Survival Analysis with SAS/STAT Procedures

The typical goal in survival analysis is to characterize the distribution of the survival time for a given population, to compare the survival distributions among different groups, or to study the relationship between the survival time and some concomitant variables.

A first step in the analysis of a set of survival data is to use PROC LIFETEST to compute and plot the estimate of the distribution of the survival time. In many applications, there are often several survival curves to compare. For example, you want to compare the survival experiences of patients who receive different treatments for their disease. The association between covariates and the survival time variable can be investigated by computing estimates of the survival distribution function within strata defined by the covariates. In particular, if the proportional hazards model is appropriate, the estimates of the log(-log(SURVIVAL)) plotted against the log(TIME) variable should give approximately parallel lines, where SURVIVAL is the survival distribution estimate and TIME is the failure time variable. Additionally, these lines should be approximately straight if the Weibull model is appropriate.

Statistics that test for association between failure time and covariates can be used to select covariates for further investigation. The LIFETEST procedure computes linear rank statistics using either Wilcoxon or log-rank scores. These statistics and their estimated covariance matrix can be used with the REG procedure with the option METHOD=RSQUARE to find the subset of variables that produce the largest joint test statistic for association. An illustration of this methodology is given in Example 49.1 of Chapter 49, The LIFETEST Procedure.

Another approach to examining the relationship between the concomitant variables and survival time is through a regression model in which the survival time has a distribution that depends on the concomitant variables. The regression coefficients can be interpreted as describing the direction and strength of the relationship of each explanatory variable on the effect of the survival time.

In many biological systems, the Cox model might be a reasonable description of the relationship between the distribution of the survival time and the prognostic factors. You use PROC PHREG to fit the Cox regression model. The regression coefficient is interpreted as the increase of the log-hazard ratio resulting in the increase of one unit in the covariate. However, the underlying hazard function is left unspecified and, as in any other model, the results can be misleading if the proportional hazards assumptions do not hold.

Accelerated failure time models are popular for survival data of physical systems. In many cases, the underlying survival distribution is known empirically. You use PROC LIFEREG to fit these parametric models. Also, PROC LIFEREG can accommodate data with left-censored or interval-censored observations, which are not allowed in PROC PHREG.

A common technique for checking the validity of a regression model is to embed it in a larger model and use the likelihood ratio test to check whether the reduction to the actual model is valid. Other techniques include examining the residuals. Both PROC LIFEREG and PROC PHREG produce predicted values, residuals, and other computed values that can be used to assess the model adequacy.