[R-sig-ME] Observation-level random effect to model

[Ryan King]

Yes, you are correct.  If a lognormal distribution bothers you, you
can definitely combine conjugate RE for over-dispersion and log-normal
RE elsewhere in the predictor.  In the poisson example, you could
imagine doing random-effect negative-binomial regression.
http://arxiv.org/abs/1101.0990 implemented that using SAS NLMIXED.

Heuristically, you end up with non-zero residuals because the
assumption of normality for the subject-level effect adds an L2
penalty away from the maximum-likelihood estimate, which would be the
entire residual.

Ryan King
Dept Health Studies
University of Chicago

On Mon, Mar 21, 2011 at 8:09 AM,
<r-sig-mixed-models-request at r-project.org> wrote:
> Message: 1
> Date: Mon, 21 Mar 2011 12:51:39 +0100
> From: "M.S.Muller" <m.s.muller at rug.nl>
> To: r-sig-mixed-models at r-project.org
> Subject: [R-sig-ME] Observation-level random effect to model
>        overdispersion
> Message-ID: <7520f6c92de64.4d8749db at rug.nl>
> Content-Type: text/plain; charset=us-ascii
>
> Dear all,
>
> I'm trying to analyze some strongly overdispersed Poisson-distributed data using R's mixed effects model function "lmer". Recently, several people have suggested incorporating an observation-level random effect, which would model the excess variation and solve the problem of underestimated standard errors that arises with overdispersed data. It seems to be working, but I feel uneasy using this method because I don't actually understand conceptually what it is doing. Does it package up the extra, non-Poisson variation into a miniature variance component for each data point? But then I don't understand how one ends up with non-zero residuals and why one can't just do this for any analyses (even with normally-distributed data) in which one would like to reduce noise.
>
> I may be way off base here, but does this approach model some kind of mixture distribution that's a combination of Poisson and whatever distribution the extra variation is? I've read that people often use a negative binomial distribution (aka Poisson-gamma) to model overdispersed count data in which they assume that the process is Poisson (so they use a log link) but the extra variation is a gamma distribution (in which variance is proportional to square of the mean). The frequently referred to paper by Elston et al (2001) describes modeling a Poisson-lognormal distribution in which overdispersion arises from errors taking on a lognormal distribution. Is the approach of using the observation-level random effect doing something similar, and simply assuming some kind of Poisson-normal mixed distribution? Does this approach therefore assume that the observation-level variance is normally distributed?
>
> If anyone could give me any guidance on this, I would appreciate it very much.
>
> Martina Muller