dBetaBin: Beta-Binomial Distribution

Description Usage Arguments Details Value References Examples

View source: R/Beta.R

Description

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

Usage

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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.

Available at: http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2538541&tool=pmcentrez&rendertype=abstract .

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

Available at: http://linkinghub.elsevier.com/retrieve/pii/0167947396900158 .

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

Examples

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#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.