[R] AFT-model with time-varying covariates and left-truncation

[Göran Broström]

On Thu, Jan 28, 2010 at 2:32 PM, Philipp Rappold
<philipp.rappold at gmail.com> wrote:
> Dear Prof. Broström,
> Dear R-mailinglist,
>
> first of all thanks a lot for your great effort to incorporate time-varying
> covariates into aftreg. It works like a charm so far and I'll update you
> with detailled benchmarks as soon as I have them.
>
> I have one more questions regarding Accelerated Failure Time models (with
> aftreg):
>
> You mention that left truncation in combination with time-varying covariates
> only works if "...it can be assumed that the covariate values during the
> first non-observable interval are the same as at the beginning of the first
> interval under observation.". My question is: Is there a way to use an AFT
> model where one has no explicit assumption about what values the covariates
> have before the subject enters the study (see example below if unclear)? For
> me personally it would already be a great help to know if this is
> statistically feasible in general, however I'm also interested if it can me
> modelled with aftreg.

The AFT model with time-fixed acceleration factor  a  is S(t; a) =
S_0(at) for some S_0.
With a time-varying  a = a(t), this becomes  S(t; a) = S_0(\int_0^t a(s) ds),
and in order to evaluate that you need the full history of  a  at each  t > 0.

> EXAMPLE (to make sure we're talking about the same thing):
> Suppose I want to model the lifetime of two wearparts A and B with
> "temperature" as a covariate. For some reason, I can only observe the
> temperature at three distinct times t1, t2, t3 where they each have a
> certain "age" (5 hours, 6 hours, 7 hours respectively). Of course, I have a
> different temperature for each part at each observation t1, t2, t3.
> Unfortunately at t1 both parts have not been used for the first time and
> already have a certain age (5 hours) and I cannot observe what the
> temperature was before (at ages 1hr, 2hr, ...).

The important thing here is whether you have left-truncated
_lifetimes_ or not. Your example is about missing observation(s) on a
covariate, which is a different problem. But a problem. And not only
for the AFT model, but for the PH model as well.

Göran

> Thanks a lot for your help!
>
> All the best
> Philipp
>
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-- 
Göran Broström