predint (version 2.2.1)

rqbinom: Sampling of overdispersed binomial data with constant overdispersion

Description

rqbinom samples overdispersed binomial data with constant overdispersion from the beta-binomial distribution such that the quasi-binomial assumption is fulfilled.

Usage

rqbinom(n, size, prob, phi)

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 (\(\pi\))

phi

dispersion parameter (\(\Phi\))

Details

It is assumed that the dispersion parameter (\(\Phi\)) is constant for all \(i=1, ... I\) clusters, such that the variance becomes $$var(y_i)=\Phi n_i \pi (1-\pi).$$ For the sampling \((a+b)_i\) is defined as $$(a+b)_i=(\Phi-n_i)/(1-\Phi)$$ where \(a_i=\pi (a+b)_i\) and \(b_i=(a+b)_i-a_i\). Then, the binomial proportions for each cluster are sampled from the beta distribution $$\pi_i \sim Beta(a_i, b_i)$$ and the numbers of success for each cluster are sampled to be $$y_i \sim Bin(n_i, \pi_i).$$ In this parametrization \(E(\pi_i)=\pi\) and \(E(y_i)=n_i \pi\). Please note, the quasi-binomial assumption is not in contradiction with the beta-binomial distribution if all cluster sizes are the same.

Examples

# Sampling of example data
set.seed(456)
qb_dat1 <- rqbinom(n=10, size=50, prob=0.1, phi=3)
qb_dat1

set.seed(456)
qb_dat2 <- rqbinom(n=3, size=c(40, 50, 60), prob=0.1, phi=3)
qb_dat2