Applied Mixed Models in Medicine E-Book

Helen Brown

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Statistics in Practice
• Sprache: Englisch
• Veröffentlichungsjahr: 2014

Seitenzahl: 796

Beschreibung

A fully updated edition of this key text on mixed models, focusing on applications in medical research The application of mixed models is an increasingly popular way of analysing medical data, particularly in the pharmaceutical industry. A mixed model allows the incorporation of both fixed and random variables within a statistical analysis, enabling efficient inferences and more information to be gained from the data. There have been many recent advances in mixed modelling, particularly regarding the software and applications. This third edition of Brown and Prescott's groundbreaking text provides an update on the latest developments, and includes guidance on the use of current SAS techniques across a 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 incomplete block designs and the analysis of bilateral 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 the new version of SAS, including new graphics for model diagnostics and the procedure PROC MCMC. * Supported by a website featuring computer code, data sets, and further material. This third edition will appeal to applied statisticians working in medical research and the pharmaceutical industry, as well as teachers and students of statistics courses in mixed models. The book will also be of great value to a broad range of scientists, particularly those working in the medical and pharmaceutical areas.

Leseprobe

Table of Contents

Statistics in Practice

Title Page

Copyright

Preface to third edition

Mixed models notation

About the Companion Website

Chapter 1: Introduction

1.1 The use of mixed models

1.2 Introductory example

1.3 A multi-centre hypertension trial

1.4 Repeated measures data

1.5 More about mixed models

1.6 Some useful definitions

Chapter 2: Normal mixed models

2.1 Model definition

2.2 Model fitting methods

2.3 The Bayesian approach

2.4 Practical application and interpretation

2.5 Example

Chapter 3: Generalised linear mixed models

3.1 Generalised linear models

3.2 Generalised linear mixed models

3.3 Practical application and interpretation

3.4 Example

Chapter 4: Mixed models for categorical data

4.1 Ordinal logistic regression (fixed effects model)

4.2 Mixed ordinal logistic regression

4.3 Mixed models for unordered categorical data

4.4 Practical application and interpretation

4.5 Example

Chapter 5: Multi-centre trials and meta-analyses

5.1 Introduction to multi-centre trials

5.2 The implications of using different analysis models

5.3 Example: a multi-centre trial

5.4 Practical application and interpretation

5.5 Sample size estimation

5.6 Meta-analysis

5.7 Example: meta-analysis

Chapter 6: Repeated measures data

6.1 Introduction

6.2 Covariance pattern models

6.3 Example: covariance pattern models for normal data

6.4 Example: covariance pattern models for count data

6.5 Random coefficients models

6.6 Examples of random coefficients models

6.7 Sample size estimation

Chapter 7: Cross-over trials

7.1 Introduction

7.2 Advantages of mixed models in cross-over trials

7.3 The AB/BA cross-over trial

7.4 Higher order complete block designs

7.5 Incomplete block designs

7.6 Optimal designs

7.7 Covariance pattern models

7.8 Analysis of binary data

7.9 Analysis of categorical data

7.10 Use of results from random effects models in trial design

7.11 General points

Chapter 8: Other applications of mixed models

8.1 Trials with repeated measurements within visits

8.2 Multi-centre trials with repeated measurements

8.3 Multi-centre cross-over trials

8.4 Hierarchical multi-centre trials and meta-analysis

8.6 Different variances for treatment groups in a simple between-patient trial

8.7 Estimating variance components in an animal physiology trial

8.8 Inter- and intra-observer variation in foetal scan measurements

8.9 Components of variation and mean estimates in a cardiology experiment

8.10 Cluster sample surveys

8.11 Small area mortality estimates

8.12 Estimating surgeon performance

8.13 Event history analysis

8.14 A laboratory study using a within-subject 4 × 4 factorial design

8.15 Bioequivalence studies with replicate cross-over designs

8.16 Cluster randomised trials

8.17 Analysis of bilateral data

8.18 Incomplete block designs

Chapter 9: Software for fitting mixed models

9.1 Packages for fitting mixed models

9.2

PROC MIXED

9.3 Using to fit mixed models to non-normal data

9.4

SAS

Glossary

References

Index

Statistics in Practice

End User License Agreement

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Guide

Cover

Table of Contents

Preface to third edition

Begin Reading

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 3.1

Figure 3.2

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 7.1

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 8.7

Figure 8.8

Figure 8.9

Figure 8.10

Figure 8.11

Figure 8.12

Figure 8.13

Figure 8.14

Figure 8.15

List of Tables

Table 1.1

Table 2.1

Table 2.2

Table 3.1

Table 3.2

Table 3.3

Table 3.4

Table 3.5

Table 3.6

Table 3.7

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5

Table 4.6

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 6.1

Table 6.2

Table 6.3

Table 6.4

Table 6.5

Table 6.6

Table 6.7

Table 7.1

Table 7.2

Table 7.3

Table 7.4

Table 7.5

Table 7.6

Table 7.7

Table 7.8

Table 7.9

Table 7.10

Table 7.11

Table 12

Table 8.1

Table 8.2

Table 8.3

Table 8.4

Table 8.5

Table 8.6

Table 8.7

Table 8.8

Table 8.9

Table 8.10

Table 8.11

Table 8.12

Table 8.13

Table 8.14

Table 8.15

Statistics in Practice

Series Advisors

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CRP-Santé, Luxembourg

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Marian Scott

University of Glasgow, UK

Industry, Commerce and Finance

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University of Maryland, USA

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Vic Barnett

Nottingham Trent University, UK

Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study.

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Applied Mixed Modelsin Medicine

Third Edition

Helen Brown

The Roslin InstituteUniversity of Edinburgh, UK

Robin Prescott

Centre for Population Health SciencesUniversity of Edinburgh, UK

This edition first published 2015

© 2015 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Brown, Helen, 1962- , author.

Applied mixed models in medicine / Helen Brown, Robin Prescott. -- Third edition.

p. ; cm.

Includes bibliographical references and index.

ISBN 978-1-118-77825-8 (cloth)

I. Prescott, Robin, author. II. Title.

[DNLM: 1. Medicine. 2. Statistics as Topic--methods. 3. Models, Statistical. WA 950]

R853.S7

610.72'7--dc23

2014026811

Preface to third edition

Analysis of variance and regression has for many years been the mainstay of statistical modelling. These techniques usually have as a basic assumption that the residual or error terms are independently and identically distributed. Mixed models are an important approach to modelling, which allows us to relax the independence assumption and take into account more complicated data structures in a flexible way. Sometimes, this interdependence of observations is modelled directly in a mixed model. For example, if a number of repeated measurements are made on a patient, then mixed models allow us to specify a pattern for the correlation between these measurements. In other contexts, such as the cross-over clinical trial, specifying that patient effects are normally distributed, rather than fixed as in the classical approach, induces observations on the same patient to be correlated.

There are many benefits to be gained from using mixed models. In some situations, the benefit will be an increase in the precision of our estimates. In others, we will be able to make wider inferences. We will sometimes be able to use a more appropriate model that will give us greater insight into what underpins the structure of the data. However, it is only the availability of software in versatile packages such as SAS® that has made these techniques widely accessible. It is now important that suitable information on their use becomes available so that they may be applied confidently on a routine basis.

Our intention in this book is to put all types of mixed models into a general framework and to consider the practical implications of their use. We aim to do this at a level that can be understood by applied statisticians and numerate scientists. Greatest emphasis is placed on skills required for the application of mixed models and interpretation of the results. An in-depth understanding of the mathematical theory underlying mixed models is not essential to gain these, but an awareness of the practical consequences of fitting different types of mixed models is necessary. While many publications are available on various aspects of mixed models, these generally relate to specific types of model and often differ in their use of terminology. Such publications are not always readily comprehensible to the applied statisticians who will be the most frequent users of the methods. An objective of this book is to help overcome this deficit.

Examples given will primarily relate to the medical field. However, the general concepts of mixed models apply equally to many other areas of application, for example, social sciences, agriculture, veterinary science and official statistics. (In the social sciences, mixed models are often referred to as ‘multi-level’ models.) Data are becoming easier to collect, with the consequence that datasets are now often large and complex. We believe that mixed models provide useful tools for modelling the complex structures that occur in such data.

The third edition of this book retains the structure of the first two, but there are further changes to reflect the continued evolution of SAS. This edition fully incorporates features of SAS up to version 9.3. Compared to what was available at the time of the previous edition, enhancements to SAS include improved graphical facilities. Importantly, there is also a new procedure, PROC MCMC, which facilitates Bayesian analysis. This has led to extensive changes in our coverage of Bayesian methods. SAS 9.3 and later versions now provide output both in text format from the output window and, additionally, as an HTML file in the results viewer. There have been accompanying minor changes in the details of outputs and graphs, such as labelling. Our approach to reporting SAS outputs in this edition has been to change our presentation from earlier editions only when we wish to highlight features that have changed substantially and, importantly, to facilitate the reader's use of mixed models, whatever their version of SAS.

During the drafting of this edition, SAS 9.4 became available. It is not fully incorporated into this book because its new features are focused more on the SAS high performance procedures than on improvements to the SAS/STAT procedures. These high performance procedures ‘provide predictive modelling tools that have been specially developed to take advantage of parallel processing in both multithread single-machine mode and distributed multi-machine mode’. Typically, the high performance procedures such as PROC HPLMIXED have a greatly reduced range of options compared to PROC MIXED and, consequently, are peripheral to the aims of this book. We do, however, consider some of the small modifications to improve procedures such as GLIMMIX and MCMC that are available in SAS/STAT® 12.1 and later versions.

Chapter 1 provides an introduction to the capabilities of mixed models, defines general concepts and gives their basic statistical properties. Chapter 2 defines models and fitting methods for normally distributed data. Chapter 3 first introduces generalised linear models that can be used for the analysis of data that are binomial or Poisson or from any other member of the exponential family of distributions. These methods are then extended to incorporate mixed models concepts under the heading of generalised linear mixed models. The fourth chapter examines how mixed models can be applied when the variable to be analysed is categorical. The main emphasis in these chapters, and indeed in the whole book, is on classical statistical approaches to inference, based on significance tests and confidence intervals. However, the Bayesian approach is also introduced in Chapter 2, since it has several potential advantages and its use is becoming more widespread. Although the overall emphasis of the book is on the application of mixed models techniques, these chapters can also be used as a reference guide to the underlying theory of mixed models.

Chapters 5–7 consider the practical implications of using mixed models for particular designs. Each design illustrates a different feature of mixed models.

Multi-centre trials and meta-analyses are considered in Chapter 5. These are examples of hierarchical data structures, and the use of a mixed model allows for any additional variation in treatment effects occurring between centres (or trials) and hence makes results more generalisable. The methods shown can be applied equally to any type of hierarchical data.

In Chapter 6, the uses of covariance pattern models and random coefficients models are described using the repeated measures design. These approaches take into account the correlated nature of the repeated observations and give more appropriate treatment effect estimates and standard errors. The material in this chapter will apply equally to any situation where repeated observations are made on the same units.

Chapter 7 considers cross-over designs where each patient may receive several treatments. In this design, more accurate treatment estimates are often achieved by fitting patient effects as random. This improvement in efficiency can occur for any dataset where a fixed effect is ‘crossed’ with a random effect.

In Chapter 8, a variety of other designs and data structures is considered. These either incorporate several of the design aspects covered in Chapters 5–7 or have structures that have arisen in a more unplanned manner. They help to illustrate the broad scope of application of mixed models. This chapter includes two new sections. We have added a section on the analysis of bilateral data, a common structure in some areas of medical research, but one that we had not previously addressed. There is also a substantial new section on incomplete block designs.

Chapter 9 gives information on software available for fitting mixed models. Most of the analyses in the book are carried out using PROC MIXED in SAS, supplemented by PROC GENMOD, PROC GLIMMIX, and PROC MCMC. This chapter introduces the basic syntax for these procedures. This information should be sufficient for fitting most of the analyses described, but the full SAS documentation should be referenced for those who wish to use more complex features. The SAS code used for most of the examples is supplied within the text. In addition, the example datasets and SAS code may be obtained electronically from www.wiley.com/go/brown/applied_mixed.

This book has been written to provide the reader with a thorough understanding of the concepts of mixed models, and we trust it will serve well for this purpose. However, readers wishing to take a shortcut to the fitting of normal mixed models should read Chapter 1 for an introduction, Section 2.4 for practical details, and the chapter relevant to their design. To fit non-normal or categorical mixed models, Section 3.3 or Section 4.4 should be read in addition to Section 2.4. In an attempt to make this book easier to use, we have presented at the beginning of the text a summary of the notation we have used, while at the end, we list some key definitions in a glossary.

Our writing of this book has been aided in many ways. The first edition evolved from a constantly changing set of course notes that accompanied a 3-day course on the subject, run regularly over the previous 6 years. The second edition was helped by many individuals who were kind enough to comment on the first edition, including the identification of some errors that had slipped in, and by further participants at our courses who have contributed to discussions and have thereby helped to shape our views. This process has continued with the third edition. We are also grateful to many other colleagues who have read and commented on various sections of the manuscript and especially to our colleagues who have allowed us to use their data. We hope that readers will find the resulting book a useful reference in an interesting and expanding area of statistics.

Helen BrownRobin PrescottEdinburgh

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/brown/applied_mixed

This website includes SAS codes and datasets for most of the examples. In the future, updates and further materials may be added.

Chapter 1Introduction

At the start of each chapter, we will ‘set the scene’ by outlining its content. In this introductory chapter, we start Section 1.1 by describing some situations where a mixed models analysis will be particularly helpful. In Section 1.2, we describe a simplified example and use it to illustrate the idea of a statistical model. We then introduce and compare fixed effects and random effects models. In the next section, we consider a more complex ‘real-life’ multi-centre trial and look at some of the variety of models that could be fitted (Section 1.3). This example will be used for several illustrative examples throughout the book. In Section 1.4, the use of mixed models to analyse a series of observations (repeated measures) is considered. Section 1.5 broadens the discussion on mixed models and looks at mixed models with a historical perspective of their use. In Section 1.6, we introduce some technical concepts: containment, balance and error strata.

We will assume in our presentation that the reader is already familiar with some of the basic statistical concepts as found in elementary statistical textbooks.