This chapter introduces the essential features of event occurrence data, explaining how and why they create the need for new analytic methods. Section 9.1 provides a context by describing three studies of event occurrence, each conducted in a different discipline. Section 9.2 identifies the three major methodological features of each study: a well-defined “event” whose occurrence is being explored; a clearly identified “beginning of time”; and a substantively meaningful metric for clocking time. Section 9.3 concludes by highlighting the hallmark feature of event occurrence data that makes the new statistical methods we soon discuss necessary—the problem known as censoring.

Change is constant in everyday life. Infants crawl and then walk, children learn to read and write, teenagers mature in myriad ways, and the elderly become frail and forgetful. Beyond these natural processes and events, external forces and interventions instigate and disrupt change: test scores may rise after a coaching course, drug abusers may remain abstinent after residential treatment. By charting changes over time and investigating whether and when events occur, researchers reveal the temporal rhythms of our lives. This book is concerned with behavioral, social, and biomedical sciences. It offers a presentation of two of today's most popular statistical methods: multilevel models for individual change and hazard/survival models for event occurrence (in both discrete- and continuous-time). Using data sets from published studies, the book takes you step by step through complete analyses, from simple exploratory displays that reveal underlying patterns through sophisticated specifications of complex statistical models.

Recently, methodologists have shown how two disparate conceptual arenas-individual growth modeling and covariance structure analysis-can be integrated. The integration brings the flexibility of covariance analysis to bear on the investigation of systematic interindividual differences in change and provides another powerful data-analytic tool for answering questions about the relationship between individual true change and potential predictors of that change. The individual growth modeling framework uses a pair of hierarchical statistical models to represent (a) within-person true status as a function of time and (b) between-person differences in true change as a function of predictors. This article explains how these models can be reformatted to correspond, respectively, to the measurement and structural components of the general LISREL model with mean structures and illustrates, by means of worked example, how the new method can be applied to a sample of longitudinal panel data. Questions about correlates and predictors of individual change over time are concerned with the detection of systematic interindividual differences in change, that is, whether individual change in a continuous outcome is related to selected characteristics of a person's background, environment, treatment, or training. Examples include the following: Do the rates at which students learn differ by attributes of the academic programs in which they are enrolled? Are longitudinal changes in children's psychosocial adjustment related to health status, gender, and home background? Questions like these can be answered only when continuous data are available longitudinally on many individuals, that is, when both time points and individuals have been sampled representatively. Traditionally, researchers have sampled individual status at only two points in time, a strategy that has proven largely inadequate because two waves of data contain only min

Educational researchers frequently ask whether and, if so, when events occur. Until relatively recently, however, sound statistical methods for answering such questions have not been readily available. In this article, by empirical example and mathematical argument, we demonstrate how the methods of discrete-time survival analysis provide educational statisticians with an ideal framework for studying event occurrence. Using longitudinal data on the career paths of 3,941 special educators as a springboard, we derive maximum likelihood estimators for the parameters of a discrete-time hazard model, and we show how the model can befit using standard logistic regression software. We then distinguish among the several types of main effects and interactions that can be included as predictors in the model, offering data analytic advice for the practitioner. To aid educational statisticians interested in conducting discrete-time survival analysis, we provide illustrative computer code ( SAS, 1989 ) for fitting discrete-time hazard models and for recapturing fitted hazard and survival functions.