# [R] GLMMs with Gauss-Hermite quadrature

Jim Lindsey james.lindsey at luc.ac.be
Tue May 29 11:37:07 CEST 2001

```There has recently been some discussion of GLMMs on this list.
Unfortunately, I deleted the messages so that this is from memory.

The original question was about doing a homework problem concerning
mixed logistic regression in R for a course based on SAS. Clearly, the
R function to use is my glmm because it uses essentially the same
algorithm as SAS (the SAS Gauss-Hermite used an adaptive method that
can reduce the number of quadrature points for the same precision).

Although I have made available this function, I never recommend its
use because I think this type of model is very artificial, except
possibly in some animal breeding situations. I have never been able to
understand why GLMMs, with their normal mixing distribution, are
popular. If someone must use mixed logistic regression in this sense,
I strongly recommend using a specialized program such as Sabre or
Egret for a variety of reasons.

At least two points came up in the discussion that I would like to

Restrictions

The main restrictions to glmm come from glm itself: 1) linear
regression and 2) exponential family conditional distribution plus 3)
the fact that the normal mixing distribution is imposed on them.
The latter is typical of the sort of general models that statisticians
love to develop that have no real mechanistic interpretation.

Troels pointed out that I have more general functions available that
lift these restrictions. gnlmm removes the two imposed by glm. Any
nonlinear model can be fitted with a wide variety of conditional
distributions. However, the mixing distribution is still normal and in
addition, in nonlinear models, a random "intercept" may often be
unnatural. However, I also have gnlmix which removes all restrictions.
It has nonlinear regression with a wide variety of conditional
distributions where any one nonlinear parameter can be random with a
mixing distribution chosen from a wide selection. Romberg integration is
used so that it is essentially exact, if rather slow. It would be
simple to extend it to say two nonlinear random parameters but time
would quickly become rather exhorbitant.

Approximations

When discussing GLMMs using Gauss-Hermite integration, the question of
approximations is essentially a red herring. No matter the number of
quadrature points, the likelihood is always exact (as exact as any
likelihood can be on a digital computer). It is a finite mixture. The
approximation question arises in the sense that this finite mixture is
more or less close to a Gaussian mixing distribution, which is a
completely artificial choice in the first place. It is quite possible
for the model with very few quadrature points to fit better than one
with sufficient for a very close approximation to the normal mixing
distribution, indicating that normal mixing is not a good choice.

It is false to say that the properties of this approximation, in the
latter sense are unknown. GLMMs using Gauss-Hermite go back at least
to an unpublished tech report by Don Pierce in the mid 70s and the
published paper by John Hinde in 1982. Since then, there is a vast
literature on the subject of the approximation in the second sense
above, especially for the model in question, mixed logistic
regression, including work by Alan Agresti, Murray Aitkin, David
Brillinger, Bruce Lindsay, etc.  (There is also another literature on
approximations replacing Gauss-Hermite, such as Breslow and Clayton.)
The most recent published reference that I am aware of is earlier this
year, but I have refereed others that are not yet in print. 15 to 20
quadrature points gives an extremely close numerical approximation to
the normal mixing distribution for most data sets, for what that is
worth.
Jim
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