I have a sample of animals and a categorical variable of $L$ behaviors $i=1,…L$. I computed the proportions of this sample corresponding to each behavior category $p_{i}$, $sum p_{i}=1$.

- Can I use a common method of confidence interval for a proportion (e.g. Clopper-Pearson) to compute the CI for each of the $p_{i}$?
- Is it possible to test that proportion $p_{i}$ is significantly different than $p_{j}$? for instance if their confidence intervals do not overlap?
- Is it possible to test that $p_{i}$, $i=1,…n$ are equal? for instance conduct a chi-squared test of goodness of fit:
`chisq.test(c(p1*n,p2*n,p3*n), c(1/3,1/3,1/3))`

, if L=3 and my sample size is n.

Thank you

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#### Best Answer

Yes, individual intervals like that are possible, since each (without regard for the others) is just binomial ("in this category" vs "not in this category")

Yes, it is possible, but you don't test it by looking at whether the individual intervals overlap. In particular, you can construct an interval for the difference in proportions that takes account of the negative correlation between sample proportions.

Yes, a straight chi-square goodness of fit will test that.

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