# dBetaBin: Beta-Binomial Distribution In Amalan-ConStat/R-fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD

## Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Beta-Binomial Distribution.

## Usage

 1 dBetaBin(x,n,a,b) 

## Arguments

 x vector of binomial random variables n single value for no of binomial trials a single value for shape parameter alpha representing as a b single value for shape parameter beta representing as b

## Details

Mixing beta distribution with binomial distribution will create the Beta-Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values

P_{BetaBin}(x)= {n \choose x} \frac{B(a+x,n+b-x)}{B(a,b)}

a,b > 0

x = 0,1,2,3,...n

n = 1,2,3,...

The mean, variance and over dispersion are denoted as

E_{BetaBin}[x]= \frac{na}{a+b}

Var_{BetaBin}[x]= \frac{(nab)}{(a+b)^2} \frac{(a+b+n)}{(a+b+1)}

over dispersion= \frac{1}{a+b+1}

Defined as B(a,b) is the beta function.

## Value

The output of dBetaBin gives a list format consisting

pdf probability function values in vector form

mean mean of the Beta-Binomial Distribution

var variance of the Beta-Binomial Distribution

over.dis.para over dispersion value of the Beta-Binomial Distribution

## References

Young-Xu, Y. & Chan, K.A., 2008. Pooling overdispersed binomial data to estimate event rate. BMC medical research methodology, 8(1), p.58.

Trenkler, G., 1996. Continuous univariate distributions. Computational Statistics & Data Analysis, 21(1), p.119.

Hughes, G., 1993. Using the Beta-Binomial Distribution to Describe Aggregated Patterns of Disease Incidence. Phytopathology, 83(9), p.759.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 #plotting the random variables and probability values col<-rainbow(5) a<-c(1,2,5,10,0.2) plot(0,0,main="Beta-binomial probability function graph",xlab="Binomial random variable", ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5)) for (i in 1:5) { lines(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85) points(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16) } dBetaBin(0:10,10,4,.2)$pdf #extracting the pdf values dBetaBin(0:10,10,4,.2)$mean #extracting the mean dBetaBin(0:10,10,4,.2)$var #extracting the variance dBetaBin(0:10,10,4,.2)$over.dis.para #extracting the over dispersion value #plotting the random variables and cumulative probability values col<-rainbow(4) a<-c(1,2,5,10) plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable", ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1)) for (i in 1:4) { lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i]) points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i]) } pBetaBin(0:10,10,4,.2) #acquiring the cumulative probability values 

Amalan-ConStat/R-fitODBOD documentation built on Oct. 1, 2018, 7:13 p.m.