Longitudinal Data Analysis: A Practical Guide for...

A PRACTICAL GUIDE FOR RESEARCHERS IN AGING, HEALTH, AND SOCIAL SCIENCES

Longitudinal Data Analysis

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EDITED BY

Jason T. Newsom Portland State University

Richard N. Jones Harvard Medical School

Scott M. Hofer University of Victoria, Canada

A PRACTICAL GUIDE FOR RESEARCHERS IN AGING, HEALTH, AND SOCIAL SCIENCES

Longitudinal Data Analysis



MATLAB® and Simulink® are trademarks of The MathWorks, Inc. and are used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® and Simulink® software or related products does not constitute endorse-ment or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® and Simulink® software.

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Library of Congress Cataloging‑in‑Publication Data

Longitudinal data analysis : a practical guide for researchers in aging, health, and social sciences / edited by Jason T. Newsom, Richard N. Jones, Scott M. Hofer.

p. cm. -- (Multivariate applications series ; 18)Includes bibliographical references and index.ISBN 978-0-415-87414-4 (hardback)1. Social sciences--Research. 2. Social sciences--Longitudinal studies.

3. Longitudinal method. 4. Aging--Research--Longitudinal studies. 5. Health--Research--Longitudinal studies. I. Newsom, Jason T. II. Jones, Richard N. III. Hofer, Scott M. IV. Title. V. Series.

H62.L645 2011300.72--dc22 2011005332

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vii

Contents

Preface .................................................................................................................ixAcknowledgments ......................................................................................... xiii

1. Existing Longitudinal Data Sets for the Study of Health and Social Aspects of Aging ................................................................... 1Nathalie Huguet, Shayna D. Cunningham, and Jason T. Newsom

2. Weighting and Complex Sampling Design Adjustments in Longitudinal Studies ......................................................................... 43Shayna D. Cunningham and Nathalie Huguet

3. Missing Data and Attrition ................................................................... 71Du Feng, Zhen Cong, and Merril Silverstein

4. Measurement Issues in the Analysis of Within-Person Change ..............................................................97Daniel E. Bontempo, Frederick M.E. Grouzet, and Scott M. Hofer

5. Basic Longitudinal Analysis Approaches for Continuous and Categorical Variables .................................................................... 143Jason T. Newsom

6. Mediation Analysis With Longitudinal Data .................................. 181David L. Roth and David P. MacKinnon

7. Growth Models With Multilevel Regression .................................. 217Benjamin A. Shaw and Jersey Liang

8. Structural Equation Modeling Approaches to Longitudinal Data ............................................................................. 243Michael J. Rovine and Siwei Liu

9. Latent Growth Curve Models ............................................................. 271Richard N. Jones

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viii Contents

10. Dynamic Structural Equation Models of Change .......................... 291Archana Jajodia

11. Using Discrete-Time Survival Analysis to Study Event Occurrence ................................................................................... 329Suzanne E. Graham, John B. Willett, and Judith D. Singer

Author Index .................................................................................................. 373

Subject Index ................................................................................................. 381

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ix

Preface

Researchers do their best work and scientific progress proceeds most swiftly when state-of-the-art tools are available and used correctly. Often, researchers do not follow optimal practices when conducting analyses because they are not well informed about the latest develop-ments or best-performing statistical tests. Statistical learning often ends for most researchers at the completion of graduate school. But new sta-tistical developments occur at a sometimes rapid pace, and knowledge about existing statistical approaches is always expanding. We hope that this book will improve the practice of data analysis for research in aging, health, and the social sciences by at least limiting the knowledge gap between new developments and common practice among researchers. The authors of each chapter have provided excellent summaries of the state of knowledge for the topics covered, targeting advanced graduate students or practicing researchers who wish to learn about analyzing data from longitudinal studies.

An increasingly important area of statistical methodology that is miss-ing from graduate training in many social science departments is the spe-cific treatment of analysis of longitudinal data. Statistical courses may or may not cover basic within-subject analyses, such as repeated measures ANOVA, and more advanced approaches may be covered in only one portion of advanced courses on structural equation or multilevel models. Courses that discuss the range of statistical approaches available for ana-lyzing longitudinal studies are rare in graduate training; yet, such dis-cussions are essential for developing a conceptual framework with a full complement of skills for investigating longitudinal hypotheses. The lack of training specific to longitudinal data analysis is likely responsible, in part, for what we see as a serious disconnect between increasing demand for longitudinal research and the skill set of many applied researchers.

Why Aging, Health, and Social Sciences?

The choice of substantive focus of the examples in this book came about for a variety of reasons. The editors and most of the chapter authors have a connection to gerontology, an interdisciplinary field that has traditionally seeded an intersection of health and social sciences in order to address questions about aging. The kernel of this project began in discussions

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x Preface

between Newsom, Jones, and colleagues in the measurement, statistics, and research design interest group at the Annual Scientific Conference of the Gerontological Society of America. That group started with the objec-tive of increasing awareness within the society of new statistical devel-opments and how they could be applied to research problems in aging, health, and the social sciences. Although our training is in psychology (Newsom and Hofer) and public health (Jones), our substantive work has focused primarily on aging and development related to psychology and health. The majority of our recent projects have centered around longitu-dinal research questions and longitudinal methodology. Thus, this book was of great interest for us as we imagined ourselves in need of such a resource 10 years ago when we were figuring things out. Although many of the examples in this book concern health or social science questions related to aging, readers from particular disciplines, such as psychology, sociology, public health, economics, and medicine, will find the examples to be of interest and the analyses to be fully relevant to their work.

Overview of the Book

In putting together this book, we have attempted to cover a wide range of analytic methods for longitudinal studies, running the gamut from sim-ple two-wave analyses to some of the most sophisticated approaches that exist today. We believe that the chapters assembled here will prove to be a well-rounded resource for researchers in aging, health, and the social sciences who wish to learn about available techniques for analyzing lon-gitudinal data. Inevitably, however, choices must be made and topics must be left out. To address the unavoidable incompleteness on a given topic, each chapter includes a short list of recommended readings at the end. Researchers can use these carefully selected readings as a next place to turn for additional information. Omitting some topics from the book was also inescapable. The book does not include information about analysis of longitudinal cohort designs, repeated cross-sectional surveys, or random-ized control trials among others. Admittedly, the selection of topics also is biased toward larger sample studies collected over longer periods rather than shorter-term repeated-measurement designs such as daily diary studies. Daily diary and similar short-term designs should undoubtedly be considered longitudinal studies nonetheless. Shorter-term longitudi-nal studies have some special analytic considerations, but much of what is presented in this book will be relevant for researchers analyzing this type of data.

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Preface xi

The book begins with an overview of publicly available longitudinal data sets that may be useful to researchers in aging, health, and the social sciences (Huguet, Cunningham, and Newsom). There is a wealth of exist-ing, publicly available studies that have been underexploited in compar-ison to their potential scientific contributions. Chapter 2 (Cunningham and Huguet) introduces the reader to sample weighting and complex sampling design adjustments that many researchers are not exposed to as part of their graduate training. Population-based samples require adjust-ments and special analytic procedures to produce correct statistical tests. This chapter outlines the importance of complex sampling design adjust-ments for population-based samples and considers issues related to their use with longitudinal studies. Many of the other examples in this book omit complex sampling design adjustments even though they are needed for correct statistical tests. They have been omitted intentionally through-out the book, however, in order to simplify the illustrations and formu-las. Chapter 3 (Feng, Cong, and Silverstein) focuses on missing data and attrition, a topic with many recent developments and great importance for longitudinal research with aging populations. Missing data is usually handled with listwise deletion for convenience and simplicity, but unbi-ased and more precise estimates are often possible in many cases with modern missing data approaches. Chapter 4 (Bontempo, Grouzet, and Hofer) considers measurement issues pertaining to longitudinal research and underscores the need for comparable measurement over time before any investigation of change can have validity. Chapter 5 (Newsom) dis-cusses traditional analyses, such as analysis of variance and regression, that provide information about average change over two or three waves. Chapter 6 (Roth and MacKinnon) explores mediation analysis to under-stand causal processes over time. Mediation analysis has received a great deal of attention by researchers and statisticians, but far fewer works con-sider mediation analysis in the context of a longitudinal design. Chapter 7 (Shaw and Liang) provides a well-rounded introduction to growth curve models using the multilevel regression (or hierarchical linear model-ing) approach, an analysis of individual trajectories of change that is fast becoming an essential tool for longitudinal research. In Chapter 8, Rovine and Liu discuss a number of ways to investigate longitudinal hypotheses with structural equation modeling. In Chapter 9, Jones illustrates latent growth curve models, a structural equation modeling approach to eval-uating individual trajectories of change. Chapter 10 (Jajodia) discusses dynamic structural equation models of change, an alternative approach to growth models based on latent difference scores. The book concludes with an overview of current approaches to survival (event history) analy-sis in Chapter 11 (Graham, Willett, and Singer) designed for data that have censored times of observation.

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xii Preface

Software implementation has not been the focus of examples in this book, partly because a good foundation in the underlying concepts and statistical issues makes implementation a relatively simple task regardless of the software program. User-friendly programming as well as extensive official and unofficial documentation is currently widely available on the web. Nevertheless, researchers often need a bridge between the abstract description and the concrete implementation on the computer. We have therefore provided two such bridges. At the end of each chapter, the authors describe any special considerations for implementing the analy-ses in the most common software packages. These remarks include infor-mation about particular options available, the preferred strategies, and certain precautions of which the user should be aware. We have also con-structed a companion Web site (http://www.psypress.com/ longitudinal-data-analysis) with syntax for each of the examples appearing in this book, illustrating all examples with multiple statistical packages.

In addition to a list of recommended readings, a list of two or three pub-lished articles with examples illustrating the analyses appears at the end of each chapter. These examples can serve as additional illustrations of how the analysis can be implemented to investigate relevant hypotheses and will provide useful examples of how results are typically reported in publication.

We hope that this book will work as a whole and that researchers will integrate the information gleaned in all of the chapters to advance best practices in their analysis of longitudinal data.

Jason T. NewsomPortland State University, Portland, Oregon

Richard N. JonesInstitute for Aging Research, Hebrew SeniorLife, Harvard Medical School,

Boston, Massachusetts

Scott M. HoferUniversity of Victoria, Victoria, British Columbia, Canada

For MATLAB® and Simulink® product information, please contact:

The MathWorks, Inc.3 Apple Hill DriveNatick, MA, 01760-2098 USATel: 508-647-7000Fax: 508-647-7001E-mail: [email protected]: www.mathworks.com

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xiii

Acknowledgments

We would first like to acknowledge the outstanding contributions of the authors to this book. All of the authors impressed us with their dedica-tion to high quality and their thoughtful decisions of what to include and, almost as important, what not to include. We learned a great deal from each chapter.

Financial support for this project came from a grant to Sociometrics Corporation from the National Institute on Aging (R44 AG029194-02, Raghupathy). In addition to this support, each of the editors has received recent support for longitudinal projects that have provided examples or oth-erwise informed our understanding of longitudinal research (R01AG022957 and R01AG034211, Newsom; P60AG08812, Jones; R01AG026453, Hofer).

We are grateful to Shayna Cunningham of Sociometrics Corporation for her support and guidance throughout the project as well as her excel-lent contributions to two of the chapters. Nathalie Huguet has provided invaluable contributions in assembling and replicating the software examples for the Web site and stepping into various other roles with her expertise to make this project a success. We are forever grateful to our editor Debra Riegert for taking on this project and for her patience and flexibility in its development. Reviewers of the book proposal, Joop Hox (Utrecht University), Kai S. Cortina (the University of Michigan), and Duane Alwin (Pennsylvania State University), provided advice that con-tinued to have an impact throughout the writing of the book. Routledge editorial assistants, Andrea Zekus and Erin Flaherty, graciously guided us through paperwork and manuscript preparation. We are indebted to Jennifer Reed, who assisted with clerical support for the final manuscript.

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329

11Using Discrete-Time Survival Analysis to Study Event Occurrence

Suzanne E. GrahamUniversity of New Hampshire

John B. WillettandJudith D. SingerHarvard University

Researchers who study aging, health, and social sciences often ask questions about whether and when events occur, and how variation in the timing of event occurrence is related to critical features of the individuals, such as their life experiences, their social backgrounds, and educational levels. For example, researchers might ask whether, and if so when, individu-als experience major life events, such as getting married or having a first child, and whether the timing of these events is related to educational levels. Or, we might ask whether and when individuals are most likely to experience declines in their cognitive functioning, and whether such declines are associated with individuals’ participation in social networks.

In this chapter, we describe how an important statistical technique, sur-vival analysis (also known as event-history analysis), provides an effective framework for the analysis of event occurrence over time using longitudinal data. The application of survival analysis is very flexible and can be used with longitudinal data collected under many different research designs. The design can be experimental or observational, prospective or retrospective. Time can be measured in a unit that makes the most sense for the research question in hand—from seconds to years to decades. And, although the use of the term “survival analysis” may imply on first glance that the focus is on the occurrence of negative events that can be “survived,” these models are entirely appropriate for modeling the occurrence of positive events too (e.g., birth of a first child, achieving certain professional milestones).

We begin our presentation by introducing an example of longitudinal data on event occurrence, which we subsequently use to illustrate the

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330 Longitudinal Data Analysis

concepts developed throughout the chapter. This example focuses on timing of first divorce among Wisconsin high school graduates from the class of 1957 who were followed longitudinally between 1957 and 2004 as part of the Wisconsin Longitudinal Study (WLS). Next, we describe important decisions that must be made before embarking on a survival analysis, including defining event occurrence, identifying the beginning of time, and selecting a metric for measuring time. Within this discus-sion, we emphasize that decisions about timing determine the type of survival analysis that one conducts. Although many researchers who use survival analysis do so using continuous-time methods such as Cox regression, we argue that when time is measured in discrete-time points, as it is in the WLS study, then the more appropriate way to model event occurrence is with discrete-time methods.

We continue our presentation by pointing to a critical problem faced by researchers studying event occurrence—how to handle censored cases, the individuals for which the event of interest is not observed. Our dis-cussion highlights why traditional methods of analysis, such as ordinary least squares regression analysis and analysis of variance, are ill-suited and inappropriate for analyzing data when there are censored cases. Next, we describe methods for exploring longitudinal data on event occurrence and, while doing so, introduce its most important components—hazard and survivor functions. We then formally introduce the discrete-time survival analysis model, define its parameters, and fit it to the WLS data. We further describe how the model may be expanded to address research questions about the impact of educational level, marriage age, and gender on the probability of a first divorce. As we do so, we discuss how survival analysis facilitates investigation of whether the effects of these substan-tive predictors are constant or whether they change over time. Finally, we consider alternative ways of parameterizing time by using polynomial specifications of time to analyze discrete-time data, and we finish our discussion by comparing our results with an analysis of the same data using the Cox proportional hazards method for continuous-time data.

Description of Example: Time to First Divorce Using Data From the Wisconsin Longitudinal Study

Because we describe survival analysis within the context of real data, we now describe the example used in this chapter. The WLS is a publicly avail-able data set that includes longitudinal data on a random sample of 10,317 men and women who graduated from Wisconsin high schools in 1957. Survey data were collected from the original respondents or their parents in

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Using Discrete-Time Survival Analysis to Study Event Occurrence 331

1957, 1964, 1975, 1992, and 2004. In addition, respondents’ siblings, spouses, and spouses of siblings were also surveyed at various time points. Detailed questionnaires include information about family, schooling, labor market experiences, social participation, psychological characteristics, physical and mental health and well-being, and morbidity and mortality.

The example presented in this chapter focuses on whether and, if so, when a Wisconsin high school graduate of 1957 is most likely to experi-ence a first divorce over the course of his or her lifetime. Given the long period of time that graduates are studied, we can investigate the probabil-ity of divorce not only in the early years of marriage, but also in the later years of marriage among individuals who have been married for decades.

Of course, not all 1957 graduates are equally likely to experience a first divorce—many individual characteristics and life experiences increase or decrease the chances of divorce. In this example, we study just three potential predictors of time to first divorce: (a) education level—whether the individual has a bachelor’s degree by the beginning of the first marriage (DEGREE = 1 for yes and 0 for no); (b) age at first marriage, in years (AGE); and (c) gender (FEMALE = 1 for women and 0 for men).

The analytic sample used for the analyses presented in this chapter consists of the 7860 WLS individuals (a) who experienced a first mar-riage; (b) for whom we have information available about whether and if so when they divorced a first time; and (c) for whom we have data avail-able on the predictors listed above.

Defining Event Occurrence, Identifying the “Beginning of Time,” and Specifying a Metric for Time

Every survival analysis should begin with a definition of event occur-rence, which represents an individual’s transition from one “state” to another “state.” These states must be mutually exclusive (nonoverlap-ping) and exhaustive (of all possible states). In many applications, each person can occupy only two possible states—for example, employed or not employed, alive or dead. In the WLS example we use in this chapter, the two states to be considered are either in a first marriage (state 1) or divorced from a first marriage (state 2). Of course, divorce is not the only way that a marriage can end—death of either an individual or spouse also may end a marriage. Although it is analytically possible to consider situations in which individuals can occupy three or more states, using competing risks survival analysis, in our example here, we consider just two (and below we describe how we can treat the individuals who die as censored cases). In addition, individuals often marry more than one time

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and also divorce more than once—states can be occupied multiple times. Researchers can study how individuals progress in and out of states—that is, into a first marriage, out of a first marriage, into a second marriage, out of a second marriage, and so on, using multiple-spell (or multiple event) survival analysis. However, because the analysis of single spells is the most common one that researchers study, our presentation here focuses just on that—of those who experience a first marriage (the vast majority of individuals in this data set), who experiences a first divorce, and when do they do so?

After defining event occurrence, we must identify the “beginning of time,” the moment when everyone in the population occupies one, and only one, of the possible states. We essentially need to “start the clock” when no one in the population has experienced the event but everyone is at least (theoretically) eligible to do so. In the language of survival analy-sis, we start the clock when everyone in the population is at risk of experi-encing the event. In our divorce example, individuals are only “at risk” of experiencing a first divorce when they are in a first marriage. Therefore, the beginning of time here is the date of the first marriage.

Once the beginning of time is identified, we must select the unit in which its passage will be recorded. Sometimes, we can record time using narrow precise units. For example, in the current study it is theoretically possible to know the number of days that a first marriage lasts if we have informa-tion on the date of the marriage and the date of the divorce. However, the public release WLS data set provides only the years of the first marriage and first divorce, and the restricted use WLS data set provides dates in months (but not days). We distinguish between data recorded in narrow precise units and those recorded in wider intervals by calling the former continuous time and the latter discrete time.

Despite the advantages of measuring time as precisely as possible, many researchers find themselves with discrete-time data. Three related factors underlie this phenomenon. First, some events can occur only at discrete points in time. High school graduates, for example, can enter college only at a small number of registration periods. College graduation occurs only at preset times during the year. Second, although some events can the-oretically occur across a wide range of times, many individuals do not experience them this way. Although employees with term contracts might be able to quit their jobs at any point in time, most leave when contracts expire (see, e.g., Singer, Davidson, Graham, & Davidson, 1998). Third, data collection constraints—especially in retrospective studies—often force researchers to use intervals to record the passage of time.

Distinguishing between continuous- and discrete-time data is more than a methodological detail. Almost every feature of survival analysis—parameter definition, model construction, estimation, and testing—depends on the metric of time. Although the earliest descriptive methods

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for event occurrence (e.g., life-table methods) were developed for discrete-time data, modern methods of analysis (e.g., Cox regression—also known as proportional hazards modeling) assume that time is recorded on a continuous scale. This emphasis arises from the fact that research-ers in medicine and engineering—the areas in which survival analysis methods were originally developed—can usually record event occurrence precisely, in days.

Unfortunately, continuous-time methods break down when event times are highly discretized due to a problem known as “ties” (Cox & Oakes, 1984). With continuous-time data, the probability that two or more indi-viduals share an identical event time (are “tied”) is infinitesimally small. Because the probability of a tie is small, actual ties are few, and those that do occur can be treated as little more than a methodological nuisance. Although there are approaches for handling such ties, these solutions fail as the number of ties increases (Singer & Willett, 2003).

With discrete-time data, ties are pervasive. In the current WLS exam-ple, if time is measured in years (the most finely grained possibility with the publicly available data), all of the 1034 individuals who experienced a first divorce share identical event times with other individuals. Given that continuous-time methods can break down in the presence of such a large number of ties, we begin our presentation of survival analysis methods by discussing discrete-time methods. After doing so, we conduct parallel analyses using the most common continuous-time method, Cox regres-sion, and then compare our results.

The Censoring Problem

Researchers studying event occurrence face an analytic difficulty not encountered by those studying other kinds of outcomes: What should be done with the individuals who do not experience the target event during the period of data collection? In the WLS analysis, for example, most of the individuals who married did not divorce by 2004, the year of the last full survey. Obviously, we would not want to set these people aside, but to include them in a computation of summary statistics (i.e., computing the mean number of years of marriage before a first divorce), it seems that we would need to assign them a value of the outcome. But what value is appropriate? How can we impute an outcome that is, by its very nature, unknown? Does assigning them a value even make sense, given that most of the individuals will not divorce at all? These questions highlight chal-lenges in studying event occurrence—No matter when data collection begins and how long it lasts, some sample members are likely to have

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unknown event times. This problem is called censoring, and people with unknown event times are censored observations.

The amount of censoring in a study is related to two factors: (a) the rate at which events occur and (b) the length of data collection. If the event is common and data collection is sufficiently long, most people will expe-rience the event during data collection and the sample will contain few censored cases. For example, in studies of when individuals are most likely to die, over a long enough time period, everyone in the sample will die. However, if the event is rare or data collection is curtailed, cen-soring will be widespread. In the WLS divorce example, even though data collection occurred over decades, the target event (first divorce) is not common.

Among the 7860 WLS individuals in our analytic sample, 6826 are cen-sored. In addition to individuals being censored if they were still in a first marriage in 2004 (end of data collection), there are two other main reasons for censoring here. First, individuals cannot still be married if either they or their spouses are dead. Therefore, individuals who either died or experienced a death of a spouse are censored in the year of the death. Second, it is common in large longitudinal studies such as these to lose participants due to attrition. In the WLS study, nearly a third of the sample had missing data for the 2004 survey responses; however, many of these individuals responded to earlier surveys, so individuals with miss-ing data for 2002 who were still known to be in a first marriage in 1992, for example, are censored in 1992.

An important type of distinction is whether censoring is noninforma-tive or informative. A noninformative censoring mechanism operates independent of event occurrence and the risk of event occurrence. If censoring is under an investigator’s control, determined in advance by design, then it is noninformative. This is the case when censoring occurs because data collection ends, not because of actions taken by study par-ticipants or because of unanticipated study results, which may then have an impact on research design (e.g., a drug treatment cut short because of declining health for some individuals). If censoring is noninformative, we can assume that all individuals who remain in the study after the censor-ing date are representative of everyone who would have remained in the study had censoring not occurred.

If censoring occurs because individuals have experienced the event or are likely to do so in the future, the censoring mechanism is informative. Consider a study of cognitive decline in later years. If participants are lost to follow-up because they were cognitively unable to respond to the sur-vey questions, the censoring mechanism is informative. If they were lost to follow-up because they moved and the researchers could not locate them (and there are no systematic differences between participants who move and those who do not), then censoring is noninformative.

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No statistical method can produce unbiased analyses of event occur-rence if the censoring mechanism is informative. The problem, of course, is that when censoring is not determined by design, we have no way of knowing whether it is due to a random occurrence (e.g., the individual just stopped participating in the study) or to impending event occurrence (e.g., a treatment is no longer effective, so an individual stops participating). The validity of survival analysis rests on the assumption that censoring is noninformative, either because it occurs at random or because it occurs at a time dictated by design. In the WLS example presented here, we assume that censoring mechanisms are noninformative.

Exploring Longitudinal Data on Event Occurrence

Censoring makes standard statistical tools inappropriate even for simple analyses of event occurrence data because a censored event time pro-vides only partial information: It tells you only that the individual did not experience the target event by the time of censoring. Traditional statisti-cal methods provide no ready way of simultaneously analyzing observed and censored event times, but survival methods do.

The fundamental tool for summarizing the sample distribution of event occurrence is the life table, which tracks the event histories of the sample of individuals from the beginning of time (when no one has yet experienced the target event) through the end of data collection. Table 11.1 presents a life table for the WLS data. Recall that the “beginning of time” is defined as the beginning of the first marriage, and individuals in the WLS were followed from high school graduation in 1957 through 2008 (with the last full survey of graduates conducted in 2004). Regardless of the chronologi-cal year in which an individual got married, time in a first marriage begins to accumulate forward from that date. So, for example, an individual who married in 1957 is in his/her first year of marriage between 1957 and 1958 and an individual who married in 1970 is similarly in his/her first year of marriage between 1970 and 1971—both of them are counted as starting a first period of marriage, even though the chronological time differs. We account for differences in chronological time later in the analysis, when we consider the impact of marriage age on the risk of first divorce.

The life table is divided into a series of rows indexing time intervals (identified in columns 1 and 2). Although we have information on the year that an individual first married and the year that divorced individ-uals first divorced, and can thus measure time in yearly intervals, here we present intervals in 4-year time periods to simplify our presentation. Additionally, given that no individuals who were married over 44 years

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got divorced, we truncate the distribution at 44 years (i.e., eleven 4-year time periods). Using 4-year time intervals rather than years as time inter-vals reduces the number of intervals from 44 (i.e., individuals who were married for 44 years) to 11 (44/4 = 11 intervals). Extensive exploratory work not presented in this chapter revealed similar patterns of risk of a first divorce, whether time is measured in discrete years or 4-year time periods. In addition, selecting 4-year time periods to clock time has meth-odological advantages because, as we describe below, the discrete-time survival model that we fit is more parsimonious.

The next three columns of the life table provide information on the number of individuals who entered the time period (column 3), who experienced a first divorce during the time period (column 4), and who were censored at the end of the time period (column 5). Taken together, these columns provide a narrative history of event occurrence over time. At the “beginning of time” (i.e., beginning of the marriage), 7860 individu-als were in a first marriage and were therefore “at risk” of experiencing

Table 11.1

Life Table Describing the Number of 4-Year Time Intervals Before a First Divorce Among the WLS High School Graduates

Number of Graduates Proportion of

Time Period

Time Interval, in Years

In First Marriage at

the Beginning of the Time

Period

Divorced During

the Time Period

Censored at the End of

the Time Period

Graduates in First Marriage

at the Beginning of

the Time Period Who Divorced

During the Time Period

All Graduates

Still Married at the End of the Time

Period

0 [0,1) 7860 — — — 1.0001 [1, 5) 7860 150 6 0.0191 0.98092 [5,9) 7704 253 26 0.0328 0.94873 [9,13) 7425 258 74 0.0347 0.91574 [13,17) 7093 190 204 0.0268 0.89125 [17,21) 6699 43 217 0.0064 0.88556 [21,25) 6439 17 108 0.0026 0.88317 [25,29) 6314 12 209 0.0019 0.88158 [29,33) 6093 20 382 0.0033 0.87869 [33,37) 5691 36 639 0.0063 0.8730

10 [37,41) 5016 43 820 0.0086 0.865511 [41,45) 4153 12 2061 0.0029 0.8630

↑Risk set

↑Hazard function

↑Survivor function

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a first divorce at some later point in time. These 7860 individuals entered the first year of marriage (beginning of the first time period). During the first 4 years of marriage, 150 individuals divorced and 6 were censored, leaving 7704 (7860 – 156) to enter the next 4-year interval. In other words, 150 individuals divorced and 6 were censored before celebrating a fourth anniversary of their first marriage, leaving 7704 individuals to begin a fifth year of marriage. Of these individuals, 253 divorced and 26 were censored by the end of their eighth year of marriage, leaving 7425 individ-uals to enter a ninth year of marriage (i.e., the third interval). Of the 7425 individuals who celebrated an eighth anniversary of their first marriage, 258 divorced and 74 were censored by the beginning of the next (fourth) time period.

Collectively, the number of individuals who enter each successive time period and who are therefore at risk of divorcing during that time period are referred to as the risk set—those eligible to experience the event dur-ing the interval. Each period’s risk set is the prior period’s risk set minus those individuals who experienced a divorce in the prior time period and those individuals who were censored at the end of the prior period. An essential feature of the risk set’s definition is that it is inherently irre-versible: Once an individual experiences the event (or is censored) in one time period, he or she drops out of the risk set in all future time periods. Irreversibility is crucial, for it ensures that everyone remains in the risk set only up to, and including, the last moment of eligibility.

How can the risk of event occurrence be summarized when some indi-viduals have censored event times? In discrete-time survival analysis, the fundamental quantity representing the risk of event occurrence in each time period is called the hazard probability, defined here as the conditional probability that an individual will experience a first divorce during a particular 4-year time period, given that he or she did not already divorce. The hazard probability that individual i will experience a divorce in time period j is denoted h(tij) and can be expressed as:

h t T j T jij i i( ) [ | ]= = ≥Pr (11.1)

where T represents a discrete random variable whose values Ti indicate the time period j when an individual i experiences a first divorce. The set of discrete-time hazard probabilities expressed as a function of time is known as the population discrete-time hazard function.

This definition of hazard connects in a straightforward way to the sum-mary of event occurrence presented in the life table. Specifically, column 6 of the life table presents the proportion of WLS high school graduates still in a first marriage at the beginning of each time period and who divorced by the end of the time period. For example, among the 7860 graduates who entered a first marriage, .0191 (n = 150) divorced before their fourth

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anniversary. Of the 7704 graduates who were married for at least 4 years, .0328 (n = 253) of them divorced before their eighth anniversary. These proportions are maximum likelihood estimates of the discrete-time haz-ard probabilities in Equation 11.1 (Singer & Willett, 1993, 2003).

A valuable way of examining the estimated discrete-time hazard func-tion is to graph its values over time, as shown in the top panel of Figure 11.1. Here, we plot the discrete-time hazard probabilities as a series of points joined together by line segments, as suggested by Singer and Willett (2003). Observation of the estimated hazard function helps us to (a) identify espe-cially risky time periods—when individuals are most likely to experience a first divorce—and (b) characterize the shape of the hazard function—determining whether risk increases, decreases, or remains constant over time. In this study, the estimated hazard function increases rapidly from the end of year 4 to the end of year 8, continues to rise less rapidly in the next period, then peaking at 12 years at .0347 (i.e., of the WLS graduates

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

4 8 12 16 20 24 28 32 36 40 44Years since beginning of first marriage

Years since beginning of first marriage

Sam

ple h

azar

d pr

obab

ility

Sam

ple s

urvi

val p

roba

bilit

y

0.84

0.860.88

0.90

0.92

0.94

0.96

0.98

1.00

4 8 12 16 20 24 28 32 36 40 44

Figure 11.1Sample hazard and survivor functions for time to first divorce, with time measured in 4-year intervals.

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in first marriages for at least 8 years, .0347 divorce in the next 4-year time period). Then, the estimated hazard function declines rather dramatically, and the conditional risk of divorce between 20 and 32 years of marriage is almost 0. Interestingly, between years 32 and 40 there is a second increase in the risk of a first divorce, with a second peak at 40 years. Although this second peak in risk of first divorce is much smaller than the peak in ear-lier years, it represents a somewhat surprising increase in risk of divorce among individuals later in life.

The survivor function provides another way of describing the distribution of event occurrence over time. The survival probability, S(tij), is defined as the probability that an individual i will survive past time period j. So, in the divorce example, the survival probability is defined as the prob-ability that a graduate who experienced a first marriage remains married during each time period. When there is no censoring, then the survival probability can be estimated directly by simply computing the proportion of individuals in the data set who have not experienced the target even by the end of time period j. However, because censoring occurs during each time period in the WLS data set, we must use an indirect approach to estimate values of the survivor function in each time period. In this case, the estimated survival probability for the time period j is the estimated survival probability for the previous time period multiplied by one minus the estimated hazard probability for that time period:

ˆ ) ˆ( )[ ˆ( )]S t S t h tj j j( = −−1 1 (11.2)

The survival probability at the start of the first marriage is 1.0, because at this point no divorces have occurred. Then, at the end of the fourth year of marriage, we estimate that .9809 of the Wisconsin high school graduates of the class of 1957 have not yet divorced. At the end of the 16th year of mar-riage (i.e., end of the fourth 4-year time period), .8912 of the graduates have not experienced a first divorce. And, at the end of the 44th year of marriage, we estimate the survival probability to be .8630. These estimated survival probabilities, shown in the last column of Table 11.1, are italicized to indi-cate that they were calculated using the indirect approach from Equation 11.2, which takes censoring into consideration.

As with the estimated hazard function, graphing the values of the esti-mated survivor function over time allows us to visually identify trends in event occurrence. The survivor function for the WLS data is presented in the bottom panel of Figure 11.1. Notice the relatively shallow slope between years 0 and 4, identifying a period when the probability of a first divorce is relatively low. Between years 4 and 16, the estimated survivor function drops more steeply, reflecting the somewhat larger estimated hazard of a first divorce during those periods. By the end of the 16th year of a first marriage, we estimate that approximately 89% of the graduates have not

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yet divorced. The rest of the estimated survivor function is relatively flat, although there is another slight dip between years 36 and 40, reflecting the slightly elevated risk of a first divorce during these time periods.

Having characterized the distribution of event times using the hazard and survivor functions, we often want to identify the distribution’s center. Were there no censoring, all event times would be known, and we could compute a sample mean. But because of censoring, another estimate of central tendency is preferred: the median lifetime. The estimated median lifetime identifies the time when the value of the estimated survivor func-tion is .5. By examining the estimated survivor function from the WLS data in Figure 11.1, we see that the median lifetime is never reached—that is, less than half of the population is predicted to divorce by the end of 44 years of marriage. If a median lifetime was reached, we could compute its estimated value as

Estimatedmedianlifetime = + −

−

+

mS t

S t S tm

m m

ˆ( ) .ˆ( ) ˆ( )

5

1

((( ) )m m+ −1

(11.3)

wherem represents the time interval when the sample survivor function is

just above .5S (tm) represents the value of the sample survivor function in that intervalS (tm+1) represents the value of the sample survivor function for the fol-

lowing interval (when it must be just below .5)

An Initial Discrete-Time Survival Analysis Model

As described above, the estimated hazard and survivor functions obtained from the life table are maximum likelihood estimates of the population hazard and survivor functions. The maximum likelihood estimates are obtained by fitting an initial hazard model (Singer & Willett, 1993, 2003):

logit ( 1 1 2 2 11 11h t D D Dij ij ij ij) = + + +α α α� (11.4)

where h(tij) represents the hazard probability of a first divorce for individ-ual i during time period j. The α parameters are multiple intercept terms describing the risk of a first divorce during each time period, up until time period 11 (which ends at the end of 44 years), in which the person either divorces or is censored. D1ij, D2ij,… are a sequence of dummy vari-ables, which take on a value of 1 during the jth time period (i.e., D1ij will

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take on a value of 1 when j = 1, and 0 otherwise). This initial,* or uncon-ditional, hazard model can be easily expanded to include the effects of substantive predictors, as we show below.

In order to fit the model, we need to restructure our data set, from what we refer to as a person-level data set, which contains one record for each person in the study, to a person-period data set, which contains one record for each time period that an individual is at risk of a first divorce. Table 11.2 illustrates the conversion from a person-oriented data set to a person-period data set using three WLS individuals. The first two indi-viduals have known divorce times—the first graduate divorced during the second 4-year time period after marriage (i.e., divorced between 4 and 8 years after marriage), and the second graduate divorced between 12 and 16 years after marriage (i.e., in time period 4). The third graduate

* The “initial” hazard model is sometimes referred to as a “baseline” model. We choose to use the term “initial” here to distinguish between other uses of the term “baseline”—specifically, when a survival analysis model includes substantive predictors, the “baseline” hazard function is the hazard function when all of the predictors’ values are simultane-ously 0. A similar terminology is used in Cox regression.

Table 11.2

Conversion of a Person-Level Data Set Into a Person-Period Data Set for Three WLS High School Graduates Who Entered a First Marriage

“Person-Level” Data Set

ID LENGTH CENSOR1025 2 01050 4 01078 7 1

“Person-Period” Data Set

ID PERIOD D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 DIVORCE1025 1 1 0 0 0 0 0 0 0 0 0 0 01025 2 0 1 0 0 0 0 0 0 0 0 0 11050 1 1 0 0 0 0 0 0 0 0 0 0 01050 2 0 1 0 0 0 0 0 0 0 0 0 01050 3 0 0 1 0 0 0 0 0 0 0 0 01050 4 0 0 0 1 0 0 0 0 0 0 0 11078 1 1 0 0 0 0 0 0 0 0 0 0 01078 2 0 1 0 0 0 0 0 0 0 0 0 01078 3 0 0 1 0 0 0 0 0 0 0 0 01078 4 0 0 0 1 0 0 0 0 0 0 0 01078 5 0 0 0 0 1 0 0 0 0 0 0 01078 6 0 0 0 0 0 1 0 0 0 0 0 01078 7 0 0 0 0 0 0 1 0 0 0 0 0

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(ID 1078) married in 1977 and was not yet divorced by the end of data collection in 2004, so was censored at the end of the seventh time period. The person-oriented data set describes these graduates’ event histories using two variables: an event time (here LENGTH, the period in which the individual experienced a first divorce or was censored) and a cen-soring indicator (CENSOR = 0 for individuals who divorced and 1 for individuals who did not divorce). The person-period data set includes a period variable, PERIOD, which specifies the time period j that the record describes. The particular time period described in the record is also iden-tified through the set of time indicator variables, D1 through D11. Finally, the person-period data set includes an event indicator, DIVORCE, which indicates whether a first divorce occurred during that time period (0 = no divorce, 1 = divorce). For each person, the event indicator must be 0 in every record except the last. Noncensored individuals (like individual 1078) experience the event in their last period, whereas censored indi-viduals never experience a first divorce, so DIVORCE remains 0 for all of their records. The person-period data set is easily expanded to include both time-invariant and time-varying substantive predictors.

To obtain maximum likelihood estimates of the population param-eters in the discrete-time hazard model in Equation 11.4, we use logistic regression to regress the event indicator (DIVORCE) on the time indica-tors (D1 through D11). Any logistic regression routine from any statistical software package can be used (i.e., PROC LOGISTIC in SAS, LOGISTIC REGRESSION in SPSS, LOGIT in STATA). For the results presented in this chapter, we used PROC LOGISTIC in SAS.

The third column of Table 11.3 presents parameter estimates from this first fitted model. As a group, the estimated αs are maximum likelihood estimates of the initial logit hazard function. The amount and direction of variation in their values describe the shape of this function and tell us whether risk increases, decreases, or remains steady over time. The increasing parameter estimates from periods 1 to 3 indicate an increase in the risk of first divorce during these periods, followed by a decreasing risk through period 7, and then a slight increase through period 10 and finally a decrease in period 11.

A better way to interpret these parameter estimates is to compute esti-mated hazard probabilities by taking the antilogit of α :

ˆ ) ˆh t

ej j

( =+ −1

1 α

(11.5)

Here e is the mathematical constant approximately equal to 2.718 and is the complimentary function of the natural logarithm. The estimated haz-ard probabilities are presented in the last column of Table 11.3. Not sur-prisingly, their numerical values (and their interpretation) are identical to

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the estimates presented in Table 11.1. For example, we estimate that 1.91% of the individuals who marry will experience a first divorce within the first 4 years after marriage, and 3.28% of the individuals married at least 4 years will divorce before their eighth wedding anniversary.

Including the Effect of a Substantive Predictor

The initial discrete-time hazard model can be easily extended to include the effects of our substantive predictors—whether the individual received a bachelor’s degree before marriage (DEGREE), the individual’s age at first marriage (AGE), and gender (FEMALE). We begin by considering the effect of just one dichotomous predictor, DEGREE, addressing the ques-tion of whether the population hazard and survivor functions differ for individuals who obtained a bachelor’s degree before marriage, compared with those individuals who did not.

To estimate the main effect of DEGREE, we fit the following model:

logit ( 1 1 2 2 11 11h t D D D DEGREEij ij ij ij i) = + + + +α βα α� 1 (11.6)

The population parameter β1 represents the hypothesized influence of the predictor DEGREE on the logit hazard function. Specifically, β1 quantifies the difference in logit hazard between the individuals with degrees and those without degrees. Its estimated value, reported in Model B of Table 11.4,

Table 11.3

Parameter Estimates and Fitted Hazard Probabilities from an Initial Discrete-Time Hazard Model Fitted to the WLS Data (n = 7860)

Time Period Predictor

Parameter Estimate (��j)

Fitted Hazard ˆ ˆh t ej

j( ) (1 / (1 ))= + -a

1 D1 −3.9396 0.01912 D2 −3.3827 0.03283 D3 −3.3243 0.03474 D4 −3.5927 0.02685 D5 −5.0421 0.00646 D6 −5.9343 0.00267 D7 −6.2637 0.00198 D8 −5.7159 0.00339 D9 −5.0568 0.0063

10 D10 −4.7506 0.008611 D11 −5.8438 0.0029

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Table 11.4

Parameter Estimates, Asymptotic Standard Errors, and Goodness-of-Fit Statistics from a Series of Hazard Models in Which Time, Bachelor’s Degree Receipt Prior to Marriage, and Their Interaction Predict the Conditional Probability That a Wisconsin 1957 High School Graduate Experiences a First Divorce (n = 7860)

Predictor Model A Model B Model C Model D

Time period1 −3.9396*** −3.8764*** −3.8955*** −3.8831***

(0.0824) (0.0831) (0.0876) (0.0868)2 −3.3827*** −3.3189*** −3.3295*** −3.3578***

(0.0639) (0.0648) (0.0677) (0.0674)3 −3.3243*** −3.2603*** −3.2684*** −3.2356***

(0.0634) (0.0643) (0.0670) (0.0646)4 −3.5927*** −3.5285*** −3.4553*** −3.4671***

(0.0735) (0.0743) (0.0749) (0.0744)5 −5.0421*** −4.9761*** −4.8927*** −4.9125***

(0.1530) (0.1534) (0.1549) (0.1537)6 −5.9343*** −5.8671*** −5.8192*** −5.8401***

(0.2429) (0.2431) (0.2504) (0.2444)7 −6.2637*** −6.1969*** −6.2717*** −6.3110***

(0.2889) (0.2892) (0.3165) (0.2967)8 −5.7159*** −5.6499*** −6.1428*** −6.0112***

(0.2240) (0.2242) (0.3018) (0.2542)9 −5.0568*** −4.9919*** −5.2143*** −5.2436***

(0.1672) (0.1675) (0.1966) (0.1896)10 −4.7506*** −4.6873*** −4.6085*** −4.6072***

(0.1532) (0.1535) (0.1551) (0.1544)11 −5.8438*** −5.7907*** −5.8438*** −5.7063***

(0.2891) (0.2892) (0.2891) (0.2892)

Bachelor’s degree receipt

DEGREE −0.5164*** −2.9196**(0.1057) (1.0538)

DEGREE × PERIOD 4.0984**(1.2795)

DEGREE × PERIOD2 −1.9261***(0.4928)

DEGREE × PERIOD3 0.3136***(0.0716)

DEGREE × PERIOD4 −0.0161***(0.0035)

DEGREE × PERIOD_1 −0.3352(0.2595)

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is −0.5164 (p < .001). Because the estimate is negative, we conclude that the logit hazard function for individuals who had college degrees when they were first married is lower than the logit hazard function for individu-als with no degrees. Specifically, the estimated logit hazard for the high school graduates with degrees is 0.5164 below the estimated logit haz-ard for the graduates without degrees, at all points in time (top panel of Figure 11.2).

To understand the estimated effect of DEGREE more meaningfully, we graph estimated hazard functions for individuals who did and did not have college degrees at the start of their marriages (middle panel of Figure 11.2). Focusing on the peaks in risk of first divorce earlier and later in marriage, we find that the estimated hazard of first divorce during the

Table 11.4 (continued)

Parameter Estimates, Asymptotic Standard Errors, and Goodness-of-Fit Statistics from a Series of Hazard Models in Which Time, Bachelor’s Degree Receipt Prior to Marriage, and Their Interaction Predict the Conditional Probability That a Wisconsin 1957 High School Graduate Experiences a First Divorce (n = 7860)

Predictor Model A Model B Model C Model D

DEGREE × PERIOD_2 −0.4134*(0.2061)

DEGREE × PERIOD_3 −0.4380*(0.2060)

DEGREE × PERIOD_4 −1.7339***(0.4162)

DEGREE × PERIOD_5 −2.0686*(1.0114)

DEGREE × PERIOD_6 −1.1230(1.0305)

DEGREE × PERIOD_7 0.0492**(0.7754)

DEGREE × PERIOD_8 1.4794**(0.4508)

DEGREE × PERIOD_9 0.7438*(0.3739)

DEGREE × PERIOD_10 −2.0235*(1.0124)

Goodness of fit−2LL (n of parameters) 9,996.13 (11) 9,968.8 (12) 9,922.56 (21) 9,925.6 (16)AIC 10,018.1 9,992.8 9,964.6 9,957.6

* p < .05. ** p < .01. *** p < .001.

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–7.00–6.50–6.00–5.50–5.00–4.50–4.00–3.50–3.00–2.50–2.00

0 4 8 12 16 20

β1

24 28 32 36 40 44Years since beginning of �rst marriage

Estim

ated

logi

t haz

ard

No degreeDegree

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0 4 8 12 16 20 24 28 32 36 40 44Years since beginning of �rst marriage

Estim

ated

haz

ard

No degreeDegree

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0 4 8 12 16 20 24 28 32 36 40 44Years since beginning of �rst marriage

Estim

ated

surv

ival

pro

babi

lity

No degreeDegree

ˆ

Figure 11.2Fitted logit hazard, hazard, and survivor functions displaying the time-invariant main effect of degree receipt before first marriage on the conditional probability of a first divorce.

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first peak risk period between 8 and 12 years of marriage is .0370 for indi-viduals with no degree and .0224 for individuals with degrees. During the smaller peak late in marriage, the estimated risk of a first divorce is .091 for the no degree group and .055 for the degree group.

In standard logistic regression analysis, a useful way to interpret parameter estimates is by antilogging them to obtain an estimated odds ratio. Antilogging −.5164, we obtain an estimated odds ratio of .60, which indicates that the estimated odds of a first divorce are lower for the degree group. Because it is generally easier to interpret odds ratios greater than 1.0, we can compute the reciprocal of the estimated odds ratio, obtaining 1.67 (i.e., 1/.6)—thus, the estimated odds of a first divorce for a high school graduate with no college degree are 1.67 times the estimated odds of a first divorce for a high school graduate with a college degree.

Finally, the estimated survivor functions presented in the bottom panel of Figure 11.2 reveal the cumulative effects of the period-to-period risk of divorce for the two groups. The differences are initially quite small, but by the end of the fourth time period (i.e., at 16 years), the difference in the estimated survival rate is nearly 5 percentage points (.88 for the individuals with no degree compared with .93 for the indi-viduals with degrees). At the end of 44 years, we estimate that 91% of the degree group and 85% of the no degree group have not yet experi-enced a first divorce.

The estimated logit hazard functions presented in Figure 11.2 display a constant DEGREE effect over time, but a natural question that may arise in examining these estimated logit hazard functions is whether the effect of DEGREE is necessarily constant over time. The model presented in Equation 11.6 imposes a constraint known as the proportionality assump-tion, which stipulates that a predictor’s effect on logit hazard does not vary over time. But is it not possible, even likely, that a predictor’s effect will change across time, particularly when we are modeling event occur-rence over the course of entire human lifetimes? How might we relax the possibly unrealistic constraints imposed by a model that does not allow effects to vary over time? In using ordinary least squares regression analy-sis and analysis of variance, data analysts are accustomed to investigating interactions between substantive predictors—such interactions provide a way for us to determine whether the effect of one predictor varies by lev-els of another. In discrete-time survival analysis, we adopt an equivalent approach to determining whether a substantive predictor’s effect varies over time by adding statistical interactions between the predictor and time to our model.

How do we add such statistical interactions to our discrete-time hazard model? Given that we have parameterized time in this model as a series of dummy variables indexing each discrete-time period, one possible

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interaction involves the cross product between degree status and each dummy variable:

logit ( 1 1 11 11 1 1 11h t D D DEGREED DEGREEij ij ij i ij) [ ] [= + + + + +α α β β� � ii ijD11 ] (11.7)

This is the most general possible interaction between degree receipt and time, but it is not the only possibility. Other options involve interactions between degree receipt and linear time, quadratic time, cubic time, and so forth—such interactions result in models with fewer parameters, which is advantageous in maximum likelihood estimation. To understand how interactions with polynomial versions of time can be included in our model, recall that the person-period data set includes a period vari-able, PERIOD, which specifies the time period j that the record describes. We can use this time variable to investigate, for example, the interaction between DEGREE and linear time, fitting the model:

logit ( 1 1 2 2 11 11h t D D D DEGREE

DEGREE

ij ij ij ij i

i

) [ ]= + + + +

+

α α α β

β

� 1

2 PPERIODj (11.8)

Table 11.4 contains parameter estimates, asymptotic standard errors, and goodness-of-fit statistics from two alternate models that include inter-actions between the substantive predictor DEGREE and time. Model C models the interaction between DEGREE and general time from Equation 11.7, with one minor change: there is no term representing an interaction between DEGREE and the time indicator variable for the 11th period. During the 11th period, the few individuals who divorced did not have college degrees so a DEGREE effect cannot be modeled during this period. Model D represents a more parsimonious option, with DEGREE interact-ing with quartic time (including lower polynomial functions of time):

logit ( 1 1 2 2 11 11h t D D D DEGREE

DEGREE

ij ij ij ij i

i

) [ ]= + + + +

+

α α α β

β

� 1

2 PPERIOD DEGREEPERIOD

DEGREEPERIOD DEGREEPERI

j i j

i j i

+

+ +

β

β β

32

43

5 OODj4

(11.9)

To determine whether the proportionality assumption has been violated in the main effects model, we compare the goodness-of-fit statistics of the main effects model (Model B) and the interaction with time models (Models C and D), using the −2LL goodness-of-fit statistic presented in the bottom part of the table. The “−2log-likelihood,” or “−2LL” statistic, is a common fit statistic generated when statistical models are fitted using maximum likeli-hood estimation, as we have done here. While it is difficult to interpret the

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absolute magnitude of this statistic, it can be regarded as a “badness-of-fit” index that functions similarly to the “sum-of-squared-error (SSE)” statistic in ordinary least squares regression analysis. The worse the overall fit of the model, the larger is the value of the −2LL statistic. As a result, the −2LL statistic proves useful in comparisons of nested models that have been fit-ted to the same data. Then, differences in the −2LL statistic between fitted (nested) models have a χ2 distribution, with degrees of freedom equal to the difference in the number of parameters between the models (see Singer and Willett, 2003, for a definition of the −2LL statistic and examples of its use in comparing nested models).

In testing interaction with time models in the WLS example, we begin by first comparing the difference in the −2LL statistics between the main effects model (Model B) and the general interaction with time model (Model C) with a χ2 distribution with 9 degrees of freedom (i.e., 21 parameters esti-mated in Model C versus 12 parameters estimated in Model B). If the inter-action with time model is sufficiently superior to the main effects model, we can justify the additional parameters, and we adopt the nonproportional Model C in lieu of Model B. Here, the difference in the deviance statistics between Models B and C is 46.24. As the critical value from a χ2 distribution with 9 degrees of freedom is 16.92 at α = .05 level (and 23.59 at α = .005 level), we conclude that the proportionality assumption has been violated and the effect of DEGREE on logit hazard is not constant over time. However, a comparison between Models C and D shows that the interaction with time may be modeled in a more parsimonious way—here the difference in −2LL is only 3.04, which does not exceed the critical value of 11.07 from a χ2 distri-bution with 5 degrees of freedom (α = .05). Interactions between DEGREE and lower polynomial powers of time (i.e., linear, quadratic, and cubic) did not fit as well as Model D and are not presented here.

Given the complexity of Model D, its interpretation is best accomplished graphically. The top panel of Figure 11.3 presents fitted logit hazard func-tions for individuals with (dashed line) and without (solid line) college degrees at the time of marriage. In contrast with the estimated logit hazard functions from Model B, presented in the top panel of Figure 11.2, here we clearly see differences in the effect of DEGREE over time. In the first two time periods, at years 4 and 8, the difference in logit hazard between the two groups is quite small. Between 8 years and 24 years, the logit hazards for the degree recipients are smaller, representing a lower risk of a first divorce. Then, at period 7 (28 years), the pattern actually reverses, and the degree group has a higher logit hazard of first divorce, but at year 40, it is lower again. Essentially, the peaks in logit hazard come somewhat earlier for the WLS graduates with degrees, compared with the no degree group.

To better understand the differences in risk of first divorce over time for individuals with and without college degrees, we turn to the estimated hazard and survivor functions from Model D, presented in the middle and

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bottom panels of Figure 11.3. The estimated hazard probability of a first divorce for the degree holders is largest in the second time period—here, .0284 of the graduates who were married at least 4 years divorced by the end of 8 years of marriage. At this point, the risk for the no degree group is only slightly larger (.0336). However, after the second time period, the risk of a first divorce plummets downward for the degree group, while still

–7

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–4

–3

–2

–1


Estim

ated

logi

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0.0000.0050.0100.0150.0200.0250.0300.0350.040

0 4 8 12 16 20 24 28 32 36 40 44

0 4 8 12 16 20 24 28 32 36 40 44


Estim

ated

haz

ard

0.840.860.880.900.920.940.960.981.00

0 4 8 12 16 20 24 28 32 36 40 44Years since beginning of first marriage

Estim

ated

surv

ival

pro

babi

lity

No degreeDegree

No degreeDegree

No degreeDegree

Figure 11.3Fitted logit hazard, hazard, and survivor functions displaying the time-varying effect of receipt of a bachelor’s degree before first marriage on the conditional probability of a first divorce.

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increasing for the no degree group—for these graduates, the peak comes at the third time period (12 years), with an estimated hazard probability of .0378. The second lower peak in the risk of first divorce exists for both groups, but at different times—at period 9 (36 years) for the degree holders and at period 10 (40 years) for those individuals with no college degree. At the end of the 11th time period (44 years), we estimate that 85% of the high school graduates with no degree have not yet experienced a first divorce, compared with 91% of the graduates with degrees. These numbers are the same as those estimated in the main effects model, so in this example the time-varying effect of DEGREE does not have an impact on the probability of “surviving” a first divorce through 44 years; however, the time-varying effect of DEGREE reveals how the relative risk of having a college degree on the probability of first divorce changes in important ways over the years.

Including Additional Substantive Predictors and Selecting and Interpreting a “Final” Model

Does the age when an individual first marries have an impact on the risk of a first divorce? Are women less likely to experience a first divorce than men? To answer these questions, we now fit additional discrete-time hazard models to the WLS data. The taxonomy of fitted models is presented in Table 11.5. For comparative purposes, Models A and D from Table 11.4, dis-cussed above, are included here as well. Then, we build on Model D by first investigating the effect of AGE, controlling for DEGREE (Model E), and then testing the effect of FEMALE, controlling for AGE and DEGREE (Model F). Although many models were fitted to explore the effects of these predictors, to save space not all are presented here. Instead, we present the “best” mod-els representing final decisions from each phase of the analysis. Specifically, we compared a model containing only the main effect of AGE to other mod-els including interactions between AGE and time, finding that the effect of AGE does vary with time and the interaction cannot be parameterized more parsimoniously than an interaction with general time without substantial worsening of model fit. In addition, a comparison of Model E and Model D shows that the inclusion of the AGE by general time interaction results in a significant improvement of model fit over the DEGREE only model, Model D (difference in −2LL = 195.9, df = 11, p < .001). We conducted similar analyses when testing the effect of FEMALE, but here we found a main effect only (Model F; difference in −2LL = 9.5, df = 1, p < .01). Finally, we tested two-way interactions between the substantive predictors, finding none.

Interpretation of the effect of FEMALE in Model F is relatively straight-forward because we can antilog its parameter estimate to obtain a fitted

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Table 11.5

Parameter Estimates, Asymptotic Standard Errors, and Goodness-of-Fit Statistics from a Taxonomy of Fitted Hazard Models in Which Time, Bachelor’s Degree Receipt Prior to Marriage, Age at First Marriage, and Respondent Sex Predict the Conditional Probability of Experiencing a First Divorce (n = 7860)

Predictor Model A Model D Model E Model F

Time period

1 −3.9396*** −3.8831*** −3.7010*** −3.4072***(0.0824) (0.0868) (0.5138) (0.5361)

2 −3.3827*** −3.3578*** −2.6579*** −2.3410***(0.0639) (0.0674) (0.4231) (0.4458)

3 −3.3243*** −3.2356*** −1.1882* −0.7972(0.0634) (0.0646) (0.4982) (0.5237)

4 −3.5927*** −3.4671*** 2.7050*** 3.2077***(0.0735) (0.0744) (0.7754) (0.7985)

5 −5.0421*** −4.9125*** 6.0169** 6.5492***(0.1530) (0.1537) (1.9608) (1.9687)

6 −5.9343*** −5.8401*** −10.6411*** −10.4628***(0.2429) (0.2444) (1.0452) (1.0562)

7 −6.2637*** −6.3110*** −8.1525*** −7.8867***(0.2889) (0.2967) (1.6666) (1.6963)

8 −5.7159*** −6.0112*** −10.0007*** −9.7519***(0.2240) (0.2542) (1.3303) (1.3443)

9 −5.0568*** −5.2436*** −7.3733*** −7.0467***(0.1672) (0.1896) (1.1735) (1.1886)

10 −4.7506*** −4.6072*** −2.5461† −2.0834(0.1532) (0.1544) (1.4382) (1.4565)

11 −5.8438*** −5.7063*** 8.7381* 9.2340*(0.2891) (0.2892) (4.2958) (4.2730)

Substantive predictors (bachelor’s degree before marriage, age at marriage, gender)

DEGREE −2.9196** −2.4168* −2.4101*(1.0538) (1.0996) (1.0994)

DEGREE × PERIOD 4.0984** 3.1231* 3.1099*(1.2795) (1.3353) (1.3348)

DEGREE × PERIOD2 −1.9261*** −1.3476** −1.3397**(0.4928) (0.5152) (0.5150)

DEGREE × PERIOD3 0.3136*** 0.2095** 0.2081**(0.0716) (0.0746) (0.0745)

DEGREE × PERIOD4 −0.0161*** −0.0105** −0.0104**(0.0035) (0.0035) (0.0036)

AGE × PERIOD_1 −0.0081 −0.0164(0.0229) (0.0237)

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odds ratio. Specifically, the estimated odds ratio is .812, and taking the inverse of .812, we find that the estimated odds of a first divorce for men are 1.23 times the estimated odds for women, controlling for age of first marriage and whether the individual had a college degree at marriage.

Given their interactions with time, the effects of DEGREE and AGE are much more difficult to understand. And while we could use the parameter estimates from Model F presented in Table 11.5 to algebraically calculate the estimated effects of each predictor during each of the 11 time peri-ods, we find that plotting graphs of fitted hazard and survivor functions for prototypical individuals is a powerful tool for communicating such

Table 11.5 (continued)

Parameter Estimates, Asymptotic Standard Errors, and Goodness-of-Fit Statistics from a Taxonomy of Fitted Hazard Models in Which Time, Bachelor’s Degree Receipt Prior to Marriage, Age at First Marriage, and Respondent Sex Predict the Conditional Probability of Experiencing a First Divorce (n = 7860)

Predictor Model A Model D Model E Model F

AGE × PERIOD_2 −0.0319† −0.0411*(0.0190) (0.0198)

AGE × PERIOD_3 −0.0947*** −0.1073***(0.0230) (0.0238)

AGE × PERIOD_4 −0.2919*** −0.3099***(0.0377) (0.0385)

AGE × PERIOD_5 −0.5293*** −0.5487***(0.0991) (0.0993)

AGE × PERIOD_6 0.1956*** 0.1920***(0.0377) (0.0380)

AGE × PERIOD_7 0.0809 0.0737(0.0693) (0.0705)

AGE × PERIOD_8 0.1738** 0.1672**(0.0527) (0.0532)

AGE × PERIOD_9 0.0975† 0.0876†

(0.0504) (0.0509)AGE × PERIOD_10 −0.0966 −0.1128†

(0.0674) (0.0681)AGE × PERIOD_11 −0.7175** −0.7352***

(0.2218) (0.2205)FEMALE −0.2086**

0.0677Goodness of fit−2LL (n of parameters) 9,996.13 (11) 9,925.6 (16) 9,729.7 (27) 9,720.2 (28)AIC 10,018.1 9,957.6 9,783.7 9,776.2

† p < .10. * p < .05. ** p < .01. *** p < .001.

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complex findings. To simplify plots, it is helpful to limit the number of prototypical values selected for plotting. Because the gender effect is rela-tively straightforward, with women at lower risk of a first divorce across their lifetimes compared with men, in our plot we focus on exploring the complex effects of degree status and age at first marriage by constructing a plot for women only. Thus, in Figure 11.4, we present fitted hazard and survivor functions for women with and without degrees at marriage, who were younger (19 years) versus older (27 years) when they married. These values of marriage age correspond with the sample mean marriage age (23 years) minus and plus one sample standard deviation (approximately 4 years). Because no women in this sample who married at age 19 had col-lege degrees at the time of marriage, there is no corresponding plot for a woman with these values of AGE and DEGREE.

0.0000.0050.0100.0150.0200.0250.0300.0350.0400.0450.0500.0550.060


Estim

ated

haz

ard

No degree, married at 19

Has degree, married at 27


0.8000.8200.8400.8600.8800.9000.9200.9400.9600.9801.000

Estim

ated

surv

ival

pro

babi

lity


Has degree, married at 27



Figure 11.4Fitted hazard and survivor functions for prototypical women who were (a) aged 19 years at the time of marriage, with no college degree; (b) aged 27 years at marriage, with no college degree; and (c) aged 27 years at marriage, with a college degree. Estimated hazard and sur-vival probabilities were obtained by the substitution of predictor values into fitted Model F.

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By comparing the solid lines in Figure 11.4, we see the effect of age at first marriage on the probability of a first divorce for women with no college degree. The lighter solid line represents a woman who married young, at age 19, compared with the bold solid line for a woman who married older. The woman who married earlier has a much higher risk of a first divorce in the first five time periods, through 20 years of marriage, and then again in the last two time periods. Furthermore, her risk of a first divorce peaks in the fourth period, at .0527, after which her risk of a first divorce decreases rapidly to nearly 0 by the sixth period (24 years). The effect of having a college degree at the time of first marriage, controlling for marriage age, can be seen in comparing the two bold lines. The bold solid line, representing the woman with no degree, is above the dashed line, representing the woman with a degree, over the early to mid-years of marriage, showing that in these periods the woman with no college degree has a larger risk of first divorce. However, the direction of this relationship reverses in the eighth and ninth time periods—so, in years 28 through 36 of marriage, a woman with no degree is at a lower risk of a first divorce. The direction of the effect changes yet again in the last two time periods. Examining the estimated survivor functions, there is a marked effect of age at first marriage on the probability that a woman will not have divorced by her 44th year of marriage—the estimated survival probability for a woman with no college degree who married at age 19 is .82, which is much lower than the estimates for women who were older when they married (.92 for a woman with a degree and .91 for a woman with no degree).

Alternative Parameterizations of Time

So far in this chapter, we have used a completely general specification for TIME in our discrete-time hazard models, by including 11 dummy vari-ables to identify the discrete-time periods in the hazard model as pre-dictors. This specification avoids constraints on the shape of the baseline hazard, and it is also (a) easily interpretable—each parameter represents the population value of logit hazard in the time period j for the baseline group (i.e., when all other predictors’ values are 0); (b) informative—patterns in the values of the parameters describe the temporal shape of hazard; and (c) consistent with life-table estimates—as we described above, fitted values from a model with no substantive predictors repro-duce estimates obtained using the life-table approach.

The use of a completely general specification for TIME is an analytic decision, not an integral feature of the model. Nothing about the model or

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its estimation requires adoption of this, or any other, particular specifica-tion for TIME. Why might we consider alternative specifications? Efron (1988) and Fahrmeir and Wagenpfeil (1996) point out that the completely general specification of TIME lacks parsimony if there are many time peri-ods (as we have here), and yields fitted hazard functions that can fluctuate erratically across consecutive time periods because of sampling variation. If goodness-of-fit statistics across models with alternative specifications of time are very close, it is reasonable to prefer the more parsimonious specification.

There are an infinite number of possible ways to reparameterize time. Here we focus on a subset of model specifications, which we find par-ticularly useful: an ordered set of polynomial representations of time (i.e., time squared, time cubed, etc.). We focus on these not because one of them always emerges as best, but because, taken together, they encompass a wide array of alternatives. If there is reason to believe that a logged ver-sion of time, for example, may be superior, its performance could also be evaluated using the strategies we present here.

The completely general specification of TIME is the model in Equation 11.4 introduced above, containing the dummy variables D1 to D11 to repre-sent TIME. Because its fitted hazard function is identical to the life-table estimates, this model has the lowest (best) −2LL statistic in comparison to any other model without substantive predictors. However, there are 11 time predictors for this model—it would be preferable if we could model the main effect of time more parsimoniously (i.e., fewer predictors), if such a model exists.

On the other end of the goodness-of-fit spectrum is a model that elimi-nates the effect of TIME altogether:

logit ( 0h t ONEj) = α (11.10)

By eliminating all predictors, this constant model constrains logit hazard to be identical across all 11 time periods. Because the constant model con-tains only one parameter, it has the largest (worst) −2LL statistic of any model. Notice that in specifying the constant model, we associate α0 with an explicit predictor labeled ONE, which takes on the value 1 for every record in the person-period data set. Although the use of an explicit con-stant like this may be unfamiliar, all computer programs that include an intercept in any statistical model by default use such a predictor. Because the general specification has no single intercept, the use of ONE facili-tates programming. By including ONE to include an explicit intercept, the “default” intercept can be excluded from all fitted discrete-time hazard models, allowing comparison of model fit using the −2LL statistic across these nested models (Singer & Willett, 2003).

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Successive polynomial representations of time begin with a linear rep-resentation of time:

logit ( 0h t ONE PERIODj) = +α α1 (11.11)

In this model, we make use of the PERIOD time variable from our person-period data set, as we did when examining interactions between substan-tive predictors and time. To model time as a quadratic function, we fit the model:

logit ( 02h t ONE PERIOD PERIODj) = + +α α α1 1 (11.12)

Successive models add higher order polynomial functions of time. Table 11.6 presents −2LL statistics from eight models with different param-eterizations of time fitted to the WLS data. The worst fitting model, as described above, is the constant model from Equation 11.10 (−2LL = 10,783.84), and the best fitting model (by definition) is the completely general model (−2LL = 9996.13). As the representations of time extend from a linear representation of time to higher order polynomial versions of time, goodness of fit improves. As we examine these increasingly com-plex polynomial functions of time, we ask two questions about goodness of fit that are addressed in the last two columns of Table 11.6: (a) Is the

Table 11.6

Comparison of Alternative Smooth Polynomial Representations for the Main Effect of TIME in an Initial Discrete-Time Hazard Model for the WLS Data (n = 7860)

Difference in −2LL Compared to

Representation for TIME n Parameters −2LL

Model in Row Above

General Model

Constant 1 10,783.84 — 787.71 (10)Linear 2 10,320.46 463.38 (1) 324.33 (9)PERIOD2 3 10,314.95 5.51 (1) 318.82 (8)PERIOD3 4 10,137.95 177.00 (1) 141.82 (7)PERIOD4 5 10,063.88 74.07 (1) 67.75 (6)PERIOD5 6 10,011.05 52.83 (1) 14.92 (5)PERIOD6 7 10,008.59 2.46 (1) 12.46 (4)General 11 99,96.13 — —

Note: Differences in −2LL that exceed the critical value from a χ2 distribution at the .05 level with df as noted in parentheses are noted in bold.

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fit of the current model superior to the fit of the prior model (i.e., the one listed in the row immediately above)? (b) Is the fit of the current model as good as that of the completely general model? We answer the first question by comparing the difference in the −2LL statistics of suc-cessive models to the critical value from a χ2 distribution with 1 degree of freedom (3.84). Differences over 3.84 are indicated in bold print. We find that all polynomial specifications of time through the fifth order represent improvements in fit over prior models, but the improvement in fit of the sixth-order polynomial representation of time compared with the fifth is not large enough to warrant the additional parameter in the model. In addressing the second question about whether the fit is as good as the completely general model, we find that the general model is superior to all models listed in Table 11.6; however, at the .01 level we would conclude that the fifth- and sixth-order polynomial specifica-tions of time are nearly as good as the completely general model (the .01 critical values from χ2 distributions with 5 and 4 df are 15.086 and 13.277, respectively).

Based on our analysis of goodness of fit, it may be reasonable to use a fifth-order representation of time, given that it fits nearly as well as less parsimonious representations of time. Before making a final decision, it is useful to examine fitted logit hazard and hazard functions from the com-pletely general model (dashed lines) and fifth-order polynomial represen-tation of time model (solid lines) in Figure 11.5. Parameter estimates from the completely general model can be found in Table 11.5. The equation of the fitted fifth-order polynomial model is

logit ( 2ˆ ) . . . .h t PERIOD PERIOD PERIOj = − − + −3 9046 0 8725 1 1949 0 4052 DD

PERIOD PERIOD

3

4 5+ −0 0484 0019. . (11.13)

The fitted logit hazard function from this model closely resembles the general model at all points in time, although the 40-year peak in fitted logit hazard is slightly higher for the fifth-order polynomial model. This difference is also evident in the plot of the estimated hazard functions (the bottom panel of Figure 11.5). In addition, here we also see a slightly higher estimate of the 12-year peak in hazard. Nevertheless, the fifth-order polynomial model represents an attractive alternative given its more par-simonious specification and parameter estimates that are quite close to those from the completely general model. This model could be expanded to include the effects of our substantive predictors, using the same strate-gies described above.

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Conducting the Analysis With Continuous-Time Survival Analysis

Researchers can often record event occurrence using a fine-grained time metric. For example, researchers who study death can record the time when someone dies to the minute. In the WLS divorce example discussed in this chapter, even though the researchers recorded marriage and divorce months and years, theoretically, it would be possible to record

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Figure 11.5Fitted logit hazard and hazard functions for time to first divorce, using alternative specifi-cations for the main effect of time.

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actual dates of marriage and divorce, allowing us to measure marriage length in days. In the concluding section of this chapter, we consider how our analyses of the WLS data would change if we adopted a continuous-time framework. We begin by describing the ways in which definitions of survivor and hazard functions must change when we assume that time is measured on a continuum. Next, we discuss one way to explore the data, using the Kaplan–Meier method for estimating survival functions in con-tinuous time. Finally, we use Cox regression (Cox, 1972) to investigate the effect of whether an individual has a college degree on risk of first divorce. As we proceed, we compare our results to those obtained using discrete-time survival analysis.

In continuous time, the definition of the survivor function is essentially the same as in discrete time, with the fundamental difference being the respecification of the random variable used to represent time, which is now continuous. Now we let T be a continuous random variable whose values Ti indicate the precise instant when individual i experiences the target event. In continuous time, the survival probability for individual i at time tj is the probability that his or her event time Ti will exceed tj:

S t T tj i j( ) Pr[ ]= > (11.14)

Individual i’s set of survival probabilities over time is still known as his or her survivor function. As expected, the initial value for the continuous-time survivor function at time t0 is 1.

The top panel of Figure 11.6 presents a Kaplan–Meier estimated sur-vivor function for the WLS first divorce data, where time is now mea-sured in years rather than the 4-year intervals used above. We use years here because this is the finest time metric available on the public release data set—researchers who have access to the restricted use data set could use months to clock time. The Kaplan–Meier method used to obtain the estimated survivor function in Figure 11.6 is well suited for describing the distribution of event times recorded in a continuous metric, although we cannot use it to estimate predictors’ effects (Singer & Willett, 2003). Kaplan–Meier survival functions are easily obtained using routines in all major statistical packages. Not surprisingly, the overall shape of the Kaplan–Meier estimated survivor function is very similar to the esti-mated survivor function based on discrete-time data in the bottom panel of Figure 11.1. The function gradually declines over the first 16 years of marriage, indicating a steadily decreasing probability that a Wisconsin high school graduate from the class of 1957 who got married has not yet divorced. After 16 years, the survivor function levels out, and then after 32 years it begins to decline again, reflecting the slight increase in risk of a first divorce during this later period of life, although the rate of decline

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in the survival function in this period is much smaller than in the first period. A comparison of the numerical values of the estimated survival probabilities between the two approaches reveals almost identical esti-mates (and, when time is clocked in years in the discrete-time method described above, the estimated survival probabilities are identical).

While the survivor function is easily estimated and interpreted in con-tinuous-time survival analysis, the hazard function is not. The hazard function assesses the risk—at a particular moment—that an individual who has not yet done so will experience the target event. In discrete time, the moments are time periods, allowing us to express hazard as a condi-tional probability. In continuous time, we must alter our definition of haz-ard because the concept of probability falls apart for a continuous random variable like T—in truly continuous time, the probability that T takes on any specific value tj is 0. We can no longer define hazard as a (conditional) probability because it would be 0 at all values of tj. Instead, hazard is defined as a rate, the conditional probability of event occurrence per unit time, as the width of the units of time (Δt) moves closer and closer to 0:

h t tT t t t T t

tj

i j j i j( )

[ ( , ) ]= →

+ ≥limit as

is in the interval∆

∆∆

0Pr

(11.15)

0.840.860.880.900.920.940.960.981.00

Estim

ated

surv

ival

pro

babi

lity

0.000.020.040.060.080.100.120.140.160.180.20

0 4 8 12 16 20 24 28 32 36 40 44

0 4 8 12Years since beginning of first marriage


16 20 24 28 32 36 40 44

Estim

ated

cum

ulat

ive h

azar

d

Figure 11.6Kaplan–Meier estimates of the survivor and cumulative hazard functions.

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Because the definitions of hazard differ in continuous and discrete time, their interpretations differ as well. Most important, continuous-time hazard is not a probability. Instead, it is a rate assessing the conditional probability of event occurrence per unit of time.

A major limitation of survival analysis of continuous-time data is its inability to estimate the continuous-time hazard function well. From dis-crete time, we know how meaningful the hazard function is and why it is essential to examine its behavior graphically over time. For example, in the WLS analysis presented above, we learned about two time peri-ods (early in marriage and then later) in which the risk of a first divorce gradually increases and then decreases. A function that can help us at least gain some understanding of the underlying hazard function is the cumulative hazard function. Denoted by H(tij), the cumulative hazard func-tion assesses, at each point in time, the total amount of accumulated risk that individual i has faced from the beginning of time until the present. Because H(tij) cumulates hazard, examination of its changing level over time tells us about the shape of the underlying hazard function.

The concept of cumulative hazard is somewhat elusive because cumu-lative hazard is measured in a metric difficult to quantify. We can make general statements—if individual i has been relatively risk-free, cumula-tive hazard will be “low,” and if individual i has faced great and per-sistent risk, cumulative hazard will be “high”; but we cannot say exactly which values are “high” and which are “low.” Cumulative hazard is not a probability nor is it a rate. Because a value of cumulative hazard has little meaning on its own, we rarely tabulate the function, preferring to explore its behavior graphically over time.

The estimated cumulative hazard function for the WLS data is presented in the bottom panel of Figure 11.6. By definition, the cumulative hazard function begins at 0 and increases over time—sometimes rapidly, some-times slowly. There is a slight increase in the cumulative hazard function between years 0 and 4, followed by a more rapid increase between years 4 and 16, representing the relatively high risk of a first divorce during this period (and corresponding with the first peak in the sample hazard function in Figure 11.1). Between years 30 and 32, the cumulative hazard function is essentially flat, representing very little risk of a first divorce. Then, the function begins to rise gradually between years 32 and 40, rep-resenting another increase in first divorce risk (and corresponding with the second “peak” in Figure 11.1).

Are individuals who have college degrees when they first marry at less risk of a first divorce than individuals who do not have college degrees? Although it is possible (and quite helpful) to obtain estimates of survival functions for values of categorical predictors (i.e., DEGREE = 0, 1 in this example) using the Kaplan–Meier approach, this approach can neither esti-mate the effect of the predictor on hazard nor explore the way survivor

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functions may vary by levels of continuous predictors (such as AGE). Thus, we need to consider how to fit survival analysis models with both cate-gorical and continuous predictors to our continuous-time data, as we did above with discrete-time data. The many different types of continuous-time survival analysis models can be roughly categorized by whether they are parametric or nonparametric (or semiparametric) models. Parametric models make assumptions about the underlying distribution of event times (e.g., Weibull, Gompertz, gamma, and log-logistic), and in some fields parametric estimation is the dominant mode of analysis. In many other fields, including most of the social, behavioral, and medical sciences, nonparametric methods, like the Cox regression model described below, are more popular. The fundamental advantage of nonparametric methods is that we need not make constraining assumptions about the distribution of event times. This flexibility is important because few researchers have a sound basis for preferring one distribution over another, and adopting an incorrect assumption can lead to erroneous conclusions.

To model the effect of DEGREE on the risk of a first divorce using Cox regression, we begin by fitting the model:

log ) log ( )H t H t DEGREEij j i( = +0 1β (11.16)

or, alternatively (and equivalently):

log ) log ( )h t h t DEGREEij j i( = +0 1β (11.17)

Here, log H0(tij) and log h0(tij) represent completely general baseline log cumulative hazard and hazard functions, which can take on any shape necessary to describe the distribution of event occurrence in the popula-tion adequately. The baseline function represents the value of log cumu-lative hazard or log hazard when all predictors in the model (here, just DEGREE) are set to 0. The population parameter β1 represents the hypoth-esized influence of the predictor DEGREE on the log cumulative hazard function or log hazard. As with the discrete-time hazard model described above, the Cox regression model can easily be expanded to include addi-tional predictors, both continuous and categorical.

An important feature of the Cox regression model is that the shape of the baseline hazard function is irrelevant. When we fit a Cox model to data, we evaluate only the predictors’ effects—We do not investigate the shape of the baseline hazard function at all because Cox’s method of par-tial likelihood used to fit the model invokes a conditioning argument that eliminates it entirely from consideration (for an explanation of the partial likelihood method, see, e.g., Cox, 1975; Singer & Willett, 2003). The irrel-evance of the baseline hazard function has one unfortunate consequence: A fitted model does not provide predicted values of hazard.

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The Cox regression model has a serious problem—its sensitivity to ties. Because ties are inevitable in real data, statisticians have developed sev-eral modifications to Cox’s basic approach. Of the various approaches for dealing with ties, the most compelling is the exact method (Kalbfleisch & Prentice, 1980; Peto, 1972). Most statistical packages implement approxima-tions to the exact solution. Simulation studies (Hertz-Picciotto & Rockhill, 1997) suggest that the Efron approximation is the best, so it is used here.

Model A in Table 11.7 presents the results from fitting the Cox regres-sion model to the WLS data. The estimated DEGREE effect of −0.5068 (p < .001) tells us that the estimated log hazard function for an individ-ual with a college degree is 0.5068 below the function for an individual with no college degree. In other words, as we found in the discrete-time analysis presented above, individuals with college degrees are less likely to experience a first divorce than individuals without college degrees. To make this parameter estimate more understandable, we can antilog it to

Table 11.7

Parameter Estimates, Asymptotic Standard Errors, and Goodness-of-Fit Statistics from Proportional and Nonproportional Hazards Cox Regression Models Fitted to the WLS Data (n = 7860)

Predictor Model A Model B Model C Model D Model E Model F

DEGREE −0.5068**** −0.5052 −0.6205**** −0.3171 0.94437* −1.7240*(0.1048) (0.3469) (0.1751) (0.3035) (0.4000) (0.7821)

DEGREE** −0.0007(ln)YEAR (0.1452)DEGREE × YEAR

0.0089 −0.0384 −0.4086*** 0.7074**

(0.0107) (0.0411) (0.0965) (0.2737)DEGREE × YEAR2

0.0012 0.0266*** −0.1010***

(0.0010) (0.0063) (0.0295)DEGREE × YEAR3

−0.00045*** 0.0047***

(0.00011) (0.0012)DEGREE × YEAR4

−0.000067***

(0.000016)Goodness of fit

−2LL (n of parameters)

18,288.43 (1)

18,288.43 (2)

18,287.76 (2)

18,286.36 (3)

18,268.72 (4)

18,244.08 (5)

AIC 18,290.43 18,292.43 18,291.76 18,292.36 18,276.72 18,254.08

Note: Efron’s method for handling ties.* p < .05. ** p < .01. *** p < .001.

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obtain a hazard ratio—the effect of DEGREE on raw hazard. Here, the estimated hazard ratio is 0.602—The estimated hazard of divorce for an individual with a degree is 0.602 that of the estimated hazard for an indi-vidual without a degree. As we did above when interpreting estimated odds ratios, we can take the reciprocal of the hazard ratio, changing the reference group for the comparative statement—that is, the estimated haz-ard of divorce for the no degree individual is 1.66 (1/.6) times the esti-mated hazard for the individual with a degree.

Although we can interpret the effect of predictors in Cox regression using similar strategies to discrete time, there is a fundamental difference because we do not estimate the baseline hazard function and can there-fore make only comparative, not absolute, statements about hazard. We can say that the hazard of a first divorce is higher for individuals without college degrees compared with individuals with degrees, but we cannot say how high, or low, either function is.

Although we cannot directly estimate hazard functions using Cox regression, we can obtain nonparametric estimates of the survivor and cumulative hazard functions. These functions are presented in the top two panels of Figure 11.7, and show the same general pattern of risk as described above. For both groups, the estimated survival probability declines mod-erately until year 16, where it levels off, reflecting little chance of divorce after that, until later in marriage, when the estimated survival function declines again, this time less steeply. The elevated risk of a first divorce for the degree group is seen in the relative position of the estimated functions in the two graphs—The estimated survivor function for the degree group is above that of the no degree group, and the estimated cumulative hazard function for the no degree group is above the degree group.

As with the discrete-time method described above, proportionality is an important assumption of the Cox regression model, which is some-times referred to as the Cox proportional hazards model. This assumption is illustrated in the bottom panel of Figure 11.7, which presents estimated log cumulative hazard functions for the degree and no degree groups. The proportionality assumption has constrained these two functions to be equidistant from each other, but our discrete-time analysis presented above suggests that the effect of DEGREE is not constant over time, so the separation between these functions should be allowed to vary over time.

There are many approaches to evaluating the proportionality assump-tion in Cox regression (see, e.g., Hosmer & Lemeshow, 2002; Singer & Willett, 2003). One useful strategy for categorical predictors is to graph the difference in the sample log cumulative hazard probabilities over time, as is shown in Figure 11.8. The top panel presents sample log cumulative hazard functions for the degree and no degree groups. Visual inspection of the plot reveals that these functions are, in fact, not parallel—there is an effect of DEGREE, but it is not constant over time. The bottom panel

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0.840.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0 4 8 12 16 20 24 28 32 36 40 44Years since beginning of rst marriage(a)

(b)

(c)

Estim

ated

surv

ival

pro

babi

lity

DEGREE= 1

DEGREE= 0

0.000.010.020.030.040.050.060.070.080.090.10

Estim

ated

cum

ulat

ive h

azar

d

DEGREE= 0

DEGREE= 1

–4.00

–3.50

–3.00

–2.50

–2.00

–1.50

–1.00

–0.50

0.00

0 5 10 15 20 25 30 35 40 45 50Years since beginning of rst marriage

Estim

ated

log

cum

ulat

ive h

azar

d

0 4 8 12 16 20 24 28 32 36 40 44Years since beginning of rst marriage

DEGREE= 0

DEGREE= 1

Figure 11.7Estimated survivor functions (a), cumulative hazard functions (b), and log cumulative hazard functions (c), showing the effect of DEGREE on the risk of a first divorce (from Model A of Table 11.7).

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of Figure 11.8 shows that the difference in sample log cumulative hazard varies over time. The pattern of the differences over time is not easily dis-cerned, but it is clear that it is complex. As with the discrete-time approach described earlier in this chapter, one way to address nonproportionality is to include interactions with time in the Cox regression model. Interactions with time may be relatively simple interactions with linear time or the log of time (Models C and B), or more complex interactions with polynomial versions of time (Models D, E, and F). In evaluating these models, compari-son of the −2LL goodness-of-fit statistics shows that Models B, C, and D do not fit significantly better than Model A (main effect model). It is not until we investigate interactions with cubic and quartic time that we find large improvements in model fit (comparison of Models E and A: χ2 = 19.71, df = 3, p < .001; comparison of Models F and E: χ2 = 24.64, df = 1, p < .001).

–8.00

–7.00

–6.00

–5.00

–4.00

–3.00

–2.00

–1.00

0.00

0 4 8 12 16 20 24 28 32 36 40 44Years since beginning of first marriage(a)

(b)

Estim

ated

log

cum

ulat

ive h

azar

d

–0.20

–0.10

0.00

0.100.20

0.30

0.40

0.500.60

0.70


Diff

eren

ce

Figure 11.8Sample log cumulative hazard functions (a) and the difference in log cumulative hazard functions (b), by degree status.

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Does the parameterization in Model F fully characterize the interaction with time? In discrete time, we could answer this question by compar-ing its fit with that of a fully interactive model (in which the predictor’s effect was allowed to differ in each discrete-time period). In continuous time, no such fully interactive model exists. So, although we can conclude that Model F, where DEGREE interacts with quartic time, is preferable to the other options in Table 11.7, there are an infinite number of other possibilities (both smooth and piecewise). One way to determine whether our interaction with time model is sufficient is to obtain residuals from the fitted model; graphs of residuals may be helpful indicators of model misspecification. A full discussion of this approach is beyond the scope of this chapter. Interested readers are referred to discussions in Singer and Willett (2003) and Hosmer and Lemeshow (2002).

How can we interpret the interaction with time in Model 6? In the con-text of discrete-time survival analysis, we suggested that graphs of fitted hazard functions are the best way to understand a complex interaction like the one we have here. However, there are no easily available approaches to construct plots of hazard functions (or even survivor functions) derived from nonproportional Cox regression models. Therefore, the best way to understand the interaction is to return to the sample functions presented in Figure 11.8. If interactions with time are of primary interest in your work, we suggest that understanding their nature is best accomplished by using the discrete-time survival methods described earlier in this chapter.

Concluding Comments

Which type of survival analysis should you choose? We offer the following practical suggestions to consider in determining whether to use discrete-time or continuous-time methods. The primary consideration, of course, should be how finely time has been measured. As we discuss earlier in this chapter, when time is measured in very narrow, precise units, continuous-time methods work well. When time is measured in wider intervals, as is common in much social science research, then continuous-time meth-ods may not work well because of the problem of ties. Nevertheless, many researchers who measure time in wide intervals use continuous-time methods, most frequently the Cox regression model described in this chap-ter. And, as we demonstrated here, even though years are not particularly narrow time intervals and ties are pervasive in the WLS data, our general results were the same using both discrete-time survival analysis and Cox regression. However, the two methods differ dramatically in the kinds of interpretation that we can glean from the results. In particular, although

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estimates of the hazard function are easily obtained using discrete-time survival analysis, they are not readily obtained using Cox regression. Therefore, discrete-time survival analysis is a preferable method for researchers who want a clear understanding of the underlying hazard function so that they can estimate the actual risk of event occurrence dur-ing particular periods of time for different groups of people and go beyond making the comparative statements about risk provided by estimated haz-ard ratios from Cox regression. As a final consideration about the type of model to choose, it is important to consider the possibility that substan-tive predictors may interact with time. Given the challenges in interpret-ing Cox regression models that contain interactions between substantive predictors and time, we recommend using discrete-time survival analysis as a way to more effectively understand such interactions.

We conclude by briefly mentioning some of the many ways that both discrete-time and Cox regression models can be extended. First, both models can incorporate time-varying predictors, that is, predictors whose values vary over time. For example, in the WLS analysis, we could have modeled degree status as a time-varying predictor, which would take on the value of 0 during all periods when a WLS individual did not have a college degree, and 1 for all periods after receipt of college degree. As with more traditional methods of analysis, such as OLS regression, the models discussed in this chapter can also be extended to include nonlinear effects of substantive predictors and interactions between substantive predictors. Finally, competing risks survival analysis (Singer & Willett, 2003) extends the model to situations where individuals can occupy three or more states (i.e., marriage ending through divorce, death, or death of a spouse), and multiple-spell (or multiple event) methods can be used when research-ers want to study how individuals progress in and out of multiple states (Willett & Singer, 1995).

Comments on Statistical Software

Discrete-time survival analysis: Logistic regression is used to fit discrete-time survival analysis models (e.g., PROC LOGISTIC in SAS, LOGISTIC REGRESSION in SPSS, LOGIT in STATA).Continuous-time survival analysis: All of the major statistical packages can conduct many different types of continuous-time survival analysis. For example, the Kaplan–Meier estimates of the survivor function are obtained using PROC LIFETEST in SAS (using the method = km option), KM in SPSS, and STS in STATA. Cox regression models are fitted using PROC PHREG in SAS, COXREG in SPSS, and STCOX in STATA.

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Recommended Readings

Allison, P. D. (2010). Survival analysis using SAS: A practical guide (2nd ed.). Cary, NC: SAS Institute Inc.

Hosmer, D. W., & Lemeshow, S. (2002). Applied survival analysis: Regression modeling of time to event data. New York: John Wiley & Sons, Inc.

Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence. New York: Oxford University Press.

Example Articles

Corning, A. F., & Malofeeva, E. V. (2004). The application of survival analysis to the study of psychotherapy termination. Journal of Counseling Psychology, 51, 354–367.

Donaldson, M. L., & Johnson, S. M. (2010). The price of misassignment: The role of teaching assignments in Teach for America teachers’ exit from low-income schools and the teaching profession. Educational Evaluation and Policy Analysis, 32, 299–323.

Foster, E. M. (2000). Does the continuum of care reduce inpatient length of stay? Evaluation and Program Planning, 23, 53–65.

Herzog W., Schellberg, D., & Deter, H. C. (1997). First recovery in anorexia nervosa patients in the long-term course: A discrete-time survival analysis. Journal of Consulting and Clinical Psychology, 65, 169–177.

Singer, J. D., Davidson, S. M., Graham, S. E., & Davidson, H. S. (1998). Physician retention in community and migrant health centers: Who stays and for how long? Medical Care, 36, 1198–1213.

References

Cox, D. R. (1972). Regression models and life tables (with discussion). Journal of Royal Statistical Society, B34, 187–220.

Cox, D. R. (1975). Partial likelihood. Biometrika, 62, 269–276.Cox, D. R., & Oakes, D. (1984). Analysis of survival data. London: Chapman & Hall.Efron, B. (1988). Logistic regression, survival analysis, and the Kaplan-Meier

curve. Journal of the American Statistical Association, 83, 414–425.Fahrmeir, L., & Wagenpfeil, S. (1996). Smoothing hazard functions and time-varying

effects in discrete duration and competing risks models. Journal of the American Statistical Association, 91, 1584–1594.

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Hertz-Picciotto, I., & Rockhill, B. (1997). Validity and efficiency of approximation methods for tied survival times in Cox regression. Biometrics, 53, 1151–1156.

Hosmer, D. W., & Lemeshow, S. (2002). Applied survival analysis: Regression modeling of time to event data. New York: John Wiley & Sons, Inc.

Kalbfleisch, J. D., & Prentice, R. L. (1980). The statistical analysis of failure time data. New York: Wiley.

Peto, R. (1972). Discussion of Professor Cox’s paper. Journal of the Royal Statistical Society, Series B, 34, 205–207.

Singer, J. D., Davidson, S. M., Graham, S. E., & Davidson, H. S. (1998). Physician retention in community and migrant health centers: Who stays and for how long? Medical Care, 36, 1198–1213.

Singer, J. D., & Willett, J. B. (1993). It’s about time: Using discrete-time survival analysis to study duration and the timing of events. Journal of Educational Statistics, 18, 155–195.

Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence. New York: Oxford University Press.

Willett, J. B., & Singer, J. D. (1995). It’s déjà-vu all over again: Using multiple-spell discrete-time survival analysis. Journal of Educational and Behavioral Statistics, 20, 41–67.

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