Log-rank test

The logrank test, or 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 logrank test can also be viewed as a time-stratified Cochran–Mantel–Haenszel test. The logrank test, or 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 logrank 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 logrank test by Richard and Julian Peto. 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. Let 1 , … , J {displaystyle 1,ldots ,J} be the distinct times of observed events in either group. Let N 1 , j {displaystyle N_{1,j}} and N 2 , j {displaystyle N_{2,j}} be the number of subjects 'at risk' (who have not yet had an event or been censored) at the start of period j {displaystyle j} in the groups, respectively. Let O 1 , j {displaystyle O_{1,j}} and O 2 , j {displaystyle O_{2,j}} be the observed number of events in the groups at time j {displaystyle j} . Finally, define N j = N 1 , j + N 2 , j {displaystyle N_{j}=N_{1,j}+N_{2,j}} and O j = O 1 , j + O 2 , j {displaystyle O_{j}=O_{1,j}+O_{2,j}} . The null hypothesis is that the two groups have identical hazard functions, H 0 : h 1 ( t ) = h 2 ( t ) {displaystyle H_{0}:h_{1}(t)=h_{2}(t)} . Hence, under H 0 {displaystyle H_{0}} , for each group i = 1 , 2 {displaystyle i=1,2} , O i , j {displaystyle O_{i,j}} follows a hypergeometric distribution with parameters N j {displaystyle N_{j}} , N i , j {displaystyle N_{i,j}} , O j {displaystyle O_{j}} . This distribution has expected value E i , j = O j N i , j N j {displaystyle E_{i,j}=O_{j}{frac {N_{i,j}}{N_{j}}}} and variance V i , j = E i , j ( N j − N i , j N j ) ( N j − O j N j − 1 ) {displaystyle V_{i,j}=E_{i,j}left({frac {N_{j}-N_{i,j}}{N_{j}}} ight)left({frac {N_{j}-O_{j}}{N_{j}-1}} ight)} . For all j = 1 , … , J {displaystyle j=1,ldots ,J} , the logrank statistic compares O i , j {displaystyle O_{i,j}} to its expectation E i , j {displaystyle E_{i,j}} under H 0 {displaystyle H_{0}} . It is defined as By the central limit theorem, the distribution of Z {displaystyle Z} converges to that of a standard normal distribution as J {displaystyle J} approaches infinity and therefore can be approximated by the standard normal distribution for a sufficiently large J {displaystyle J} . 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. If the two groups have the same survival function, the logrank statistic is approximately standard normal. A one-sided level α {displaystyle alpha } test will reject the null hypothesis if Z > z α {displaystyle Z>z_{alpha }} where z α {displaystyle z_{alpha }} is the upper α {displaystyle alpha } quantile of the standard normal distribution. If the hazard ratio is λ {displaystyle lambda } , there are n {displaystyle n} total subjects, d {displaystyle d} is the probability a subject in either group will eventually have an event (so that n d {displaystyle nd} 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 ( log ⁡ λ ) n d 4 {displaystyle (log {lambda }),{sqrt {frac {n,d}{4}}}} and variance 1. For a one-sided level α {displaystyle alpha } test with power 1 − β {displaystyle 1-eta } , the sample size required is n = 4 ( z α + z β ) 2 d log 2 ⁡ λ {displaystyle n={frac {4,(z_{alpha }+z_{eta })^{2}}{dlog ^{2}{lambda }}}} where z α {displaystyle z_{alpha }} and z β {displaystyle z_{eta }} are the quantiles of the standard normal distribution. Suppose Z 1 {displaystyle Z_{1}} and Z 2 {displaystyle Z_{2}} are the logrank statistics at two different time points in the same study ( Z 1 {displaystyle Z_{1}} earlier). Again, assume the hazard functions in the two groups are proportional with hazard ratio λ {displaystyle lambda } and d 1 {displaystyle d_{1}} and d 2 {displaystyle d_{2}} are the probabilities that a subject will have an event at the two time points where d 1 ≤ d 2 {displaystyle d_{1}leq d_{2}} . Z 1 {displaystyle Z_{1}} and Z 2 {displaystyle Z_{2}} are approximately bivariate normal with means log ⁡ λ n d 1 4 {displaystyle log {lambda },{sqrt {frac {n,d_{1}}{4}}}} and log ⁡ λ n d 2 4 {displaystyle log {lambda },{sqrt {frac {n,d_{2}}{4}}}} and correlation d 1 d 2 {displaystyle {sqrt {frac {d_{1}}{d_{2}}}}} . 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.