[R-sig-ME] models with overdispersion and autocorr.

[Highland Statistics Ltd.]

>Hi there
>
>I'm trying to fit repeated count models at several locations.  The 
>locations are a random effect as there are >50, and we're not 
>interested in the actual location values.
>But the data needs to be fitted with a quasi or zero-inflated model, 
>and there is autocorrelation through time.
>
>I can't find a function which allows quasi/zi AND autocorrelation to 
>be fitted.  Is there one?  Or is there another way the model could 
>be constructed to avoid the problem?
>
>Thanks,

Alison,
I guess you want the following:

Y_ijk ~ ZIP(mu_ijk, pi_ijk)              (or its NB cousin)
Y_ijk = observation i at time j at location k

logit(pi_ijk) = X_ijk* beta     + b_k + epsilon_ijk
log(mu_ijk) = X'_ijk * beta' +  b'_k  + eta_ijk

Then this can be solved in RBugs. So..MCMC stuff. The b_k and b'_k 
are the random intercepts for location. The epsilon and eta can be 
used to introduce some extra auto-regressive correlation. See also 
Ntzoufras (2009) for ZIP code, or Chapter 23 in Zuur et al (2009) for 
a simple auto-correlation Poisson GLM example. In fact, most of the 
ingredients are in Ntzoufras (2009).

But such a model would only do correlation between observations from 
the same location. Things get a bit more nasty if you also have 
correlation between locations (if your birds fly from one location to 
nearby locations in the same year)....and it becomes even more nasty 
if your birds fly from one location to another location the next 
year. I guess you could try to add some spatial correlations via the 
epsilon and the eta..using some of the spatial correlation functions 
described in Chapter 5 of Pinheiro and Bates (2000). That would be 
very nice to try...:-).  Try to visualise a very big correlation 
matrix for the entire data set. Which values would be most correlated?

The ZIP above can be fitted in RBugs.......but I guess that you want 
to do this in the context of a GAM? Then you need to program the 
spline into X*beta. See Wood (2006).


It is not impossible to solve this problem...but it will keep you 
busy for a while. We have a book scheduled for the end of 2009 in 
which all this stuff is applied; "Analysing Ecological Data; 
Practical Solutions When Things Get Complicated". Not that this is of 
much use to you right now.

Have fun..:-).

Alain




Dr. Alain F. Zuur
First author of:

1. Analysing Ecological Data (2007).
Zuur, AF, Ieno, EN and Smith, GM. Springer. 680 p.



2. Mixed effects models and extensions in ecology with R. (2009).
Zuur, AF, Ieno, EN, Walker, N, Saveliev, AA, and Smith, GM. Springer.



3. A Beginner's Guide to R (2009).
Zuur, AF, Ieno, EN, Meesters, EHWG. Springer





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