# Survival & Failure Time Analysis Introductory Overview - Comparing Samples

General Introduction. You can compare the survival or failure times in two or more samples. In principle, because survival times are not normally distributed, nonparametric tests that are based on the rank ordering of survival times should be applied. Nonparametrics and Distributions offers a wide range of nonparametric tests that can be used in order to compare survival times; however, the tests in that module cannot "handle" censored observations.

Available Tests. Survival Analysis contains five different (mostly nonparametric) tests for censored data: Gehan's generalized Wilcoxon test, the Cox-Mantel test, the Cox's F test , the log-rank test, and Peto and Peto's generalized Wilcoxon test. A nonparametric test for the comparison of multiple groups is also available. Most of these tests are accompanied by appropriate z-values (values of the standard normal distribution); these z-values can be used to test for the statistical significance of any differences between groups. However, note that most of these tests will only yield reliable results with fairly large samples sizes; the small sample "behavior" is less well understood.

Choosing a Two Sample Test. There are no widely accepted guidelines concerning which test to use in a particular situation. Cox's F test tends to be more powerful than Gehan's generalized Wilcoxon test when:

1. Sample sizes are small (i.e., N per group less than 50);

2. If samples are from an exponential or Weibull distribution;

3. If there are no censored observations (see Gehan & Thomas, 1969).

Lee, Desu, and Gehan (1975) compared Gehan's test to several alternatives and showed that the Cox-Mantel test and the log-rank test are more powerful (regardless of censoring) when the samples are drawn from a population that follows an exponential or Weibull distribution; under those conditions there is little difference between the Cox-Mantel test and the log-rank test. Lee (1980) discusses the power of different tests in greater detail.

Multiple Sample Test. The multiple-sample test implemented in Survival Analysis is an extension (or generalization) of Gehan's generalized Wilcoxon test, Peto and Peto's generalized Wilcoxon test, and the log-rank test. First, a score is assigned to each survival time using Mantel's procedure (Mantel, 1967); next a Chi-square value is computed based on the sums (for each group) of this score.

N = Number of cases

K = Number of groups

Sj = Sum of scores for jth group

If only two groups are specified, then this test is equivalent to Gehan's generalized Wilcoxon test, and the computations will default to that test in this case.

Unequal Proportions of Censored Data. When comparing two or more groups it is very important to examine the number of censored observations in each group. Particularly in medical research, censoring can be the result of, for example, the application of different treatments: patients who get better faster or get worse as the result of a treatment may be more likely to drop out of the study, resulting in different numbers of censored observations in each group. Such systematic censoring may greatly bias the results of comparisons.