[S] Re: Why is AIC labeled Cp in step()?

Trevor Hastie (trevor@stat.stanford.edu)
Fri, 27 Feb 1998 11:01:32 -0800 (PST)

The reason a distinction was made is that drop1() and add1() use a
quadratic (Gaussian) approximation to the log-likelihood, and the
comparisons are based on Chi-squared approximations. Having selected a
candidate using drop1() and add1(), step() then fits the model by ML
and reports the AIC (which is on the deviance scale, rather than
Chi-Squared). Leaving both named AIC might have been even more confusing.

A final detail, which is explained in "Statistical Models in S", is
that Cp is defined as "RSS + 2p.sigma^2", which differs from the
original definition. This version estimates total prediction
error. Likewise AIC is defined to be "Deviance +2p.phi", with phi the
dispersion parameter for the GLM, and also measures prediction error
using the log-likelihood loss.

Trevor Hastie

> Subject: [S] Why is AIC labeled Cp in step()?
> Hello all,
> I'm teaching a linear models course to stat grad students. Since I
> know they will eventually run into people who use stepwise methods of
> model selection, I'm trying to explain how those methods work. (All the
> better to express my deep distrust of their black-box-iness) I realize
> that in normal theory linear models AIC and Cp are equivalent measures,
> but I don't understand why the step(), add1() and drop1() functions give
> a column labeled Cp which is really AIC.
> Quoting from the _Guide to Stat & Math Analysis_ p 7-13:
> " The Cp statistic (actually what is shown is the AIC statistic, the
> likelihood version of the Cp statistic - the two are related by the
> equation AIC=sigmahat^2(Cp+n)) provides a convenient criterion for
> determining whether a model is improved by dropping a term."
> I also see in V&R MASS2 p 220-2 that their stepAIC() does actually print
> AIC at the top of the column.
> My concern is that people see the column labeled Cp and think that a
> good model will have values close to p (# of parameters). Is there some
> reason that this column has been mislabeled? Could we get it changed?
> Jim Robison-Cox ____________
> Department of Math Sciences | | phone: (406)994-5340
> 2-214 Wilson Hall \ BZN, MT | FAX: (406)994-1789
> Montana State University | *_______|
> Bozeman, MT 59717 \_| e-mail: jimrc@math.montana.edu
Trevor Hastie, Associate Professor trevor@stat.stanford.edu
Phone: 650-725-2231 Fax: 650-725-8977
address: room 104, Department of Statistics, Sequoia Hall
370 Serra Mall, Stanford University, CA 94305-4065
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