A student asked me recently whether the log rank test for time to event data *assumes* that the hazard ratio between the two groups is constant over time, as is assumed in Cox’s famous proportional hazards model. The BMJ ‘Statistics at square one’ Survival Analysis article for example says the test assumes:

That the risk of an event in one group relative to the other does not change with time. Thus if linoleic acid reduces the risk of death in patients with colorectal cancer, then this risk reduction does not change with time (the so called proportional hazards assumption ).

https://www.bmj.com/about-bmj/resources-readers/publications/statistics-square-one/12-survival-analysis

Personally I would not say the log rank test *assumes* proportional hazards. Under the null hypothesis that the (true) survival curves in the two groups are the same, or equivalently that the hazard functions are identical in the two groups, the log rank test would only wrongly reject 5% of the time. Of course under this null the hazards are proportional (indeed identical).

When this null does not hold, if the hazard ratio is constant over time, the log rank test is the most powerful test. When it is not constant over time it is not optimal in terms of power, but the non-constant hazard ratio does not invalidate the test per se. It just means that there may be alternative methods of analysis that might be preferable (see my recent PSI event slides here).