Beta distribution of proportions
Probability mass function, distribution function and random generation for the reparametrized beta distribution.
dprop(x, size, mean, prior = 0, log = FALSE) pprop(q, size, mean, prior = 0, lower.tail = TRUE, log.p = FALSE) qprop(p, size, mean, prior = 0, lower.tail = TRUE, log.p = FALSE) rprop(n, size, mean, prior = 0)
x, q |
vector of quantiles. |
size |
non-negative real number; precision or number of binomial trials. |
mean |
mean proportion or probability of success on each trial;
|
prior |
(see below) with |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x]. |
p |
vector of probabilities. |
n |
number of observations. If |
Beta can be understood as a distribution of x = k/φ proportions in φ trials where the average proportion is denoted as μ, so it's parameters become α = φμ and β = φ(1-μ) and it's density function becomes
f(x) = (x^(φμ+π-1) * (1-x)^(φ(1-μ)+π-1))/B(φμ+π, φ(1-μ)+π)
where π is a prior parameter, so the distribution is a posterior distribution after observing φμ successes and φ(1-μ) failures in φ trials with binomial likelihood and symmetric Beta(π, π) prior for probability of success. Parameter value π = 1 corresponds to uniform prior; π = 1/2 corresponds to Jeffreys prior; π = 0 corresponds to "uninformative" Haldane prior, this is also the re-parametrized distribution used in beta regression. With π = 0 the distribution can be understood as a continuous analog to binomial distribution dealing with proportions rather then counts. Alternatively φ may be understood as precision parameter (as in beta regression).
Notice that in pre-1.8.4 versions of this package, prior was not settable
and by default fixed to one, instead of zero. To obtain the same results as in
the previous versions, use prior = 1 in each of the functions.
Ferrari, S., & Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799-815.
Smithson, M., & Verkuilen, J. (2006). A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychological Methods, 11(1), 54-71.
x <- rprop(1e5, 100, 0.33) hist(x, 100, freq = FALSE) curve(dprop(x, 100, 0.33), 0, 1, col = "red", add = TRUE) hist(pprop(x, 100, 0.33)) plot(ecdf(x)) curve(pprop(x, 100, 0.33), 0, 1, col = "red", lwd = 2, add = TRUE) n <- 500 p <- 0.23 k <- rbinom(1e5, n, p) hist(k/n, freq = FALSE, 100) curve(dprop(x, n, p), 0, 1, col = "red", add = TRUE, n = 500)
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