Contents
Cover
Half Title page
Title page
Copyright page
Dedication
Preface
Preface to First Edition
Acknowledgments
Part I: Introduction to Longitudinal and Clustered Data
Chapter 1: Longitudinal and Clustered Data
1.1 Introduction
1.2 Longitudinal and Clustered Data
1.3 Examples
1.4 Regression Models for Correlated Responses
1.5 Organization of the Book
1.6 Further Reading
Chapter 2: Longitudinal Data: Basic Concepts
2.1 Introduction
2.2 Objectives of Longitudinal Analysis
2.3 Defining Features of Longitudinal Data
2.4 Example: Treatment of Lead-Exposed Children Trial
2.5 Sources of Correlation in Longitudinal Data
2.6 Further Reading
Part II: Linear Models for Longitudinal Continuous Data
Chapter 3: Overview of Linear Models for Longitudinal Data
3.1 Introduction
3.2 Notation and Distributional Assumptions
3.3 Simple Descriptive Methods of Analysis
3.4 Modeling the Mean
3.5 Modeling the Covariance
3.6 Historical Approaches
3.7 Further Reading
Chapter 4: Estimation and Statistical Inference
4.1 Introduction
4.2 Estimation: Maximum Likelihood
4.3 Missing Data Issues
4.4 Statistical Inference
4.5 Restricted Maximum Likelihood (REML) Estimation
4.6 Further Reading
Chapter 5: Modeling the Mean: Analyzing Response Profiles
5.1 Introduction
5.2 Hypotheses Concerning Response Profiles
5.3 General Linear Model Formulation
5.4 Case Study
5.5 One-Degree-of-Freedom Tests for Group by Time Interaction
5.6 Adjustment for Baseline Response
5.7 Alternative Methods of Adjusting for Baseline Response
5.8 Strengths and Weaknesses of Analyzing Response Profiles
5.9 Computing: Analyzing Response Profiles Using PROC MIXED in SAS
5.10 Further Reading
Chapter 6: Modeling the Mean: Parametric Curves
6.1 Introduction
6.2 Polynomial Trends in Time
6.3 Linear Splines
6.4 General Linear Model Formulation
6.5 Case Studies
6.6 Computing: Fitting Parametric Curves Using PROC MIXED in SAS
6.7 Further Reading
Chapter 7: Modeling the Covariance
7.1 Introduction
7.2 Implications of Correlation among Longitudinal Data
7.3 Unstructured Covariance
7.4 Covariance Pattern Models
7.5 Choice among Covariance Pattern Models
7.6 Case Study
7.7 Discussion: Strengths and Weaknesses of Covariance Pattern Models
7.8 Computing: Fitting Covariance Pattern Models Using PROC MIXED in SAS
7.9 Further Reading
Chapter 8: Linear Mixed Effects Models
8.1 Introduction
8.2 Linear Mixed Effects Models
8.3 Random Effects Covariance Structure
8.4 Two-Stage Random Effects Formulation
8.5 Choice among Random Effects Covariance Models
8.6 Prediction of Random Effects
8.7 Prediction and Shrinkage
8.8 Case Studies
8.9 Computing: Fitting Linear Mixed Effects Models Using PROC MIXED in SAS
8.10 Further Reading
Chapter 9: Fixed Effects versus Random Effects Models
9.1 Introduction
9.2 Linear Fixed Effects Models
9.3 Fixed Effects versus Random Effects: Bias-Variance Trade-off
9.4 Resolving the Dilemma of Choosing Between Fixed and Random Effects Models
9.5 Longitudinal and Cross-sectional Information
9.6 Case Study
9.7 Computing: Fitting Linear Fixed Effects Models Using PROC GLM in SAS
9.8 Computing: Decomposition of Between-Subject and Within-Subject Effects Using PROC MIXED in SAS
9.9 Further Reading
Chapter 10: Residual Analyses and Diagnostics
10.1 Introduction
10.2 Residuals
10.3 Transformed Residuals
10.4 Aggregating Residuals
10.5 Semi-Variogram
10.6 Case Study
10.7 Summary
10.8 Further Reading
Part III: Generalized Linear Models for Longitudinal Data
Chapter 11: Review of Generalized Linear Models
11.1 Introduction
11.2 Salient Features of Generalized Linear Models
11.3 Illustrative Examples
11.4 Ordinal Regression Models
11.5 Overdispersion
11.6 Computing: Fitting Generalized Linear Models Using PROC GENMOD in SAS
11.7 Overview of Generalized Linear Models
11.8 Further Reading
Chapter 12: Marginal Models: Introduction and Overview
12.1 Introduction
12.2 Marginal Models for Longitudinal Data
12.3 Illustrative Examples of Marginal Models
12.4 Distributional Assumptions for Marginal Models
12.5 Further Reading
Chapter 13: Marginal Models: Generalized Estimating Equations (GEE)
13.1 Introduction
13.2 Estimation of Marginal Models: Generalized Estimating Equations
13.3 Residual Analyses and Diagnostics
13.4 Case Studies
13.5 Marginal Models and Time-Varying Covariates
13.6 Computing: Generalized Estimating Equations Using PROC GENMOD in SAS
13.7 Further Reading
Chapter 14: Generalized Linear Mixed Effects Models
14.1 Introduction
14.2 Incorporating Random Effects in Generalized Linear Models
14.3 Interpretation of Regression Parameters
14.4 Overdispersion
14.5 Estimation and Inference
14.6 A Note on Conditional Maximum Likelihood
14.7 Case Studies
14.8 Computing: Fitting Generalized Linear Mixed Models Using PROC GLIMMIX in SAS
14.9 Further Reading
Chapter 15: Generalized Linear Mixed Effects Models: Approximate Methods of Estimation
15.1 Introduction
15.2 Penalized Quasi-Likelihood
15.3 Marginal Quasi-Likelihood
15.4 Cautionary Remarks on the Use of PQL and MQL
15.5 Case Studies
15.6 Computing: Fitting GLMMs Using PROC GLIMMIX in SAS
15.7 Basis of PQL and MQL Approximations
15.8 Further Reading
Chapter 16: Contrasting Marginal and Mixed Effects Models
16.1 Introduction
16.2 Linear Models: A Special Case
16.3 Generalized Linear Models
16.4 Simple Numerical Illustration
16.5 Case Study
16.6 Conclusion
16.7 Further Reading
Part IV: Missing Data and Dropout
Chapter 17: Missing Data and Dropout: Overview of Concepts and Methods
17.1 Introduction
17.2 Hierarchy of Missing Data Mechanisms
17.3 Implications for Longitudinal Analysis
17.4 Dropout
17.5 Common Approaches for Handling Dropout
17.6 Bias of Last Value Carried Forward Imputation
17.7 Further Reading
Chapter 18: Missing Data and Dropout: Multiple Imputation and Weighting Methods
18.1 Introduction
18.2 Multiple Imputation
18.3 Inverse Probability Weighted Methods
18.4 Case Studies
18.5 “Sandwich” Variance Estimator Adjusting for Estimation of Weights
18.6 Computing: Multiple Imputation Using PROC MI in SAS
18.7 Computing: Inverse Probability Weighted (IPW) Methods in SAS
18.8 Further Reading
Part V: Advanced Topics for Longitudinal and Clustered Data
Chapter 19: Smoothing Longitudinal Data: Semiparametric Regression Models
19.1 Introduction
19.2 Penalized Splines for a Univariate Response
19.3 Case Study
19.4 Penalized Splines for Longitudinal Data
19.5 Case Study
19.6 Fitting Smooth Curves to Individual Longitudinal Data
19.7 Case Study
19.8 Computing: Fitting Smooth Curves Using PROC MIXED in SAS
19.9 Further Reading
Chapter 20: Sample Size and Power
20.1 Introduction
20.2 Sample Size for a Univariate Continuous Response
20.3 Sample Size for a Longitudinal Continuous Response
20.4 Sample Size for a Longitudinal Binary Response
20.5 Summary
20.6 Computing: Sample Size Calculation Using Pseudo-Data
20.7 Further Reading
Chapter 21: Repeated Measures and Related Designs
21.1 Introduction
21.2 Repeated Measures Designs
21.3 Multiple Source Data
21.4 Case Study 1: Repeated Measures Experiment
21.5 Case Study 2: Multiple Source Data
21.6 Summary
21.7 Further Reading
Chapter 22: Multilevel Models
22.1 Introduction
22.2 Multilevel Data
22.3 Multilevel Linear Models
22.4 Multilevel Generalized Linear Models
22.5 Summary
22.6 Further Reading
Appendix A Gentle Introduction to Vectors and Matrices
Appendix B Properties of Expectations and Variances
Appendix C Critical Points for a 50:50 Mixture of Chi-Squared Distributions
References
Index
Applied Longitudinal Analysis
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Library of Congress Cataloging-in-Publication Data:
Fitzmaurice, Garrett M., 1962-
Applied longitudinal analysis / Garrett M. Fitzmaurice, Nan M. Laird, James H. Ware. — 2nd ed.
p. cm.
ISBN 978-0-470-38027-7 (hardback)
1. Longitudinal method. 2. Regression analysis. 3. Multivariate analysis. 4. Medical statistics. I. Laird, Nan M., 1943- II. Ware, James H., 1941- III. Title.
QA278.F575 2011
519.5’3—dc22
2011012197
To Laura, Kieran, and Aidan
— G.M.F.
To Joel, Richard, and Lily
— N.M.L.
To Janice, Cameron, and Jake
— J.H.W.
Preface
The first edition of Applied Longitudinal Analysis was designed to serve as a textbook for a course on modern statistical methods for longitudinal data analysis, and subsequently, as a reference resource for students and researchers. The book was targeted at a broad audience: graduate students in statistics, statisticians working in the health sciences, pharmaceutical industry, and governmental health-related agencies, as well as researchers and graduate students from a variety of substantive fields. In the seven years that have elapsed since publication of the first edition, Applied Longitudinal Analysis has been used extensively in university classrooms throughout the United States and abroad. We are grateful to many colleagues, course instructors, students, and readers who have offered constructive suggestions on how the book could be improved. This feedback has been invaluable and helped shape the content of the second edition.
The feedback we received has encouraged us to retain the general structure and format of the first edition while taking the opportunity to introduce a number of new and important topics. Although there is much new material in this second edition, the principles that guided us in writing the first edition have not changed. Our primary goal is to present a rigorous and comprehensive description of modern statistical methods for the analysis of longitudinal data that is accessible to a wide range of readers. A strong emphasis is placed on the application of these methods to longitudinal data and the interpretation of results. Although the methods are presented in the setting of numerous applications to actual data sets drawn from studies in health-related fields, reflecting our own research interests in the health sciences, they apply equally to other areas of application, for example, education, psychology, and other branches of the behavioral and social sciences.
How does this edition differ from its predecessor? The major changes in this edition have resulted from the addition of six new chapters:
In addition the chapter on residual analyses and diagnostics (Chapter 10) has been revised to include material on recently developed model-checking techniques based on cumulative sums of residuals and the chapters that review generalized linear models (Chapter 11) and generalized linear mixed effects models (Chapter 14) have been updated to include new material on models for ordinal data and on methods for handling overdispersion. Finally, extra problem sets have been added to many of the chapters.
As in the first edition, the prerequisites for a course based on this book are an introductory course in statistics and a strong background in regression analysis. Some previous exposure to generalized linear models (e.g., logistic regression) would be helpful, although these models are reviewed in detail in the text. An understanding of matrix algebra or calculus is not assumed. Although we do not assume a high level of mathematical preparation, we have written this book for the motivated reader who is willing to consider mathematical ideas. The more technical or mathematical sections of the book are signposted with asterisks and may be omitted at first reading without loss of continuity.
The methods described in this book require the use of appropriate statistical software. As before, we include illustrative SAS commands for performing the analyses presented throughout the text at the end of many chapters, with basic descriptions of their usage. Because many of the analyses we discuss can be performed using alternative software packages (e.g., R, S-Plus, Stata, and SPSS), this book can be supplemented with any one of them. Readers are encouraged to perform and verify the results of analyses using statistical software of their choice. Programming statements and computer output for selected examples, prepared using SAS, Stata, and R, can be downloaded from the website: www.biostat.harvard.edu/~fitzmaur/ala2e. Because statistical software is constantly evolving, we will endeavor to update the website as new procedures become available in the major statistical software packages. The thirty-two real data sets used throughout the text and problem sets to illustrate the applications of longitudinal methods also can be downloaded from the website.
We hope this second edition of Applied Longitudinal Analysis provides a broader foundation in modern methods for the analysis of longitudinal data and will prove a worthy successor to the first edition. The original impetus for writing this book arose from teaching a graduate-level course on “Applied Longitudinal Analysis” at the Harvard School of Public Health. We are especially grateful to the students who have participated in the course since its inception almost twenty years ago; we have learned much from these extraordinary students. The collection of individuals who gave us useful feedback on the first edition is far too long to list. However, we would like to thank the many friends and colleagues who have helped us with this project. A special word of thanks to Amy Herring and Russell Localio. We thank Amy for her many helpful and constructive suggestions on how the book could be improved. We thank Russell for reading a draft of the new chapters and for providing invaluable feedback and suggestions that improved their content. Thanks also to Nick Horton, Stu Lipsitz, and Caitlin Ravichandran for their helpful suggestions and insightful comments on several chapters. Finally, we thank Steve Quigley and Susanne Steitz-Filler of Wiley, for their advice and encouragement during all stages of this project.
GARRETT M. FITZMAURICE
NAN M. LAIRD
JAMES H. WARE
Boston, Massachusetts
May, 2011
Preface to First Edition
Our goal in writing this book is to provide a rigorous and systematic description of modern methods for analyzing data from longitudinal studies. In recent years there have been remarkable developments in methods for longitudinal analysis. Despite these important advances, the methods have been somewhat slow to move into the mainstream. Applied Longitudinal Analysis bridges the gap between theory and application by presenting a comprehensive account of these methods in a way that is accessible to a wide range of readers.
The impetus for this book arose from teaching a graduate-level course on “Applied Longitudinal Analysis” at the Harvard School of Public Health. As course instructors, we were frustrated by the lack of a suitable textbook that adequately covered modern statistical methods for longitudinal analysis at a level accessible to a broad audience of researchers and graduate students in the health and medical sciences. We envision this book as a textbook for such a course and, subsequently, as a reference resource for researchers and graduate students. It is also suitable for graduate students in statistics and for statisticians already working in the health sciences, governmental health-related agencies, and the pharmaceutical industry. It is intended to allow a diverse group of statisticians, researchers, and graduate students in substantive fields to master modern methods for longitudinal data analysis.
The scope of this book is broad, covering methods for the analysis of diverse types of longitudinal data arising in the health sciences. The methods are presented in the setting of numerous applications to real data sets. Our main emphasis is on the practical rather than the theoretical aspects of longitudinal analysis. Twenty-five real data sets, drawn from studies in health-related fields, are used throughout the text and problem sets to illustrate the applications of longitudinal methods. These data sets can be downloaded from the website for the book: www.biostat.harvard.edu/~fitzmaur/ala. Although the methods are applied to data sets drawn from the health sciences, they apply equally to other areas of application, for example, education, psychology, and other branches of the behavioral and social sciences.
Because longitudinal data are a special case of clustered data, albeit with a natural ordering of the measurements within a cluster, we include also a description of modern methods for analyzing clustered data, more broadly defined. Indeed, one of our goals is to demonstrate that methods for longitudinal analysis are, more or less, special cases of more general regression methods for clustered data. As a result a comprehensive understanding of longitudinal data analysis provides the basis for a broader understanding of methods for analyzing the wide range of clustered data that commonly arises in studies in the biomedical and health sciences.
The prerequisites for a course based on this book are an introductory course in statistics and a strong background in regression analysis. Some previous exposure to generalized linear models (e.g., logistic regression) would be helpful, although these models are reviewed in the text. An understanding of matrix algebra or calculus is not assumed; the reader will be gently introduced to only those aspects of vector and matrix notation necessary for understanding the matrix representation of regression models for longitudinal data. Because vectors and matrices are used to simplify notation, the reader is required to attain some basic facility with the addition and multiplication of vectors and matrices. Although we do not assume a high level of mathematical preparation, a willingness to read and consider mathematical ideas is required. More technical or mathematical sections of the book are marked with asterisks and may be omitted at first reading without loss of continuity.
To use the methods described in this book, appropriate statistical software is required. In general, the methods available via commercially available software lag behind the recent advances in statistical methods; longitudinal data analysis is not exceptional in this regard. Recently the introduction of new programs for analyzing multivariate and longitudinal data has made these methods far more accessible to practitioners and students. We use SAS, which is widely available, to perform the analyses presented throughout the text. Illustrative SASSASwww.biostat.harvard.edu/~fitzmaur/alaSASS-PlusStataBMDP5-V