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 was first proposed by Nathan Mantel and was named the logrank test by Richard and Julian Peto.
[1][2][3] The logrank 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.
Consider two groups of patients, e.g., treatment vs. control.
be the distinct times of observed events in either group.
be the number of subjects "at risk" (who have not yet had an event or been censored) at the start of period
be the observed number of events in the groups at time
The null hypothesis is that the two groups have identical hazard functions,
follows a hypergeometric distribution with parameters
, the logrank statistic compares
It is defined as By the central limit theorem, the distribution of each
converges to that of a standard normal distribution as
approaches infinity and therefore can be approximated by the standard normal distribution for a sufficiently large
An improved approximation can be obtained by equating this quantity to Pearson type I or II (beta) distributions with matching first four moments, as described in Appendix B of the Peto and Peto paper.
[2] If the two groups have the same survival function, the logrank statistic is approximately standard normal.
test will reject the null hypothesis if
quantile of the standard normal distribution.
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 logrank statistic is approximately normal with mean
, the sample size required is
are the quantiles of the standard normal distribution.
are the logrank statistics at two different time points in the same study (
Again, assume the hazard functions in the two groups are proportional with hazard ratio
are the probabilities that a subject will have an event at the two time points where
are approximately bivariate normal with means
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.
The logrank test is based on the same assumptions as the Kaplan-Meier survival curve—namely, that censoring is unrelated to prognosis, the survival probabilities are the same for subjects recruited early and late in the study, and the events happened at the times specified.
Deviations from these assumptions matter most if they are satisfied differently in the groups being compared, for example if censoring is more likely in one group than another.