Does the log rank test assume proportional hazards?

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 ).

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).

Non-proportional hazards – an introduction to their possible causes and interpretation

I had the pleasure today to participate in a PSI event on non-proportional hazards and applications in immuno-oncology. Non-proportional hazards are increasingly encountered in clinical trials, and there remain important questions about how to analyse trials when non-proportional hazards could occur. These include questions about how to formulate an appropriate hypothesis test for assessing evidence of benefit of the new treatment over the control and how to best quantify the treatment effect. The talks were really very interesting, and lead me to believe there is still lots of important work to be done in this area.

A video of the day’s talks will soon be available on the PSI website (upon which I will add a link). In the meantime, here are the slides of my talk in case they are of interest, where I discuss some of the subtleties involved in interpreting changes in hazards and hazard ratios over time, which are complicated by the ubiquitous presence of frailty effects. I’ve posted on this topic previously quite a lot – for those interested see the related posts below.

What can we infer from proportional hazards?

Colleagues and I recently wrote a letter to the editor regarding difficulties of interpreting period specific hazard ratios from randomised trials as representing solely changes in treatment effect over time, as discussed in a previous post. The authors whose paper we were writing about responded to our letter. One of their points was the following:

Note that in a randomized controlled trial, if the proportional hazards assumption holds and there are no unobserved confounders (i.e., the patient population is homogeneous and there is no differential treatment effect across different subpopulations), a HR generated by Cox regression with treatment alone as a single covariate does have a causal interpretation. The hazard functions in this case do not depend on any confounder.

It is fairly implausible in any setting that a patient’s hazard does not depend on any of their characteristics. Indeed it is because we often have some understanding of what these variables are that they are used in stratified randomisation schemes and in the statistical analysis of the trial’s data.

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