rbbinom() samples beta-binomial data according to Menssen and Schaarschmidt (2019).
Usage
rbbinom(n, size, prob, rho)
Value
a data.frame with two columns (succ, fail)
Arguments
n
defines the number of clusters (\(i\))
size
integer vector defining the number of trials per cluster (\(n_i\))
prob
probability of success on each trial (\(\pi\))
rho
intra class correlation (\(\rho\))
Details
For beta binomial data with \(i=1, ... I\) clusters, the variance is
$$var(y_i)= n_i \pi (1-\pi) [1+ (n_i - 1) \rho]$$
with \(\rho\) as the intra class correlation coefficient
$$\rho = 1 / (1+a+b).$$
For the sampling \((a+b)\) is defined as
$$(a+b)=(1-\rho)/\rho$$
where \(a=\pi (a+b)\) and \(b=(a+b)-a\). Then, the binomial proportions
for each cluster are sampled from the beta distribution
$$\pi_i \sim Beta(a, b)$$
and the number of successes for each cluster are sampled to be
$$y_i \sim Bin(n_i, \pi_i).$$
In this parametrization \(E(\pi_i)=\pi=a/(a+b)\) and \(E(y_i)=n_i \pi\).
Please note, that \(1+ (n_i-1) \rho\) is a constant if all cluster sizes are
the same and hence, in this special case, also the quasi-binomial assumption is
fulfilled.
References
Menssen M, Schaarschmidt F.: Prediction intervals for overdispersed binomial data
with application to historical controls. Statistics in Medicine. 2019;38:2652-2663.
tools:::Rd_expr_doi("10.1002/sim.8124")