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Statistics Notes BMJ: first published as 10.1136/bmj.328.7447.1073 on 29 April 2004. Downloaded from The logrank test J Martin Bland, Douglas G Altman

We often wish to compare the survival experience of Department of 1.0 Health Sciences, two (or more) groups of individuals. For example, the Astrocytoma University of York, table shows survival times of 51 adult patients with Glioblastoma York YO10 5DD 0.8 recurrent malignant gliomas1 tabulated by type of J Martin Bland tumour and indicating whether the patient had died or professor of health 0.6 statistics was still alive at analysis—that is, their survival time was Survival proportion 2 Cancer Research censored. As the figure shows, the survival curves dif- 0.4 UK/NHS Centre fer, but is this sufficient to conclude that in the popula- for Statistics in Medicine, Institute tion patients with anaplastic astrocytoma have worse 0.2 of Health Sciences, survival than patients with glioblastoma? Oxford OX3 7LF 3 We could compute survival curves for each group 0 Douglas G Altman 0 52 104 156 208 and compare the proportions surviving at any specific professor of statistics time. The weakness of this approach is that it does not Time (weeks) in medicine provide a comparison of the total survival experience of Survival curves for women with glioma by diagnosis Correspondence to: Professor Bland the two groups, but rather gives a comparison at some arbitrary time point(s). In the figure the difference in BMJ 2004;328:1073 survival is greater at some times than others and eventu- ally becomes zero. We describe here the logrank test, the 2 − 1 = 1. From a table of the
2 distribution we get most popular method of comparing the survival of P < 0.01, so that the difference between the groups is groups, which takes the whole follow up period into statistically significant. There is a different method of account. It has the considerable advantage that it does calculating the test statistic,4 but we prefer this not require us to know anything about the shape of the Weeks to death or approach as it extends easily to several groups. It is censoring in 51 survival curve or the distribution of survival times. also possible to test for a trend in survival across adults with The logrank test is used to test the null hypothesis 1 ordered groups.4 Although we have shown how the recurrent gliomas that there is no difference between the populations in calculation is made, we strongly recommend the use of (A=astrocytoma, the probability of an event (here a death) at any time statistical software. G=glioblastoma) point. The analysis is based on the times of events The logrank test is based on the same assumptions AG

(here deaths). For each such time we calculate the http://www.bmj.com/ as the Kaplan Meier survival curve3—namely, that cen- 610 observed number of deaths in each group and the soring is unrelated to prognosis, the survival probabili- 13 10 number expected if there were in reality no difference ties are the same for subjects recruited early and late in 21 12 between the groups. The first death was in week 6, the study, and the events happened at the times speci- 30 13 when one patient in group 1 died. At the start of this fied. Deviations from these assumptions matter most if 31* 14 week, there were 51 subjects alive in total, so the risk of 37 15 they are satisfied differently in the groups being death in this week was 1/51. There were 20 patients in 38 16 compared, for example if censoring is more likely in group 1, so, if the null hypothesis were true, the 47* 17 one group than another. expected number of deaths in group 1 is 20 × 1/51 = 49 18 on 27 September 2021 by guest. Protected copyright. The logrank test is most likely to detect a difference 0.39. Likewise, in group 2 the expected number of 50 20 between groups when the risk of an event is deaths is 31 × 1/51 = 0.61. The second event 63 24 consistently greater for one group than another. It is occurred in week 10, when there were two deaths. 79 24 unlikely to detect a difference when survival curves 80* 25 There were now 19 and 31 patients at risk (alive) in the cross, as can happen when comparing a medical with a 82* 28 two groups, one having died in week 6, so the surgical intervention. When analysing survival data, the 82* 30 probability of death in week 10 was 2/50. The survival curves should always be plotted. 86 33 expected numbers of deaths were 19 × 2/50 = 0.76 Because the logrank test is purely a test of 98 34* and 31 × 2/50 = 1.24 respectively. significance it cannot provide an estimate of the size of 149* 35 The same calculations are performed each time an the difference between the groups or a confidence 202 37 event occurs. If a survival time is censored, that interval. For these we must make some assumptions 219 40 individual is considered to be at risk of dying in the 40 about the data. Common methods use the hazard ratio, week of the censoring but not in subsequent weeks. 40* including the Cox proportional hazards model, which This way of handling censored observations is the 46 we shall describe in a future Statistics Note. same as for the Kaplan-Meier survival curve.3 48 From the calculations for each time of death, the Competing interests: None declared. 70* total numbers of expected deaths were 22.48 in group 76 1 and 19.52 in group 2, and the observed numbers of 1 Rostomily RC, Spence AM, Duong D, McCormick K, Bland M, Berger 81 MS. Multimodality management of recurrent adult malignant gliomas: 82 deaths were 14 and 28. We can now use a
2 test of the results of a phase II multiagent chemotherapy study and analysis of 91 null hypothesis. The test statistic is the sum of (O – cytoreductive surgery. Neurosurgery 1994;35:378-8. 112 2 2 Altman DG, Bland JM. Time to event (survival) data. BMJ E) /E for each group, where O and E are the totals of 1998;317:468-9. 181 the observed and expected events. Here (14 − 22.48)2 3 Bland JM, Altman DG. Survival probabilities. The Kaplan-Meier method. *Censored survival + − 2 BMJ 1998;317:1572. / 22.48 (28 19.52) / 19.52 = 6.88. The degrees of 4 Altman DG. Practical statistics for medical research. London: Chapman & time (still alive at freedom are the number of groups minus one, i.e. Hall, 1991: 371-5. follow up).

BMJ VOLUME 328 1 MAY 2004 bmj.com 1073