[R-sig-ME] lmer: ML and REML estimation

dave fournier otter at otter-rsch.com
Fri Mar 27 19:01:12 CET 2009

  Maybe I'm missing something here. The way I see it,
in its most general form a mixed model is simply one
where some of the parameters are random variables with a prior.
So say the likelihood looks like


where the vector of parameters u have a prior p(u).
If the u are normally distributed with a vector of
std devs sigma this becomes


Integrating over u gives

  F(x,sigma) = \Int F(x,u)p(u,sigma) du

so the MLE's xhat,sigamhat are found by maximizing
F wrt x and sigma.

Now if we are willing to take a Bayesian point of view
we could also integrate over x and get the marginal
likelihood for the sigma alone. We may need a prior on x,
or assume a locally uniform prior. In any event we obtain

 G(sigma) = \Int F(x,u)p(u,sigma) du dx

The integral can be approximated using the Laplace approximation
at the mode of  F(x,u)p(u,sigma).  Now I'm not claiming that this
approach always works well, but it is a simple recipe for producing
an estimation procedure which may give better estimates for sigma.
Using AD Model Builders Random effects module which is freely available
at http://admb-project.org it is simple to
change the model specification to carry out this procedure. As I said
before it seemed to work well in a nonlinear fisheries model I looked

David A. Fournier
P.O. Box 2040,
Sidney, B.C. V8l 3S3
Phone/FAX 250-655-3364

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