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
The difference in this simple example is slight, but
it is magnified when using my data. My understanding is
that it is necessary to use the complete likelihood functions
for both the Gamma model and the 'lognormal' model (no constants
removed, etc.) in order to accurately compare using AIC.
Can someone point out my error, or explain this discrepancy?
Thanks in advance!
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