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