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Introduction to Survival Analysis Procedures

Background

Survival data consist of a response variable that measures the duration of time until a specified event occurs (event time, failure time, or survival time) and possibly a set of independent variables thought to be associated with the failure time variable. These independent variables (concomitant variables, covariates, or prognostic factors) can be either discrete, such as sex or race, or continuous, such as age or temperature. The system that gives rise to the event of interest can be biological, as for most medical data, or physical, as for engineering data. The purpose of survival analysis is to model the underlying distribution of the failure time variable and to assess the dependence of the failure time variable on the independent variables.

An intrinsic characteristic of survival data is the possibility for censoring of observations, that is, the actual time until the event is not observed. Such censoring can arise from withdrawal from the experiment or termination of the experiment. Because the response is usually a duration, some of the possible events may not yet have occurred when the period for data collection has terminated. For example, clinical trials are conducted over a finite period of time with staggered entry of patients. That is, patients enter a clinical trial over time and thus the length of follow-up varies by individuals; consequently, the time to the event may not be ascertained on all patients in the study. Additionally, some of the responses may be lost to follow-up (for example, a participant may move or refuse to continue to participate) before termination of data collection. In either case, only a lower bound on the failure time of the censored observations is known. These observations are said to be right censored. Thus, an additional variable is incorporated into the analysis indicating which responses are observed event times and which are censored times. More generally, the failure time may only be known to be smaller than a given value (left censored) or known to be within a given interval (interval censored). There are numerous possible censoring schemes that arise in survival analyses. The monograph by Maddala (1983) discusses several related types of censoring situations, and the text by Kalbfleisch and Prentice (1980) also discusses several censoring schemes. Data with censored observations cannot be analyzed by ignoring the censored observations because, among other considerations, the longer-lived individuals are generally more likely to be censored. The method of analysis must take the censoring into account and correctly use the censored observations as well as the uncensored observations.

Another characteristic of survival data is that the response cannot be negative. This suggests that a transformation of the survival time such as a log transformation may be necessary or that specialized methods may be more appropriate than those that assume a normal distribution for the error term. It is especially important to check any underlying assumptions as a part of the analysis because some of the models used are very sensitive to these assumptions.

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