PropBeta: Beta distribution of proportions

In twolodzko/extraDistr: Additional Univariate and Multivariate Distributions

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.

See Also

beta, binomial

Examples

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)
                       

twolodzko/extraDistr documentation built on Dec. 4, 2023, 8:56 p.m.