I'd like a clarification about the Cox proportional hazard coefficients reported on this page. For example, doesn't the hazard ratio (HR) for gender in that analysis of 0.41 (rounded from 0.407) mean that being female reduces the hazard of experiencing an end point by 59% compared to being male, (1-0.41) x 100? And doesn't the inverse of that HR, 2.46, mean that being male increase the probability of experiencing an end point by a factor of 2.46? Maybe I am wrong, but I am trying to understand how to interpret these coefficients.


Your understanding is a useful approximation in some circumstances, but it doesn't hold in general. It depends on what you mean by "the hazard of experiencing an end point." Survival analysis examines how quickly events occur, not simply whether they occur.

The value of $exp(\beta_i)$ reported after Cox modeling is the instantaneous hazard ratio (HR) associated with covariate $i$ for individuals at risk for the event. Under the proportional hazards (PH) assumption of Cox regression, any HR is constant over time (even if the value of the covariate has been changing over time). In other words, at any time $t$, the HR between 2 groups represents the relative risk of an event occurring in a short period of time around $t$ among those in the 2 groups who are still at risk.

So if you are restricting your interest to a short period of time around any particular time, your interpretation of the "hazard of experiencing an endpoint" for gender on the page you linked* is a useful approximation. The HR of 0.41 is the ratio of the risk of the event for women versus men over any short period of time. And in that case you are correct that the reciprocal of that HR, 2.46, is the corresponding ratio for men versus women.

But things get complicated over longer periods of time; after all, "In the long run we are all dead" and experience that event. This paper goes into details showing that a HR does not readily translate into relative risk (RR), expressed in that paper as the relative risk reduction.

What's going on is that the relative numbers at risk in the 2 groups are changing over time even if the HR is constant. In your example, even if you started with equal numbers of men and women, at later points in time there will be fewer men than women still at risk. At a sufficiently late time there will be many more women than men at risk. So even if the HR of 0.41 still favors women over men among those at risk, the number of events among women around later times $t$ will eventually start to exceed those among men until all men and women have had the event.

The plot below shows how one simple definition of relative risk (RR, ratio of numbers of events in two groups, starting with equal numbers at $t=0$, plotted on a logarithmic axis) changes over time. This is for exponential survival in the reference group ($S(t)=exp(-\lambda_0 t)$; time on the plot is expressed in units of $\lambda_0 t$)) and HR values of 1.5, 2, 3, and 4. At $t=0$ each RR equals the HR. Soon, however, the RR values drop well below the corresponding HR.

enter image description here

*On the page that you linked there is a hazard ratio (HR) for gender of about 0.41, but I could find no information there about what the reference group was. Was the reference group males, in which case the HR represents the ratio for females versus males (as you implicitly assumed), or was it females, in which case the HR is for males versus females? Knowing the reference group is important in all regressions involving categorical predictors; unfortunately, reference groups aren't always specified clearly. For simplicity, I used your assumption that the HR of 0.41 is for females versus males.

  • $\begingroup$ Thank you for your help. With regard to point 2, isn't it more correct to consider the cumulative ratio as Relative Risk rather than Hazard Ratio? Shouldn't the Hazard Ratio be the risk at any istaneous point, whereas RR is the "whole" relative risk? I think that the summary table built by statistical software after Cox model calculation release RR as expBi.. $\endgroup$ – Alex ME Mar 16 at 20:30
  • $\begingroup$ @AlexME I've edited my answer. The distinction between '"whole" relative risk' (RR) and HR is very important; I hope that this version makes it clearer. What's reported as $exp(\beta_i)$ by Cox software, however, is specifically the HR, not a RR. Software for survival analysis typically provides methods for estimating RR (however you wish to define it), based on the HR values and the baseline survival curve. But there's no simple general relation between HR and RR values. $\endgroup$ – EdM 11 hours ago

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