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OK, so this isn't the greatest implementation, but I think it serves our purpose. The beta binomial regression makes the assumption that the data are generated from a binomial distribution $$y_i\vert p_i, \kappa, a, b \sim \operatorname{Binomial}(p_i, n)$$ Each person's probability of success is assumed to be iid Beta  p_i \vert \kappa, a, b \sim \...

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Most of the answers here seem to cover two approaches - Bayesian and the order statistic. I'd like to add a viewpoint from the binomial, which I think the easiest to grasp. The intuition for a beta distribution comes into play when we look at it from the lens of the binomial distribution. The difference between the binomial and the beta is that the former ...

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I think you might make progress by asking your audience to assume that these values are distributed on the range [0,5] in the set {(0:10)/2} with a beta-binomial distribution. The beta-binomial distribution arose from a different process than your situation but it is an ordered discrete distribution. Ben Bolker has a nice discussion of simulation using the ...

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One approach is to use bootstrapping: library(simpleboot) library(boot) set1 <- as.data.frame(c(3,3,2.5,2.5,4.5,3,2,4,3,3.5,3.5,2.5,3,3,3.5,3,3,4,3.5,3.5,4,3.5,3.5,4,3.5)) colnames(set1) <- "numbers" set1.boot = one.boot(set1\$numbers, mean, R=10^4) ## hist(set1.boot) boot.ci(set1.boot, type="bca") ## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS ## ...

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