When models fitted using REML are compared, the fixed structure needs to be the same between all models. However, when models are fit using ML (to compare fixed effects), does the random structure need to be the same?
from Zuur et al. (2009; PAGE 122):
To compare models with nested fixed effects (but with the same random structure), ML estimation must be used and not REML.
This suggests to me that yes, the random effects need to be the same. [Zuur et al. 2009. Mixed Effect Models and Extensions in Ecology with R. Springer.]
however, from Doug Bates SASmixed vignette:
When models are fit by maximum likelihood, ...these quality-of-fit criteria can be used to evaluate different fixed-effects specifications or different random-effects specifications or different specifications of both fixed effects and random effects
Am I misinterpreting the Zuur et al (2009) quote?