Selecting random effects in lme4: ML vs. QAICc
Richard Feldman wrote:
Thank you very much for your informative response. I guess I'm confused about how to interpret overdispersion. Using the Zuur et al. Owls data that you also present in your worked example, I ran the following models, keeping the fixed and random effects just as you present it: glmer, family = gaussian glmer, family = poisson glmer, family = quasipoisson glmmPQL, family = quasipoisson glmmADMB, family = poisson ##zeroInflation=FALSE glmmADMB, family = negative binomial ##zeroInflation=FALSE glmmADMB, family = poisson ##zeroInflation=TRUE glmmADMB, family = negative binomial ##zeroInflation=TRUE For all but the quasipoisson glmer, I calculated dispersion following your example:
rdev<-sum(residuals(model)^2) mdf<-length(fixef(model)) rdf<-nrow(Data)-mdf rdev/rdf
For the quasipoisson glmer, I extracted dispersion as:
>lme4:::sigma(model)^2
The results are as follows: # model dispersion #1 glmer.gaussian 35.534989 #2 glmer.pois 5.630751 #3 glmer.quasi -- sigma(model)^2 29.830221 #4 glmmPQL.quasi 1.076906 #5 glmmADMB.pois 7.585654 #6 glmmADMB.nbinom 1.085072 #7 glmmADMB.pois.zero 8.255389 #8 glmmADMB.nbinom.zero 1.516587 Am I right to interpret this as saying 1) the sigma(model)^2 method is inaccurate and 2) the glmmPQL.quasi and glmmADMB.nbinom are sufficiently correcting for the overdispersion? I had thought I had read you advising using the quasipoisson model to calculate the overdispersion parameter (assuming it did so correctly) needed to adjust AIC for QAIC. It would seem the dispersion parameter should come from the Poisson regression. More likely I just misread you. Also, is it a concern that the dispersion is higher in the zero-inflated models? Does this mean zero-inflation is not an issue, at least when wanting to calculate AIC values?
(1) Yes. I would agree that we don't quite know what's going on with quasi- in glmer, and that using other methods is better if possible: various people have reported odd results, Doug Bates has gone on record as saying he wouldn't really know how to interpret a quasi-likelihood GLMM anyway (I think that's a fair summary of his position), and it's not clear whether the problem is with bugs in a little-tested corner of the software or fundamental problems with the definitions of the model. That said, quasi- is also the easiest way forward ... (2) glmmPQL.quasi uses penalized quasi-likelihood, so at least it's consistent in the way it handles the random effects and the individual variance structures. PQL is known to be a bit dicey for data with small numbers (e.g. means < 5) [Breslow 2003], not that that has stopped lots of people from using it because for a long time it was the only game in town. (3) The dispersion approx. 1 in the neg binom models does look reasonable. See Venables and Ripley's section on overdispersion for some cautions on this approach ... (4) I don't know why the deviance is coming out slightly higher for the zero-inflated neg binom -- seems odd. (5) Since you've gone to all this trouble to fit the overdispersed and zero-inflated likelihood models, your best bet is to try likelihood ratio tests between nested models (e.g. #8 vs #6 to test for zero-inflation, #8 vs #7 or #6 vs #5 to test for overdispersion). The quasi- and overdispersion calculations are usually done as a way of avoiding having to the fit the more complex models at all.