| dbetabinom {emdbook} | R Documentation |
Density function and random variate generator for the beta-binomial function, parameterized in terms of probability and overdispersion
dbetabinom(x, prob, size, theta, shape1, shape2, log = FALSE) rbetabinom(n, prob, size, theta, shape1, shape2)
x |
a numeric vector of 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 |
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) Beta(N-x+theta(1-p),x+theta p)} {mbox{Beta}(theta(1-p),theta p)}
{p(x) =
(C(N,x)*Beta(N-x+theta*(1-p),x+theta*p))/
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.
A vector of probability densities or random deviates.
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.)
Ben Bolker
Morris (1997), American Naturalist 150:299-327
set.seed(100)
n <- 9
z <- rbetabinom(1000, 0.5, size=n, theta=4)
plot(table(z)/length(z),ylim=c(0,0.34),col="gray",lwd=4)
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=3,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))
}