# pCorrBin: Correlated 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 Correlated Binomial Distribution.

## Usage

 1 pCorrBin(x,n,p,cov) 

## Arguments

 x vector of binomial random variables n single value for no of binomial trials p single value for probability of success cov single value for covariance

## Details

The probability function and cumulative function can be constructed and are denoted below

The cumulative probability function is the summation of probability function values

P_{CorrBin}(x) = {n \choose x}(p^x)(1-p)^{n-x}(1+(\frac{cov}{2p^2(1-p)^2})((x-np)^2+x(2p-1)-np^2))

x = 0,1,2,3,...n n = 1,2,3,... 0 < p < 1 -∞ < cov < +∞

The Correlation is in between

\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) ≤ cov ≤ \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo}

where fo=min (x-(n-1)p-0.5)^2

The mean and the variance are denoted as

E_{CorrBin}[x]= np

Var_{CorrBin}[x]= n(p(1-p)+(n-1)cov)

Corr_{CorrBin}[x]=\frac{cov}{p(1-p)}

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further

## Value

The output of pCorrBin gives cumulative probability values in vector form.

## References

Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate discrete distributions (Vol. 444). Hoboken, NJ: Wiley-Interscience.

L. L. Kupper, J.K.H., 1978. The Use of a Correlated Binomial Model for the Analysis of Certain Toxicological Experiments. Biometrics, 34(1), pp.69-76.

Paul, S.R., 1985. A three-parameter generalization of the binomial distribution. Communications in Statistics - Theory and Methods, 14(6), pp.1497-1506.

Jorge G. Morel and Nagaraj K. Neerchal. Overdispersion Models in SAS. SAS Institute, 2012.

CBprob
  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 28 29 30 #plotting the random variables and probability values col<-rainbow(5) a<-c(0.58,0.59,0.6,0.61,0.62) b<-c(0.022,0.023,0.024,0.025,0.026) plot(0,0,main="Correlated 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,dCorrBin(0:10,10,a[i],b[i])$pdf,col = col[i],lwd=2.85) points(0:10,dCorrBin(0:10,10,a[i],b[i])$pdf,col = col[i],pch=16) } dCorrBin(0:10,10,0.58,0.022)$pdf #extracting the pdf values dCorrBin(0:10,10,0.58,0.022)$mean #extracting the mean dCorrBin(0:10,10,0.58,0.022)$var #extracting the variance dCorrBin(0:10,10,0.58,0.022)$corr #extracting the correlation dCorrBin(0:10,10,0.58,0.022)$mincorr #extracting the minimum correlation value dCorrBin(0:10,10,0.58,0.022)$maxcorr #extracting the maximum correlation value #plotting the random variables and cumulative probability values col<-rainbow(5) a<-c(0.58,0.59,0.6,0.61,0.62) b<-c(0.022,0.023,0.024,0.025,0.026) plot(0,0,main="Correlated binomial probability function graph",xlab="Binomial random variable", ylab="Probability function values",xlim = c(0,10),ylim = c(0,1)) for (i in 1:5) { lines(0:10,pCorrBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85) points(0:10,pCorrBin(0:10,10,a[i],b[i]),col = col[i],pch=16) } pCorrBin(0:10,10,0.58,0.022) #acquiring the cumulative probability values