Description Usage Arguments Details Value References See Also Examples

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

1 |

`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 |

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

The output of `pCorrBin`

gives cumulative probability values in vector form.

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.

Available at: http://www.tandfonline.com/doi/abs/10.1080/03610928508828990 .

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

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#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
``` |

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