# dbetabinom

0th

Percentile

##### Beta-binomial distribution

Density function and random variate generator for the beta-binomial function, parameterized in terms of probability and overdispersion

Keywords
distribution
##### Usage
dbetabinom(x, prob, size,  theta, shape1, shape2, log = FALSE)
rbetabinom(n, prob, size, theta, shape1, shape2)
##### Arguments
x

a numeric vector of integer values

prob

numeric vector: mean probability of underlying beta distribution

size

integer: number of samples

theta

overdispersion parameter

shape1

shape parameter of per-trial probability distribution

shape2

shape parameter of per-trial probability distribution

log

(logical) return log probability density?

n

integer number of random variates to return

##### Details

The beta-binomial distribution is the result of compounding a beta distribution of probabilities with a binomial sampling process. The density function is $$p(x) = \frac{C(N,x) \mbox{Beta}(N-x+\theta(1-p),x+\theta p)}% {\mbox{Beta}(\theta(1-p),\theta p)}%$$ The parameters shape1 and shape2 are the more traditional parameterization in terms of the parameters of the per-trial probability distribution.

##### Value

A vector of probability densities or random deviates. If x is non-integer, the result is zero (and a warning is given).

##### Note

Although the quantile (qbetabinom) and cumulative distribution (pbetabinom) functions are not available, in a pinch they could be computed from the pghyper and qghyper functions in the SuppDists package -- provided that shape2>1. As described in ?pghyper, pghyper(q,a=-shape1, N=-shape1-shape2,k=size) should give the cumulative distribution for the beta-binomial distribution with parameters (shape1,shape2,size), and similarly for qghyper. (Translation to the (theta,size,prob) parameterization is left as an exercise.)

##### References

Morris (1997), American Naturalist 150:299-327

dbeta, dbinom

• BetaBinomial
• dbetabinom
• rbetabinom
##### Examples
# NOT RUN {
set.seed(100)
n <- 9
z <- rbetabinom(1000, 0.5, size=n, theta=4)
par(las=1,bty="l")
plot(table(z)/length(z),ylim=c(0,0.34),col="gray",lwd=4,
ylab="probability")
points(0:n,dbinom(0:n,size=n,prob=0.5),col=2,pch=16,type="b")
points(0:n,dbetabinom(0:n,size=n,theta=4,
prob=0.5),col=4,pch=17,type="b")
## correspondence with SuppDists
if (require(SuppDists)) {
d1a <- dghyper(0:5,a=-5,N=-10,k=5)
d1b <- dbetabinom(0:5,shape1=5,shape2=5,size=5)
max(abs(d1a-d1b))
p1a <- pghyper(0:5,a=-5,N=-10,k=5,lower.tail=TRUE)
p1b <- cumsum(d1b)
max(abs(p1a-p1b))
}
# }

Documentation reproduced from package emdbook, version 1.3.10, License: GPL

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