Observation-level random effect to model
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