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 The LIFETEST Procedure

# Overview

A common feature of lifetime or survival data is the presence of right-censored observations due either to withdrawal of experimental units or to termination of the experiment. For such observations, you know only that the lifetime exceeded a given value; the exact lifetime remains unknown. Such data cannot be analyzed by ignoring the censored observations because, among other considerations, the longer-lived units are generally more likely to be censored. The analysis methodology must correctly use the censored observations as well as the noncensored observations. Several texts that discuss the survival analysis methodology are Collett (1994), Cox and Oakes (1984), Kalbfleish and Prentice (1980), Lawless (1982), and Lee (1992). Usually, a first step in the analysis of survival data is the estimation of the distribution of the survival times. Survival times are often called failure times, and event times are uncensored survival times. The survival distribution function (SDF), also known as the survivor function, is used to describe the lifetimes of the population of interest. The SDF evaluated at t is the probability that an experimental unit from the population will have a lifetime exceeding t, that is

S(t) = Pr(T > t)
where S(t) denotes the survivor function and T is the lifetime of a randomly selected experimental unit. The LIFETEST procedure can be used to compute nonparametric estimates of the survivor function either by the product-limit method (also called the Kaplan-Meier method) or by the life table method. Some functions closely related to the SDF are the cumulative distribution function (CDF), the probability density function (PDF), and the hazard function. The CDF, denoted F(t), is defined as 1-S(t) and is the probability that a lifetime does not exceed t. The PDF, denoted f(t), is defined as the derivative of F(t), and the hazard function, denoted h(t), is defined as f(t)/S(t). If the life table method is chosen, the estimates of the probability density function and the hazard function can also be computed. Plots of these estimates can be produced by a graphical or line printer device.

An important task in the analysis of survival data is the comparison of survival curves. It is of interest to determine whether two or more samples have arisen from identical survivor functions. PROC LIFETEST provides two rank tests and a likelihood ratio test for testing the homogeneity of survival functions across strata. The rank tests are censored-data generalizations of the Savage (exponential scores) test and the Wilcoxon test. The generalized Savage test is also known as the log-rank test, while the generalized Wilcoxon test is simply referred to as the Wilcoxon test. The likelihood ratio test is based on an underlying exponential model, whereas the rank tests are not.

Often there are prognostic variables called covariates that are thought to be related to the failure time. These variables can be used to define strata, and the resulting SDF estimates can be compared visually or by using the tests of homogeneity of strata. The variables can also be used to construct statistics to test for association between the covariates and the lifetime variable. PROC LIFETEST can compute two such test statistics: censored data linear rank statistics based on the exponential scores and the Wilcoxon scores. The corresponding tests are known as the log-rank test and the Wilcoxon test, respectively. These tests are computed by pooling over any defined strata, thus adjusting for the stratum variables. Except for a difference in the treatment of ties, these two rank tests are the same as those used to test for homogeneity over strata.

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