Cox models adjusting for current age discussion

[agalecki]

Hi Jim, Nick, Deb

Thank you, Jim and Nick for your nice presentation last week. It caught my attention that in Cox model you used age as a time-varying covariate. It prompted me to perform some literature search. Please note that your model is equivalent to a typical approach, i.e. adjusting for age at baseline.

This issue is discussed in more detail in [1] (see attached). In particular, it is stated that

"This [...model with continuous age as a time-dependent covariate] sounds like an improvement over method #1 [... standard approach with age at baseline treated as covariate], but as detailed by Allison (1995, pg 142), by definition this approach gives the exact same results as method #1..When the effect of age is modeled linearly, then age as a time-dependent covariate and age as a fixed covariate are equivalent, since if the effect per year of age is constant, then the effect of a given age difference remains the same over time"

Please note that in both approaches time-on-study (not age) is used as the time-scale. In other words, the results from both approaches should be interpreted in terms of intensity of the underlying process (i.e. developing dementia) versus time-on study (not age).

To be able to use age as the time-scale we should use approach #2 described in [1]

Looking forward to hearing your thoughts Hope it helps,

Andrzej

[1] Canchola, A.J., Stewart, S.L., Bernstein, L., West, D.W., Ross, R.K., Deapen, D., Pinder, R., Reynolds, P., Wright, W., Anton-Culver, H. and Peel, D., 2003. Cox regression using different time-scales. Western Users of SAS Software. San Francisco, California.

Canchola_et_al_CoxModelTimeScales.pdf

[agalecki]

Andrzej-

Your interpretation of the current squares with my understanding. Time-varying implies, in my understanding, an interaction between time and the covariate of interest. In our case, I think that the “time varying” part of the concept is being accounted for by the survival/hazard function which non-linearly slopes upward over time. So, I wouldn’t have called our current baseline model time-varying.

I tried fitting a model with time-varying age in stata (i.e. an age-analysis time interaction), but I didn’t get better model fit and I’m not sure that I’m generating the baseline hazard function correctly or that it doesn’t make a lot of sense. (You have to do some gymnastics to get it to work because the baseline “tvc” parameter on stcox doesn’t allow you to generate baseline hazards).

I think that, though, that a well recalibrated baseline hazard function (using population level data) that starts with baseline age is going to give us a relatively realistic picture of the problem.

I’m completely open to alternatives, but I don’t know how to execute a time-varying model that gives us meaningful baseline survival data (and thus can be turned into state transition probabilities). If anybody can show me that, I can plug it into the simulation!

Jim

[agalecki]

Hi Jim, Nick, Deb:

To follow on our email exchange I asked Nick to run some test programs, did some digging in the literature. Here are my thoughts for consideration:

1. To accommodate time-varying covariates, we are using counting style of input, with subject represented by multiple (start, stop] time intervals.

2. Our initial Cox model for dementia included (start, stop] interval-specific cognition slope as the only time-varying covariate. Data were prepared by Nick according to this specification.

a. Please note that (start, stop] data assume that time-varying covariate remains constant

within a (start, stop] interval. This is a necessary and (I believe) sensible assumption for the interval specific cognition slope.

3. Recently, our Cox model was modified by including the subject’s current age as an additional time-varying covariate. Please note the following.

a. The current age is an example of a continuously varying covariate

b. The assumption that current age covariate stays constant within (start, stop] interval is clearly not valid and for this reason, current age covariate requires modifications in our approach.

c. One simple and elegant way to go around this issue available in SAS PROC PHREG is to use programming statements. It appears though that this feature is not available in Stata.

d. Another approach is to create new analytical (star, stop] data by dividing our time intervals into very small subintervals and recalculate current age accordingly. This is illustrated in https://www.stata.com/support/faqs/statistics/estimate-cox-model/.

Moving forward I think we need to follow the approach described in 3d.

Looking forward hearing from you

Andrzej

[agalecki]

i think that a parametric approach works just fine and would be completely fine switching out the cox model for a parametric one — particularly in a case like this where we understand that the the underlying baseline hazard function is likely smooth...

i’m not entirely sure how stata generates the baseline hazard from a cox model, but somehow it does! Jim