Log-rank test

In statistics, the log-rank test is a hypothesis test to compare the survival distributions of two samples. It is a nonparametric test and appropriate to use when the data are right skewed and censored (technically, the censoring must be non-informative). It is widely used in clinical trials to establish the efficacy of a new treatment in comparison with a control treatment when the measurement is the time to event (such as the time from initial treatment to a heart attack). The test is sometimes called the Mantel–Cox test, named after Nathan Mantel and David Cox. The log-rank test can also be viewed as a time-stratified Cochran–Mantel–Haenszel test.

The test was first proposed by Nathan Mantel and was named the log-rank test by Richard and Julian Peto.[1][2][3]


The log-rank test statistic compares estimates of the hazard functions of the two groups at each observed event time. It is constructed by computing the observed and expected number of events in one of the groups at each observed event time and then adding these to obtain an overall summary across all-time points where there is an event.

Let j = 1, ..., J be the distinct times of observed events in either group. For each time , let and be the number of subjects "at risk" (have not yet had an event or been censored) at the start of period in the two groups (often treatment vs. control), respectively. Let . Let and be the observed number of events in the groups respectively at time , and define .

Given that events happened across both groups at time , under the null hypothesis (of the two groups having identical survival and hazard functions) has the hypergeometric distribution with parameters , , and . This distribution has expected value and variance .

The log-rank statistic compares each to its expectation under the null hypothesis and is defined as

Asymptotic distribution

If the two groups have the same survival function, the log-rank statistic is approximately standard normal. A one-sided level test will reject the null hypothesis if where is the upper quantile of the standard normal distribution. If the hazard ratio is , there are total subjects, is the probability a subject in either group will eventually have an event (so that is the expected number of events at the time of the analysis), and the proportion of subjects randomized to each group is 50%, then the log-rank statistic is approximately normal with mean and variance 1.[4] For a one-sided level test with power , the sample size required is where and are the quantiles of the standard normal distribution.

Joint distribution

Suppose and are the log-rank statistics at two different time points in the same study ( earlier). Again, assume the hazard functions in the two groups are proportional with hazard ratio and and are the probabilities that a subject will have an event at the two time points where . and are approximately bivariate normal with means and and correlation . Calculations involving the joint distribution are needed to correctly maintain the error rate when the data are examined multiple times within a study by a Data Monitoring Committee.

Relationship to other statistics

See also


  1. Mantel, Nathan (1966). "Evaluation of survival data and two new rank order statistics arising in its consideration.". Cancer Chemotherapy Reports. 50 (3): 163–70. PMID 5910392.
  2. Peto, Richard; Peto, Julian (1972). "Asymptotically Efficient Rank Invariant Test Procedures". Journal of the Royal Statistical Society, Series A. Blackwell Publishing. 135 (2): 185–207. doi:10.2307/2344317. JSTOR 2344317.
  3. Harrington, David (2005). "Linear Rank Tests in Survival Analysis". Encyclopedia of Biostatistics. Wiley Interscience. doi:10.1002/0470011815.b2a11047.
  4. Schoenfeld, D (1981). "The asymptotic properties of nonparametric tests for comparing survival distributions". Biometrika. 68: 316–319. doi:10.1093/biomet/68.1.316. JSTOR 2335833.
  5. Berty, H. P.; Shi, H.; Lyons-Weiler, J. (2010). "Determining the statistical significance of survivorship prediction models". J Eval Clin Pract. 16 (1): 155–165. doi:10.1111/j.1365-2753.2009.01199.x. PMID 20367827.

Further reading

This article is issued from Wikipedia - version of the 3/25/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.