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

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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.
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... A discrete time logit model is an extension of the logit model that addresses the change across time in the underlying probability of an outcome being true (Allison, 1982). It does this by treating time as discrete units or intervals rather than a continuum, with each interval reflecting a different probability of event occurrence derived from the individuals present during that interval (Allison, 1982;Singer & Willett, 1993). 16 Event histories such as the NLSY97 are ideal data sources for discrete time analysis (Singer & Willett, 1993) because the regular recurrence of follow-up interviews not only defines the length of the interval but also allows measurements to be repeated on each person. ...
... It does this by treating time as discrete units or intervals rather than a continuum, with each interval reflecting a different probability of event occurrence derived from the individuals present during that interval (Allison, 1982;Singer & Willett, 1993). 16 Event histories such as the NLSY97 are ideal data sources for discrete time analysis (Singer & Willett, 1993) because the regular recurrence of follow-up interviews not only defines the length of the interval but also allows measurements to be repeated on each person. Each set of repeated measurements enables the hazard for event occurrence during that interval to be updated. ...
... Each set of repeated measurements enables the hazard for event occurrence during that interval to be updated. When the event of interest is nonrepeatable (such as finishing a bachelor's degree), individuals experiencing an event become right censored, which means the hazard for each subsequent interval is further affected by the decreasing number of surviving individuals (Allison, 1982;Singer & Willett, 1993). This positions the discrete time model alongside survival analysis methods like Cox regression, but with specific advantages for handling time-varying regressors and right-censoring (Bahr, 2009;Roksa & Velez, 2012;Wao, 2010) and the discrete, repeated nature of longitudinal data (Richardson, 2010). ...
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... Duration or survival models have been applied to the student dropout problem to analyse survival time. In this study, survival time refers the period between the moment a student enters the Higher Education Institution (HEI), and the moment he/she drops out (Singer & Willett, 1993). ...
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... The models were fitted separately for each age-group to account for changes in baseline hazards by age and to more accurately characterize discrepancies in mortality caused by health status in each age group. The discrete time complementary log-log model, [34][35][36] a discrete analog of the Cox proportional hazards model, was employed for the proportional hazards models. All statistical analyses were conducted using SAS 9.3 (SAS Institute, Cary NC, USA) and SAS-callable SUDAAN [37] to account for the complex NHIS survey design and weights using SAS PROC SURVEYLOGISTIC. ...
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... Discrete-time logit regression can be applied when time is measured at a discrete (not continuous) time scale; thus, it accommodates multiple persons having the same apparent time the event occurs. 10,17,18 All participants with baseline PQRS data were included. For analyses of the time course of the secondary cognitive outcome measures, we used linear mixed models (LMMs) with a spatial power covariance structure of repeated observations within participant, and personspecific random intercepts. ...
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