Survival Analysis

Abstract

Analytical techniques which look at survival over time are common in the biomedical literature. A survival curve demonstrates the proportion of individuals (or things) still surviving at each point in time. A corollary concept is “hazard function.” A hazard function curve shows the slope of the survival curve at each point in time across the time window (formally the negative of the first derivative of log(S)). “Survival” not only includes patient survival but also survival of a piece of equipment, a building, or anything else that starts with 100 % and over time diminishes to a theoretical value of 0 %. Modern medical science is impatient, so waiting until all entered patients have had the opportunity to survive for, say, 5 years is “too long.” A shortcut is available which allows looking at survival when at least some of the patients have survived that long and then estimating the proportionate survival at each earlier time period. This technique, known as Kaplan-Meier survival analysis utilizes “actuarial survival” rather than actual survival. If the hazard function is not constant, the actual survival might turn out to be different than “predicted” by K-M. Comparing survival in two groups can be accomplished by the log-rank test. It is relatively robust but suffers the same limitation as K-M. Contributing factors to survival can be analyzed with the Cox proportional hazard test, which creates a comparison of two or more exponential functions which have linear exponents. The Cox model has a number of very important assumptions which must be met for conclusions to be valid.