Survival Analysis

Survival time is defined as the time to the occurrence of a specific event, which may be the development of a disease, response to a treatment, relapse, or death. Survival analysis has been extended to fields beyond biomedical studies to include electrical engineering, sociology, and marketing. An example of survival time in sociology might be the duration of first marriages.

A common complication of survival data are censored observations. A censored observation is one where the given event of interest wasn’t recorded, either because the subject was lost to the study, or because the study ended before the event occurred. The emphasis of the procedures in the chapter are those that can handle censored observations.

The distribution of survival times are described using the survivorship function (or survivor function) and the hazard function. The survivorship function S(t) is defined as the probability that an individual survives longer than t. The hazard function h(t) gives the conditional failure rate. It’s the probability of failure during a small time interval assuming that the individual has survived to the beginning of the interval.

The Kaplan-Meier procedure computes the Kaplan-Meier product limit estimates of the survival functions. This method of estimating the survival functions can handle censored data and doesn’t require any assumptions about the form of the survival function. It’s appropriate for small and large data sets. The survivorship and hazard functions can be plotted.

The Two-Sample Survival Tests procedure computes five nonparametric tests for comparing two survival distributions: Gehan-Wilcoxon Test, Cox-Mantel Test, Logrank Test, Peto-Wilcoxon Test, and Cox's F Test. These tests are based on the ranks of the survival times and work for censored or uncensored observations.

The Multi-Sample Survival Tests procedure computes three nonparametric tests for comparing three or more survival distributions: Gehan-Wilcoxon Test, Logrank Test, and the Peto-Wilcoxon Test. These tests can be used to compare survival times for censored data.

The Mantel-Haenzel Test is used to compare survival experience between two groups when adjustments for other prognostic factors are needed. It’s often used in clinical and epidemiologic studies as a method of controlling the effects of confounding variables.

The Proportional Hazards Regression procedure computes Cox’s proportional hazards regression for survival data. It can be used to establish the statistical relationship between survival time and independent variables or covariates measured on the subjects. The reports include a regression coefficient table, likelihood test for the overall model, and a variance-covariance matrix of the coefficients. The regression coefficients can be used to compute relative risk, a measure of the effect of a factor on a subject’s survival time.