[R-sig-ME] conditional vs. marginal coefficients in GLMM [was: Shrinkage of ORs in a glmm]

David Atkins datkins at u.washington.edu
Tue Mar 8 05:48:32 CET 2011

Hi Lorenz--

Building on David Duffy's post, there is, in fact, a notable difference 
between the fixed-effects generated from a GLMM and the fixed-effects 
from a GEE -- the former are conditional on the random-effects (ie, they 
do not "average over" the random-effects) whereas the latter are marginal.

The best description of this that I have found is the article by 
Heagerty and Zeger in Statistical Science (2000, I think).

The GLMM are conditional because:

-- there is a non-identity link function, and
-- the random-effects are part of the linear predictor

because of these, the random-effects have zero mean on the scale of the 
linear predictor, but *not* on the scale of the outcome.  Raudenbush and 
Bryk (2002) also discuss this in their book.

To the list generally, I am curious as to how folks have addressed this 
in applied settings -- that is, as far as I can tell, many times we *do* 
want to interpret GLMM fixed-effects as if they were marginal.  Heagerty 
and Zeger present formula for converting between the two, but not sure 
whether that would be possible to use with glmer() output or not.

Would be happy to provide an example with real data if that were helpful 
to clarify the issues.

cheers, Dave

On Mon, 7 Mar 2011, lorenz.gygax at art.admin.ch wrote:

 > Dear Mixed-modelers,
 > in a recently submitted paper, we used a glmm to estimate the risk of
 > claw injuries in dairy cows on a set of 36 farms that differed in the
 > type of flooring and were visited three times.
 > So far, we have worked with glmmPQL but may now switch to glmer because
 > we were asked to calculate LR-tests by one of the reviewers. In
 > addition, we are asked that we shrink our estimated odds-ratios at the
 > population level. So far we have calculated these as e to the power of
 > the estimated parameters.
 > I have thought that shrinkage happens automatically in a mixed-effects
 > model but this does not seem to be the case depending on the numerical
 > implementation (see comment of the reviewer below). Is this additional
 > shrinkage indeed necessary and is there an easy implementation on how
 > this shrinkage can be done in e.g. lme4?

If I understand this correctly, his comments are wrt marginal versus
mixed-effects estimates for fixed effects.  Fitzmaurice gives some nice
numerical examples in lecture notes online (towards the end)


Using PQL is also complicated by its problems with bias.

Simplest might be to fit a GEE and compare the results, but taking your
predictions from a logistic mixed model, converting these to absolute
risks and looking at risk differences for the different flooring
for animals across the range of farms would address the practical
outcomes.  The _attributable risk_ would be the predicted
total fall in (annual) injuries if one changed all farms to the best floor
covering.  This obviously can be specific to your universe of 36 farms, or
extended to a larger population of farms assuming a particular
distribution of fixed and random effects. This type of number can also be
plugged into a cost-effectiveness analysis.

Cheers, David Duffy.

| David Duffy (MBBS PhD)                                         ,-_|\
| email: davidD at qimr.edu.au  ph: INT+61+7+3362-0217 fax: -0101  /     *
| Epidemiology Unit, Queensland Institute of Medical Research   \_,-._/
| 300 Herston Rd, Brisbane, Queensland 4029, Australia  GPG 4D0B994A v

Dave Atkins, PhD
Research Associate Professor
Department of Psychiatry and Behavioral Science
University of Washington
datkins at u.washington.edu

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