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, [EMAIL PROTECTED] 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) Oxford OX1 3TG, UK Fax: +44 1865 272595 ______________________________________________ [email protected] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
