logrank

Logrank is a nonparametric statistical test used to compare the survival distributions of two or more groups using time-to-event data. It is commonly applied in clinical trials to determine whether treatments have different effects on survival, while accounting for censored observations.

The test operates on data organized by observed event times. At each event time t, for each group i, n_i(t) denotes the number at risk just before t, and d(t) the total number of events at t. The total at risk is N(t) = sum_i n_i(t). Under the null hypothesis that all groups share the same survival curve, the expected number of events in group i at t is E_i(t) = d(t) * [n_i(t) / N(t)], and the observed events are O_i(t). The logrank statistic aggregates the differences O_i(t) − E_i(t) over all event times and groups. A common formulation uses the overall statistic X^2 = sum_{i=1}^g (O_i − E_i)^2 / Var_i, where g is the number of groups and Var_i is the variance of the differences for group i. Under the null hypothesis, X^2 follows a chi-squared distribution with g − 1 degrees of freedom. For two groups, this reduces to the familiar one-degree-of-freedom test.

Assumptions include non-informative censoring and independence between groups. The logrank test is most powerful against alternatives consistent with proportional hazards, though it does not require a specific parametric survival model. It remains a standard tool for comparing survival curves in medical research and other fields dealing with time-to-event data. Variants and related tests, such as Wilcoxon (Breslow) and Tarone-Ware, weight early or late events differently, offering robustness to certain alternative patterns.