[R] polynomials REML and ML in nlme

[Douglas Bates]

Berton Gunter wrote:
> Duncan:
> 
> Bates and Pinheiro do show explicitly and in detail that the restricted log
> likelihood depends on the parameterization: "An important difference between
> the likelihood function and the restricted likelihood is that the former is
> invariant to one-to-one reparametrizations of the fixed effects ... while
> the latter is not." (p.76)
> 
> So that answers both your questions: With different fixed effect
> parameterizations you get different restricted log likelihoods even for the
> same random effects specifications. So you cannot compare models and you see
> differences of the sort you saw. That's just the way REML works.
> 
> I believe the the basic issue can be summarized as: the restricted log
> likelihood is a nonlinear function of the fixed effects specification. Hence
> linear transformations (reparametrizations) of the fixed effects change the
> log likelihood. But if you want more than that, get the book (or another
> reference of your choice). You could also try googling "REML".
> 
> All the above is subject to correction by Doug Bates, of course.

Doug has nothing to add.  Good explanation.  Thanks.