# BB: The Beta-Binomial Distribution In PROreg: Patient Reported Outcomes Regression Analysis

## Description

Density and random generation for the beta-binomial distribution.

## Usage

 ```1 2``` ```dBB(m,p,phi) rBB(k,m,p,phi) ```

## Arguments

 `k` number of simulations. `m` maximum socre number in each beta-binomial observation.. `p` probability parameter of the beta-binomial distribution. `phi` dispersion parameter of the beta-binomial distribution.

## Details

The beta-binomial distribution consists of a finite sum of Bernoulli dependent variables whose probability parameter is random and follows a beta distribution. Assume that we have y_j a set of variables, j=1,...,m, with m integer, that conditioned on a random variable u, are independent and follow a Bernoulli distribution with probability parameter u. On the other hand, the random variable u follows a beta distribution with parameter p/phi and (1-p)/phi. Namely,

y_j \sim Ber(u), u \sim Beta(p/phi,(1-p)/phi),

where 0<p<1 and phi>0. The first and second order marginal moments of this distribution are defined as

E[y_j]=p, Var[y_j]=p(1-p),

and correlation between observations is defined as

Corr[y_j,y_k]=phi/(1+phi),

where j,k=1,...,m are different. Consequently, phi can be considered as a dispersion parameter.

If we sum up all the variables we will define a new variable which follows a new distribution that is called beta-binomial distribution, and it is defined as follows. The variable y follows a beta-binomial distribution with parameters m, p and phi if

y|u \sim Bin(m,u), u\sim Beta(p/phi,(1-p)/phi).

## Value

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

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

## Author(s)

J. Najera-Zuloaga

D.-J. Lee

I. Arostegui

## References

Arostegui I., Nunez-Anton 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 15 16 17``` ```set.seed(12) # We define m <- 10 p <- 0.4 phi <- 1.8 # We perform k beta-binomial simulations for those parameters. k <- 100 bb <- rBB(k,m,p,phi) bb dd <- dBB(m,p,phi) # We are going to plot the histogram of the created variable, # and using dBB() function we are going to fit the distribution: hist(bb,col="grey",breaks=seq(-0.5,m+0.5,1),probability=TRUE, main="Histogram",xlab="Beta-binomial random variable") lines(seq(0,m),dd,col="red",lwd=4) ```