Logrank 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.

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 ${\displaystyle j}$, let ${\displaystyle N_{1j}}$ and ${\displaystyle N_{2j}}$ be the number of subjects "at risk" (have not yet had an event or been censored) at the start of period ${\displaystyle j}$ in the two groups (often treatment vs. control), respectively. Let ${\displaystyle N_{j}=N_{1j}+N_{2j}}$. Let ${\displaystyle O_{1j}}$ and ${\displaystyle O_{2j}}$ be the observed number of events in the groups respectively at time ${\displaystyle j}$, and define ${\displaystyle O_{j}=O_{1j}+O_{2j}}$.

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