# Interpretation of Log-rank test with "close" survival curves

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:
Upper figure:

• number at risk = 6477
• number of failures = 592 (light blue), 2757 (medium blue), 2412 (dark blue)

Lower figure:

• 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 survival and survminer packages for the computation.

I would be grateful for any help!

Best, Florian

• what are the sample sizes (starting at risk number and total number of failures) in each case? As with all statistical tests the pvalue is a function of the effect size and the precision (in this case precision corresponds to the number of failures in each group) Jan 19 at 14:05
• Just to comment that PH assumption does not look to be an issue here... Jan 19 at 14:25
• @George Savva Thank you for your response! Indeed, the sample sizes are a good point. I edited the question and added the number at risk and number of failures for each group. Jan 19 at 15:13
• @ocram Thanks for your response as well. To be honest, I'm a little confused. I thought that crossing curves indicate non-proportionality, which would be the case in both figures. Even though that's not the discussion I wanted to come up with here, I'm still interested in your thoughts about that. I would be grateful, if you could give me further advice. Jan 19 at 15:23
• Some overlap might occur due to variability. In particular, if the two population survival functions are exactly the same (and are thus proportional), then it is not surprising to see some crossing in the sample survival functions. Likewise, population survival functions are always close at the beginning, and thus some crossing is also expected at the beginning in the sample survival functions. Jan 19 at 15:56