Re: st: stcox in case the ph-assumption is rejected

[Maarten Buis <maartenlbuis@gmail.com>]

On Sat, Jan 7, 2012 at 4:54 PM, Yuval Arbel wrote:
> Marteen,
>
> I don't see why -stpm2- does not solve my problem. After all -stpm2-
> somewhat relaxes the PH assumption.

Unfortunatley, that is incorrect. You seem to be mistaking a Cox model
for a exponential model: an exponential model assumes that the
baseline hazard function (and the hazard ratios) is constant over
time, a Cox model leaves the shape of the baseline hazard completely
free, in fact it does not even estimate it, it only asumes that the
hazard ratios (the effects of the explanatory variables) are constant
over time. This is called the proportional hazard assumption. In this
respect -stcox- is extremely similar to -stpm2- with the
-scale(hazard) option. Both are part of the general form:

h_i(t) = h_0(t)*exp(b1*x1_i +b2*x2_i ...)

So the hazard of observation i at time t is some baseline hazard
function that depends on time and a multiplier that depends on the
characteristics (the xs) of observation i. -stcox- and -stpm2- differ
with respect to the baseline hazard: -stcox- leaves the baseline
hazard completely free(*), -stpm2- uses a very flexible paramteric
function to approximate the the baseline hazard. In principle one
could say that -stcox- is a bit more flexible in the baseline hazard
as -stpm2-, in practice it is a difference between a very very
flexible baseline hazard function (-stcox-) and a very flexible
baseline hazard function (-stpm2-) So it is no surprise that you find
very similar results. In fact on page 278 of (Lambert and Royston
2009) the authors of -stpm2- note :

"The estimated hazard ratios and their 95% confidence intervals are
very similar to the Cox model, and in fact, there is no difference up
to four decimal places. We have yet to find an example of a
proportional hazards model where there is a large difference in the
estimated hazard ratios between these two models."

Notice that the efects of the xs in both models (in the default
parametrization) do not depend on the time: if x1 increases by 1 unit
the baseline hazard will increase by a factor exp(b1). This is what is
meant with the proportional hazard assumption, and both models make
that assumption. You can relax the proportional hazard assumption by
adding an interaction term between (some function of) time and an x,
which is what the -tvc()- option does, or you can allow the different
groups as represented by an x to have their own baseline hazard, which
is what the -stratify()- option does. To use your analogy with fixed
effects regression, I would say that the stratify option is closest to
fixed effects regression.

Hope this helps,
Maarten (again, _not_ Marteen)

(*) See for example section 7 of
<http://www.maartenbuis.nl/wp/survival.pdf> on how -stcox- can
estimate hazard ratios without estimating the baseline hazard
function.

Paul Lambert and Patrick Royston (2009) Further development of
flexible parametric models for survival analysis. The Stata Journal
9(2):265-290.

--------------------------
Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen
Germany


http://www.maartenbuis.nl
--------------------------
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