# BB: The Beta-Binomial Distribution In idaejin/HRQoL: Health Related Quality of Life Analysis

## Description

Density and random generation for the beta-binomial distribution.

## Usage

 ```1 2``` ```dBB(y,n,p,phi) rBB(k,n,p,phi) ```

## Arguments

 `n` the maximum score of the beta-binomial trials. `k` the number of simulations. `y` the number of successes in n beta-binomial trials. `p` the probabilily parameter of the beta-binomial distribution. `phi` the dispersion parameter of the beta-binomial distribution.

## Details

The beta-binomial distribution is defined as a mixture between a binomial distribution and a beta distribution. It assumes that conditioned on some random components θ, with beta distribution with parameters p/φ and p/(1-φ), the response variable y follows a binomial distribution with probability parameter θ,

y|θ \sim Bin(n,θ), θ \sim Beta(p/φ,(1-p)/φ).

The expectation, variance and density function of this distribution can be explicitly calculated:

E[y]=np,

Var[y]=np(1-p)[1+(n-1)φ/(1+φ)].

Consequently, p is called the probability parameter and φ is called the dispersion parameter of the beta-binomial distribution. Hence, the response variable y follows a beta-binomial distribution of parameters n, p and φ,

y \sim BB(n,p,φ).

## Value

`dBB` gives the density of a beta-binomial distribution with the defined `n`, `p` and `phi` parameters.

`rBB` generates `k` random observations based on a beta-binomial distribution with the defined `n`, `p` and `phi` parameters.

## Author(s)

Josu Najera-Zuloaga

Dae-Jin Lee

## References

Arostegui I., Nuñez-Antón V. & Quintana J. M. (2006): Analysis of short-form-36 (SF-36): The beta-binomial distribution approach, Statistics in Medicine, 26, 1318-1342.

The `rbeta` and `rbinom` functions of package `<stats>`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```set.seed(12) # We define n <- 10 #maximum number of scores in the binomial trials p <- 0.4 #probability parameter of the beta-binomial distribution phi <- 1.8 #dispersion parameter of the beta-binomial distribution # We perform k beta-binomial simulations for those parameters. k <- 100 bb <- rBB(k,n,p,phi) # Plot the histogram of the created variable, # dBB() function fits beta-binomial distribution: hist(bb,col="grey",breaks=seq(-0.5,n+0.5,1),probability=TRUE,main="Histogram",xlab="beta-binomial random variable") lines(c(0:n),dBB(0:n,n,p,phi),col="red",lwd=4) ```