# PropBeta: Beta distribution of proportions In extraDistr: Additional Univariate and Multivariate Distributions

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```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 ≤ 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/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.

## 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.

`beta`, `binomial`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```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) ```