pCorrBin: Correlated Binomial Distribution

Description Usage Arguments Details Value References See Also Examples

View source: R/CorrBin.R

Description

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

Usage

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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.

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.

See Also

CBprob

Examples

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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

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