Logrank Test
A Logrank Test is a non-parametric hypothesis test that is used for comparing two independent samples at different failure times.
- AKA: Log-Rank Test, Mantel-Cox Test.
- It is usually used to compare hazard and survival functions from two groups.
- Example(s)
- See: Log-rank Test Statistic, Julian Peto, Statistics, Hypothesis Test, Survival Analysis, Nonparametric, Censoring (Statistics), Clinical Trials, Nathan Mantel, David Cox (Statistician), Cochran–Mantel–Haenszel Statistics, Richard Peto.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Log-rank_test Retrieved:2016-12-17.
- 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.
- 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.
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] (...)
- 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
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
- 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.