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

[Nick Cox <njcoxstata@gmail.com>]

Maarten gave the reference in the post you are replying to.

Nick

On Sun, Jan 8, 2012 at 6:08 PM, Yuval Arbel <yuval.arbel@gmail.com> wrote:
> 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.
>>
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