Praise from the Experts “This is a revision of an already excellent text. The authors take time to explain and provide motivation for the calculations being done. The examples are information rich, and I can see them serving as templates for a wide variety of applications. Each is followed by an interpretation section that is most helpful. Nonlinear and generalized linear mixed models are addressed, as are Bayesian methods, and some helpful suggestions are presented for dealing with convergence problems. Those familiar with the previous release will be excited to learn about the new features in PROC MIXED. “The MIXED procedure has had a great influence on how statistical analyses are performed. It has allowed us to do correct analyses where we have previously been hampered by computational limitations. It is hard to imagine anyone claiming to be a modern professional data analyst without knowledge of the methods presented in this book. The mixed model pulls into a common framework many analyses of experimental designs and observational studies that have traditionally been treated as being different from each other. By describing the three model components X, Z, and the error term e, one can reproduce and often improve on the analysis of any designed experiment. “I am looking forward to getting my published copy of the book and am sure it will be well worn in no time.” David A. Dickey Professor of Statistics, North Carolina State University “SAS for Mixed Models, Second Edition addresses the large class of statistical models with random and fixed effects. Mixed models occur across most areas of inquiry, including all designed experiments, for example. “This book should be required reading for all statisticians, and will be extremely useful to scientists involved with data analysis. Most pages contain example output, with the capabilities of mixed models and SAS software clearly explained throughout. I have used the first edition of SAS for Mixed Models as a textbook for a second-year graduate-level course in linear models, and it has been well received by students. The second edition provides dramatic enhancement of all topics, including coverage of the new GLIMMIX and NLMIXED procedures, and a chapter devoted to power calculations for mixed models. The chapter of case studies will be interesting reading, as we watch the experts extract information from complex experimental data (including a microarray example). “I look forward to using this superb compilation as a textbook.” Arnold Saxton Department of Animal Science, University of Tennessee “With an abundance of new material and a thorough updating of material from the first edition, SAS for Mixed Models, Second Edition will be of inordinate interest to those of us engaged in the modeling of messy continuous and categorical data. It contains several new chapters, and its printed format makes this a much more readable version than its predecessor. We owe the authors a tip of the hat for providing such an invaluable compendium.” Timothy G. Gregoire J. P. Weyerhaeuser Professor of Forest Management School of Forestry and Environmental Studies, Yale University “Because of the pervasive need to model both fixed and random effects in most efficient experimental designs and observational studies, the SAS System for Mixed Models book has been our most frequently used resource for data analysis using statistical software. The second edition wonderfully updates the discussion on topics that were previously considered in the first edition, such as analysis of covariance, randomized block designs, repeated measures designs, split-plot and nested designs, spatial variability, heterogeneous variance models, and random coefficient models. If that isn’t enough, the new edition further enhances the mixed model toolbase of any serious data analyst. For example, it provides very useful and not otherwise generally available tools for diagnostic checks on potentially influential and outlying random and residual effects in mixed model analyses. “Also, the new edition illustrates how to compute statistical power for many experimental designs, using tools that are not available with most other software, because of this book’s foundation in mixed models. Chapters discussing the relatively new GLIMMIX and NLMIXED procedures for generalized linear mixed model and nonlinear mixed model analyses will prove to be particularly profitable to the user requiring assistance with mixed model inference for cases involving discrete data, nonlinear functions, or multivariate specifications. For example, code based on those two procedures is provided for problems ranging from the analysis of count data in a split-plot design to the joint analysis of survival and repeated measures data; there is also an implementation for the increasingly popular zero-inflated Poisson models with random effects! The new chapter on Bayesian analysis of mixed models is also timely and highly readable for those researchers wishing to explore that increasingly important area of application for their own research.” Robert J. Tempelman Michigan State University “We welcome the second edition of this book, given a multitude of scientific and software evolutions in the field of mixed models. Important new developments have been incorporated, including generalized linear mixed models, nonlinear mixed models, power calculations, Bayesian methodology, and extended information on spatial approaches. “Since mixed models have been developing in a variety of fields (agriculture, medicine, psychology, etc.), notation and terminology encountered in the literature is unavoidably scattered and not as streamlined as one might hope. Faced with these challenges, the authors have chosen to serve the various applied segments. This is why one encounters randomized block designs, random effects models, random coefficients models, and multilevel models, one next to the other. “Arguably, the book is most useful for readers with a good understanding of mixed models theory, and perhaps familiarity with simple implementations in SAS and/or alternative software tools. Such a reader will encounter a number of generic case studies taken from a variety of application areas and designs. Whereas this does not obviate the need for users to reflect on the peculiarities of their own design and study, the book serves as a useful starting point for their own implementation. In this sense, the book is ideal for readers familiar with the basic models, such as a mixed model for Poisson data, looking for extensions, such as zero-inflated Poisson data. “Unavoidably, readers will want to deepen their understanding of modeling concepts alongside working on implementations. While the book focuses less on methodology, it does contain an extensive and up-to-date reference list. “It may appear that for each of the main categories (linear, generalized linear, and nonlinear mixed models) there is one and only one SAS procedure available (MIXED, GLIMMIX, and NLMIXED, respectively), but the reader should be aware that this is a rough rule of thumb only. There are situations where fitting a particular model is easier in a procedure other than the one that seems the obvious choice. For example, when one wants to fit a mixed model to binary data, and one insists on using quadrature methods rather than quasi-likelihood, NLMIXED is the choice.” Geert Verbeke Biostatistical Centre, Katholieke Universiteit Leuven, Belgium Geert Molenberghs Center for Statistics, Hasselt University, Diepenbeek, Belgium “Publication of this second edition couldn’t have come at a better time. Since the release of the first edition, a number of advances have been made in the field of mixed models, both computationally and theoretically, and the second edition captures many if not most of these key developments. To that end, the second edition has been substantially reorganized to better explain the general nature and theory of mixed models (e.g., Chapter 1 and Appendix 1) and to better illustrate, within dedicated chapters, the various types of mixed models that readers are most likely to encounter. This edition has been greatly expanded to include chapters on mixed model diagnostics (Chapter 10), power calculations for mixed models (Chapter 12), and Bayesian mixed models (Chapter 13). “In addition, the authors have done a wonderful job of expanding their coverage of generalized linear mixed models (Chapter 14) and nonlinear mixed models (Chapter 15)—a key feature for those readers who are just getting acquainted with the recently released GLIMMIX and NLMIXED procedures. The inclusion of material related to these two procedures enables readers to apply any number of mixed modeling tools currently available in SAS. Indeed, the strength of this second edition is that it provides readers with a comprehensive overview of mixed model methodology ranging from analytically tractable methods for the traditional linear mixed model to more complex methods required for generalized linear and nonlinear mixed models. More importantly, the authors describe and illustrate the use of a wide variety of mixed modeling tools available in SAS—tools without which the analyst would have little hope of sorting through the complexities of many of today’s technology-driven applications. I highly recommend this book to anyone remotely interested in mixed models, and most especially to those who routinely find themselves fitting data to complex mixed models.” Edward F. Vonesh, Ph.D. Senior Baxter Research Scientist Statistics, Epidemiology and Surveillance Baxter Healthcare Corporation SAS Press SAS for ® Mixed Models Second Edition Ramon C. Littell, Ph.D. George A Milliken, Ph.D. Walter W. Stroup, Ph.D. Russell D. Wolfinger, Ph.D. Oliver Schabenberger, Ph.D. The correct bibliographic citation for this manual is as follows: Littell, Ramon C., George A. Milliken, Walter W. Stroup, Russell D. Wolfinger, and Oliver Schabenberger. 2006. SAS® for Mixed Models, Second Edition. Cary, NC: SAS Institute Inc. SAS® for Mixed Models, Second Edition Copyright © 2006, SAS Institute Inc., Cary, NC, USA ISBN-13: 978-1-59047-500-3 ISBN-10: 1-59047-500-3 All rights reserved. Produced in the United States of America. For a hard-copy book: No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, or otherwise, without the prior written permission of the publisher, SAS Institute Inc. For a Web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. U.S. Government Restricted Rights Notice: Use, duplication, or disclosure of this software and related documentation by the U.S. government is subject to the Agreement with SAS Institute and the restrictions set forth in FAR 52.227-19, Commercial Computer Software-Restricted Rights (June 1987). SAS Institute Inc., SAS Campus Drive, Cary, North Carolina 27513. 1st printing, February 2006 SAS Publishing provides a complete selection of books and electronic products to help customers use SAS software to its fullest potential. For more information about our e-books, e-learning products, CDs, and hard-copy books, visit the SAS Publishing Web site at support.sas.com/pubs or call 1-800-727-3228. SAS® and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. ® indicates USA registration. Other brand and product names are registered trademarks or trademarks of their respective companies. Contents Preface ix Chapter 1 Introduction 1 1.1 Types of Models That Produce Data 1 1.2 Statistical Models 2 1.3 Fixed and Random Effects 4 1.4 Mixed Models 6 1.5 Typical Studies and the Modeling Issues They Raise 7 1.6 A Typology for Mixed Models 11 1.7 Flowcharts to Select SAS Software to Run Various Mixed Models 13 Chapter 2 Randomized Block Designs 17 2.1 Introduction 18 2.2 Mixed Model for a Randomized Complete Blocks Design 18 2.3 Using PROC MIXED to Analyze RCBD Data 22 2.4 Introduction to Theory of Mixed Models 42 2.5 Example of an Unbalanced Two-Way Mixed Model: Incomplete Block Design 44 2.6 Summary 56 Chapter 3 Random Effects Models 57 3.1 Introduction: Descriptions of Random Effects Models 58 3.2 Example: One-Way Random Effects Treatment Structure 64 3.3 Example: A Simple Conditional Hierarchical Linear Model 75 3.4 Example: Three-Level Nested Design Structure 81 3.5 Example: A Two-Way Random Effects Treatment Structure to Estimate Heritability 88 3.6 Summary 91 Chapter 4 Multi-factor Treatment Designs with Multiple Error Terms 93 4.1 Introduction 94 4.2 Treatment and Experiment Structure and Associated Models 94 4.3 Inference with Mixed Models for Factorial Treatment Designs 102 4.4 Example: A Split-Plot Semiconductor Experiment 113 iv Contents 4.5 Comparison with PROC GLM 130 4.6 Example: Type × Dose Response 135 4.7 Example: Variance Component Estimates Equal to Zero 148 4.8 More on PROC GLM Compared to PROC MIXED: Incomplete Blocks, Missing Data, and Estimability 154 4.9 Summary 156 Chapter 5 Analysis of Repeated Measures Data 159 5.1 Introduction 160 5.2 Example: Mixed Model Analysis of Data from Basic Repeated Measures Design 163 5.3 Modeling Covariance Structure 174 5.4 Example: Unequally Spaced Repeated Measures 198 5.5 Summary 202 Chapter 6 Best Linear Unbiased Prediction 205 6.1 Introduction 206 6.2 Examples of BLUP 206 6.3 Basic Concepts of BLUP 210 6.4 Example: Obtaining BLUPs in a Random Effects Model 212 6.5 Example: Two-Factor Mixed Model 219 6.6 A Multilocation Example 226 6.7 Location-Specific Inference in Multicenter Example 234 6.8 Summary 241 Chapter 7 Analysis of Covariance 243 7.1 Introduction 244 7.2 One-Way Fixed Effects Treatment Structure with Simple Linear Regression Models 245 7.3 Example: One-Way Treatment Structure in a Randomized Complete Block Design Structure—Equal Slopes Model 251 7.4 Example: One-Way Treatment Structure in an Incomplete Block Design Structure—Time to Boil Water 263 7.5 Example: One-Way Treatment Structure in a Balanced Incomplete Block Design Structure 272 7.6 Example: One-Way Treatment Structure in an Unbalanced Incomplete Block Design Structure 281 7.7 Example: Split-Plot Design with the Covariate Measured on the Large-Size Experimental Unit or Whole Plot 286 7.8 Example: Split-Plot Design with the Covariate Measured on the Small-Size Experimental Unit or Subplot 297 7.9 Example: Complex Strip-Plot Design with the Covariate Measured on an Intermediate-Size Experimental Unit 308 7.10 Summary 315 Contents v Chapter 8 Random Coefficient Models 317 8.1 Introduction 317 8.2 Example: One-Way Random Effects Treatment Structure in a Completely Randomized Design Structure 320 8.3 Example: Random Student Effects 326 8.4 Example: Repeated Measures Growth Study 330 8.5 Summary 341 Chapter 9 Heterogeneous Variance Models 343 9.1 Introduction 344 9.2 Example: Two-Way Analysis of Variance with Unequal Variances 345 9.3 Example: Simple Linear Regression Model with Unequal Variances 354 9.4 Example: Nested Model with Unequal Variances for a Random Effect 366 9.5 Example: Within-Subject Variability 374 9.6 Example: Combining Between- and Within-Subject Heterogeneity 393 9.7 Example: Log-Linear Variance Models 402 9.8 Summary 411 Chapter 10 Mixed Model Diagnostics 413 10.1 Introduction 413 10.2 From Linear to Linear Mixed Models 415 10.3 The Influence Diagnostics 424 10.4 Example: Unequally Spaced Repeated Measures 426 10.5 Summary 435 Chapter 11 Spatial Variability 437 11.1 Introduction 438 11.2 Description 438 11.3 Spatial Correlation Models 440 11.4 Spatial Variability and Mixed Models 442 11.5 Example: Estimating Spatial Covariance 447 11.6 Using Spatial Covariance for Adjustment: Part 1, Regression 457 11.7 Using Spatial Covariance for Adjustment: Part 2, Analysis of Variance 460 11.8 Example: Spatial Prediction—Kriging 471 11.9 Summary 478 Chapter 12 Power Calculations for Mixed Models 479 12.1 Introduction 479 12.2 Power Analysis of a Pilot Study 480 12.3 Constructing Power Curves 483 12.4 Comparing Spatial Designs 486