[R-sig-ME] AIC and BIC with ML and REML

[Gorjanc Gregor]

Hi!

A time ago I forgot the details about the calculation about how AIC and BIC are calculated and I did
a search. All went fine that day, i.e., AIC = - 2 * l + 2 * p, where l is log-likelihood value and p is a number
of parameters in the model. Similar for BIC. However, I have read a paper today where they state that when
we fit a model using ML we should use

AIC = - 2 * l + 2 * (p + q),

while the following should be used in the case of REML

AIC = - 2 * l + 2 * q,

where l is as above, p is a number of free parameters in the fixed part of the model and q is the number of
variance components for the random part of the model. Is this correct? I checked the calculations
with lmer() and it seems that only the first approach is taken. Can anyone comment on this?

Thank you!

mu <- 10
a <- c(0, 3)
b <- rnorm(n=20, sd=3)

facA <- gl(n=2, k=100)
facB <- gl(n=20, k=10)

y <- rnorm(n=length(facA), mean=(mu + a[facA] + b[facB]), sd=3)

library(package="lme4")

lmer(y ~ facA + (1 | facB))
##  AIC  BIC logLik deviance REMLdev
## 1054 1067 -522.9     1050    1046
##
## (1054 - 2 * 522.9) / 2 = 4.1 --> 4 parameters, mu, a2, sigma^2_b, and sigma^2_e --> p = 2, q = 2

lmer(y ~ facA + (1 | facB), REML=FALSE)
##   AIC  BIC logLik deviance REMLdev
##  1058 1071 -524.9     1050    1046
##
## (1058 - 2 * 524.9) / 2 = 4.1  --> 4 parameters, mu, a2, sigma^2_b, and sigma^2_e --> p = 2, q = 2

Lep pozdrav / With regards,
    Gregor Gorjanc
----------------------------------------------------------------------
University of Ljubljana       PhD student
Biotechnical Faculty          www: http://gregor.gorjanc.googlepages.com
Department of Animal Science  blog: http://ggorjan.blogspot.com
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