The Discrete Distribution Used for the Log‐rank Test Can Be Inaccurate

Abstract

Abstract When the number of tumors is small, a significance level for the Cox‐Mantel (log‐rank) test Z is often computed using a discrete approximation to the permutation distribution. For j = 0,…, J let N j (t) be the number of animals in group j alive and tumor‐free at the start of time t. Make a 2 × (1+ J ) table for each time t of the number of animals R j (t) with newly palpated tumor out of the total N j (t) at risk. There are a total of say K tables, one for each distinct time t with observed death or newly palpated tumor. The usual discrete approximation to the permutation distribution of Z is defined by taking tables to be independent with fixed margins N j (t) and Σ R j (t) for all t. However, the N j (t) are random variables for the actual permutation distribution of Z , resulting in dependence among the tables. Calculations for the exact permutation distribution are explained, and examples are given where the exact significance level differs substantially from the usual discrete approximation. The discrepancy arisis primarily because permutations with different Z ‐scores under the exact distribution can be equal for the discrete approximation, inflating the approximate P ‐value.