AFT-model with time-varying covariates and left-truncation
On Thu, Jan 28, 2010 at 2:32 PM, Philipp Rappold
<philipp.rappold at gmail.com> wrote:
Dear Prof. Brostr?m, 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. G?ran
Thanks a lot for your help! All the best Philipp
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G?ran Brostr?m