[R-sig-ME] RE : Generalized mixed effects models

[Douglas Bates]

Questions like this are better sent to the R-SIG-Mixed-Models at R-project.org
mailing list as there are many people who read that list and are able to
respond faster than I am.

I have taken the liberty of cc:'ing the list on this reply.


On Fri, Aug 22, 2014 at 10:43 AM, PLUQUET Thibault <
Thibault.PLUQUET at ensae-paristech.fr> wrote:

> Good afternoon,
>
> I'm a French student. I'm sorry to disturb you. I know for generalized
> linear mixed models we , first, have to do the penalized iteratively
> reweighted least squares, bu why ? I understad why for linear mixed models
> but for generalized linear mixed models , I don't understand why we have
> the matrix W and the formula of the discrepancy function .
>

Evaluating the likelihood for generalized linear mixed models requires
integrating the conditional distribution of the random effects evaluated at
the current parameter values.  For a linear mixed model this integral has a
closed form, because of the properties of the multivariate Gaussian
distribution and the fact that the conditional mean of the response vector
is a linear function of the random effects.  In a generalized linear mixed
model the conditional mean response is a nonlinear function (the inverse
link) of the linear predictor and hence a nonlinear function of the random
effects and the conditional distribution of the response given the random
effects is not Gaussian.  There is no closed form expression for the
integral over the random effects, in general.

There have been many different approaches to approximating this integral.
In the lme4 package we use a Laplacian approximation at the conditional
mode of the random effects or an adaptive Gauss-Hermite quadrature. Both
approaches require determining the conditional mode of the random effects
given the observed data and the current values of the model parameters. We
use the penalized iteratively re-weighted least squares (PIRLS) algorithm
to determine the conditional mode.  The weights are part of the PIRLS
algorithm.

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