How to Read (and Generate) Kaplan-Meier Survival Curves
Product-limit survival estimation, right-censoring and the Mantel-Cox log-rank test, explained for preclinical and clinical survival data.
The Kaplan-Meier estimator computes survival probability S(t) over time using the non-parametric product-limit formula.
**Understanding the Math:** At each event time t_i, d_i is the number of deaths/events and n_i is the number of subjects at risk just before t_i. The probability of surviving past t_i is (1 - d_i/n_i), multiplied by the previous survival probability S(t_{i-1}).
**Right-Censored Data (Tick Marks):** When an animal or patient completes the study without an event, or is lost to follow-up, they are marked as censored (status = 0). Censoring reduces the denominator n_{i+1} for subsequent time points without reducing S(t).
**The Mantel-Cox Log-Rank Test:** Compares two or more survival curves across all time points by calculating expected events E_{1j} = n_{1j} (d_j / n_j) under the null hypothesis of equal hazard rates. The test statistic follows a chi-squared distribution with k-1 degrees of freedom.
**Crucial Publishing Rules:** 1. Always report the median survival time with 95% confidence intervals. 2. Do not truncate the x-axis before the last observation. 3. Verify proportional hazards assumption before using standard log-rank tests.