logLik.lm()
Dear Prof. Ripley:
I gather you disagree with the observation in Burnham and Anderson
(2002, ch. 2) that the "complexity penalty" in the Akaike Information
Criterion is a bias correction, and with this correction, they can use
"density = exp(-AIC/2)" to compute approximate posterior probabilities
comparing even different distributions?
They use this even to compare discrete and continuous distributions,
which makes no sense to me. However, with a common dominating measure,
it seems sensible to me. They cite a growing literature on "Bayesian
model averaging". What I've seen of this claims that Bayesian model
averaging produces better predictions than predictions based on any
single model, even using these approximate posteriors ("Akaike weights")
in place of full Bayesian posteriors.
I don't have much experience with this, but so far, I seem to have
gotten great, informative answers to my clients' questions. If there
are serious deficiencies with this kind of procedure, I'd like to know.
Comments?
Best Wishes,
Spencer Graves
###############
REFERENCE: Burnam and Anderson (2002) Model Selection and Multimodel
Inference, 2nd ed. (Springer)
Prof Brian Ripley wrote:
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't use 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