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Survival Analysis

Educational researchers are often interested in studying longitudinal processes such as college completion, student dropout, or teacher promotion. Survival analysis is a method of statistical modeling that allows researchers to analyze longitudinal data where the outcome is the time to an event of interest (e.g., time to graduation, time to dropout, and time to promotion). Two distinguishing features of survival analysis that separate it from traditional logistic regression analysis are the ability to naturally incorporate time into the model and the ability to handle incomplete data (i.e., censored data). Survival models can accommodate time measured continuously (continuous-time survival analysis) or time measured discretely (discrete-time survival analysis); however, in educational research, time is typically measured discretely (i.e., per semester, per academic year). For that reason, discrete-time survival analysis is the focus here.

Censoring

The most common form of censoring, right censoring, occurs when the outcome of an event of interest is unknown for an individual. That is, the event did not occur during the time frame under consideration or the individual leaves prior to the end of the study. Censoring results in missing data—that is, there is incomplete knowledge about occurrence or nonoccurrence of the event of interest for that individual. Omitting censored individuals from analysis will potentially bias the results. Survival analysis naturally allows for the incorporation of censored data.

Censoring is classified into two types: informative and noninformative. An important assumption of the survival analysis model is noninformative censoring (i.e., censoring that occurs for random and not systematic reasons). For example, treating dropouts as censored when studying college completion is most likely a violation of this assumption. Specifically, students who leave the college are likely to have systematically different characteristics than students who did not graduate during the time frame studied.

Hazard and Survivor Functions

The hazard function consists of conditional probabilities that an individual will experience the event by a particular time period, given that the individual did not experience the event by a previous time period. These probabilities are sometimes called hazard probabilities or hazards. For a given time period, the hazard probabilities are computed by dividing the number of individuals who experienced the event by the number of individuals who were at risk of experiencing the event. Individuals at risk of experiencing the event are those who have not already experienced the event or who have not been censored for that time period.

The survivor function cumulates hazard probabilities across time and contains survival probabilities that represent the probability that an individual has not experienced the event, or survived the event, by a particular point in time. In many educational applications, it is actually the opposite of the survivor function that is of interest. For example, a researcher may be interested in the probability of a teacher being promoted; however, the survivor function gives probabilities of a teacher “surviving” their current status without promotion (i.e., remaining at the teacher’s current rank). Subtracting the survivor probabilities from one will often give more informative probabilities, the probability of “not surviving.”

Graphs of Survivor and Hazard Curves

Survivor and hazard probabilities can be plotted against time. The hazard plot shows the probability of an event for each point in time, whereas the survivor plot shows the probability that an individual has experienced the event by a particular point in time. Examples of hazard and survivor plots are given in Figure 1. The shape of the hazard plot will correspond to the values of the hazard probabilities at each time period; namely, the hazard plot may take on a variety of shapes depending on the hazard probabilities at each time period. For the example in Figure 1, the hazard probabilities increase to a particular time period and then decrease. Conversely, the survivor plot has a more distinctive shape. It is a decreasing function that begins at a survival probability of one because no one has experienced the event and decreases as individuals experience the event over time. The survivor plot gives a quick look at the proportion of individuals who have experienced the event at or before a particular time period.

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