What test to use to compare observed and expected frequencies when expected frequencies for each subject are independent from each other?

Question

I conducted a study where I presented subjects with a treatment that they could either respond to with a match or non-match. I will use whales as an example. Whales can breach the water in several ways, just the tail, blowhole, head etc. However, in this example each whale prefers to breach the water in different ways. For example, whale 1 breaches with his tail 75% of the time, blowhole 10% of the time etc. Whale two breaches with his head 60% of the time, tail 30 % of the time etc.

I'm interested in seeing whether the whales will respond to a simulated whale breach with the same whale breach (e.g. if I simulate a tail breach the whale does a tail breach). However, since breaches happen at different probabilities for each whale I'm under the impression that my expected value should be the proportion that they naturally do the breach that I'm showing them. So if I show whale 1 from above a tail breach I expect that there is a 75% chance it will tail breach regardless. I simulate multiple tail breaches over a certain amount of time and record the whale's breaching behaviour. I then multiply the expected proportion of tail breaches (0.75) by the number of responses (e.g. 10) and get 7.5 as my expected value. I then do this for every other whale using their specific proportions as they relate to the breach I showed them (sometimes tail, sometimes head etc.). Some whales did not breach in response however showed interest in the stimuli in other ways e.g. came close to the simulated breach. These whales have an expected value of 0 but I don't know if it's appropriate to throw these data out considering they were aware of the stimulus and had the opportunity to match the breach.

I have data that look like this:

Obs     Exp
1       1.1
0       0
0       0.08
0       0.44
0       0
0       0.63
23      2.38
0       0.17
0       0.45
3       0.4
0       0
0       0
0       0.33

Observed is the number of matching breaches during the treatment. I want to know if the whales matched the breach more often than expected by chance. After helpful comments below, it seems a chi-square test is not appropriate here. What test would be appropriate here? Fisher Exact?

Answer 1

I would stop thinking about this as a chi-square problem or one involving observed and expected in this way.

First, let me try to summarize your problem. You think that the type of breach that the whale sees affects the type of breach they perform. Each whale may have different patterns of responses and each whale is shown multiple breaches.

If that is correct, then this seems like a regression problem with dependent errors and one solution would be to use a multilevel model and, since the response variable is a count, probably you need a nonlinear MLM. In R, look into the nlme package.