BetaBinomial: The beta-binomial distribution.

Description

Density, distribution function, quantile function and random generation for the beta-binomial distribution with parameters size, prob, theta, shape1, shape2. This distribution corresponds to an overdispersed binomial distribution.

Usage

dbetabinom(x, size, prob, theta, shape1, shape2, log = FALSE)
pbetabinom(
  q,
  size,
  prob,
  theta,
  shape1,
  shape2,
  lower.tail = TRUE,
  log.p = FALSE
)
qbetabinom(
  p,
  size,
  prob,
  theta,
  shape1,
  shape2,
  lower.tail = TRUE,
  log.p = FALSE
)
rbetabinom(n, size, prob, theta, shape1, shape2)

Value

dbetabinom gives the density, pbetabinom gives the distribution function, qbetabinom gives the quantile function and rbetabinom

generates random deviates.

Arguments

x, q: Vector of quantiles.
size: Number of trials.
prob: Probability of success on each trial.
theta: Aggregation parameter (theta = 1 / (shape1 + shape2)).
shape1, shape2: Shape parameters.
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.

Details

Be aware that in this implementation theta = 1 / (shape1 + shape2). prob and theta, or shape1 and shape2 must be specified. if theta = 0, use *binom family instead.

Examples

Run this code

# Compute P(25 < X < 50) for X following the Beta-Binomial distribution
# with parameters size = 100, prob = 0.5 and theta = 0.35:
sum(dbetabinom(25:50, size = 100, prob = 0.5, theta = 0.35))

# When theta tends to 0, dbetabinom outputs tends to dbinom outputs:
sum(dbetabinom(25:50, size = 100, prob = 0.5, theta = 1e-7))
sum(dbetabinom(25:50, size = 100, shape1 = 1e7, shape2 = 1e7))
sum(dbinom(25:50, size = 100, prob = 0.5))

# Example of binomial and beta-binomial frequency distributions:
n   <- 15
q   <- 0:n
p1  <- dbinom(q, size = n, prob = 0.33)
p2  <- dbetabinom(q, size = n, prob = 0.33, theta = 0.22)
res <- rbind(p1, p2)
dimnames(res) <- list(c("Binomial", "Beta-binomial"), q)
barplot(res, beside = TRUE, legend.text = TRUE, ylab = "Frequency")

# Effect of the aggregation parameter theta on probability density:
thetas <- seq(0.001, 2.5, by = 0.001)
density1 <- rep(sum(dbinom(25:50, size = 100, prob = 0.5)), length(thetas))
density2 <- sapply(thetas, function(theta) {
    sum(dbetabinom(25:50, size = 100, prob = 0.5, theta = theta))
})
plot(thetas, density2, type = "l",
     xlab = expression("Aggregation parameter ("*theta*")"),
     ylab = "Probability density between 25 and 50 (size = 100)")
lines(thetas, density1, lty = 2)