pAddBin: Additive Binomial Distribution

In fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD

Additive Binomial Distribution

Description

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

Usage

pAddBin(x,n,p,alpha)

Arguments

x	vector of binomial random variables.
n	single value for no of binomial trials.
p	single value for probability of success.
alpha	single value for alpha parameter.

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_{AddBin}(x)= {n \choose x} p^x (1-p)^{n-x}(\frac{alpha}{2}(\frac{x(x-1)}{p}+\frac{(n-x)(n-x-1)}{(1-p)}-\frac{alpha n(n-1)}{2})+1)

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

n = 1,2,3,...

0 < p < 1

-1 < alpha < 1

The alpha is in between

\frac{-2}{n(n-1)}min(\frac{p}{1-p},\frac{1-p}{p}) ≤ alpha ≤ (\frac{n+(2p-1)^2}{4p(1-p)})^{-1}

The mean and the variance are denoted as

E_{Addbin}[x]=np

Var_{Addbin}[x]=np(1-p)(1+(n-1)alpha)

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

Value

The output of pAddBin 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: doi: 10.1080/03610928508828990

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

Examples

#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="Additive 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,dAddBin(0:10,10,a[i],b[i])$pdf,col = col[i],lwd=2.85)
  points(0:10,dAddBin(0:10,10,a[i],b[i])$pdf,col = col[i],pch=16)
}

dAddBin(0:10,10,0.58,0.022)$pdf     #extracting the probability values
dAddBin(0:10,10,0.58,0.022)$mean    #extracting the mean
dAddBin(0:10,10,0.58,0.022)$var     #extracting the variance

#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="Additive 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,pAddBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85)
points(0:10,pAddBin(0:10,10,a[i],b[i]),col = col[i],pch=16)
}

pAddBin(0:10,10,0.58,0.022)       #acquiring the cumulative probability values

fitODBOD documentation built on Jan. 15, 2023, 5:11 p.m.