Likelihood ratio tests using ML vs. REML I am using Mixed effects models (nlme package in R) to choose the model with the best random and fixed effects.  I am following the procedure of Zurr et al. (2009) and read about "testing on the boundary" and the effects that it has on p values.  From what I have read, it looks as though this is only something that is a problem when using REML, but I am not sure.
My questions are:  


*

*When performing a LRT between 2 nested models which only differ in fixed effects (thus using ML instead of REML), do the p values have to be adjusted for testing on the boundary?

*If this is a problem only for REML estimation, why?  Does it have to do with the REML estimation itself, or the fact that it is usually used with the LRT when comparing models that differ in random effects?
 A: Regarding your first question, if the two models only differ in the fixed effects, no parameters are on the boundary in any of the models (since all coefficients can take values on the complete real line, not just on the positive numbers, like variances must).
For example, in the smaller model, one parameter (a regression coefficient) is set to zero, but in the larger models it can be both greater or less than zero, so there is no problem.
Regarding your second question, parameters on the boundary are a problem for likelihood ratio test in general (not just for mixed-effects models). The asymptotics break down when the parameter(s) in one of the models are on the boundary of the parameter space. So yes, this is a problem for ML estimation (too).
In mixed-effects models the parameter is often a variance. In the smaller model it is set to zero, but in the larger model it can only be greater than zero. So in the smaller model, it is on the boundary of the parameter space (which goes from zero to positive infinity), and the asymptotics break down.
