Joerg Heining <[email protected]> asks:
> right now I am concentrating my studies on the analysis of duration data.
> While working through the Stata manual (version 8) on "Survival Analysis and
> Epidemiological Tables" I found and remark which I could not understand so
> far. On p. 204 the manual says that the relationship between the coefficents
> of a PH and an AFT Weibull model does only hold if the ancillary parameter
> is a constant. It does not hold when the ancillary parameter is parametrized
> in terms of covariates, e.g. using the strata option. Can anyone explain
> this to me?
Consider the following PH Weibull model, run on the cancer data:
. streg drug, dist(weib) nohr
[stuff deleted]
------------------------------------------------------------------------------
_t | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | -1.20743 .251076 -4.81 0.000 -1.69953 -.7153303
_cons | -2.595273 .6680031 -3.89 0.000 -3.904535 -1.286011
-------------+----------------------------------------------------------------
/ln_p | .4660248 .1454774 3.20 0.001 .1808943 .7511554
-------------+----------------------------------------------------------------
p | 1.593647 .2318396 1.198289 2.119447
1/p | .6274917 .0912859 .4718211 .8345235
------------------------------------------------------------------------------
The manual claims that the mapping from PH to AFT is one-to-one, and
examination of Table 1 in [ST] streg shows that if I take the coefficients of
the above PH model and divide them by -p, I'll get the coefficients of the AFT
model. Namely,
[-1.20743, -2.595273] / -1.593647 = [.75765, 1.6285]
and running the AFT model confirms this:
. streg drug, dist(weib) time
[stuff deleted]
------------------------------------------------------------------------------
_t | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
drug | .7576525 .1594317 4.75 0.000 .445172 1.070133
_cons | 1.628512 .2760891 5.90 0.000 1.087387 2.169637
-------------+----------------------------------------------------------------
/ln_p | .4660248 .1454774 3.20 0.001 .1808943 .7511554
-------------+----------------------------------------------------------------
p | 1.593647 .2318396 1.198289 2.119447
1/p | .6274917 .0912859 .4718211 .8345235
------------------------------------------------------------------------------
The point is once you have fit a PH Weibull model, it would not be necessary
to rerun the estimation in order to get the AFT model coefficients, but I
usually rerun it anyway because my computer can fit a Weibull AFT model
faster than I can divide by hand.
When you stratify, however, you are stipulating that p is not constant, but
instead parameterized as
ln(p) = some linear combination of covariates in your data
As such, the above by-hand conversion from PH to AFT is impossible. There
is no single p to divide by. Instead, p varies with the data.
No worries, though. Just re-estimate in this case.
--Bobby
[email protected]
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