Goodness of fit for possibly dependent data

Question

My data consists of about a hundred animal visits to three different feeding zones, and I'm trying to determine if there was a preference for one zone over the others.

Each animal visits between one and about a hundred zones (think hummingbirds and feeders).

The problem is that 1) the sequence of zones a given animal visits are likely not independent because each individual animal may have its own preferences, and 2) the visits of different animals are likely not independent, because it is possible (likely) that, say, the first animal and the fifth are the same (they are not identifiable), so the two sequences of visits to zones by the first and fifth may be highly correlated.

If everything was independent, then I could use a chi square test.

Any suggestions?

Edits based on the comments: The data is like:

bird1<-c(1,2,1,1,3,2,1,2)
bird2<-c(3,2,1,2,1,2,3,2,1,3,2,3,2,1)
bird3<-c(1)

...

birdn<-c(2,3,1)

where 1,2 & 3 are labels for the zones visited before the bird left the area.

Now any bird may have its own preferences. And bird1 may be bird3, because once they leave the area, if they come back we have no way of knowing it was the same bird (once it is out of eyesight).

The null hypothesis would be that the three zones are equally preferred; the alternative hypothesis would be that some zone is preferred.

If everything was independent, I could do a chi-square test comparing the observed frequencies to c(1/3,1/3,1/3), but there may be dependence here, and simulations may prove difficulty because there is no obvious model for the dependence.

Answer 1

Quite what the data are is not clear, so it would help to be told what data you might feed into a chi-square test (for example). Real or realistic examples would help greatly.

At first sight, this does seem unduly pessimistic. Usually for something like a chi-square test the benchmark or reference situation (null hypothesis, in more traditional terms) is independence and whether the real data show something else is precisely the question. So, on this story, any kinds of dependence are interesting, not a problem.

If this misreads the problem, then the answer is likely to be that you need to set up simulation models with plausible assumptions to give an idea of what you might expect.

Identifiability of individuals is presumably a known problem in your field.

(UPDATE) As birds are not individually identifiable with certainty, how much do you lose by collapsing to a table of visits to each zone and testing that against independence? Chi-square could be a stretch here because expected frequencies are all low, but you could get a handle on sampling distribution by simulation.