[R] logLik.lm()

[Prof Brian Ripley]

Please try to read the whole paragraph.  When you have done that, read my 
exposition in my PRNN book.

Hint 1: AIC is about maximized likelihood, and one of the models being 
compared is not being fitted by ML.

Hint 2: Differences in AIC for non-nested models are subject to large
sampling fluctuations (and Mr Dick was worrying about differences of the 
order of 0.06). 


On Fri, 27 Jun 2003, Ravi Varadhan wrote:

> This is not a typical R posting, but I was quite surprised to read 
> Prof. Ripley's comment about the inappropriate use of AIC to 
> compare "non-nested" models. As he says, While it is indeed true that 
> Akaike's (1973) develops AIC for nested models, i.e. models which can 
> be obtained by various restrictions on parameters, it is not at all 
> obvious to me that it can't be used for non-nested cases. 

That it is not obvious to you does not make it true.  You have to prove
that methods work, not that they don't work (a common mistake).

> To quote Stone (1977, JRSS B): "Akaike's derivation of AIC was for 
> heirarchical models but, as he finally remarked, this restriction is 
> unnecessary."  I don't know where Akaike made this remark - I couldn't 
> see it in his 1973 paper - but AIC has indeed been used in various 
> situations where the models are non-nested. From the motivation of AIC 
> as an unbiased estimator of the Kullback-Leibler divergence of asssumed 
> model from the "true" model, it is not clear that the models have to be 
> nested. 
> 
> Any thoughts or comments on this issue?
> 
> Best,
> Ravi.
> 
> 
> ----- Original Message -----
> From: Prof Brian Ripley <ripley at stats.ox.ac.uk>
> Date: Wednesday, June 25, 2003 2:59 pm
> Subject: Re: [R] logLik.lm()
> 
> > Your by-hand calculation is wrong -- you have to use the MLE of 
> > sigma^2.
> > sum(dnorm(y, y.hat, sigma * sqrt(16/18), log=TRUE))
> > 
> > Also, this is an inappropriate use of AIC: the models are not 
> > nested, and
> > Akaike only proposed it for nested models.  Next, the gamma GLM is 
> > not a
> > maximum-likelihood fit unless the shape parameter is known, so you 
> > can'tuse AIC with such a model using the dispersion estimate of shape
> > 
> > The AIC output from glm() is incorrect (even in that case, since the
> > shape is always estimated by the dispersion).
> > 
> > On Wed, 25 Jun 2003, Edward Dick wrote:
> > 
> > > Hello,
> > > 
> > > I'm trying to use AIC to choose between 2 models with
> > > positive, continuous response variables and different error
> > > distributions (specifically a Gamma GLM with log link and a
> > > normal linear model for log(y)). I understand that in some
> > > cases it may not be possible (or necessary) to discriminate
> > > between these two distributions. However, for the normal
> > > linear model I noticed a discrepancy between the output of
> > > the AIC() function and my calculations done "by hand."
> > > This is due to the output from the function logLik.lm(),
> > > which does not match my results using the dnorm() function
> > > (see simple regression example below).
> > > 
> > > x <- c(43.22,41.11,76.97,77.67,124.77,110.71,144.46,188.05,171.18,
> > >        
> > 204.92,221.09,178.21,224.61,286.47,249.92,313.19,332.17,374.35)> y 
> > <- c(5.18,12.47,15.65,23.42,27.07,34.84,31.03,30.87,40.07,57.36,
> > >        47.68,43.40,51.81,55.77,62.59,66.56,74.65,73.54)
> > > test.lm <- lm(y~x)
> > > y.hat <- fitted(test.lm)
> > > sigma <- summary(test.lm)$sigma
> > > logLik(test.lm)
> > > # `log Lik.' -57.20699 (df=3)
> > > sum(dnorm(y, y.hat, sigma, log=T))
> > > # [1] -57.26704
> > 
> > -- 
> > 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)
> > Oxford OX1 3TG, UK                Fax:  +44 1865 272595
> > 
> > ______________________________________________
> > R-help at stat.math.ethz.ch mailing list
> > https://www.stat.math.ethz.ch/mailman/listinfo/r-help
> > 
> 
> 

-- 
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)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595