Background: Note: My data set and R code are included below text
I wish to use AIC to compare two mixed effects models generated using the
lme4 package in R. Each model has one fixed effect and one random effect. The fixed effect differs between models, but the random effect remains the same between models. I've found that if I use
model2 has the lower AIC score, but if I use
model1 has the lower AIC score.
Support for using ML:
Zuur et al. (2009; p. 122) suggest that "To compare models with nested fixed effects (but with the same random structure), ML estimation must be used and not REML." This indicates to me that I ought to use ML since my random effects are the same in both models, but my fixed effects differ. [Zuur et al. 2009. Mixed Effect Models and Extensions in Ecology with R. Springer.]
Support for using REML:
However, I notice that when I use ML, the residual variance associated with the random effects differs between the two models (
model1 = 136.3;
model2 = 112.9), but when I use REML, it is the same between models (model1=model2=151.5). This implies to me that I ought instead to use REML so that the random residual variance remains the same between models with the same random variable.
Doesn't it make more sense to use REML than ML for comparisons of models where the fixed effects change and the random effects remain the same? If not, can you explain why or point me to other literature that explains more?
# Model2 "wins" if REML=TRUE: REMLmodel1 = lmer(Response ~ Fixed1 + (1|Random1),data,REML = TRUE) REMLmodel2 = lmer(Response ~ Fixed2 + (1|Random1),data,REML = TRUE) AIC(REMLmodel1,REMLmodel2) summary(REMLmodel1) summary(REMLmodel2) # Model1 "wins" if REML=FALSE: MLmodel1 = lmer(Response ~ Fixed1 + (1|Random1),data,REML = FALSE) MLmodel2 = lmer(Response ~ Fixed2 + (1|Random1),data,REML = FALSE) AIC(MLmodel1,MLmodel2) summary(MLmodel1) summary(MLmodel2)
Response Fixed1 Fixed2 Random1 5.20 A A 1 32.50 A A 1 6.57 A A 2 24.77 A B 3 41.69 A B 3 34.29 A B 4 1.80 A B 4 10.00 A B 5 15.56 A B 5 4.44 A C 6 21.65 A C 6 9.20 A C 7 4.11 A C 7 12.52 B D 8 0.25 B D 8 27.34 B D 9 11.54 B E 10 0.86 B E 10 0.68 B E 11 4.00 B E 11