PropBeta: Beta distribution of proportions

Description

Probability mass function, distribution function and random generation for the reparametrized beta distribution.

Usage

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)

Arguments

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; 0 < mean < 1.
prior: (see below) with prior = 0 (default) the distribution corresponds to re-parametrized beta distribution used in beta regression. This parameter needs to be non-negative.
log, log.p: logical; if TRUE, probabilities p are given as log(p).
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Beta can be understood as a distribution of $x = k/\phi$ proportions in $\phi$ trials where the average proportion is denoted as $\mu$, so it's parameters become $\alpha = \phi\mu$ and $\beta = \phi(1-\mu)$ and it's density function becomes

$$ f(x) = \frac{x^{\phi\mu+\pi-1} (1-x)^{\phi(1-\mu)+\pi-1}}{\mathrm{B}(\phi\mu+\pi, \phi(1-\mu)+\pi)} $$

where $\pi$ is a prior parameter, so the distribution is a posterior distribution after observing $\phi\mu$ successes and $\phi(1-\mu)$ failures in $\phi$ trials with binomial likelihood and symmetric $\mathrm{Beta}(\pi, \pi)$ prior for probability of success. Parameter value $\pi = 1$ corresponds to uniform prior; $\pi = 1/2$ corresponds to Jeffreys prior; $\pi = 0$ corresponds to "uninformative" Haldane prior, this is also the re-parametrized distribution used in beta regression. With $\pi = 0$ the distribution can be understood as a continuous analog to binomial distribution dealing with proportions rather then counts. Alternatively $\phi$ 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.

References

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.

Examples

Run this code


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)