[R] model selection, stepAIC(), and coxph() (fwd)

[Prof Brian Ripley]

On Mon, 8 May 2006, Thomas Lumley wrote:

> On Sat, 6 May 2006, Chad Reyhan Bhatti wrote:
>
>> Hello,
>>
>> My question concerns model selection, stepAIC(), add1(), and coxph().
>>
>> In Venables and Ripley (3rd Ed) pp389-390 there is an example of using
>> stepAIC() for the automated selection of a coxph model for VA lung cancer
>> data.
>>
>> A statistics question:  Can partial likelihoods be interpreted in the same
>> manner as likelihoods with respect to information based criterion and
>> likelihood ratio tests?  It seems that they should be treated as
>> quasilikelihoods which would make stepAIC() invalid and would require the
>> use of add1() with a F-test for the reduction in deviance.
>
> Since this is a question about the MASS book you would be better off
> contacting the authors.
>
> They do (as usual) know what they are doing.  The Cox model is an
> unusually (perhaps uniquely) well-behaved semiparametric model, and the
> partial likelihood really does behave this way.
>
> - For data without ties in the survival time the partial likelihood is
> (proportional to) the marginal likelihood of the ranks, so it is a
> perfectly good parametric likelihood. (Kalbfleisch & Prenctice,
> Biometrika, 1973)
>
> - The chi^2 distribution (rather than F distribution) for the likelihood
> ratio test is justified by the marginal likelihood, or by martingale
> arguments (eg the book by Fleming and Harrington), or in more modern times
> by empirical process arguments or as a semiparametric profile likelihood.
> However, the only technically hard part is showing weak convergence -- the
> original paper by Cox showed that the variance of the partial score and
> the Hessian of the partial likelihood were the same, which is the key fact
> for the chi^2 rather than F test to be valid (if one of them is)
>
> - The same arguments suggest AIC will be appropriate for comparing
> different subsets of variables in the same way that it is for generalized
> linear models. I don't have a reference here.

Therneau & Grambsch (2000) use AIC defined in this way and I got the idea 
directly from one of those authors (and I forget which).

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
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