I have a question about the interpretation of the log-rank test, when the survival curves (in the figures below hazard curves) are close together.
Below you can find two figures with survival curves for three groups. In both cases, the Log-rank test indicates a significant difference in the probability of the event at any time point according to the recommended interpretation in this BMJ article. In the figure, the vertical dashed lines indicate the median hazard time for each group.
The numbers at risk and number of failures are:
- number at risk = 6477
- number of failures = 592 (light blue), 2757 (medium blue), 2412 (dark blue)
- number at risk = 3493
- number of failures = 225 (light blue), 1707 (medium blue), 1378 (dark blue)
I know that the interpretation of the Log-rank test is not really appropriate, when looking at the figures above as the different curves are crossing and this violates the proportional hazards assumption. But regardless of the reduced interpretability of the test results, shouldn't the lower Log-rank test indicate kind of a "less significant" difference between the groups? When looking at the lower figure, I would expect that there is no difference between them. I also tried alternative approaches to test the difference such as Gehan-Breslow test, which puts higher weights on earlier events but all tests return the same results (p < 0.0001). I used R and the
survminer packages for the computation.
I would be grateful for any help!