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Overview

A mixed model allows the incorporation of both fixed and random variables within a statistical analysis. This enables efficient inferences and more information to be gained from the data. The application of mixed models is an increasingly popular way of analysing medical data, particularly in the pharmaceutical industry. There have been many recent advances in mixed modelling, particularly regarding the software and applications. This new edition of a groundbreaking text discusses the latest developments, from updated SAS techniques to the increasingly wide range of applications.

Presents an overview of the theory and applications of mixed models in medical research, including the latest developments and new sections on bioequivalence, cluster randomised trials and missing data. Easily accessible to practitioners in any area where mixed models are used, including medical statisticians and economists. Includes numerous examples using real data from medical and health research, and epidemiology, illustrated with SAS code and output. Features new version of SAS, including the procedure PROC GLIMMIX and an introduction to other available software. Supported by a website featuring computer code, data sets, and further material, available at: http://www.chs.med.ed.ac.uk/phs/mixed/. This much-anticipated second edition is ideal for applied statisticians working in medical research and the pharmaceutical industry, as well as teachers and students of statistics courses in mixed models. The text will also be of great value to a broad range of scientists, particularly those working in the medical and pharmaceutical areas.

Disc. problems in use of mixed models; when conventional, fixed effect models might be prefered; theory; software.

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Editorial Reviews

From the Publisher
"…a valuable mixed model resource for most applied statisticians working in the medical environment." (Biometrics, June 2007)

"…useful for practitioners and applied statisticians working in medical science." (Journal of the American Statistical Association, September 2007)

"…takes a practical rather than theoretical approach and requires understanding of only basic statistics." (MAA Reviews, October 30, 2006)

“This second edition gives an overview of the theory of mixed models and its application to real data in medical research.” (Zentralblatt MATH, April 2007)

Doody's Review Service
Reviewer: Sharon M. Homan, PhD (Kansas Health Institute)
Description: Mixed models, also known as multilevel models in the social sciences, allow both fixed and random variables within a statistical analysis. The general linear model assumption that the error terms are independent and identically distributed is relaxed in mixed models, so that observations can be correlated (e.g., repeated measures, cross-over trial, etc.). This second edition describes current methods and advanced SAS techniques for applying mixed models. The first edition was published in 1999.
Purpose: The authors present the theory and application of mixed models in medical research, including the latest developments in bioequivalence, cross-over trials, and cluster randomized trials. Their purpose is to make mixed modeling easily accessible to practitioners such as medical statisticians and economists. The book is well written, thorough, and highly applicable. The examples, complete with SAS code and output, are outstanding. The authors meet their objectives.
Audience: This is written for applied statisticians working in medical research and the pharmaceutical industry, as well as teachers and students of statistics courses in mixed models. The authors are experienced statisticians and highly credible scholars from the U.K. The book is part of the Statistics in Practice international series of books that provide statistical support for professionals and researchers.
Features: Beginning by describing the capabilities of mixed models, the authors introduce readers to the general linear model for fitting normally distributed data, and then extend the general linear model to general linear mixed models. The authors then examine how mixed models can be applied with categorical outcome variables. Chapters 5 to 7 are devoted to practical application of mixed models using particular designs. Chapter 8 includes a new section on bioequivalence studies and cluster randomized trials. Chapter 9 concludes by describing software options, including SAS and the PROC GLIMIX and PROC GENMOD procedures. The reference pages on mixed model notation, the glossary of terms, and the detailed SAS programming code and annotation greatly enhance the book's usefulness.
Assessment: This is an excellent resource for biostatisticians and medical researchers. It provides the reader with a thorough understanding of the concepts of mixed models. There are many social science texts on mixed modeling (also called multi-level modeling) but few that clearly link mixed models to clinical research designs. The second edition uses the 9th version of SAS and expands the coverage of categorical outcomes.
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Product Details

  • ISBN-13: 9780471965541
  • Publisher: Wiley, John & Sons, Incorporated
  • Publication date: 1/28/2000
  • Series: Statistics in Practice Series , #29
  • Edition number: 1
  • Pages: 428
  • Product dimensions: 6.18 (w) x 9.19 (h) x 1.13 (d)

Meet the Author

Helen Brown, Principal Statistician, NHS Scotland, Edinburgh, UK

Robin Prescott, Medical Statistics Unit, University of Edinburgh Medical School, UK

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Table of Contents


Preface to Second Edition     xiii
Mixed Model Notations     xvii
Introduction     1
The Use of Mixed Models     1
Introductory Example     3
Simple model to assess the effects of treatment (Model A)     3
A model taking patient effects into account (Model B)     6
Random effects model (Model C)     6
Estimation (or prediction) of random effects     11
A Multi-Centre Hypertension Trial     12
Modelling the data     13
Including a baseline covariate (Model B)     13
Modelling centre effects (Model C)     16
Including centre-by-treatment interaction effects (Model D)     16
Modelling centre and centre-treatment effects as random (Model E)     17
Repeated Measures Data     18
Covariance pattern models     19
Random coefficients models     20
More about Mixed Models     22
What is a mixed model     22
Why use mixed models     23
Communicating results     24
Mixed models in medicine     25
Mixed models in perspective     25
Some Useful Definitions     27
Containment     27
Balance     28
Error strata     30
Normal Mixed Models     33
Model Definition     33
The fixed effects model     34
The mixed model     36
The random effects model covariance structure     38
The random coefficients model covariance structure     41
The covariance pattern model covariance structure     43
Model Fitting Methods     45
The likelihood function and approaches to its maximisation     45
Estimation of fixed effects     49
Estimation (or prediction) of random effects and coefficients     50
Estimation of variance parameters     52
The Bayesian Approach     56
Introduction     57
Determining the posterior density     58
Parameter estimation, probability intervals and p-values     59
Specifying non-informative prior distributions     61
Evaluating the posterior distribution     66
Practical Application and Interpretation     70
Negative variance components     70
Accuracy of variance parameters     74
Bias in fixed and random effects standard errors     75
Significance testing     76
Confidence intervals     79
Model checking     79
Missing data     81
Example     83
Analysis models     83
Results     85
Discussion of points from Section 2.4     87
Generalised Linear Mixed Models     107
Generalised Linear Models     108
Introduction     108
Distributions     109
The general form for exponential distributions     111
The GLM definition     112
Fitting the GLM     115
Expressing individual distributions in the general exponential form     117
Conditional logistic regression     119
Generalised Linear Mixed Models     120
The GLMM definition     120
The likelihood and quasi-likelihood functions     121
Fitting the GLMM     124
Practical Application and Interpretation     128
Specifying binary data     128
Uniform effects categories     129
Negative variance components     130
Fixed and random effects estimates     130
Accuracy of variance parameters and random effects shrinkage     131
Bias in fixed and random effects standard errors     132
The dispersion parameter     133
Significance testing     135
Confidence intervals     136
Model checking     136
Example     137
Introduction and models fitted     137
Results     139
Discussion of points from Section 3.3     142
Mixed Models for Categorical Data     153
Ordinal Logistic Regression (Fixed Effects Model)     153
Mixed Ordinal Logistic Regression     158
Definition of the mixed ordinal logistic regression model     158
Residual variance matrix     159
Alternative specification for random effects models     161
Likelihood and quasi-likelihood functions     162
Model fitting methods     162
Mixed Models for Unordered Categorical Data     163
The G matrix     165
The R matrix     166
Fitting the model     166
Practical Application and Interpretation     166
Expressing fixed and random effects results     167
The proportional odds assumption     167
Number of covariance parameters     167
Choosing a covariance pattern     168
Interpreting covariance parameters     168
Checking model assumptions     168
The dispersion parameter     168
Other points     168
Example     169
Multi-Centre Trials and Meta-Analyses     183
Introduction to Multi-Centre Trials     183
What is a multi-centre trial?     183
Why use mixed models to analyse multi-centre data?     184
The Implications of using Different Analysis Models     184
Centre and centre-treatment effects fixed     184
Centre effects fixed, centre-treatment effects omitted     185
Centre and centre treatment effects random     186
Centre effects random, centre-treatment effects omitted     187
Example: A Multi-Centre Trial     188
Practical Application and Interpretation     195
Plausibility of a centre-treatment interaction     195
Generalisation     195
Number of centres     196
Centre size     196
Negative variance components     196
Balance     197
Sample Size Estimation     197
Normal data     197
Non-normal data     201
Meta-Analysis     203
Example: Meta-analysis     204
Analyses      204
Results     205
Treatment estimates in individual trials     206
Repeated Measures Data     215
Introduction     215
Reasons for repeated measurements     215
Analysis objectives     216
Fixed effects approaches     216
Mixed models approaches     217
Covariance Pattern Models     218
Covariance patterns     219
Choice of covariance pattern     223
Choice of fixed effects     225
General points     226
Example: Covariance Pattern Models for Normal Data     228
Analysis models     229
Selection of covariance pattern     229
Assessing fixed effects     231
Model checking     231
Example: Covariance Pattern Models for Count Data     237
Analysis models     238
Analysis using a categorical mixed model     242
Random Coefficients Models     245
Introduction     245
General points     247
Comparisons with fixed effects approaches     249
Examples of Random Coefficients Models     249
A linear random coefficients model     250
A polynomial random coefficients model     252
Sample Size Estimation     267
Normal data     267
Non-normal data     269
Categorical data     270
Cross-Over Trials     271
Introduction     271
Advantages of Mixed Models in Cross-Over Trials     272
The AB/BA Cross-Over Trial     272
Example: AB/BA cross-over design     275
Higher Order Complete Block Designs     279
Inclusion of carry-over effects     279
Example: four-period, four-treatment cross-over trial     279
Incomplete Block Designs     284
The three-treatment, two-period design (Koch's design)     284
Example: two-period cross-over trial     285
Optimal Designs     287
Example: Balaam's design     287
Covariance Pattern Models     290
Structured by period     290
Structured by treatment     290
Example: four-way cross-over trial     291
Analysis of Binary Data     299
Analysis of Categorical Data     303
Use of Results from Random Effects Models in Trial Design     307
Example     308
General Points     308
Other Applications of Mixed Models     311
Trials with Repeated Measurements within Visits     311
Covariance pattern models     312
Example     317
Random coefficients models     323
Example: random coefficients models     325
Multi-Centre Trials with Repeated Measurements     330
Example: multi-centre hypertension trial     331
Covariance pattern models     332
Multi-Centre Cross-Over Trials     337
Hierarchical Multi-Centre Trials and Meta-Analysis     338
Matched Case-Control Studies     339
Example     339
Analysis of a quantitative variable     340
Check of model assumptions     342
Analysis of binary variables     343
Different Variances for Treatment Groups in a Simple Between-Patient Trial     351
Example     352
Estimating Variance Components in an Animal Physiology Trial     355
Sample size estimation for a future experiment     356
Inter- and Intra-Observer Variation in Foetal Scan Measurements     361
Components of Variation and Mean Estimates in a Cardiology Experiment     353
Cluster Sample Surveys     365
Example: cluster sample survey      365
Small Area Mortality Estimates     367
Estimating Surgeon Performance     371
Event History Analysis     372
Example     373
A Laboratory Study Using a Within-Subject 4 x 4 Factorial Design     375
Bioequivalence Studies with Replicate Cross-Over Designs     378
Example     333
Cluster Randomised Trials     392
Example: a trial to evaluate integrated care pathways for treatment of children with asthma in hospital     392
Example: Edinburgh randomised trial of breast screening     395
Software for Fitting Mixed Models     401
Packages for Fitting Mixed Models     401
Basic use of PROC Mixed     403
Using SAS to Fit Mixed Models to Non-Normal Data     423
PROC GLIMMIX     423
PROC GENMOD     428
Glossary     431
References     435
Index     441
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