testing.- 2.5 Case Studies.- 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.- 3.5 Example.- 3.6 The RANDOM and REPEATED Statements.- 3.7 Testing and Estimating Contrasts of Fixed Effects.- 3.8 PROC MIXED versus PROC GLM.- 3.9 Tests for the Need of Random Effects.- 3.10 Comparing Non-Nested Covariance Structures.- 3.11 Estimating the Random Effects.- 3.12 General Guidelines for Model Construction.- 3.13 Model Checks and Diagnostic Tools ?.- 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.- 5 Linear Mixed Models and Missing Data.- 5.1 Introduction.- 5.2 Missing Data.- 5.3 Approaches to Incomplete Data.- 5.4 Complete Case Analysis.- 5.5 Simple Forms of Imputation.- 5.6 Available Case Methods.- 5.7 Likelihood-Based Ignorable Analysis and PROC MIXED.- 5.8 How Ignorable Is Missing At Random ? ?.- 5.9 The Expectation-Maximization Algorithm ?.- 5.10 Multiple Imputation ?.- 5.11 Exploring the Missing Data Process.- 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.