Log-rank test for comparing three survival curves?

Question

I have a plot of three survival curves that are stratified by a categorical variable (a=1,2,or 3). The log-rank test can be used to test the global null hypothesis that all these curves are the same, but what if I want to test curve 1 vs curve 2 or curve 2 vs curve 3?

Answer 1

To expand on the comment from @EmmaJean:

A simple way to proceed would be to do each of the 3 pairwise comparisons of the groups (2 vs 1, 3 vs 1, 3 vs 2). Combined with a correction for multiple comparisons, you would in principle be OK.

A potentially more powerful approach would be to use all of the data together in a single survival regression model, for example a Cox model, with the group membership as a categorical predictor variable. The log-rank test is directly related to the score test for a Cox model. A Cox model will provide estimates of hazard ratios among the groups, not just yes/no difference evaluations, and could be extended to incorporate information about other outcome-associated covariates of interest. You then could perform post-modeling tests of pairwise differences among the (covariate-adjusted) regression coefficients for the groups of interest, for example via the tools of the emmeans package in R.