Survival Analysis

Survival analysis, generally speaking, is the modeling of time-to-event data. Traditionally, survival analysis has been used to study the survival of [Page 1477]biological organisms, including human beings and laboratory animals. In this instance, survival analysis seeks to determine the amount of time until the death of the organism. The death is considered the event, and the time until the event is what is modeled. In such instances, survival analysis can determine the proportion of organisms alive at a certain point in time. Furthermore, factors to increase (or shorten) the life span of the organism can be determined. Survival analysis has been applied to many other fields to model data that are not specifically the “life” and “death” of an organism. For instance, survival analysis has been used in engineering and manufacturing to determine the life span of a mechanical process. Death, in this case, is the failure of the mechanism. The social sciences have applied survival analysis to determine the time to events such as marriage, childbirth, and so on. As such, survival analysis is a tool that can be used in a variety of situations, where what is of interest is the time until a particular event can happen.

Regardless of the application, what is central in survival analysis is that the event, be it the death of an organism, the failure of a machine, or the failure of a marriage, is unambiguous. Death is clearly unambiguous, as is marriage. However, the failure of a machine is more ambiguous as failure could be considered partial. Therefore, it is important to define the event in clear terms, and in a way that it is measurable in an absolute way and not open to interpretation. Furthermore, it is assumed that the event can only happen once to a member of the population. Again, in the case of the death of an organism, this is clear. In the case where a machine fails, it can be less clear, depending on the nature of the definition of failure. Many different parts of the machine could fail, and therefore a clear definition of the event should be determined so that the event can only happen once. In the examples of social sciences, this assumption makes it obvious that saying the time to marriage is insufficient; certainly people get married more than once in today's society. However, the event could be defined as first marriage. Doing so also renders the research question more interesting: Given that people marry multiple times, considering the time until marriage in general might be less informative than the time until one gets married for the first time. Precision in the definition of the event can lead to making the second assumption more tenable.

To conduct a survival analysis it is essential to understand the concept of censored observations. Censored observations include observations for which the event has not yet occurred. There are other types of censored observations, but the case where the censorship is caused by not having experienced the event is simplest, and as such will be used for the illustration here. Consider the case where the survival of an organism is of interest. If all the organisms in our sample have died, then looking for a way to predict the survival of an organism is simple: compute the median survival time. This will give us a good indication of the survival time of the organism. (Note that the mean survival time is not likely to be as useful as the median, because survival time data are rarely normally distributed and often are skewed.) In these cases, there is no need for advanced statistical tools. However, if some of the organisms have died while others have not, then a predictive model can be useful. It is important to note, however, that the reason that some organisms have not died (i.e., experienced the event) is because of a limitation of time in the study and that given enough time all organisms would die.

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