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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Probability and Statistics
- Sprache: Englisch
Praise for the First Edition ". . . [this book] should be on the shelf of everyone interested in . . . longitudinal data analysis." --Journal of the American Statistical Association Features newly developed topics and applications of the analysis of longitudinal data Applied Longitudinal Analysis, Second Edition presents modern methods for analyzing data from longitudinal studies and now features the latest state-of-the-art techniques. The book emphasizes practical, rather than theoretical, aspects of methods for the analysis of diverse types of longitudinal data that can be applied across various fields of study, from the health and medical sciences to the social and behavioral sciences. The authors incorporate their extensive academic and research experience along with various updates that have been made in response to reader feedback. The Second Edition features six newly added chapters that explore topics currently evolving in the field, including: * Fixed effects and mixed effects models * Marginal models and generalized estimating equations * Approximate methods for generalized linear mixed effects models * Multiple imputation and inverse probability weighted methods * Smoothing methods for longitudinal data * Sample size and power Each chapter presents methods in the setting of applications to data sets drawn from the health sciences. New problem sets have been added to many chapters, and a related website features sample programs and computer output using SAS, Stata, and R, as well as data sets and supplemental slides to facilitate a complete understanding of the material. With its strong emphasis on multidisciplinary applications and the interpretation of results, Applied Longitudinal Analysis, Second Edition is an excellent book for courses on statistics in the health and medical sciences at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for researchers and professionals in the medical, public health, and pharmaceutical fields as well as those in social and behavioral sciences who would like to learn more about analyzing longitudinal data.
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Seitenzahl: 1367
Veröffentlichungsjahr: 2012
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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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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, MassachusettsMay, 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 SAS commands are included at the end of many of the chapters, with basic descriptions of their usage. Programming statements and computer output for the examples, prepared using SAS, can be downloaded from the website: www.biostat.harvard.edu/~fitzmaur/ala. We selected SAS because all of the analyses we discuss can be performed using its procedures. Many of the methods can be carried out using alternative software packages (e.g., S-Plus and Stata) or special purpose programs (e.g., BMDP5-V) and this book can be supplemented with any one of them. Readers are encouraged to perform and verify the results of analyses using software of their choice. Because statistical software is constantly evolving, we anticipate that all of the methods we discuss will soon be available within most of the major statistical packages.
Throughout the text references have been kept to an absolute minimum. Instead, at the end of each chapter we include suggestions for further readings that provide more in-depth coverage of certain topics. We also include “bibliographic notes” that highlight key references in the mainstream statistical literature. Although many of our readers may find the latter references to be too technical, they are included to give due credit to those who have contributed to the statistical methods described in each chapter.
Finally, we would like to thank the many friends and colleagues who have helped us to write this book. A special word of thanks to Misha Salganik, for preparation of the diagrams and many helpful suggestions for improvement of graphical displays. We are especially grateful to Joe Hogan and Russell Localio, for reading a first draft and providing invaluable feedback, comments, and suggestions that improved the book. We would also like to thank Rino Bellocco, Brent Coull, Nick Horton, Sharon-Lise Normand, Misha Salganik, Judy Singer, S. V. Subramanian, and Florin Vaida, for their insightful comments on several chapters. We are grateful to the students who have participated in the course on “Applied Longitudinal Analysis” at the Harvard School of Public Health since its inception; they have provided the impetus and motivation for writing this book. We gratefully acknowledge support from grant GM 29745 from the National Institutes of Health. The first author gratefully acknowledges support from the Junior Faculty Sabbatical Program at the Harvard School of Public Health; the support provided by a sabbatical created a unique opportunity to begin writing this book. Last, but not least, we thank Steve Quigley and Susanne Steitz of Wiley, for their advice and encouragement during all stages of this project.
GARRETT M. FITZMAURICE NAN M. LAIRD JAMES H. WARE
Boston, Massachusetts March, 2004
Acknowledgments
Throughout this book we have used data sets drawn from published studies in health-related fields to exemplify important concepts in the analysis of longitudinal and clustered data. We are grateful to the following investigators for sharing their data with us: Graham Bentham, Doug Dockery, Brian Flay, Robert Greenberg, Keith Henry, Aviva Must, Elena Naumova, George Rhoads, Jan Schouten, Linda Van Matter, and Gwen Zahner.
We also thank the following publishers for permission to reproduce published data sets in print and electronic format: The American Statistical Association, Blackwell Publishing, Brooks/Cole (a division of Thomson Learning), CRC Press, Elsevier, Iowa State Press, Oxford University Press, and SAS Institute, Inc.
Finally, in all data sets used throughout this book, the original subject identification (ID) numbers have been deleted and replaced with new subject ID numbers, to ensure that the data sets cannot be linked to the original records.
Part I
Introduction to Longitudinal and Clustered Data
Chapter 1
Longitudinal and Clustered Data
1.1 INTRODUCTION
Research on statistical methods for the design and analysis of human investigations expanded explosively in the second half of the twentieth century. Beginning in the early 1950s, the U.S. government shifted a substantial part of its research support from military to biomedical research. The legislative foundation for the modern National Institutes of Health (NIH), the Public Health Service Act, was passed in 1944 and NIH grew rapidly throughout the 1950s and 1960s. During these “golden years” of NIH expansion, the entire NIH budget grew from $8 million in 1947 to more than $1 billion in 1966. The NIH sponsored many of the important epidemiologic studies and clinical trials of that period, including the influential Framingham Heart Study (Dawber et al., 1951; Dawber, 1980).
The typical focus of these early studies was morbidity and, especially, mortality. Investigators sought to identify the causes of early death and to evaluate the effectiveness of treatments for delaying death and morbidity. In the Framingham Heart Study, participants were seen at two-year intervals. Survival outcomes during successive two-year periods were treated as independent events and modeled using multiple logistic regression. The successful use of multiple logistic regression in this setting, and the recognition that it could be applied to case-control data, led to widespread use of this methodology beginning in the 1960s. The analysis of time-to-event data was revolutionized by the seminal 1972 paper of D. R. Cox, describing the proportional hazards model (Cox, 1972). This paper was followed by a rich and important body of work that established the conceptual basis and the computational tools for modern survival analysis.
Although the design of the Framingham Heart Study and other cohort studies called for periodic measurement of the patient characteristics thought to be determinants of chronic disease, interest in the levels and patterns of change of those characteristics over time was initially limited. As the research advanced, however, investigators began to ask questions about the behavior of these risk factors. In the Framingham Heart Study, for example, investigators began to ask whether blood pressure levels in childhood were predictive of hypertension in adult life. In the Coronary Artery Risk Development in Young Adults (CARDIA) Study, investigators sought to identify the determinants of the transition from normotensive or normocholesterolemic status in early adult life to hypertension and hypercholesterolemia in middle age (Friedman et al., 1988). In the treatment of arthritis, asthma, and other diseases that are not typically life-threatening, investigators began to study the effects of treatments on the level and change over time in measures of severity of disease. Similar questions were being posed in every disease setting. Investigators began to follow populations of all ages over time, both in observational studies and clinical trials, to understand the development and persistence of disease and to identify factors that alter the course of disease development.
This interest in the temporal patterns of change in human characteristics came at a period when advances in computing power made new and more computationally intensive approaches to statistical analysis available at the desktop. Thus, in the early 1980s, Laird and Ware proposed the use of the EM algorithm to fit a class of linear mixed effects models appropriate for the analysis of repeated measurements (Laird and Ware, 1982); Jennrich and Schluchter (1986) proposed a variety of alternative algorithms, including Fisher-scoring and Newton–Raphson algorithms. Later in the decade, Liang and Zeger introduced the generalized estimating equations in the biostatistical literature and proposed a family of generalized linear models for fitting repeated observations of binary and counted data (Liang and Zeger, 1986; Zeger and Liang, 1986). Many other investigators writing in the biomedical, educational, and psychometric literature contributed to the rapid development of methodology for the analysis of these “longitudinal” data. The past 30 years have seen considerable progress in the development of statistical methods for the analysis of longitudinal data. Despite these important advances, methods for the analysis of longitudinal data have been somewhat slow to move into the mainstream. This book bridges the gap between theory and application by presenting a comprehensive description of methods for the analysis of longitudinal data accessible to a broad range of readers.
1.2 LONGITUDINAL AND CLUSTERED DATA
The defining feature of longitudinal studies is that measurements of the same individuals are taken repeatedly through time, thereby allowing the direct study of change over time. The primary goal of a longitudinal study is to characterize the change in response over time and the factors that influence change. With repeated measures on individuals, one can capture within-individual change. Indeed, the assessment of within-subject changes in the response over time can only be achieved within a longitudinal study design. For example, in a cross-sectional study, where the response is measured at a single occasion, one can only obtain estimates of between-individual differences in the response. That is, a cross-sectional study may allow comparisons among sub-populations that happen to differ in age, but it does not provide any information about how individuals change during the corresponding period.
To highlight this important distinction between cross-sectional and longitudinal study designs, consider the following simple example. Body fatness in girls is thought to increase just before or around menarche, leveling off approximately 4 years after menarche. Suppose that investigators are interested in determining the increase in body fatness in girls after menarche. In a cross-sectional study design, investigators might obtain measurements of percent body fat on two separate groups of girls: a group of 10-year-old girls (a pre-menarcheal cohort) and a group of 15-year-old girls (a post-menarcheal cohort). In this cross-sectional study design, direct comparison of the average percent body fat in the two groups of girls can be made using a two-sample (unpaired) t-test. This comparison does not provide an estimate of the change in body fatness as girls age from 10 to 15 years. The effect of growth or aging, an inherently within-individual effect, simply cannot be estimated from a cross-sectional study that does not obtain measures of how individuals change with time. In a cross-sectional study the effect of aging is potentially confounded with possible cohort effects. Put in a slightly different way, there are many characteristics that differentiate girls in these two different age groups that could distort the relationship between age and body fatness. On the other hand, a longitudinal study that measures a single cohort of girls at both ages 10 and 15 can provide a valid estimate of the change in body fatness as girls age. In the longitudinal study the analysis is based on a paired t-test, using the difference or change in percent body fat within each girl as the outcome variable. This within-individual comparison provides a valid estimate of the change in body fatness as girls age from 10 to 15 years. Moreover, since each girl acts as her own control, changes in percent body fat throughout the duration of the study are estimated free of any between-individual variation in body fatness.
A distinctive feature of longitudinal data is that they are clustered. In longitudinal studies the clusters are composed of the repeated measurements obtained from a single individual at different occasions. Observations within a cluster will typically exhibit positive correlation, and this correlation must be accounted for in the analysis. Longitudinal data also have a temporal order; the first measurement within a cluster necessarily comes before the second measurement, and so on. The ordering of the repeated measures has important implications for analysis. There are, however, many studies in the health sciences that are not longitudinal in this sense but which give rise to data that are clustered or cluster-correlated. For example, clustered data commonly arise when intact groups are randomized to health interventions or when naturally occurring groups in the population are randomly sampled. An example of the former is group-randomized trials. In a group-randomized trial, also known as a cluster-randomized trial, groups of individuals, rather than each individual alone, are randomized to different treatments or health interventions. Data on the health outcomes of interest are obtained on all individuals within a group. Alternatively, clustered data can arise from random sampling of naturally occurring groups in the population. Families, households, hospital wards, medical practices, neighborhoods, and schools are all instances of naturally occurring clusters in the population that might be the primary sampling units in a study. Finally, clustered data can arise when data on the health outcome of interest are simultaneously obtained either from multiple raters or from different measurement instruments.
In all these examples of clustered data, we might reasonably expect that measurements on units within a cluster are more similar than the measurements on units in different clusters. The degree of clustering can be expressed in terms of correlation among the measurements on units within the same cluster. This correlation invalidates the crucial assumption of independence that is the cornerstone of so many standard statistical techniques. Instead, statistical models for clustered data must explicitly describe and account for this correlation. Because longitudinal data are a special case of clustered data, albeit with a natural ordering of the measurements within a cluster, this book includes a description of modern methods of analysis for clustered data, more broadly defined. Indeed, one of the goals of this book is to demonstrate that methods for the analysis of longitudinal data are, more or less, special cases of more general regression methods for clustered data. As a result a comprehensive understanding of methods for the analysis of longitudinal data 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 examples described above consider only a single level of clustering, for example, repeated measurements on individuals. More recently investigators have developed methodology for the analysis of multilevel data, in which observations may be clustered at more than one level. For example, the data may consist of repeated measurements on patients clustered by clinic. Alternatively, the data may consist of observations on children nested within classrooms, nested within schools. Although the analysis of multilevel data is not the primary focus of this book, multilevel data are discussed in Chapter 22.
Interest in the analysis of longitudinal and multilevel data continues to grow. New and more flexible models have been developed and advances in computation, such as Markov chain Monte Carlo (MCMC) methods, have allowed greater flexibility in model specification. Moreover, improvements in statistical software packages, especially SAS, Stata, SPSS, R, and S-Plus, have made these models much more accessible for use in routine data analysis. Despite these advances, however, methods for the analysis of longitudinal data are not widely used and are seen to be accessible only to statisticians with specialized expertise.
We believe that the methodology for the analysis of longitudinal data can be much more widely understood and applied. It is our hope that this book will help make that possible. It provides a comprehensive introduction to methods for the analysis of longitudinal data, written for a reader with a basic knowledge of statistics and a strong background in regression analysis. The book does not require a high level of mathematical preparation but does assume a willingness to read and consider mathematical ideas.
1.3 EXAMPLES
To highlight some of the distinctive features of longitudinal and clustered data, we introduce four examples drawn from studies in the biomedical sciences. These four examples will be used later in the book to illustrate different analytic approaches. Additional examples, also drawn from studies in the biomedical and health sciences, will be introduced in later chapters of the book.
1.3.1 Treatment of Lead-Exposed Children (TLC) Trial
Exposure to lead can produce cognitive impairment, especially among young children and infants. A young child exposed to high levels of lead may experience various adverse health effects, including hyperactivity, hearing or memory loss, learning disabilities, and damage to the brain and nervous system. Although the use of lead as an additive in gasoline has been discontinued, at least in the United States, resulting in a dramatic reduction in airborne lead levels, a small percentage of children continue to be exposed to lead at levels that can produce impairment. Much of this exposure is due to deteriorating lead-based paint (e.g., chipping and peeling paint) in older homes. Lead was used as a pigment and drying agent in “alkyd” oil-based paint. While the United States government banned the use of lead-based paint in housing in 1978, many homes built before 1978 contain lead-based paint. When lead-based paint deteriorates, it becomes lead paint chips, which can be eaten by young children, and lead-contaminated paint dust, which can be ingested by young children during normal teething and hand-to-mouth behavior. The U.S. Centers for Disease Control and Prevention (CDC) has concluded that children with blood lead levels above 10 micrograms per deciliter (μg/dL) of whole blood are at risk of adverse health effects.
Lead poisoning in children is treatable in the sense that there are medical interventions, known as chelation treatments, that can help a child to excrete the lead that has been ingested. Until recently chelation treatment of children with high levels of blood lead was administered by injection and required hospitalization. A new chelating agent, succimer, enhances urinary excretion of lead and has the distinct advantage that it can be given orally, rather than by injection. In the 1990s the Treatment of Lead-Exposed Children (TLC) Trial Group conducted a placebo-controlled, randomized trial of succimer in children with confirmed blood lead levels of 20 to 44 μg/dL, levels well above the CDC’s threshold for concern about the adverse health effects of exposure to lead (Treatment of Lead-Exposed Children (TLC) Trial Group, 2000; Rogan et al., 2001). The children were aged 12 to 33 months at enrollment and lived in deteriorating inner city housing. The mean age of the children at randomization was 2 years and the mean blood lead level was 26 μg/dL. Children received up to three 26-day courses of succimer or placebo and were followed for 3 years.
Table 1.1 presents data on blood lead levels at baseline, week 1, week 4, and week 6 for 10 randomly selected children from the study. The mean blood lead levels at each measurement occasion for a random subset of 100 children, broken down by treatment group, are presented in Table 1.2
