Dear Dr. Bates,

Thank you for you very instructive response to Andrew Digby's question. I've been struggling in figuring out how to deal with correlation and variance structure in a binary GLMM and you response certainly helped. The only thing I'm still trying to get my head around is the correlation among observations on the same subject. In your response, you mentioned that in a GLMM "the conditional distribution of the response given the random effects has independent components." But isn't one of the points of using a mixed model to account for the correlation among observations on the same subject? My understanding was that a mixed model assumes a compound symmetry correlation structure but what if that correlation changes with time? Wouldn't that be relevant / applicable to a GLMM?

Sorry if this is confusing, I'm not a statistician...

Thanks again,

Eduardo



On Tue, Aug 7, 2012 at 5:21 PM, Andrew Digby <andrewdigby@mac.com> wrote:
> 
> Despite lots of investigation, I haven't found any R packages might be suitable for the following problem. I'd be very grateful for suggestions.
> 
> I have three-way nested data, with a series of measures (obs) taken in quick succession (equal time spacing) from each subject on different days. The measures taken on the same day are temporally correlated, so I'd like to use an AR1 correlation structure for those, but treat subjects and days as nested random factors (random intercept) since there is little temporal correlation between days. The response is binary.
> 
> So I need a GLMM with a correlation structure. I've tried using GEE, but the R packages can't cope with multilevel nested data. The only R function I've found that can do this is glmmPQL.

Before you look for an R function, you should first check whether
there is indeed a statistical model with the properties that you
mention.  In the standard definition of a generalized linear mixed
model, and the only one that makes sense to me, the conditional
distribution of the response given the random effects has independent
components.

Aspects of linear mixed models that depend on being able to model the
variance-covariance of the response separately from the mean don't
carry over to generalized linear mixed models.  One of the fundamental
properties of GLMs and GLMMs is that the variance does depend on the
mean.

> m <- glmmPQL(y ~ f1 * f2 * f3 + (1|subj/day), correlation=corAR1(form =~obsno|subj/day))
> 
> f1 - f3 are fixed factors
> 
> However, PQL estimation is not recommended for binary response data. With no AIC and unreliable p values, model selection seems impossible! So my question is:
> 
> 1) are there any other functions which are suitable for a GLMM with multilevel nested random effects and a AR1 correlation structure? Or is MCMC the only option?
> 2) to make things more complicated, I'd also like to include a varFunc variance structure to cope with heterogeneity. Is this possible in ML methods in R? I'd also like to extend to a multinomial response at a later stage.
> 
> GEE seems the best bet, but I come unstuck with the three-way nested factors.
> 
> Thanks for your help.
> 
> Note: I originally posted this on R-help, but it was suggested that this list might be more appropriate.
> 
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