Nothing
R/CorrBin.R
In fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD
Defines functions
summary.fitCB
print.fitCB
fitCorrBin.default
fitCorrBin
.EstMLECorrBin
EstMLECorrBin
NegLLCorrBin
pCorrBin
dCorrBin
Documented in
dCorrBin
EstMLECorrBin
fitCorrBin
NegLLCorrBin
pCorrBin
#' Correlated Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Correlated Binomial Distribution.
#'
#' @usage
#' dCorrBin(x,n,p,cov)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param p single value for probability of success.
#' @param 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.
#'
#' \deqn{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)) }
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{0 < p < 1}
#' \deqn{-\infty < cov < +\infty }
#'
#' The Correlation is in between
#' \deqn{\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) \le correlation \le \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo} }
#' where \eqn{fo=min [(x-(n-1)p-0.5)^2] }
#'
#' The mean and the variance are denoted as
#' \deqn{E_{CorrBin}[x]= np}
#' \deqn{Var_{CorrBin}[x]= n(p(1-p)+(n-1)cov)}
#' \deqn{Corr_{CorrBin}[x]=\frac{cov}{p(1-p)}}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{dCorrBin} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of Correlated Binomial Distribution.
#'
#' \code{var} variance of Correlated Binomial Distribution.
#'
#' \code{corr} correlation of Correlated Binomial Distribution.
#'
#' \code{mincorr} minimum correlation value possible.
#'
#' \code{maxcorr} maximum correlation value possible.
#'
#' @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="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
#'
#' @export
dCorrBin<-function(x,n,p,cov)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,n,p,cov))) | any(is.infinite(c(x,n,p,cov))) | any(is.nan(c(x,n,p,cov))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if at any chance the binomial random variable is greater than binomial trial value
#if so providing an error message and stopping the function progress
if(max(x) > n )
{
stop("Binomial random variable cannot be greater than binomial trial value")
}
#checking if any random variable or trial value is negative if so providig an error message
#and stopping the function progress
else if(any(x<0) | n<0)
{
stop("Binomial random variable or binomial trial value cannot be negative")
}
else
{
correlation<-cov/(p*(1-p))
#checking the probability value is inbetween zero and one
if( p <= 0 | p >= 1 )
{
stop("Probability value doesnot satisfy conditions")
}
else
{
value<-NULL
#creating the necessary limits for correlation, the left hand side and right hand side limits
left.h<-(-2/(n*(n-1)))*min(p/(1-p),(1-p)/p)
right.h<-(2*p*(1-p))/(((n-1)*p*(1-p))+0.25-min(((0:n)-(n-1)*p-0.5)^2))
# checking if the correlation output satisfies conditions mentioned above
if(correlation < -1 | correlation > 1 | correlation < left.h | correlation > right.h)
{
stop("Correlation cannot be greater than 1 or Lesser than -1 or it cannot be greater than Maximum Correlation or Lesser than Minimum Correlation")
}
else
{
#constructing the probability values for all random variables
y<-0:n
value1<-NULL
for(i in 1:length(y))
{
value1[i]<-((choose(n,y[i]))*(p^y[i])*((1-p)^(n-y[i]))*
(1+(cov/(2*(p^2)*((1-p)^2)))*(((y[i]-n*p)^2)+(y[i]*(2*p-1))-(n*(p^2)))))
}
check1<-sum(value1)
#checking if the sum of all probability values leads upto one
#if not providing an error message and stopping the function progress
if(check1 < 0.9999 | check1 >1.0001 | any(value1 < 0) | any(value1 >1))
{
stop("Input parameter combinations of probability of success and covariance does
not create proper probability function")
}
else
{
#for each random variable in the input vector below calculations occur
for (i in 1:length(x))
{
value[i]<-((choose(n,x[i]))*(p^x[i])*((1-p)^(n-x[i]))*
(1+(cov/(2*(p^2)*((1-p)^2)))*(((x[i]-n*p)^2)+(x[i]*(2*p-1))-(n*(p^2)))))
}
# generating an output in list format consisting pdf,mean and variance
return(list("pdf"=value,"mean"=n*p,"var"=n*(p*(1-p)+(n-1)*cov),
"corr"=cov/(p*(1-p)),"mincorr"=left.h,"maxcorr"=right.h))
}
}
}
}
}
}
#' Correlated Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Correlated Binomial Distribution.
#'
#' @usage
#' pCorrBin(x,n,p,cov)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param p single value for probability of success.
#' @param 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.
#'
#' \deqn{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)) }
#' \eqn{x = 0,1,2,3,...n}
#' \eqn{n = 1,2,3,...}
#' \eqn{0 < p < 1}
#' \eqn{-\infty < cov < +\infty }
#'
#' The Correlation is in between
#' \deqn{\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) \le cov \le \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo} }
#' where \eqn{fo=min (x-(n-1)p-0.5)^2 }
#'
#' The mean and the variance are denoted as
#' \deqn{E_{CorrBin}[x]= np}
#' \deqn{Var_{CorrBin}[x]= n(p(1-p)+(n-1)cov)}
#' \deqn{Corr_{CorrBin}[x]=\frac{cov}{p(1-p)}}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{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: \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="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
#'
#' @export
pCorrBin<-function(x,n,p,cov)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative proability function
#values are calculated
for(i in 1:length(x))
{
ans[i]<-sum(dCorrBin(0:x[i],n,p,cov)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of Correlated Binomial distribution
#'
#' This function will calculate the negative log likelihood value when the vector of binomial random
#' variables and vector of corresponding frequencies are given with the input parameters.
#'
#' @usage
#' NegLLCorrBin(x,freq,p,cov)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param p single value for probability of success.
#' @param cov single value for covariance.
#'
#' @details
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{-\infty < cov < +\infty}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{NegLLCorrBin} will produce a single numeric value.
#'
#' @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
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' NegLLCorrBin(No.D.D,Obs.fre.1,.5,.03) #acquiring the negative log likelihood value
#'
#' @export
NegLLCorrBin<-function(x,freq,p,cov)
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
correlation<-cov/(p*(1-p))
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,freq,p,cov))) | any(is.infinite(c(x,freq,p,cov))) |
any(is.nan(c(x,freq,p,cov))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if any of the random variables of frequencies are less than zero if so
#creating a error message as well as stopping the function progress
if(any(c(x,freq) < 0) )
{
stop("Binomial random variable or frequency values cannot be negative")
}
#checking the probability value is inbetween zero and one or covariance is greater than zero
else if( p <= 0 | p >= 1)
{
stop("Probability value doesnot satisfy conditions")
}
else
{
value<-NULL
#creating the necessary limits for correlation, the left hand side and right hand side limits
left.h<-(-2/(n*(n-1)))*min(p/(1-p),(1-p)/p)
right.h<-(2*p*(1-p))/(((n-1)*p*(1-p))+0.25-(min(((0:n)-(n-1)*p-0.5)^2)))
# checking if the correlation output satisfies conditions mentioned above
if(correlation < -1 | correlation > 1 | correlation < left.h | correlation > right.h)
{
stop("Correlation cannot be greater than 1 or Lesser than -1 or it cannot be greater than Maximum Correlation or Lesser than Minimum Correlation")
}
else
{
#constructing the probability values for all random variables
y<-0:n
value1<-NULL
for(i in 1:length(y))
{
value1[i]<-((choose(n,y[i]))*(p^y[i])*((1-p)^(n-y[i]))*
(1+(cov/(2*(p^2)*((1-p)^2)))*(((y[i]-n*p)^2)+(y[i]*(2*p-1))-(n*(p^2)))))
}
check1<-sum(value1)
#checking if the sum of all probability values leads upto one
#if not providing an error message and stopping the function progress
if(check1 < 0.9999 | check1 >1.0001 | any(value1 < 0) | any(value1 >1))
{
stop("Input parameter combinations of probability of success and covariance does
not create proper probability function")
}
else
{
for (i in 1:sum(freq))
{
value[i]<-log(1+((cov/(2*(p^2)*((1-p)^2)))*(((data[i]-n*p)^2)+(data[i]*(2*p-1))-(n*(p^2)))))
}
#calculating the negative log likelihood value and representing as a single output value
return(-(sum(log(choose(n,data[1:sum(freq)]))) +
log(p)*sum(data[1:sum(freq)]) + log(1-p)*sum(n-data[1:sum(freq)]) + sum(value)))
}
}
}
}
}
#' Estimating the probability of success and correlation for Correlated Binomial
#' Distribution
#'
#' The function will estimate the probability of success and correlation using the maximum log
#' likelihood method for the Correlated Binomial distribution when the binomial random
#' variables and corresponding frequencies are given.
#'
#' @usage
#' EstMLECorrBin(x,freq,p,cov,...)
#'
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param p single value for probability of success.
#' @param cov single value for covariance.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#' @details
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#' \deqn{0 < p < 1}
#' \deqn{-\infty < cov < +\infty}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' \code{EstMLECorrBin} here is used as a wrapper for the \code{mle2} function of \pkg{bbmle} package
#' therefore output is of class of mle2.
#'
#' @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.
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLECorrBin(x=No.D.D,freq=Obs.fre.1,p=0.5,cov=0.0050)
#'
#' bbmle::coef(parameters) #extracting the parameters
#'
#'@export
EstMLECorrBin<-function(x,freq,p,cov,...)
{
suppressWarnings2 <-function(expr, regex=character())
{
withCallingHandlers(expr, warning=function(w)
{
if (length(regex) == 1 && length(grep(regex, conditionMessage(w))))
{
invokeRestart("muffleWarning")
}
} )
}
output<-suppressWarnings2(bbmle::mle2(.EstMLECorrBin,data=list(x=x,freq=freq),
start = list(p=p,cov=cov),...),"NaN")
return(output)
}
.EstMLECorrBin<-function(x,freq,p,cov)
{
#with respective to using bbmle package function mle2 there is no need impose any restrictions
#therefor the output is directly a single numeric value for the negative log likelihood value of
#Correlated Binomial distribution
value<-NULL
n<-max(x)
data<-rep(x,freq)
for (i in 1:sum(freq))
{
value[i]<-log(1+((cov/(2*(p^2)*((1-p)^2)))*(((data[i]-n*p)^2)+(data[i]*(2*p-1))-(n*(p^2)))))
}
return(-(sum(log(choose(n,data[1:sum(freq)]))) + log(p)*sum(data[1:sum(freq)]) +
log(1-p)*sum(n-data[1:sum(freq)]) + sum(value)))
}
#' Fitting the Correlated Binomial Distribution when binomial
#' random variable, frequency, probability of success and covariance are given
#'
#' The function will fit the Correlated Binomial Distribution
#' when random variables, corresponding frequencies, probability of success and covariance are given.
#' It will provide the expected frequencies, chi-squared test statistics value, p value,
#' and degree of freedom so that it can be seen if this distribution fits the data.
#'
#' @usage
#' fitCorrBin(x,obs.freq,p,cov)
#'
#' @param x vector of binomial random variables.
#' @param obs.freq vector of frequencies.
#' @param p single value for probability of success.
#' @param cov single value for covariance.
#'
#' @details
#' \deqn{obs.freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{-\infty < cov < +\infty}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{fitCorrBin} gives the class format \code{fitCB} and \code{fit} consisting a list
#'
#' \code{bin.ran.var} binomial random variables.
#'
#' \code{obs.freq} corresponding observed frequencies.
#'
#' \code{exp.freq} corresponding expected frequencies.
#'
#' \code{statistic} chi-squared test statistics.
#'
#' \code{df} degree of freedom.
#'
#' \code{p.value} probability value by chi-squared test statistic.
#'
#' \code{corr} Correlation value.
#'
#' \code{fitCB} fitted probability values of \code{dCorrBin}.
#'
#' \code{NegLL} Negative Log Likelihood value.
#'
#' \code{AIC} AIC value.
#'
#' \code{call} the inputs of the function.
#'
#' Methods \code{summary}, \code{print}, \code{AIC}, \code{residuals} and \code{fitted}
#' can be used to extract specific outputs.
#'
#' @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
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLECorrBin(x=No.D.D,freq=Obs.fre.1,p=0.5,cov=0.0050)
#'
#' pCorrBin <- bbmle::coef(parameters)[1]
#' covCorrBin <- bbmle::coef(parameters)[2]
#'
#' #fitting when the random variable,frequencies,probability and covariance are given
#' results <- fitCorrBin(No.D.D,Obs.fre.1,pCorrBin,covCorrBin)
#' results
#'
#' #extracting the AIC value
#' AIC(results)
#'
#' #extract fitted values
#' fitted(results)
#'
#' @export
fitCorrBin<-function(x,obs.freq,p,cov)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,obs.freq,p,cov))) | any(is.infinite(c(x,obs.freq,p,cov))) |
any(is.nan(c(x,obs.freq,p,cov))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dCorrBin(x,max(x),p,cov)
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$pdf
#using the estimated probability values the expected frequencies are calculated
exp.freq<-round((sum(obs.freq)*est.prob),2)
#chi-squared test statistics is calculated with observed frequency and expected frequency
statistic<-sum(((obs.freq-exp.freq)^2)/exp.freq)
#degree of freedom is calculated
df<-length(x)-3
#p value of chi-squared test statistic is calculated
p.value<-1-stats::pchisq(statistic,df)
#checking if df is less than or equal to zero
if(df<0 | df==0)
{
stop("Degrees of freedom cannot be less than or equal to zero")
}
#checking if any of the expected frequencies are less than five and greater than zero, if so
#a warning message is provided in interpreting the results
if(min(exp.freq)<5 && min(exp.freq) > 0)
{
message("Chi-squared approximation may be doubtful because expected frequency is less than 5")
}
#checking if expected frequency is zero, if so providing a warning message in interpreting
#the results
if(min(exp.freq)==0)
{
message("Chi-squared approximation is not suitable because expected frequency approximates to zero")
}
NegLL<-NegLLCorrBin(x,obs.freq,p,cov)
names(NegLL)<-NULL
#the final output is in a list format containing the calculated values
final<-list("bin.ran.var"=x,"obs.freq"=obs.freq,"exp.freq"=exp.freq,"statistic"=round(statistic,4),
"df"=df,"p.value"=round(p.value,4),"corr"=est$corr,"fitCB"=est,"NegLL"=NegLL,
"p"=p,"cov"=cov,"AIC"=2*2+2*NegLL,"call"=match.call())
class(final)<-c("fitCB","fit")
return(final)
}
}
#' @method fitCorrBin default
#' @export
fitCorrBin.default<-function(x,obs.freq,p,cov)
{
return(fitCorrBin(x,obs.freq,p,cov))
}
#' @method print fitCB
#' @export
print.fitCB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Correlated Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated p value :",x$p," ,estimated cov value :",x$cov,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n")
}
#' @method summary fitCB
#' @export
summary.fitCB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Correlated Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated p value :",object$p," ,estimated cov value :",object$cov,"\n\t
X-squared :",object$statistic," ,df :",object$df," ,p-value :",object$p.value,"\n\t
Negative Loglikehood value :",object$NegLL,"\n\t
AIC value :",object$AIC,"\n")
}
#' @importFrom bbmle mle2
#' @importFrom stats pchisq
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Defines functions summary.fitCB print.fitCB fitCorrBin.default fitCorrBin .EstMLECorrBin EstMLECorrBin NegLLCorrBin pCorrBin dCorrBin
Documented in dCorrBin EstMLECorrBin fitCorrBin NegLLCorrBin pCorrBin
#' Correlated Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Correlated Binomial Distribution.
#'
#' @usage
#' dCorrBin(x,n,p,cov)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param p single value for probability of success.
#' @param 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.
#'
#' \deqn{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)) }
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{0 < p < 1}
#' \deqn{-\infty < cov < +\infty }
#'
#' The Correlation is in between
#' \deqn{\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) \le correlation \le \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo} }
#' where \eqn{fo=min [(x-(n-1)p-0.5)^2] }
#'
#' The mean and the variance are denoted as
#' \deqn{E_{CorrBin}[x]= np}
#' \deqn{Var_{CorrBin}[x]= n(p(1-p)+(n-1)cov)}
#' \deqn{Corr_{CorrBin}[x]=\frac{cov}{p(1-p)}}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{dCorrBin} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of Correlated Binomial Distribution.
#'
#' \code{var} variance of Correlated Binomial Distribution.
#'
#' \code{corr} correlation of Correlated Binomial Distribution.
#'
#' \code{mincorr} minimum correlation value possible.
#'
#' \code{maxcorr} maximum correlation value possible.
#'
#' @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="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
#'
#' @export
dCorrBin<-function(x,n,p,cov)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,n,p,cov))) | any(is.infinite(c(x,n,p,cov))) | any(is.nan(c(x,n,p,cov))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if at any chance the binomial random variable is greater than binomial trial value
#if so providing an error message and stopping the function progress
if(max(x) > n )
{
stop("Binomial random variable cannot be greater than binomial trial value")
}
#checking if any random variable or trial value is negative if so providig an error message
#and stopping the function progress
else if(any(x<0) | n<0)
{
stop("Binomial random variable or binomial trial value cannot be negative")
}
else
{
correlation<-cov/(p*(1-p))
#checking the probability value is inbetween zero and one
if( p <= 0 | p >= 1 )
{
stop("Probability value doesnot satisfy conditions")
}
else
{
value<-NULL
#creating the necessary limits for correlation, the left hand side and right hand side limits
left.h<-(-2/(n*(n-1)))*min(p/(1-p),(1-p)/p)
right.h<-(2*p*(1-p))/(((n-1)*p*(1-p))+0.25-min(((0:n)-(n-1)*p-0.5)^2))
# checking if the correlation output satisfies conditions mentioned above
if(correlation < -1 | correlation > 1 | correlation < left.h | correlation > right.h)
{
stop("Correlation cannot be greater than 1 or Lesser than -1 or it cannot be greater than Maximum Correlation or Lesser than Minimum Correlation")
}
else
{
#constructing the probability values for all random variables
y<-0:n
value1<-NULL
for(i in 1:length(y))
{
value1[i]<-((choose(n,y[i]))*(p^y[i])*((1-p)^(n-y[i]))*
(1+(cov/(2*(p^2)*((1-p)^2)))*(((y[i]-n*p)^2)+(y[i]*(2*p-1))-(n*(p^2)))))
}
check1<-sum(value1)
#checking if the sum of all probability values leads upto one
#if not providing an error message and stopping the function progress
if(check1 < 0.9999 | check1 >1.0001 | any(value1 < 0) | any(value1 >1))
{
stop("Input parameter combinations of probability of success and covariance does
not create proper probability function")
}
else
{
#for each random variable in the input vector below calculations occur
for (i in 1:length(x))
{
value[i]<-((choose(n,x[i]))*(p^x[i])*((1-p)^(n-x[i]))*
(1+(cov/(2*(p^2)*((1-p)^2)))*(((x[i]-n*p)^2)+(x[i]*(2*p-1))-(n*(p^2)))))
}
# generating an output in list format consisting pdf,mean and variance
return(list("pdf"=value,"mean"=n*p,"var"=n*(p*(1-p)+(n-1)*cov),
"corr"=cov/(p*(1-p)),"mincorr"=left.h,"maxcorr"=right.h))
}
}
}
}
}
}
#' Correlated Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Correlated Binomial Distribution.
#'
#' @usage
#' pCorrBin(x,n,p,cov)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param p single value for probability of success.
#' @param 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.
#'
#' \deqn{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)) }
#' \eqn{x = 0,1,2,3,...n}
#' \eqn{n = 1,2,3,...}
#' \eqn{0 < p < 1}
#' \eqn{-\infty < cov < +\infty }
#'
#' The Correlation is in between
#' \deqn{\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) \le cov \le \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo} }
#' where \eqn{fo=min (x-(n-1)p-0.5)^2 }
#'
#' The mean and the variance are denoted as
#' \deqn{E_{CorrBin}[x]= np}
#' \deqn{Var_{CorrBin}[x]= n(p(1-p)+(n-1)cov)}
#' \deqn{Corr_{CorrBin}[x]=\frac{cov}{p(1-p)}}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{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: \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="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)$maxcorr #extracting the maximum correlation value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(5)
#' a <- c(0.58,0.59,0.6,0.61,0.62)
#' b <- c(0.022,0.023,0.024,0.025,0.026)
#' plot(0,0,main="Correlated binomial probability function graph",xlab="Binomial random variable",
#' ylab="Probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:5)
#' {
#' lines(0:10,pCorrBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85)
#' points(0:10,pCorrBin(0:10,10,a[i],b[i]),col = col[i],pch=16)
#' }
#'
#' pCorrBin(0:10,10,0.58,0.022) #acquiring the cumulative probability values
#'
#' @export
pCorrBin<-function(x,n,p,cov)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative proability function
#values are calculated
for(i in 1:length(x))
{
ans[i]<-sum(dCorrBin(0:x[i],n,p,cov)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of Correlated Binomial distribution
#'
#' This function will calculate the negative log likelihood value when the vector of binomial random
#' variables and vector of corresponding frequencies are given with the input parameters.
#'
#' @usage
#' NegLLCorrBin(x,freq,p,cov)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param p single value for probability of success.
#' @param cov single value for covariance.
#'
#' @details
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{-\infty < cov < +\infty}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{NegLLCorrBin} will produce a single numeric value.
#'
#' @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
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' NegLLCorrBin(No.D.D,Obs.fre.1,.5,.03) #acquiring the negative log likelihood value
#'
#' @export
NegLLCorrBin<-function(x,freq,p,cov)
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
correlation<-cov/(p*(1-p))
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,freq,p,cov))) | any(is.infinite(c(x,freq,p,cov))) |
any(is.nan(c(x,freq,p,cov))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if any of the random variables of frequencies are less than zero if so
#creating a error message as well as stopping the function progress
if(any(c(x,freq) < 0) )
{
stop("Binomial random variable or frequency values cannot be negative")
}
#checking the probability value is inbetween zero and one or covariance is greater than zero
else if( p <= 0 | p >= 1)
{
stop("Probability value doesnot satisfy conditions")
}
else
{
value<-NULL
#creating the necessary limits for correlation, the left hand side and right hand side limits
left.h<-(-2/(n*(n-1)))*min(p/(1-p),(1-p)/p)
right.h<-(2*p*(1-p))/(((n-1)*p*(1-p))+0.25-(min(((0:n)-(n-1)*p-0.5)^2)))
# checking if the correlation output satisfies conditions mentioned above
if(correlation < -1 | correlation > 1 | correlation < left.h | correlation > right.h)
{
stop("Correlation cannot be greater than 1 or Lesser than -1 or it cannot be greater than Maximum Correlation or Lesser than Minimum Correlation")
}
else
{
#constructing the probability values for all random variables
y<-0:n
value1<-NULL
for(i in 1:length(y))
{
value1[i]<-((choose(n,y[i]))*(p^y[i])*((1-p)^(n-y[i]))*
(1+(cov/(2*(p^2)*((1-p)^2)))*(((y[i]-n*p)^2)+(y[i]*(2*p-1))-(n*(p^2)))))
}
check1<-sum(value1)
#checking if the sum of all probability values leads upto one
#if not providing an error message and stopping the function progress
if(check1 < 0.9999 | check1 >1.0001 | any(value1 < 0) | any(value1 >1))
{
stop("Input parameter combinations of probability of success and covariance does
not create proper probability function")
}
else
{
for (i in 1:sum(freq))
{
value[i]<-log(1+((cov/(2*(p^2)*((1-p)^2)))*(((data[i]-n*p)^2)+(data[i]*(2*p-1))-(n*(p^2)))))
}
#calculating the negative log likelihood value and representing as a single output value
return(-(sum(log(choose(n,data[1:sum(freq)]))) +
log(p)*sum(data[1:sum(freq)]) + log(1-p)*sum(n-data[1:sum(freq)]) + sum(value)))
}
}
}
}
}
#' Estimating the probability of success and correlation for Correlated Binomial
#' Distribution
#'
#' The function will estimate the probability of success and correlation using the maximum log
#' likelihood method for the Correlated Binomial distribution when the binomial random
#' variables and corresponding frequencies are given.
#'
#' @usage
#' EstMLECorrBin(x,freq,p,cov,...)
#'
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param p single value for probability of success.
#' @param cov single value for covariance.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#' @details
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#' \deqn{0 < p < 1}
#' \deqn{-\infty < cov < +\infty}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' \code{EstMLECorrBin} here is used as a wrapper for the \code{mle2} function of \pkg{bbmle} package
#' therefore output is of class of mle2.
#'
#' @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.
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLECorrBin(x=No.D.D,freq=Obs.fre.1,p=0.5,cov=0.0050)
#'
#' bbmle::coef(parameters) #extracting the parameters
#'
#'@export
EstMLECorrBin<-function(x,freq,p,cov,...)
{
suppressWarnings2 <-function(expr, regex=character())
{
withCallingHandlers(expr, warning=function(w)
{
if (length(regex) == 1 && length(grep(regex, conditionMessage(w))))
{
invokeRestart("muffleWarning")
}
} )
}
output<-suppressWarnings2(bbmle::mle2(.EstMLECorrBin,data=list(x=x,freq=freq),
start = list(p=p,cov=cov),...),"NaN")
return(output)
}
.EstMLECorrBin<-function(x,freq,p,cov)
{
#with respective to using bbmle package function mle2 there is no need impose any restrictions
#therefor the output is directly a single numeric value for the negative log likelihood value of
#Correlated Binomial distribution
value<-NULL
n<-max(x)
data<-rep(x,freq)
for (i in 1:sum(freq))
{
value[i]<-log(1+((cov/(2*(p^2)*((1-p)^2)))*(((data[i]-n*p)^2)+(data[i]*(2*p-1))-(n*(p^2)))))
}
return(-(sum(log(choose(n,data[1:sum(freq)]))) + log(p)*sum(data[1:sum(freq)]) +
log(1-p)*sum(n-data[1:sum(freq)]) + sum(value)))
}
#' Fitting the Correlated Binomial Distribution when binomial
#' random variable, frequency, probability of success and covariance are given
#'
#' The function will fit the Correlated Binomial Distribution
#' when random variables, corresponding frequencies, probability of success and covariance are given.
#' It will provide the expected frequencies, chi-squared test statistics value, p value,
#' and degree of freedom so that it can be seen if this distribution fits the data.
#'
#' @usage
#' fitCorrBin(x,obs.freq,p,cov)
#'
#' @param x vector of binomial random variables.
#' @param obs.freq vector of frequencies.
#' @param p single value for probability of success.
#' @param cov single value for covariance.
#'
#' @details
#' \deqn{obs.freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{-\infty < cov < +\infty}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{fitCorrBin} gives the class format \code{fitCB} and \code{fit} consisting a list
#'
#' \code{bin.ran.var} binomial random variables.
#'
#' \code{obs.freq} corresponding observed frequencies.
#'
#' \code{exp.freq} corresponding expected frequencies.
#'
#' \code{statistic} chi-squared test statistics.
#'
#' \code{df} degree of freedom.
#'
#' \code{p.value} probability value by chi-squared test statistic.
#'
#' \code{corr} Correlation value.
#'
#' \code{fitCB} fitted probability values of \code{dCorrBin}.
#'
#' \code{NegLL} Negative Log Likelihood value.
#'
#' \code{AIC} AIC value.
#'
#' \code{call} the inputs of the function.
#'
#' Methods \code{summary}, \code{print}, \code{AIC}, \code{residuals} and \code{fitted}
#' can be used to extract specific outputs.
#'
#' @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
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLECorrBin(x=No.D.D,freq=Obs.fre.1,p=0.5,cov=0.0050)
#'
#' pCorrBin <- bbmle::coef(parameters)[1]
#' covCorrBin <- bbmle::coef(parameters)[2]
#'
#' #fitting when the random variable,frequencies,probability and covariance are given
#' results <- fitCorrBin(No.D.D,Obs.fre.1,pCorrBin,covCorrBin)
#' results
#'
#' #extracting the AIC value
#' AIC(results)
#'
#' #extract fitted values
#' fitted(results)
#'
#' @export
fitCorrBin<-function(x,obs.freq,p,cov)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,obs.freq,p,cov))) | any(is.infinite(c(x,obs.freq,p,cov))) |
any(is.nan(c(x,obs.freq,p,cov))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dCorrBin(x,max(x),p,cov)
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$pdf
#using the estimated probability values the expected frequencies are calculated
exp.freq<-round((sum(obs.freq)*est.prob),2)
#chi-squared test statistics is calculated with observed frequency and expected frequency
statistic<-sum(((obs.freq-exp.freq)^2)/exp.freq)
#degree of freedom is calculated
df<-length(x)-3
#p value of chi-squared test statistic is calculated
p.value<-1-stats::pchisq(statistic,df)
#checking if df is less than or equal to zero
if(df<0 | df==0)
{
stop("Degrees of freedom cannot be less than or equal to zero")
}
#checking if any of the expected frequencies are less than five and greater than zero, if so
#a warning message is provided in interpreting the results
if(min(exp.freq)<5 && min(exp.freq) > 0)
{
message("Chi-squared approximation may be doubtful because expected frequency is less than 5")
}
#checking if expected frequency is zero, if so providing a warning message in interpreting
#the results
if(min(exp.freq)==0)
{
message("Chi-squared approximation is not suitable because expected frequency approximates to zero")
}
NegLL<-NegLLCorrBin(x,obs.freq,p,cov)
names(NegLL)<-NULL
#the final output is in a list format containing the calculated values
final<-list("bin.ran.var"=x,"obs.freq"=obs.freq,"exp.freq"=exp.freq,"statistic"=round(statistic,4),
"df"=df,"p.value"=round(p.value,4),"corr"=est$corr,"fitCB"=est,"NegLL"=NegLL,
"p"=p,"cov"=cov,"AIC"=2*2+2*NegLL,"call"=match.call())
class(final)<-c("fitCB","fit")
return(final)
}
}
#' @method fitCorrBin default
#' @export
fitCorrBin.default<-function(x,obs.freq,p,cov)
{
return(fitCorrBin(x,obs.freq,p,cov))
}
#' @method print fitCB
#' @export
print.fitCB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Correlated Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated p value :",x$p," ,estimated cov value :",x$cov,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n")
}
#' @method summary fitCB
#' @export
summary.fitCB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Correlated Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated p value :",object$p," ,estimated cov value :",object$cov,"\n\t
X-squared :",object$statistic," ,df :",object$df," ,p-value :",object$p.value,"\n\t
Negative Loglikehood value :",object$NegLL,"\n\t
AIC value :",object$AIC,"\n")
}
#' @importFrom bbmle mle2
#' @importFrom stats pchisq
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