Solved – get equal AIC, BIC and log likelihood for different models in LME framework

aicbicmaximum likelihoodmixed modelr

I have two LME models with the same interaction, one containing both main effects and one containing only one main effect, say :

$$ H\_CE = Season + Crownlevel + Season:Crownlevel , random = 1|CollectorID $$
and
$$ H\_CE = Season + Season:Crownlevel , random = 1|CollectorID $$

There are 4 levels of each, and every combination of Season, Crownlevel and CollectorID
The AIC, BIC and log likelihood of both models are completely equal. Given the formula for AIC being

$$ \mathit{AIC} = 2k – 2\ln(L)\ $$

one would expect this to be different, even if the likelihoods are exactly the same. In the end, they have a different number of parameters. Or so I thought…

Trying this toy example in R :

library(nlme)

Season <- rep(as.factor(rep(letters[1:4],each=4)),4)
Crownlevel <-rep(as.factor(rep(letters[11:14],4)),4)
CollectorID <-rep(letters[20:23],each=16)
X <-  model.matrix(~Season+Crownlevel+Season:Crownlevel)
B <- c(1,1,-2,2,0.3,0.4,0.4,2,3,1,-2,-3,-4,2,1,2)
H_CE <- X %*% B + rnorm(16*4)
KBM <- data.frame(Season,Crownlevel,H_CE,CollectorID)

model1 <- lme(H_CE~Season+Crownlevel+Season:Crownlevel,data=KBM,
       method="ML",random=~1|CollectorID)
model1e <- lme(H_CE~Season+Season:Crownlevel,data=KBM,
       method="ML",random=~1|CollectorID)

I get : 

anova(model1,model1e)
        Model df      AIC      BIC    logLik
model1      1 18 174.1834 213.0433 -69.09168
model1e     2 18 174.1834 213.0433 -69.09168

What am I missing here? Why are the numbers completely equal? It has to do something with the model specification, but I can't really see what.

The model specification in itself is faulty, I know that. But I can't explain what makes it return a different set of parameters, but exactly the same residuals, likelihood and degrees of freedom :

> all.equal(residuals(model1),residuals(model1e))
[1] TRUE

As fabians rightfully pointed out, both models are perfectly equivalent. Yet, I fail to see why in the AIC calculation the same value for the number of parameters k is used.

The k in AIC uses the df, which explains everything.

Best Answer

The models are exactly equivalent. In both models you effectively specify one parameter for each combination of levels of Season and Crownlevel - the only difference is the parameterization:

In the first model, you fit main effects for Season and Crownlevel and an interaction effect to capture the combination-specific deviations from the main effects.

In the second model, you specify only the main effect of season, and the interaction effect then captures the deviations for each crownlevel within a season. 

H_CE~Season:Crownlevel

would also yield an equivalent model, with one parameter for each combination of season and crownlevel (minus one that is non-identifiable because of the intercept, i.e. constitutes the reference category).

BTW: I don't think your model specification is faulty, which specification is better depends on the inference you want to do with your model.

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