# Survival analysis with time dependent covariates and cured fraction

I have a problem specified in this way, I'll make a fictional example, because the actual data requires quite a bit of domain knowledge to be understood.

There is a series of newborn babies (let's say 10000), affected by a rare condition that makes their chance of survival really small.

We measure the concentration of a certain protein (denoted by variable $x_1$) each day at 9 am for 20 years.

What we observe is that most of the babies die within 1 year, and almost all are gone within 3 years. However there is a very small fraction of these babies that survive all the way (and expected to live a normal life).

I'd like to model if there is a relation between the level of this protein and the chance of survival. But I have both time-dependent covariates and a very small, but crucial, "immortal" fraction. So far I found that the extended cox model can be used to model time-varying covariates, but not a cured fraction.

I found that there are parameteric mixed models that allow for cured fraction but not time-varying covariates.

How would you approach this problem?

• parametric models can deal with time-varying covariates. So I would start with reconsidering that assumption. – Maarten Buis Apr 21 '15 at 8:57
• @Maarten Thanks for the comment! Can you please recommend me a reference (book or paper) to read? – pygabriel Apr 21 '15 at 16:01
• A good introduction is: Paul Lambert (2007) "Modeling of the cure fraction in survival studies". The Stata Journal, 7(3): 351-375. link The author and others have continued to work in this area, so also look for follow up articles. – Maarten Buis Apr 22 '15 at 7:52