Thanks to both for your responses. From what you have said, suggested
reading, and other reading found along the way I feel comfortable accepting
the lognormal Poisson as a viable option in my model comparisons. Any other
reasons I should not?

One thing I notice as I work through different model scenarios is that ZIP
and ZINB models have greater than an order of magnitude or much larger
random effect variance (I have one random effect, except when running the
lognormal Poisson model of course) and the Wald test p-values are equally
larger for the fixed effects. My random effect variances are near zero for
all other families (0.01-0.12, depending on number of effects and
distribution family). The coefficient estimates are approximately the same
or slightly larger for these two compared to the other distribution
families used as well. The ZIP models tend to be my second best fit
according to AIC values followed by the negative binomial type 1. I am
having trouble trying to figure out why this is and would appreciate any
explanations. I have now ruled out considering these zero-inflated models
because exploratory analyses show that the data are not zero-inflated but
are over-dispersed, but I am still interested in knowing what causes it.

Thanks to all,

Shawn

On Sat, Jan 28, 2012 at 7:09 PM, Ben Bolker <bbolker@gmail.com> wrote:

> > Date: Thu, 26 Jan 2012 13:16:12 -0600 From: Shawn McCracken
> > <smccracken@...> To: r-sig-mixed-models@... Subject: [R-sig-ME] GLMM
> >  distribution family model comparison using Poisson w/observation
> > level random effect Message-ID:
> > <CAE+9gVGEtv6coOG1Un=pWVKWyd2s=cZQGZv3SeCgMvvENVjbWA@...>
> > Content-Type: text/plain
> >
> > Dear R mixed model users,
> >
> > I have been using package glmmADMB to run a full model and then
> > reduced model versions with the following distribution families:
> > poisson, zero-inflated poisson, poisson w/observation level random
> > effect, negative binomial, zero-inflated negative binomial, negative
> >  binomial type 1, and zero-inflated negative binomial type 1. The
> > models using poisson w/observation level random effect give the best
> >  fit according to AIC values but I am wondering if this is a fair
> > comparison since it has an additional random variable (observation)?
>
>  If you look, I believe you will find that the Poisson with
> observation-level random effect is counted as having the same number of
> parameters as the negative binomial.  The number of parameters *should* be:
>
>  Poisson           N
>  ZIP              N+1
>  Poisson w/ obs   N+1
>  NB               N+1
>  ZINB             N+2
>  NB1              N+1
>  ZINB1            N+2
>
> I just tried running this with a relatively trivial example (intercept +
> 1 continuous covariate + 1 intercept-only random effect, so N=3),
> and glmmADMB appears to agree with what I thought it should do:
>
>  poiss     ZIP LNPoiss      NB    ZINB     NB1   ZINB1
>      3       4       4       4       5       4       5
>
> The particular example I ran had a true NB2 distribution with N=500, 10
> blocks, intercept=1, slope=2, RE variance=1, overdispersion parameter
> =1.2, and this was the AIC table:
>
> > AICtab(mlist)
>        dAIC   df
> NB         0.0 4
> ZINB       0.9 5
> LNPoiss   43.4 4
> ZINB1    104.6 5
> NB1      115.4 4
> ZIP     1854.7 4
> poiss   4730.9 3
>
>  This is somewhat comforting.  The only thing I find surprising here is
> that LNPoiss (= lognormal Poisson = Poisson with observation-level
> error) is so much worse, since it has exactly the same mean-variance
> relationship as NB2.  Everything else is about as I expected.
>
>  As Mollie hints, there are two issues with applying AIC to mixed
> models (one of which also applies to ZI models): (1) boundary effects
> and (2) counting number of parameters.  There's more on this on
> http://glmm.wikidot.com/faq as well as in the paper Mollie cites.
>
> =========================
> Since no one has responded yet, I'll take a stab at this.
>
> You are correct that it's not quite fair, but it's not straightforward.
> I'm guessing that negative binomial is the 2nd runner up because it also
> is based on a Poisson distribution with a mean that comes from a
> right skewed distribution (Gamma distribution). The difference is that
> by including an observation level random effect, you have somewhere
> between 1 and nobs-1 parameters. I believe this gives more flexibility
> than the 1 extra parameter of the negative binomial model
> and ideally the model would be penalized for this, but estimating
> degrees of freedom in mixed models is not straightforward (see Box 3 of
> Bolker et al 2009 doi:10.1016/j.tree.2008.10.008). By running a glmmadmb
> example I see that the df only counts 1 parameter for the random effect
> (it's standard deviation). Approximations of df are controversial.
>
>
>
> Mollie
> >
>

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