# Survival & Failure Time Analysis Introductory Overview - Kaplan-Meier Product-Limit Estimator

Rather than classifying the observed survival times into a life table, we can estimate the survival function directly from the continuous survival or failure times. Intuitively, imagine that we create a life table so that each time interval contains exactly one case. Multiplying out the survival probabilities across the "intervals" (i.e., for each single observation) we would get for the survival function:

S(t) = Õjt= 1 [(n-j)/(n-j+1)] d( j )

In this equation, S(t) is the estimated survival function, n is the total number of cases, and Õ denotes the multiplication (geometric sum) across all cases less than or equal to t; d(j) is a constant that is either 1 if the j'th case is uncensored (complete), and 0 if it is censored. This estimate of the survival function is also called the product-limit estimator, and was first proposed by Kaplan and Meier (1958). The advantage of the Kaplan-Meier Product-Limit method over the life table method for analyzing survival and failure time data is that the resulting estimates do not depend on the grouping of the data (into a certain number of time intervals). Actually, the Product-Limit method and the life table method are identical if the intervals of the life table contain at most one observation.