Log-Rank Test

Background

Statistical tests do comparisons between different populations by determining how likely the observed differences in the data collected from those populations are due to chance only, under the assumption that those populations are the same with respect to the quantity under comparison (the null hypothesis). A p value of .05 means that, under the null hypothesis, the chance of observing such a difference in the collected data is 5%. The power of a statistical test is the chance of it correctly rejecting the null hypothesis under a specified difference between the populations under testing. For any well-designed statistical test, a larger sample size results in greater power.

Some statistical tests assume that the distribution of the quantity of interest has a certain mathematical format. For survival times, this can be Weibull, log-normal, gamma, or others. Such tests are called parametric tests. Other statistical tests do not make such distribution assumptions. These tests are called nonparametric tests. A common way to conduct a nonparametric test is to combine and then rank the data from the populations under study. The test statistic then depends only on the ranks of the data, not on the exact value of the data. Such tests are called rank tests. The log-rank test is a rank test. It does not require specification of the underlying survival time distribution. Nonparametric tests are thought to be more robust because they are not subject to bias arising from misspecification of the parametric distributions. Parametric tests, on the other hand, can be more powerful.

A survival function specifies the probability that the survival time is greater than a given number. The most common way of estimating a survival function is the Kaplan-Meier method. And the most common way to compare different estimated survival functions is the log-rank test.

History

The log-rank test was first derived by Nethan Mantel in 1966. The name log-rank was first used by Richard Peto and Julian Peto in 1972. They showed the optimality of the test under certain conditions. The “log” in the test name comes from the fact that, at a given time point, the factor used in the test, “the number of failures divided by the number of subjects at risk,” is an estimate for the change of the logarithm of the survival function at that time point. Stringent justification of the properties of the log-rank test needs modern mathematical theory.

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