On Mon, 26 Aug 2013, Luca Corlatti wrote:

>...

>For the sake of curiosity, however, I tried to re-run the model selection using the function glmer, with family=poisson, adding the >observation-level as a random factor (1|obs) to account for overdispersion, as recently suggested. In this case, however, visual inspection of >residuals for the global model were not very satisfactory. After running the model selection, the results were quite different from those >obtained with glmmadmb (not dramatically different, but still...). Given I have no deep knowledge into the !

>philosophy behind the use of glmer with (1|obs), I was wondering:



About the "philosophy behind the use of glmer with (1|obs)...". This approach is an approximate marginal quasi-likelihood solution to fitting a NB response and its variance relationship, that is, marginal to the random "obs" effects but conditional on any higher level random effects. The NB is not an exponential family distribution so it doesn't fit in with the GLM iteratively (re)-weighted least squares algorithm (with single error term) and by extension cannot be fitted directly as a GLMM. If u is the conditional expectation for a Poisson GLMM without the (1|obs) RE term, the response has conditional variance of u (assuming dispersion at the obs level is 1) whereas the NB has conditional variance u^2/k+u. If you add the (1|obs) RE term to the Poisson GLMM and as usual assume a log link then the 1st order Taylor series approximation to u is exp(nu) + obsRE.u + u where nu is the linear predictor excluding the (1|obs) term and obsRE are the random obs (in addition to the usual error, e, so this model is artificial). Then the approximate marginal variance of the response (but conditional on nu) is the same as that of the NB (i.e. 1/k=sigma^2) where the variance of obsRE is sigma^2. However, when you fit this glmer the residuals are conditioned (I assume) on obsRE and have variance u (or phi.u if the dispersion parameter is not fixed to 1), whereas the glmmadmb residuals will have variance u^2/k+u. Therefore to get equivalent residuals from glmer you could do predictions setting obsRE to zero, or possibly(?) add obsRE_hat.u_hat to the glmer ordinary residuals (this is the case for GLMMs fitted using PQL but I am not sure exactly of the details of fitting and calculation of residuals for glmer).  Pearson or deviance residuals are more complicated without using the prediction method. This also depends on whether or not phi is set to 1 for the Poisson GLMM. If phi>1 then it could be considered to have a variance function equivalent to an overdispersed NB, whatever that distribution is(?).


Dr Steven G. Candy
Senior Applied Statistician
Wildlife Conservation and Fisheries
Australian Antarctic Division
203 Channel Hwy, Kingston, Tasmania, 7050, Australia
ph: (+61) 3 62 323135


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