I want to test questions relating to whether individual ants of a certain species have personal food preferences, using a Bayesian model built up of multiple beta-binomial distributions. My problems are mostly conceptual at this point.

The model is relatively simple: suppose there are $N$ colonies of a certain ant species. A colony $i$ has $N_{i}$ individual ants. Whenever an ant forages for food, it must make a binary choice between bringing back Apples or Bananas. An ant $j_{i}$ has a preference $p_{j,i}$, which corresponds to the probability that they will pick Apples on a given foraging run, and goes foraging $n_{j,i}$ times.

The number of Apples ant $j_{i}$ brings back is drawn from a binomial distribution Binomial($p_{j,i}$, $n_{j,i}$). Each ant's preference ($p_{j,i}$) is in turn drawn from a beta distribution of the colony-wide preference, parameterized with $\mu_{i}$ and $\sigma_{i}^{2}$ (with $\mu_{i}$ being the preference toward Apples). I believe this makes a beta-binomial, right?

I'd like to test a number of theories, but they are all based around a single central question: do individual ants of a certain species have personal preferences?

I'm having a hell of a time coming up with principled ways of testing this, but here are some ways I've thought of that might (partially) get at this question:

Question: Can a colony's behavior be explained by the colony having a preference, but the individual ants do not? I.e., what is the probability of the ant data for a colony if the colony's beta distribution is set so that it's $\alpha$ = number of Apples and $\beta$ = number of Bananas (a reparametrization, I know), without regard to which ant brought back what?

In this scenario, each time an ant goes out to forage, it basically flips a coin weighted based on the colony's preference when it decides what to get. I'm assuming you could calculate the probability that this generative model explains the ants' actual behavior, given the set of $n_{j,i}$, right? I'm a little worried that this doesn't really compare two hypotheses though...

Problem: I don't know how to generalize from such colony-specific measures to the whole species. Even if I find a way of asking certain hypotheses about a single colony, I don't know how I could ask it for multiple colonies of the same species at once. Friends have suggested linking the $\mu_{i}$s of each colony by assuming they're generated from an underlying distribution (from another beta distribution?), which I'm fine doing, but don't see how that would actually solve anything. I don't really care about knowing the distribution of $\mu_{i}$s or the particular values of $\mu_{i}$.

Question: What is the 'Polarization Index' of a colony? I stumbled across this article by Rungie, Brown, Laurent & Rudrapatna (2005), which describes the Polarization Index, which seems to be very applicable to my current needs. Basically, since the highest variance a beta distribution can have is when all the constituent $p$s are either 0 or 1, and since the variance for any beta distribution is bounded, the Polarization Index is essentially "a standardized form of the variance: the variance divided by its upper bound", where 0 is when all the decision makers are making decisions at random, and 1 is when they all are completely constant. Presumably, I would just take the average PI for all the colonies of a species.

Problem: This doesn't feel completely right, since one could imagine that ants from a certain colony might have very strong individual 25%/75% preferences. In this case, they wouldn't every have a Polarization Index of 1, no matter how consistently they stuck to such preferences. Are there any other indices that capture this vague intuition?


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