# interpreting HR with time dependent covariates cox regression

I did time-dependent covariate analyses with cox regression in R. But I wonder if the interpretation is the same as for a cox model with a fixed covariate?

Context: in my analysis the physical activity score index was analyzed 5 months after their lung cancer remission period and then 5 more months after that, i.e. 10 months after remission. Some people relapse lung cancer between 5 and 10 months after remission, and others did not. I created my dataframe with the time intervals so the physical activity index score at the second line (i.e. at 10 months) changes. My baseline is 5 months so t start is 0 days.

Here is the HR that I obtained with a score index as continous variable : 0.65 (0.80-1.30)

Despite my research, I don't know if I can interpret it as a classic cox model or not.

• Please edit the question to provide more information about the data set and your study. There's a danger that the association is the other way around: those who are sick enough to relapse don't have physical activity. There's also a problem that, if the activity score changes in between 5 and 10 months, you won't know which score is appropriate to use: a Cox model is based on the predictor values in place at every event time in the data set, so there's a big risk that your model chose some incorrect values for times between 5 and 10 months (or after 10 months). Was there a score at 0 months?
– EdM
Apr 3, 2023 at 16:39
• the index score was done 5 months after their lung cancer remission period and then 5 months after, i.e. 10 months. My sample is small, in total out of 200 participants, more than half of them relapsed. I created my dataframe with the time intervals so the physical activity index score at the second line (i.e. 10 months) changes. My baseline is 5 month so t start is 0 days. if i interpret as a classical HR it would mean that for each unit increase in the physical activity index score, the risk of relapse decreases by 35%? Apr 3, 2023 at 16:59
• I did cox time dependent because some people relapse before the second score (at 10month) Apr 3, 2023 at 16:59
• To make it easier for others to find, and to avoid a loss of information if a comment gets deleted, I tried to add the critical information from your comments into the question itself. Please check that my editing of your question correctly describes your situation, and re-edit if it doesn't.
– EdM
Apr 3, 2023 at 17:17
• Thank you, my principal question is in general when we do a cox analyse time-dependent do we interpret it as a classical cox analysis ? Apr 3, 2023 at 17:25

A big difficulty here is that even a "classical" Cox analysis with all predictor values specified at time = 0 makes a very specific assumption about the association between predictor variables and outcome. It assumes that the only thing that matters is the current values of the predictor variables at each event time.

To fit the model, at each event time the values of the predictor variables of the case having the event are compared against those of all the cases at risk. There is nothing included about the past history of a predictor variable, unless that past history is somehow incorporated into a new predictor variable. If a predictor's value should change from its initial value during the course of the study, then the model will use an incorrect value for it in calculations at all subsequent event times.

To answer your question most directly: insofar as those assumptions of the Cox model are met, then an interpretation of a hazard ratio with respect to current predictor values is valid.

The problem is that situations with time-varying covariates and data sets like yours often don't meet those assumptions.

When time-varying covariates are included in a model, difficulties are compounded. The fact that you have a predictor value available at some time means that you already know that the individual is alive at that time. In your case, I suspect that there were deaths, not just relapses, before 10 months.

In your data, if there is were any larges changes in the activity index between 5 and 10 months, then it's likely that the value was different from the value at 5 months for a good deal of the intervening time. Thus all calculations based on events during that time period between 5 and 10 months could be erroneous, not just those for the cases with incorrect index values. Similarly, if the activity index is changing over time, it's likely that many calculations based on events after 10 months are also in error if they use the values at 10 months. There are ways to do joint modeling of covariate values along with survival outcomes, but I don't how well they would work with only 2 time points for your index.

There's also a problem with the direction of causality. If someone at 5 months was feeling ill due to an impending but not yet clinically detectable relapse, such an individual would probably score low on the physical activity score. In that situation, the association of a low physical activity score with faster relapse might be due to the clinical biology leading to the relapse, rather than the other way around.

Thus it will be difficult to give a reliable interpretation of the model results. It would have to very carefully stated, something like "the hazard associated with the most recently observed activity index was..." Even that type of statement would not deal with potential miscalculations due to changes in the index during intermediate times, or the problem of the direction of causality.