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Generalized Linear Mixed Models: Modern Concepts, Methods and Applications

Walter W. Stroup
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Texts in Statistical Science
[Reviewed by
Homer White
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Walter Stroup is a leading authority on generalized linear mixed models (GLMMs) for applied statisticians, especially as implemented in the SAS programming environment. He offers here a thorough, engaging and opinionated treatment of the subject, one that he says is directed to “graduate students in statistics, statistics professionals seeking to get up to speed, and researchers new to the generalized linear model thought process.”

Most people learn regression modeling in a sequence of steps that more or less increase in generality:

  1. linear regression and one and two-way ANOVA;
  2. the revelation that the procedures in Item 1 are special cases of something called the “general linear model”;
  3. logistic, Poisson, binomial regression, etc.;
  4. the further revelation that the procedures in items 2 and 3 are all special cases of something called the “generalized linear model”;
  5. introduction of the notion of random effects, as opposed to the fixed effects that constituted the predictor variables in Items 1-5 above;
  6. integration of fixed and random effects, producing “generalized linear mixed models”.

Stroup thinks that the time for this sequenced approach has passed. The distinctive feature of this book, therefore, is that it goes for full generality right from the start. Stroup believes that early awareness of the full GLMM picture will help the working statistician improve his/her ability to understand issues in experimental design and statistical modeling, even in those cases where GLMM in its fullness is not required.

Accordingly the book begins with ground-up treatments of the concept of a statistical model (Chapter One, Modeling Basics) and the design of experiments and studies (Chapter Two, Design Matters). An especially illuminating feature of Chapter Two is the author’s distinction between two techniques for moving from a study design to the construction of an appropriate linear predictor, namely: the “unit of replication” approach and the WWFD (What Would Fisher Do) approach. Chapter Three introduces the basic tools of statistical inference in GLMMs: parameter estimation and hypothesis testing.

All readers should grapple with Chapters 1–3. Readers interested in learning more of the background theory behind the inferential procedures will want to study Chapters 4–6.

Chapters 7–9 cover topics that prepare the reader for specific applications in Chapters 10–16. These later chapters — addressing rates and proportions, count data, time-to-event data, multinomial categorical and ordinal data, repeated measures and spatial variability — clearly are the manna of the text. It appears possible to jump into most of the topics in Chapters 10–16 after a perusal of Chapters 1 through 3 — I tried a bit of that myself — but at some points a study of portions of Chapters 7, 8 and 9 is necessary for full understanding and responsible use of the procedure of interest.

How well does the book succeed in its stated purpose? My own evaluation is based on my status as someone in between a “statistics professional trying to get up to speed” and a “researcher new to the generalized linear model thought process.” I found the “fully general “ GLMM approach to modeling and design issues (Chapters 1 and 2) to be quite illuminating. However, I am not sure that readers can master this material on their own without reasonable prior exposure to linear models and to issues in experimental design at the level of, say, Statistical Methods, the classic text of Snedecor and Cochran:

Another issue is software: this text is centered around SAS, and in fact the author notes that he has worked for years with Oliver Schabenberger, the developer of SAS’s POC GLIMMIX, and says that “we have been colleagues in thinking about what [the] GLMM curriculum should look like for years. The thought processes embedded in GLIMMIX reflect the way we think about GLMMs.” Hence it is best to use this text in conjunction with SAS. Prospective readers without current access to SAS will be pleased to know that a reasonable level of access to SAS is now available at no cost to students and teachers on the web: see

But as Stroup himself admits, “the statistics world seems to be moving toward R.” If the reader prefers to work with GLMMs in the free, powerful and state-of-the-art R environment, then he/she should supplement this text with some others that are built around R. I myself had good luck using Stroup’s text along with Julian Faraway’s two books Linear Models With R and Expanding the Linear Model With R, both published by CRC Press.

Homer White is Professor of Mathematics at Georgetown College, in Kentucky. A typical Jack-of-All-Trades Small-College Mathematician, he enjoys the teaching of statistics at all levels, statistical consultation, and even institutional research. His interests and occasional forays into research in the history of mathematics include the geometrical works of Leonhard Euler and the mathematics of classical India.

PART I The Big Picture
Modeling Basics
What Is a Model?
Two Model Forms: Model Equation and Probability Distribution
Types of Model Effects
Writing Models in Matrix Form
Summary: Essential Elements for a Complete Statement of the Model

Design Matters
Introductory Ideas for Translating Design and Objectives into Models
Describing "Data Architecture" to Facilitate Model Specification
From Plot Plan to Linear Predictor
Distribution Matters
More Complex Example: Multiple Factors with Different Units of Replication


Setting the Stage
Goals for Inference with Models: Overview
Basic Tools of Inference
Issue I: Data Scale vs. Model Scale
Issue II: Inference Space
Issue III: Conditional and Marginal Models


PART II Estimation and Inference Essentials

Essential Background
Fixed Effects Only
Gaussian Mixed Models
Generalized Linear Mixed Models


Inference, Part I: Model Effects
Essential Background
Approaches to Testing
Inference Using Model-Based Statistics
Inference Using Empirical Standard Error
Summary of Main Ideas and General Guidelines for Implementation


Inference, Part II: Covariance Components
Formal Testing of Covariance Components
Fit Statistics to Compare Covariance Models
Interval Estimation


PART III Working with GLMMs
Treatment and Explanatory Variable Structure

Types of Treatment Structures
Types of Estimable Functions
Multiple Factor Models: Overview
Multifactor Models with All Factors Qualitative
Multifactor: Some Factors Qualitative, Some Factors Quantitative
Multifactor: All Factors Quantitative


Multilevel Models
Types of Design Structure: Single- and Multilevel Models Defined
Types of Multilevel Models and How They Arise
Role of Blocking in Multilevel Models
Working with Multilevel Designs
Marginal and Conditional Multilevel Models


Best Linear Unbiased Prediction
Review of Estimable and Predictable Functions
BLUP in Random-Effects-Only Models
Gaussian Data with Fixed and Random Effects
Advanced Applications with Complex Z Matrices


Rates and Proportions
Types of Rate and Proportion Data
Discrete Proportions: Binary and Binomial Data
Alternative Link Functions for Binomial Data
Continuous Proportions


Overdispersion in Count Data
More on Alternative Distributions
Conditional and Marginal
Too Many Zeroes


Time-to-Event Data
Introduction: Probability Concepts for Time-to-Event Data
Gamma GLMMs
GLMMs and Survival Analysis


Multinomial Data
Multinomial Data with Ordered Categories
Nominal Categories: Generalized Logit Models
Model Comparison


Correlated Errors, Part I: Repeated Measures
Gaussian Data: Correlation and Covariance Models for LMMs
Covariance Model Selection
Non-Gaussian Case
Issues for Non-Gaussian Repeated Measures


Correlated Errors, Part II: Spatial Variability
Gaussian Case with Covariance Model
Spatial Covariance Modeling by Smoothing Spline
Non-Gaussian Case


Power, Sample Size, and Planning
Basics of GLMM-Based Power and Precision Analysis
Gaussian Example
Power for Binomial GLMMs
GLMM-Based Power Analysis for Count Data
Power and Planning for Repeated Measures