# BI: The Binomial distribution with overdispersion In idaejin/HRQoL: Health Related Quality of Life Analysis

## Description

Density and random generation for the binomial distribution with optional dispersion parameter calculation.

## Usage

 1 2 dBI(n,p,phi) rBI(k,n,p,phi) 

## Arguments

 k number of simulations. n the maximum score of the binomial trials. p the probability of scoring a success in each binomial trial. phi the dispersion parameter of the binomial distribution. If phi=1, then the simple binomial model will be performed.

## Details

The inclusion of a dispersion parameter in the variance equation of the binomial distribution relaxes the relationship that is expected between the mean and variance in binomial models,

The density function of this binomial model is calculated considering as an exponential family, where the density function has the following form

f(y)=\frac{y log(p/(1-p))+n log(1-p)}{φ}+c(y,φ),

where c(y,φ) is a function that is approximated by the deviance of the model.

## Value

dBI gives the density of a binomial distribution for those n, p and phi parameters.

rBI generates k random observations based on a binomial distribution with those n, p and phi parameters.

## Author(s)

Josu Najera Zuloaga

Dae-Jin Lee

## References

Pawitan Y. (2001): In All Likelihood: Statistical Modelling and Inference Using Likelihood, Oxford University Press.

The rbinom function of package <stats>. This function performs simulations based on a binomial distribution without dispersion parameter.
  1 2 3 4 5 6 7 8 9 10 11 12 k <- 1000 n <- 10 p <- 0.765 phi <- 4.35 #simulating y <- rBI(k,n,p,phi) y #density function d <- dBI(n,p,phi) d