# interpreting HR from the time-dependent cox regression

I have done two time-dependent cox regression. My time to event is time to disability, and exposure is disease (0/1). I have done a cox regression considering disease as time-dependent covariate. I want to know if my interpretation below is correct? if not what is the interpretation of the given HR?

1) HR = 1.5 , 95% CI =(1.15, 10.63)

2) HR = 1.15 , 95%CI = (1.11,1.42)

• The values in the question are internally inconsistent. The CI for the HR don't cover the first point estimate of HR=1.33, and it seems that you might have a misprint further down when you say "1.23 time risk." Please edit the question to provide the correct values.
– EdM
Commented Aug 26, 2022 at 16:34
• @ EdM Please see the completely new updated question. Commented Aug 27, 2022 at 10:49
• Thanks for the update. In case 2, however, the point estimate of 1.25 isn't within the 95% CI (1.11,1.15). Also, it's not clear why you have two separate HR values here. Please edit the question to provide that information, as comments are easy to overlook and can be deleted.
– EdM
Commented Aug 27, 2022 at 11:31
• @EdM I have two, because it is done the same for two population samples. I have updated in the Q. i dont know why 1.25 isn't within the 95% CI (1.11,1.15). That is the result I got. Commented Aug 27, 2022 at 12:07

A Cox model is based on whatever covariate values are in place at each event time. Prior history of covariate values isn't considered. So hazard ratios have the same interpretation regardless of whether the covariates were time-varying or not.

Be careful in jumping from "hazard" to "risk," however. The hazard at an event time in a Cox model is the probability of an event given that the individual has already survived that long. Most people think of "risk" in a more general sense, taking into account the entire survival curve. With time-varying covariates you need to know the entire covariate history to judge "risk" in that sense. It's safest to use the words "hazard" and "hazard ratio" to be specific about what you have estimated.

If your model meets the proportional hazards and other assumptions needed for a Cox model, one way to proceed with a HR of 1.25 is to say something like "among individuals still having 'normal ability,' those with cancer are 1.25 times more probable at any time to develop 'weak ability' than those without." That's a fair statement that doesn't depend on knowing details of survival curves.

You might look at an answer to a similar question, which has a link to a freely available paper that discusses the distinction between hazards and risks in more detail.

• @ EdM you are right. Thanks for the point. There was a typo. I just corrected in the updated Q the 95%CI. .Could you please help me with the interpretation of e.g one of the HR? as I mention cancer is a time-varying covariate meaning that can happen at baseline or any time during follow-up. And, the event of interest is time-to-ability decline Commented Aug 27, 2022 at 18:07
• @user358238 I added a bit to the answer. You have to be very careful in how you word such things, as the word "risk" isn't always as well defined as you might hope. Also, you don't need to call the author out with @ if you comment on an answer. As I understand the system, the author of an answer always gets notified of comments on that answer; similarly for authors of questions.
– EdM
Commented Aug 27, 2022 at 18:35
• @ EdM another question is that does it make sense to check the PH assumption when using the time-dependent cox regression? To me does not make sense to check it anymore. Is that true? Commented Aug 27, 2022 at 19:09
• @user358238 it's still important to check PH with time-dependent covariates, maybe more so. If you want to describe results in terms of hazard ratios, you want relative hazards to be constant over time--the PH assumption. Otherwise you won't know if some change in survival has to do with changes in the covariates themselves or with the passage of time. With your model of developing 'weak ability' as a function of time and of developing cancer, both of which risks tend to increase with time, that would be very important to document.
– EdM
Commented Aug 27, 2022 at 19:13
• @ EdM thanks. Good point. My exposure is cancer status. If assumption for PH is not fulfilled for this, is it an issue? other covariates are age+gender+education+... which if the PH assumption is not fulfilled, I can stratify on that specific covariate. But what if cancer status is not. Using let us say cox.zph() or scaled Schoenfeld residuals. To me it is fine if it is not fulfilled for cancer status as it is already time-varying in the cox model. Do you have any comment on this? thanks I always get very nice comment and clear explanation from you. Thanks a lot. Commented Aug 27, 2022 at 19:22