John B. Willett

Applied Longitudinal Data Analysis

PART I 1. A framework for investigating change over time 2. Exploring Longitudinal Data on Change 3. Introducing the multilevel model for change 4. Doing data analysis with the multilevel mode for change 5. Treating TIME more flexibly 6. Modelling discontinuous and nonlinear change 7. Examining the multilevel models error covariance structure 8. Modelling change using covariance structure analysis PART II 9. A Framework for Investigating Event Occurrence 10. Describing discrete-time event occurrence data 11. Fitting basic Discrete-Time Hazard Models 12. Extending the Discrete-Time Hazard Model 13. Describing Continuous-Time Event Occurrence Data 14. Fitting Cox Regression Models 15. Extending the Cox Regression Model

Using Covariance Structure Analysis to Detect Correlates and Predictors of Individual Change Over Time

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 persons 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 childrens 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

It's About Time: Using Discrete-Time Survival Analysis to Study Duration and the Timing of Events

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.

Understanding correlates of change by modeling individual differences in growth

The study of correlates of change is the investigation of systematic individual differences in growth. Our representation of systematic individual differences in growth is built up in two parts: (a) a model for individual growth and, (b) a model for the dependence of parameters in the individual growth models on individual characteristics. First, explicit representations of correlates of change are constructed for various models of individual growth. Second, for the special case of initial status as a correlate of change, properties of collections of growth curves provide new results on the relation between change and initial status. Third, the shortcomings of previous approaches to the assessment of correlates of change are demonstrated. In particular, correlations of residual change measures with exogenous individual characteristics are shown to be poor indicators of systematic individual differences in growth.

Modeling the days of our lives: Using survival analysis when designing and analyzing longitudinal studies of duration and the timing of events.

Psychologists studying whether and when events occur face unique design and analytic difficulties. The fundamental problem is how to handle censored observations, the people for whom the target event does not occur before data collection ends. The methods of survival analysis overcome these difficulties and allow researchers to describe patterns of occurrence, compare these patterns among groups, and build statistical models of the risk of occurrence over time. This article presents a unified description of survival analysis that focuses on 2 topics: study design and data analysis

The design and analysis of longitudinal studies of development and psychopathology in context: Statistical models and methodological recommendations

The utility and flexibility of recent advances in statistical methods for the quantitative analysis of developmental data--in particular, the methods of individual growth modeling and survival analysis--are unquestioned by methodologists, but have yet to have a major impact on empirical research within the field of developmental psychopathology and elsewhere. In this paper, we show how these new methods provide developmental psychopathologists with powerful ways of answering their research questions about systematic changes over time in individual behavior and about the occurrence and timing of life events. In the first section, we present a descriptive overview of each method by illustrating the types of research questions that each method can address, introducing the statistical models, and commenting on methods of model fitting, estimation, and interpretation. In the following three sections, we offer six concrete recommendations for developmental psychopathologists hoping to use these methods. First, we recommend that when designing studies, investigators should increase the number of waves of data they collect and consider the use of accelerated longitudinal designs. Second, we recommend that when selecting measurement strategies, investigators should strive to collect equatable data prospectively on all time-varying measures and should never standardize their measures before analysis. Third, we recommend that when specifying statistical models, researchers should consider a variety of alternative specifications for the time predictor and should test for interactions among predictors, particularly interactions between substantive predictors and time. Our goal throughout is to show that these methods are essential tools for answering questions about life-span developmental processes in both normal and atypical populations and that their proper use will help developmental psychopathologists and others illuminate how important contextual variables contribute to various pathways of development.

Investigating onset, cessation, relapse, and recovery: why you should, and how you can, use discrete-time survival analysis to examine event occurrence

In this article, we show how discrete-time survival analysis can address questions about onset, cessation, relapse, and recovery. Using data on the onset of suicide ideation and depression and relapse into cocaine use, we introduce key concepts underpinning the method, describe the action of the discrete-time hazard model, and discuss several types of main effects and interactions that can be included as predictors. We also comment on practical issues of data analysis and strategies for interpretation and presentation.

Some Results on Reliability for the Longitudinal Measurement of Change: Implications for the Design of Studies of Individual Growth

When changes in educational or psychological status are being measured, every subject in the sample must be observed on several chronologically successive occasions. In the pursuit of such longitudinal data, traditional researchers have been content to administer only a pre-test and a post-test (thus collecting two waves of data on each subject). More recently, however, methodologists have argued that multiwave data (i.e., more than two waves) must be collected for the effective measurement of change. Multi-wave data allows a suitable mathematical model to be fitted to each of the individual growth records as a way of summarizing the growth of each subject. Subsequent investigations of between-individual differences in growth can then be based on the results of these fits. In this article, individual growth-modeling permits the reliability of change measurement to be examined. This reliability is shown to depend upon three factors: the magnitude of the inter-individual heterogeneity in true growth, the size of the measurement-error variance, and the number of waves of data that have been collected. The paper demonstrates that dramatic increases in the reliability of change measurement can be achieved by collecting relatively few additional waves of data, a finding that has considerable import for the informed design of longitudinal studies of individual change.

Estimating the Labor Market Signaling Value of the GED

This paper tests the labor market signaling hypothesis for the General Educational Development (GED) equivalency credential. Using a unique data set containing GED test scores and Social Security Administration (SSA) earnings data, we exploit variation in GED status generated by differential state GED passing standards to identify the signaling value of the GED, net of human capital effects. Our results indicate that the GED signal increases the earnings of young white dropouts by 10 to 19 percent. We find no statistically significant effects for minority dropouts.

Family environment and glycemic control: a four-year prospective study of children and adolescents with insulin-dependent diabetes mellitus.

&NA; An onset cohort of children and adolescents with insulin‐dependent diabetes mellitus (IDDM) and their parents were studied. Aspects of family environment were evaluated at study inception, and their influence on the initial level of, and change in, glycemic control over 4 years was examined. Family measures of expressiveness, cohesiveness, and conflict were linked to differences in the longitudinal pattern of glycemic control. In particular, the encouragement to act openly and express feelings directly (expressiveness) seemed to ameliorate deterioration of glycemic control over time in both boys and girls. Boys were especially sensitive to variations in family cohesiveness and conflict; those from more cohesive and less conflicted families showed less deterioration in glycemic control. This study demonstrated the important influence of family psychosocial factors present at diabetes onset on glycemic control in children and adolescents over the first 4 years of IDDM.