Which test should be used to compare crude mortality between 15 groups? I have fifteen groups (factor levels) and I need to compare their crude mortality. Is Logrank test limited to two groups only? Which test would be correct to use in this case?
 A: In principle, you could treat your groups as a 15-level categorical predictor and perform a Cox proportional hazards regression against that categorical predictor. That has the further advantage that covariates associated with outcome could be included in the model.
Provided that the proportional hazards assumption holds, ANOVA comparing models with and without that 15-level set of groups would document whether there were any differences in survival among the groups. Then the coefficient values associated with each of the groups could be used to determine specific between-group differences.
Two cautions, however. First, unless you have something on the order of 200 deaths you will be at risk of overfitting. You typically need about 10-20 events per predictor, and each of your groups beyond the first counts as a separate predictor.
Second, if you don't have some specific comparisons among groups in mind to start you will be faced with a major problem in correcting for multiple comparisons.
Depending on the goals of your study, you might instead consider a mixed-effect Cox model, for example with the coxme package in R, treating the groups as random effects. That would provide an estimate of the overall variability in outcomes associated with different groups, potentially working better with fewer events while still providing useful information.
