I'm looking at a dataset of survival times of animals in two different groups. While there is some early mortality in the first group, the Kaplan-Meier survival curves overlap, the log-rank test and the Cox proportional hazard model are not significant, which would lead me to conclude that there is no evidence that the survival rate would be different in the two groups.
However, I can also tally the number of live and dead animals in the two groups at particular time points, and run a Fisher's exact test to see if there is an association between the group membership and the survival status. Lo and behold, some of these turn out to be significant at some time points, which would lead me to conclude that ther is an association between survival and membership in the two groups.
Which of these results should I trust? Is the Fisher's test appropriate here and if it is not, then why not?
All your insights would be highly appreciated!