SAS for Mixed Models Second Edition

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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

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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