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


From   Yuval Arbel <yuval.arbel@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: stcox in case the ph-assumption is rejected
Date   Sun, 8 Jan 2012 20:08:10 +0200

Thanks Maarten, that was very helpful.

Can you recommend on good econometric books that deal with survival
analysis, Cox Regressions and Competing-Risk Models? what is the full
reference for Lambert and Royston (2009)?

On Sun, Jan 8, 2012 at 11:46 AM, Maarten Buis <maartenlbuis@gmail.com> wrote:
> 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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-- 
Dr. Yuval Arbel
School of Business
Carmel Academic Center
4 Shaar Palmer Street,
Haifa 33031, Israel
e-mail1: yuval.arbel@carmel.ac.il
e-mail2: yuval.arbel@gmail.com

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