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.
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 groups 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 has the hypergeometric distribution with parameters , , and . This distribution has expected value and variance .
The logrank statistic compares each to its expectation under the null hypothesis and is defined as
If the two groups have the same survival function, the logrank 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, and the proportion of subjects randomized to each group is 50%, then the logrank statistic is approximately normal with mean and variance 1. For a one-sided level test with power , the sample size required is where and are the quantiles of the standard normal distribution.
Suppose and are the logrank 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. 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.