extraDistr (version 1.10.0)

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 $x=k/\varphi$ proportions in $\varphi$ trials where the average proportion is denoted as $\mu$, so it's parameters become $\alpha =\varphi \mu$ and $\beta =\varphi (1-\mu )$ and it's density function becomes

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

where $\pi$ is a prior parameter, so the distribution is a posterior distribution after observing $\varphi \mu$ successes and $\varphi (1-\mu )$ failures in $\varphi$ 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 $\varphi$ 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)