How is the non time dependent hazard ratio computed?

Question

I'm a bit confused as it relates to the hazard ratio.

It seems to me that the hazard ratio is a time varying quantity however it seems that studies report a single number.

Read this https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5045282/ and it seems to imply that a hazard ratio is assumed constant? I find that very hard to believe with real world data - what is the approximation used?

Hazard ratios change over time and are reflected in the slope of the K-M plot. Reported hazard ratios assume that the differences between groups are a constant distance apart (i.e., the K-M survival curves) and are proportional

And from wiki (https://en.wikipedia.org/wiki/Hazard_ratio) it is stated that the hazard ratio is the ratio of the hazard rates which makes sense but again points to a time varying quantity.

Answer 1

In principle, hazard ratios might be expected to vary over time. In practice they almost certainly do, at least to some extent. A proportional hazards (PH) survival model, however, specifically assumes that hazard ratios are constant over time.

The estimates of hazard ratios are then based on that assumption. For a parametric proportional hazards model (e.g., a Weibull model), the estimated hazard ratios (and associated parameters of the baseline hazard) are the values that maximize the likelihood of the data. In semi-parametric Cox PH models, the baseline hazard isn't estimated directly; the estimated hazard ratios are those that maximize the partial likelihood of the data (ignoring the likelihood associated with the baseline hazard itself). See this page for the general formula for a Cox model, expressed in terms of the log-partial-likelihood and regression coefficients $\beta$ that are the logarithms of the hazard ratios. Therneau and Grambsch go into detail.

It is important to evaluate how well the PH assumption holds once the model has been fit. Methods for testing that are described in many pages on this site, by Therneau and Grambsch, and in the R time-dependent survival vignette. In practice, the assumption often holds well enough.