Extended Cox model (Cox model with time-varying covariates)

[Yoshihiro Tanaka]

Dear STATALIST

I would like to investigate the association of baseline treatment A (binary variable) with incident stroke events adjusted for covariates X, Y, and Z.
Proportionality assumption was not satisfied in A and X by Schoenfeld residuals and log-log plot.
Therefore, I try to use covariates A and X as a time-varying covariate.
In this setting, I have no idea which following formula is correct in Stata because someone used 1 and the other used 2;
1. stcox A X Y Z, tvc (A X) nohr
2. stcox Y Z, tvc (A X) nohr

If 1 is correct, you will get two beta coefficients from main and tvc for A.
How can I summarize the result for paper?
Is it OK with me to show the result like beta (main) + beta (tvc)*_t (time)? or other choices?

Could you please answer these questions?

[Paul Dickman]

I also use expression 1 and believe it is the most transparent. It's possible expression 2 is equivalent (i.e., stcox includes the main effects of any covariate specified in tvc()) but I don't have Stata handy to test that.

Note that your tvc model is just one of many possible ways of having non-proportional hazards. The form of the non-proportional hazards is specified using the texp() option and the default is texp(_t). That is, your model assumes the HRs for A and X change as linear functions of time. If that's not appropriate then you need to do something else (e.g., change texp() or stsplit). Given your final question (on how to show the results) it appears you already understand this but I thought it was worth mentioning anyway.

Your model also assumes that the effects of X,Y,and Z are linear; that assumption should also be evaluated.

How can I summarize the result for paper?
Is it OK with me to show the result like beta (main) + beta (tvc)*_t (time)? or other choices?

Yes, that would be technically correct.

However, in medical science it is more common to report hazard ratios. You could show the corresponding formula but on the HR scale. Alternatively, you could report the estimated HRs at selected values of time or you could present a graph of the HRs as a function of time.

[Yoshihiro Tanaka]

Dear Paul Dickman,

Appreciated your clear explanations and suggestions.
Your comments are really helpful for understanding.

I don`t usually change texp() option but a simple heaviside function can be performed by specifying texp() like texp(_t <1.0).
In my case, a proportionality assumption in A changed dramatically within the first one year according to the log-log plot, and is this an alternative to compute each hazard ratio in each different time period in the following way?
stcox A X Y Z, tvc (A X) nohr texp(_t < 1.0)

Finally, in addition to summarizing results, I also would like to add stratified analysis to compare HRs in different subgroups if possible.
But, HR is time-dependent, and we need to specify the time for comparison or compare the HR-time curve graphically as you pointed out.
Do you have alternatives to compare HRs in this setting?

Best regards,
Yoshihiro Tanaka

[Paul Dickman]

is this an alternative to compute each hazard ratio in each different time period in the following way?
stcox A X Y Z, tvc (A X) nohr texp(_t < 1.0)

Yes. For A and X you will report two hazard ratios, one for the first year of follow-up and one for after 1 year.

I would need more details to be able to answer your second question (compare HRs in different subgroups). Specifically, what are your subgroups? Are they defined by variables already in your model? The standard approach to comparing HRs for your baseline treatment (A) within levels of a factor B would be fit an A by B interaction. That A is time-varying makes interpretation slightly more complicated, but it's doable.

[Yoshihiro Tanaka]

Dear Paul Dickman,

I am thinking about the stratified analysis by the levels of non-time-dependent covariate, Z.
Yes, I agree with you in terms of the comparison of HRs. Interpretation is also one of my concerns as you mentioned.
Finally, could you please tell me how to calculate interaction between A and the levels of Z?
I have no idea if I need just one interaction term (1) or need two interaction terms for main and tvc (2).

1. stcox A##Z X Y, tvc(A X)
2. stcox A##Z X Y, tvc(A##Z X)