Log-Rank Test

The log-rank test is a statistical method to compare two survival distributions—that is, to determine whether two samples may have arisen from two identical survivor functions. The log-rank test is easy to compute and has a simple heuristic justification and is therefore often advocated for use to nonstatisticians. The log-rank test can also be thought as a censored data rank test.

Suppose we obtain two samples from two populations and we are interested in the null hypothesis

H0: S1 = S2

that the survival distribution from Sample 1 is identical to the survival distribution in Sample 2.

The idea behind the log-rank test is to compare the observed number of deaths at each failure time with the expected number of deaths under the null hypothesis (i.e., assuming that the null hypothesis is true).

To do this, we consider the ordered failure times for the combined samples t1 ≤ t2 ≤ tj ≤ tk: We then divide the observation period into small intervals (I1 = 1/2t0, t1)(M − 1) I2 = 1/2t1, t2), …, Ij = 1/2tj − 1, tj), …, IK = 1/2tK − 1, tK)), each one corresponding to the survival time of the noncensored individuals. For each interval Ij (j = 1, K), dj is the number of individuals who die at tj and rj is the number of individuals who are alive and at risk just before tj.

For each table, the quantities d1j=r1j and d2j=r2j are hazard estimates.

To perform the log-rank test, we construct a 2 × 2 table at each of the failure times tj.

From this table, define

and calculate

Sample 1 under the null hypothesis of no difference between the two survival distributions, and V• = Pj Vj = Variance term for the failures in Sample 1.

[Page 616]The log-rank test is given by

This test statistic follows a chi-square distribution with 1 degree of freedom under the null hypothesis (although the tables are not really independent, the distributional result still holds).

Large values of the test statistic indicate that the observed distribution of deaths in Sample 1 diverges from the expected number of deaths if the two survival distributions were identical. Although different censoring patterns do not invalidate the log-rank test, the test can be sensitive to extreme observations in the right tail of the distribution.

The log-rank test is particularly recommended when the ratio of hazard functions in the population being compared is approximately constant. For small data set, the log-rank test can be easily calculated by hand. Most statistical software contains routines for the calculation of the log-rank test.