Logrank Test

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A Logrank Test is a non-parametric hypothesis test that is used for comparing two independent samples at different failure times.



References

2016


2015

  • (Lu Tian, 2015) ⇒ Survival Analysis (STAT331) http://web.stanford.edu/~lutian/coursepdf/unitweek3.pdf
    • The logrank test is the most commonly-used statistical test for comparing the survival distributions of two or more groups (such as different treatment groups in a clinical trial). The purpose of this unit is to introduce the logrank test from a heuristic perspective and to discuss popular extensions. Formal investigation of the properties of the logrank test will be covered in later units

      Assume that we have 2 groups of individuals, say group 0 and group 1. In group j, there are [math]\displaystyle{ n_j }[/math] i.i.d. underlying survival times with common c.d.f. denoted [math]\displaystyle{ Fj (·), \text{for} j=0,1 }[/math]. The corresponding hazard and survival functions for group j are denoted [math]\displaystyle{ h_j (·) }[/math] and [math]\displaystyle{ S_j (·) }[/math], respectively.

      As usual, we assume that the observations are subject to noninformative right censoring: within each group, the [math]\displaystyle{ T_i }[/math]and [math]\displaystyle{ C_i }[/math] are independent.

      We want a nonparametric test of H0 : F0(·) = F1(·), or equivalently, of [math]\displaystyle{ S_0(·) = S1(·) }[/math], or [math]\displaystyle{ h0(·) = h1(·) }[/math] (...)

2004

  • (Bland et al. 2004) ⇒ Bland, J. M., & Altman, D. G. (2004). The logrank test. Bmj, 328(7447), 1073 doi:10.1136/bmj.328.7447.1073.
    • QUOTE: The logrank test is used to test the null hypothesis that there is no difference between the populations in the probability of an event (here a death) at any time point. The analysis is based on the times of events (here deaths). For each such time we calculate the observed number of deaths in each group and the number expected if there were in reality no difference between the groups. The first death was in week 6, when one patient in group 1 died. At the start of this week, there were 51 subjects alive in total, so the risk of death in this week was 1/51. There were 20 patients in group 1, so, if the null hypothesis were t,rue, the expected number of deaths in group 1 is 20 × 1/51 = 0.39. Likewise, in group 2 the expected number of deaths is 31 × 1/51 = 0.61. The second event occurred in week 10, when there were two deaths. There were now 19 and 31 patients at risk (alive) in the two groups, one having died in week 6, so the probability of death in week 10 was 2/50. The expected numbers of deaths were 19 × 2/50 = 0.76 and 31 × 2/50 = 1.24 respectively.

      (...) The logrank test is most likely to detect a difference between groups when the risk of an event is consistently greater for one group than another. It is unlikely to detect a difference when survival curves cross, as can happen when comparing a medical with a surgical intervention. When analysing survival data, the survival curves should always be plotted.

      Because the logrank test is purely a test of significance it cannot provide an estimate of the size of the difference between the groups or a confidence interval. For these we must make some assumptions about the data. Common methods use the hazard ratio, including the Cox proportional hazards model, which we shall describe in a future Statistics