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• Author: Ramon C. Littell, George A. Milliken, Walter W. Stroup, Russell D. Wolfinger, Ol
• Publisher: SAS Publishing
• Publish Date: 30 March, 2006
• ISBN: 1590475003

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

left 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

Preface

The subject of mixed linear models is taught in graduate-level statistics courses and is familiar

to most statisticians. During the past 10 years, use of mixed model methodology has expanded

to nearly all areas of statistical applications. It is routinely taught and applied even in disciplines

outside traditional statistics. Nonetheless, many persons who are engaged in analyzing mixed

model data have questions about the appropriate implementation of the methodology. Also,

even users who studied the topic 10 years ago may not be aware of the tremendous new

capabilities available for applications of mixed models.

Like the first edition, this second edition presents mixed model methodology in a setting that is

driven by applications. The scope is both broad and deep. Examples are included from

numerous areas of application and range from introductory examples to technically advanced

case studies. The book is intended to be useful to as diverse an audience as possible, although

persons with some knowledge of analysis of variance and regression analysis will benefit most.

Since the first edition of this book appeared in 1996, mixed model technology and mixed model

software have made tremendous leaps forward. Previously, most of the mixed model

capabilities in the SAS System hinged on the MIXED procedure. Since the first edition, the

capabilities of the MIXED procedure have expanded, and new procedures have been developed

to implement mixed model methodology beyond classical linear models. The NLMIXED

procedure for nonlinear mixed models was added in SAS 8, and recently the GLIMMIX

procedure for generalized linear mixed models was added in SAS 9.1. In addition, ODS and

ODS statistical graphics provide powerful tools to request and manage tabular and graphical

output from SAS procedures. In response to these important advances we not only brought the

SAS code in this edition up-to-date with SAS 9.1, but we also thoroughly re-examined the text

and contents of the first edition. We rearranged some topics to provide a more logical flow, and

introduced new examples to broaden the scope of application areas.

Note to SAS 8 users: Although the examples in this book were tested using SAS 9.1, you will

find that the vast majority of the SAS code applies to SAS 8 as well. Exceptions are ODS

statistical graphics, the RESIDUAL and INFLUENCE options in the MODEL statement of

PROC MIXED, and the GLIMMIX procedure.

The second edition of SAS for Mixed Models will be useful to anyone wishing to use SAS for

analysis of mixed model data. It will be a good supplementary text for a statistics course in

mixed models, or a course in hierarchical modeling or applied Bayesian statistics. Many mixed

model applications have emerged from agricultural research, but the same or similar

methodology is useful in other subject areas, such as the pharmaceutical, natural resource,

engineering, educational, and social science disciplines. We are of the belief that almost all data

sets have features of mixed models, and sometimes are identified by other terminology, such as

hierarchical models and latent variables.

Not everyone will want to read the book from cover to cover. Readers who have little or no

exposure to mixed models will be interested in the early chapters and can progress through later

chapters as their needs require. Readers with good basic skills may want to jump into the

chapters on topics of specific interest and refer to earlier material to clarify basic concepts.

The introductory chapter provides important definitions and categorizations and delineates

mixed models from other classes of statistical models. Chapters 2–9 cover specific forms of

mixed models and the situations in which they arise. Randomized block designs with fixed

treatment and random block effects (Chapter 2) are among the simplest mixed models; they

allow us to discuss some of the elementary mixed model operations, such as best linear

unbiased prediction and expected mean squares, and to demonstrate the use of SAS mixed

model procedures in this simple setting. Chapter 3 considers models in which all effects are

random. Situations with multiple random components also arise naturally when an experimental

design gives rise to multiple error terms, such as in split-plot designs. The analysis of the

associated models is discussed in Chapter 4. Repeated measures and longitudinal data give rise

to mixed models in which the serial dependency among observations can be modeled directly;

this is the topic of Chapter 5. A separate chapter is devoted to statistical inference based on best

linear unbiased prediction of random effects (Chapter 6). Models from earlier chapters are

revisited here. Chapter 7 deals with the situation where additional continuous covariates have

been measured that need to be accommodated in the mixed model framework. This naturally

leads us to random coefficient and multi-level linear models (Chapter 8). Mixed model

technology and mixed model software find application in situations where the error structure

does not comply with that of the standard linear model. A typical example is the correlated error

model. Also of great importance to experimenters and analysts are models with independent but

heteroscedastic errors. These models are discussed in Chapter 9. Models with correlated errors

are standard devices to model spatial data (Chapter 11).

Chapters 10, 12, and 13 are new additions to this book. Diagnostics for mixed models based on

residuals and influence analysis are discussed in Chapter 10. Calculating statistical power of

tests is the focus of Chapter 12. Mixed modeling from a Bayesian perspective is discussed in

Chapter 13.

Chapters 14 and 15 are dedicated to mixed models that exhibit nonlinearity. The first of these

chapters deals with generalized linear mixed models where normally distributed random effects

appear inside a link function. This chapter relies on the GLIMMIX procedure. Mixed models

with general nonlinear conditional mean function are discussed in Chapter 15, which relies

primarily on the NLMIXED procedure.

The main text ends with Chapter 16, which provides 12 case studies that cover a wide range of

applications, from response surfaces to crossover designs and microarray analysis.

Good statistical applications require a certain amount of theoretical knowledge. The more

advanced the application, the more theoretical skills will help. While this book certainly

revolves around applications, theoretical developments are presented as well, to describe how

mixed model methodology works and when it is useful. Appendix 1 contains some important

details about mixed model theory.

Appendix 2 lists the data used for analyses in the book in abbreviated form so you can see the

general structure of the data sets. The full data sets are available on the accompanying CD and

on the companion Web site for this book (support.sas.com/companionsites). These sources

also contain the SAS code to perform the analyses in the book, organized by chapter.

We would like to extend a special thanks to the editorial staff at SAS Press. Our editor,

Stephenie Joyner, has shown a precious combination of persistence and patience that kept us on

track. Our admiration goes out to our copy editor, Ed Huddleston, for applying his thorough and

exacting style to our writing, adding perspicuity.

Writing a book of this scope is difficult and depends on the support, input, and energy of many

individuals, groups, and organizations. Foremost, we need to thank our families for their

patience, understanding, and support. Thanks to our respective employers—the University of

Florida, Kansas State University, the University of Nebraska, and SAS Institute—for giving us

degrees of freedom to undertake this project. Thanks to mixed model researchers and statistical

colleagues everywhere for adjusting those degrees of freedom by shaping our thinking through

their work. Thanks to the statisticians, analysts, and researchers who shared their data sets and

data stories and allowed us to pass them along to you. Special thanks go to Andrew Hartley for

his considerable and thoughtful commentary on Chapter 13, as well as for many of the

references in that chapter. Thanks to the many SAS users who have provided feedback about the

first edition. Providing the details of all those who have effectively contributed to this book and

by what means would require another whole volume!

As mixed model methodology blazes ahead in the coming decades and continues to provide a

wonderful and unifying framework for understanding statistical practice, we trust this volume

will be a useful companion as you apply the techniques effectively. We wish you success in

becoming a more proficient mixed modeler.

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