Nothing
R/MultiBin.R
In fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD
Defines functions
summary.fitMuB
print.fitMuB
fitMultiBin.default
fitMultiBin
.EstMLEMultiBin
EstMLEMultiBin
NegLLMultiBin
pMultiBin
dMultiBin
Documented in
dMultiBin
EstMLEMultiBin
fitMultiBin
NegLLMultiBin
pMultiBin
#' Multiplicative Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Multiplicative Binomial Distribution.
#'
#' @usage
#' dMultiBin(x,n,p,theta)
#'
#' @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 theta single value for theta.
#'
#' @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_{MultiBin}(x)= {n \choose x} p^x (1-p)^{n-x} \frac{(theta^{x(n-x)}}{f(p,theta,n)} }
#'
#' here \eqn{f(p,theta,n)} is
#' \deqn{f(p,theta,n)= \sum_{k=0}^{n} {n \choose k} p^k (1-p)^{n-k} (theta^{k(n-k)} )}
#'
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{k = 0,1,2,...,n}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' \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{dMultiBin} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of Multiplicative Binomial Distribution.
#'
#' \code{var} variance of Multiplicative Binomial Distribution.
#'
#' @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}.
#'
#' @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="Multiplicative 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,dMultiBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dMultiBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dMultiBin(0:10,10,.58,10.022)$pdf #extracting the pdf values
#' dMultiBin(0:10,10,.58,10.022)$mean #extracting the mean
#' dMultiBin(0:10,10,.58,10.022)$var #extracting the variance
#'
#'
#' #plotting 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="Multiplicative 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,pMultiBin(0:10,10,a[i],1+b[i]),col = col[i],lwd=2.85)
#' points(0:10,pMultiBin(0:10,10,a[i],1+b[i]),col = col[i],pch=16)
#' }
#'
#' pMultiBin(0:10,10,.58,10.022) #acquiring the cumulative probability values
#'
#' @export
dMultiBin<-function(x,n,p,theta)
{
#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,theta))) | any(is.infinite(c(x,n,p,theta))) |
any(is.nan(c(x,n,p,theta))) )
{
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
{
#checking the probability value is inbetween zero and one if so providig an error message
#and stopping the function progress
if( p <= 0 | p >= 1)
{
stop("Probability value doesnot satisfy conditions")
}
else
{
#constructing the probability values for all random variables
y<-0:n
value1<-NULL
func1<-
for (i in 1:length(y))
{
value1[i]<-choose(n,y[i])*(p^y[i])*((1-p)^(n-y[i]))*(theta^(y[i]*(n-y[i])))/(sum(choose(n,0:n)*(p^(0:n))*((1-p)^(n-(0:n)))*(theta^((0:n)*(n-(0:n))))))
}
check1<-sum(value1)
#checking if the theta value is less than or equal to zero if so providig an error message
#and stopping the function progress
if(theta <= 0)
{
stop("Theta parameter value cannot be zero or less than zero")
}
#checking if the sum of all probability values leads upto one
#if not providing an error message and stopping the function progress
else if(check1 < 0.9999 | check1 >1.0001 | any(value1 < 0) | any(value1 >1))
{
stop("Input parameter combinations of probability of success and theta does
not create proper probability function")
}
else
{
value<-NULL
#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]))*(theta^(x[i]*(n-x[i])))/
(sum(choose(n,0:n)*(p^(0:n))*((1-p)^(n-(0:n)))*(theta^((0:n)*(n-(0:n))))))
}
# generating an output in list format consisting pdf,mean and variance
return(list("pdf"=value,"mean"=sum((0:n)*value1),
"var"=sum(((0:n)^2)*value1)-(sum((0:n)*value1))^2))
}
}
}
}
}
#' Multiplicative Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Multiplicative Binomial Distribution.
#'
#' @usage
#' pMultiBin(x,n,p,theta)
#'
#' @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 theta single value for theta.
#'
#' @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_{MultiBin}(x)= {n \choose x} p^x (1-p)^{n-x} \frac{(theta^{x(n-x)}}{f(p,theta,n)} }
#'
#' here \eqn{f(p,theta,n)} is
#' \deqn{f(p,theta,n)= \sum_{k=0}^{n} {n \choose k} p^k (1-p)^{n-k} (theta^{k(n-k)} )}
#'
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{k = 0,1,2,...,n}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' \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{pMultiBin} 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}.
#'
#' @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="Multiplicative 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,dMultiBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dMultiBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dMultiBin(0:10,10,.58,10.022)$pdf #extracting the pdf values
#' dMultiBin(0:10,10,.58,10.022)$mean #extracting the mean
#' dMultiBin(0:10,10,.58,10.022)$var #extracting the variance
#'
#' #plotting 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="Multiplicative 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,pMultiBin(0:10,10,a[i],1+b[i]),col = col[i],lwd=2.85)
#' points(0:10,pMultiBin(0:10,10,a[i],1+b[i]),col = col[i],pch=16)
#' }
#'
#' pMultiBin(0:10,10,.58,10.022) #acquiring the cumulative probability values
#'
#' @export
pMultiBin<-function(x,n,p,theta)
{
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(dMultiBin(0:x[i],n,p,theta)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of Multiplicative Binomial distribution
#'
#' This function will calculate the negative log likelihood value when the vector of binomial random
#' variable and vector of corresponding frequencies are given with the input parameters.
#'
#' @usage
#' NegLLMultiBin(x,freq,p,theta)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param p single value for probability of success.
#' @param theta single value for theta parameter.
#'
#' @details
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' @return
#' The output of \code{NegLLMultiBin} 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}.
#'
#' @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
#'
#' NegLLMultiBin(No.D.D,Obs.fre.1,.5,3) #acquiring the negative log likelihood value
#'
#' @export
NegLLMultiBin<-function(x,freq,p,theta)
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
#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,theta))) | any(is.infinite(c(x,freq,p,theta))) |
any(is.nan(c(x,freq,p,theta))) )
{
stop("NA or Infinite or NAN values in the Input")
}
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]))*(theta^(y[i]*(n-y[i])))/
sum(choose(n,0:n)*(p^(0:n))*((1-p)^(n-(0:n)))*(theta^((0:n)*(n-(0:n)))))
}
check1<-sum(value1)
#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 if probability value is less than zero or greater than one and
#theta value greater than zero or equal to zero
#if so creating an error message as well as stopping the function progress
else if( p <= 0 | p >= 1 | theta <= 0)
{
stop("Probability or Theta parameter value doesnot satisfy conditions")
}
#checking if the sum of all probability values leads upto 1
#if not providing an error message and stopping the function progress
else if(check1 < 0.9999 | check1 >1.0001 | any(value1 < 0) | any(value1 >1))
{
stop("Input parameter combinations of probability of success and theta does
not create proper probability function")
}
else
{
k<-0:n
#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)])+
log(theta)*sum(data[1:sum(freq)]*(n-data[1:sum(freq)]))-
sum(freq)*log(sum(choose(n,k)*(p^k)*((1-p)^(n-k))*(theta^(k*(n-k)))))))
}
}
}
#' Estimating the probability of success and theta for Multiplicative Binomial
#' Distribution
#'
#' The function will estimate the probability of success and theta parameter using the
#' maximum log likelihood method for the Multiplicative Binomial distribution when the binomial random
#' variables and corresponding frequencies are given.
#'
#' @usage
#' EstMLEMultiBin(x,freq,p,theta,...)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param p single value for probability of success.
#' @param theta single value for theta parameter.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#' @details
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' @return
#' \code{EstMLEMultiBin} 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} .
#'
#' @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 <- EstMLEMultiBin(x=No.D.D,freq=Obs.fre.1,p=0.5,theta=15)
#'
#' bbmle::coef(parameters) #extracting the parameters
#'
#' @export
EstMLEMultiBin<-function(x,freq,p,theta,...)
{
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(.EstMLEMultiBin,data=list(x=x,freq=freq),
start = list(p=p,theta=theta),...),"NaN")
return(output)
}
.EstMLEMultiBin<-function(x,freq,p,theta)
{
#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
#MultiBinomial distribution
n<-max(x)
data<-rep(x,freq)
k<-0:n
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)])+
log(theta)*sum(data[1:sum(freq)]*(n-data[1:sum(freq)]))-
sum(freq)*log(sum(choose(n,k)*(p^k)*((1-p)^(n-k))*(theta^(k*(n-k)))))))
}
#' Fitting the Multiplicative Binomial Distribution when binomial
#' random variable, frequency, probability of success and theta parameter are given
#'
#' The function will fit the Multiplicative Binomial distribution
#' when random variables, corresponding frequencies, probability of success and theta parameter
#' are given. It will provide the expected frequencies, chi-squared test statistics value,
#' p value and degree of freedom value so that it can be seen if this distribution
#' fits the data.
#'
#' @usage fitMultiBin(x,obs.freq,p,theta)
#'
#' @param x vector of binomial random variables.
#' @param obs.freq vector of frequencies.
#' @param p single value for probability of success.
#' @param theta single value for theta parameter.
#'
#' @details
#' \deqn{obs.freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' @return
#' The output of \code{fitMultiBin} gives the class format \code{fitMuB} 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{fitMuB} fitted probability values of \code{dMultiBin}.
#'
#' \code{NegLL} Negative Log Likelihood value.
#'
#' \code{p} estimated probability value.
#'
#' \code{theta} estimated theta parameter 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}.
#'
#' @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 <- EstMLEMultiBin(x=No.D.D,freq=Obs.fre.1,p=0.1,theta=.3)
#'
#' pMultiBin <- bbmle::coef(parameters)[1] #assigning the estimated probability value
#' thetaMultiBin <- bbmle::coef(parameters)[2] #assigning the estimated theta value
#'
#' #fitting when the random variable,frequencies,probability and theta are given
#' results <- fitMultiBin(No.D.D,Obs.fre.1,pMultiBin,thetaMultiBin)
#' results
#'
#' #extracting the AIC value
#' AIC(results)
#'
#' #extract fitted values
#' fitted(results)
#'
#' @export
fitMultiBin<-function(x,obs.freq,p,theta)
{
#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,theta))) | any(is.infinite(c(x,obs.freq,p,theta))) |
any(is.nan(c(x,obs.freq,p,theta))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dMultiBin(x,max(x),p,theta)
#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<-NegLLMultiBin(x,obs.freq,p,theta)
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),"fitMuB"=est,
"NegLL"=NegLL,"p"=p,"theta"=theta,"AIC"=2*2+2*NegLL,"call"=match.call())
class(final)<-c("fitMuB","fit")
return(final)
}
}
#' @method fitMultiBin default
#' @export
fitMultiBin.default<-function(x,obs.freq,p,theta)
{
return(fitMultiBin(x,obs.freq,p,theta))
}
#' @method print fitMuB
#' @export
print.fitMuB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Multiplicative Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated p value :",x$p," ,estimated theta parameter :",x$theta,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n")
}
#' @method summary fitMuB
#' @export
summary.fitMuB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Multiplicative Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated p value :",object$p," ,estimated theta parameter :",object$theta,"\n\t
X-squared :",object$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.fitMuB print.fitMuB fitMultiBin.default fitMultiBin .EstMLEMultiBin EstMLEMultiBin NegLLMultiBin pMultiBin dMultiBin
Documented in dMultiBin EstMLEMultiBin fitMultiBin NegLLMultiBin pMultiBin
#' Multiplicative Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Multiplicative Binomial Distribution.
#'
#' @usage
#' dMultiBin(x,n,p,theta)
#'
#' @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 theta single value for theta.
#'
#' @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_{MultiBin}(x)= {n \choose x} p^x (1-p)^{n-x} \frac{(theta^{x(n-x)}}{f(p,theta,n)} }
#'
#' here \eqn{f(p,theta,n)} is
#' \deqn{f(p,theta,n)= \sum_{k=0}^{n} {n \choose k} p^k (1-p)^{n-k} (theta^{k(n-k)} )}
#'
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{k = 0,1,2,...,n}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' \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{dMultiBin} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of Multiplicative Binomial Distribution.
#'
#' \code{var} variance of Multiplicative Binomial Distribution.
#'
#' @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}.
#'
#' @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="Multiplicative 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,dMultiBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dMultiBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dMultiBin(0:10,10,.58,10.022)$pdf #extracting the pdf values
#' dMultiBin(0:10,10,.58,10.022)$mean #extracting the mean
#' dMultiBin(0:10,10,.58,10.022)$var #extracting the variance
#'
#'
#' #plotting 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="Multiplicative 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,pMultiBin(0:10,10,a[i],1+b[i]),col = col[i],lwd=2.85)
#' points(0:10,pMultiBin(0:10,10,a[i],1+b[i]),col = col[i],pch=16)
#' }
#'
#' pMultiBin(0:10,10,.58,10.022) #acquiring the cumulative probability values
#'
#' @export
dMultiBin<-function(x,n,p,theta)
{
#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,theta))) | any(is.infinite(c(x,n,p,theta))) |
any(is.nan(c(x,n,p,theta))) )
{
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
{
#checking the probability value is inbetween zero and one if so providig an error message
#and stopping the function progress
if( p <= 0 | p >= 1)
{
stop("Probability value doesnot satisfy conditions")
}
else
{
#constructing the probability values for all random variables
y<-0:n
value1<-NULL
func1<-
for (i in 1:length(y))
{
value1[i]<-choose(n,y[i])*(p^y[i])*((1-p)^(n-y[i]))*(theta^(y[i]*(n-y[i])))/(sum(choose(n,0:n)*(p^(0:n))*((1-p)^(n-(0:n)))*(theta^((0:n)*(n-(0:n))))))
}
check1<-sum(value1)
#checking if the theta value is less than or equal to zero if so providig an error message
#and stopping the function progress
if(theta <= 0)
{
stop("Theta parameter value cannot be zero or less than zero")
}
#checking if the sum of all probability values leads upto one
#if not providing an error message and stopping the function progress
else if(check1 < 0.9999 | check1 >1.0001 | any(value1 < 0) | any(value1 >1))
{
stop("Input parameter combinations of probability of success and theta does
not create proper probability function")
}
else
{
value<-NULL
#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]))*(theta^(x[i]*(n-x[i])))/
(sum(choose(n,0:n)*(p^(0:n))*((1-p)^(n-(0:n)))*(theta^((0:n)*(n-(0:n))))))
}
# generating an output in list format consisting pdf,mean and variance
return(list("pdf"=value,"mean"=sum((0:n)*value1),
"var"=sum(((0:n)^2)*value1)-(sum((0:n)*value1))^2))
}
}
}
}
}
#' Multiplicative Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Multiplicative Binomial Distribution.
#'
#' @usage
#' pMultiBin(x,n,p,theta)
#'
#' @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 theta single value for theta.
#'
#' @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_{MultiBin}(x)= {n \choose x} p^x (1-p)^{n-x} \frac{(theta^{x(n-x)}}{f(p,theta,n)} }
#'
#' here \eqn{f(p,theta,n)} is
#' \deqn{f(p,theta,n)= \sum_{k=0}^{n} {n \choose k} p^k (1-p)^{n-k} (theta^{k(n-k)} )}
#'
#' \deqn{x = 0,1,2,3,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{k = 0,1,2,...,n}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' \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{pMultiBin} 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}.
#'
#' @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="Multiplicative 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,dMultiBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dMultiBin(0:10,10,a[i],1+b[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dMultiBin(0:10,10,.58,10.022)$pdf #extracting the pdf values
#' dMultiBin(0:10,10,.58,10.022)$mean #extracting the mean
#' dMultiBin(0:10,10,.58,10.022)$var #extracting the variance
#'
#' #plotting 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="Multiplicative 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,pMultiBin(0:10,10,a[i],1+b[i]),col = col[i],lwd=2.85)
#' points(0:10,pMultiBin(0:10,10,a[i],1+b[i]),col = col[i],pch=16)
#' }
#'
#' pMultiBin(0:10,10,.58,10.022) #acquiring the cumulative probability values
#'
#' @export
pMultiBin<-function(x,n,p,theta)
{
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(dMultiBin(0:x[i],n,p,theta)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of Multiplicative Binomial distribution
#'
#' This function will calculate the negative log likelihood value when the vector of binomial random
#' variable and vector of corresponding frequencies are given with the input parameters.
#'
#' @usage
#' NegLLMultiBin(x,freq,p,theta)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param p single value for probability of success.
#' @param theta single value for theta parameter.
#'
#' @details
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' @return
#' The output of \code{NegLLMultiBin} 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}.
#'
#' @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
#'
#' NegLLMultiBin(No.D.D,Obs.fre.1,.5,3) #acquiring the negative log likelihood value
#'
#' @export
NegLLMultiBin<-function(x,freq,p,theta)
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
#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,theta))) | any(is.infinite(c(x,freq,p,theta))) |
any(is.nan(c(x,freq,p,theta))) )
{
stop("NA or Infinite or NAN values in the Input")
}
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]))*(theta^(y[i]*(n-y[i])))/
sum(choose(n,0:n)*(p^(0:n))*((1-p)^(n-(0:n)))*(theta^((0:n)*(n-(0:n)))))
}
check1<-sum(value1)
#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 if probability value is less than zero or greater than one and
#theta value greater than zero or equal to zero
#if so creating an error message as well as stopping the function progress
else if( p <= 0 | p >= 1 | theta <= 0)
{
stop("Probability or Theta parameter value doesnot satisfy conditions")
}
#checking if the sum of all probability values leads upto 1
#if not providing an error message and stopping the function progress
else if(check1 < 0.9999 | check1 >1.0001 | any(value1 < 0) | any(value1 >1))
{
stop("Input parameter combinations of probability of success and theta does
not create proper probability function")
}
else
{
k<-0:n
#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)])+
log(theta)*sum(data[1:sum(freq)]*(n-data[1:sum(freq)]))-
sum(freq)*log(sum(choose(n,k)*(p^k)*((1-p)^(n-k))*(theta^(k*(n-k)))))))
}
}
}
#' Estimating the probability of success and theta for Multiplicative Binomial
#' Distribution
#'
#' The function will estimate the probability of success and theta parameter using the
#' maximum log likelihood method for the Multiplicative Binomial distribution when the binomial random
#' variables and corresponding frequencies are given.
#'
#' @usage
#' EstMLEMultiBin(x,freq,p,theta,...)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param p single value for probability of success.
#' @param theta single value for theta parameter.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#' @details
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' @return
#' \code{EstMLEMultiBin} 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} .
#'
#' @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 <- EstMLEMultiBin(x=No.D.D,freq=Obs.fre.1,p=0.5,theta=15)
#'
#' bbmle::coef(parameters) #extracting the parameters
#'
#' @export
EstMLEMultiBin<-function(x,freq,p,theta,...)
{
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(.EstMLEMultiBin,data=list(x=x,freq=freq),
start = list(p=p,theta=theta),...),"NaN")
return(output)
}
.EstMLEMultiBin<-function(x,freq,p,theta)
{
#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
#MultiBinomial distribution
n<-max(x)
data<-rep(x,freq)
k<-0:n
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)])+
log(theta)*sum(data[1:sum(freq)]*(n-data[1:sum(freq)]))-
sum(freq)*log(sum(choose(n,k)*(p^k)*((1-p)^(n-k))*(theta^(k*(n-k)))))))
}
#' Fitting the Multiplicative Binomial Distribution when binomial
#' random variable, frequency, probability of success and theta parameter are given
#'
#' The function will fit the Multiplicative Binomial distribution
#' when random variables, corresponding frequencies, probability of success and theta parameter
#' are given. It will provide the expected frequencies, chi-squared test statistics value,
#' p value and degree of freedom value so that it can be seen if this distribution
#' fits the data.
#'
#' @usage fitMultiBin(x,obs.freq,p,theta)
#'
#' @param x vector of binomial random variables.
#' @param obs.freq vector of frequencies.
#' @param p single value for probability of success.
#' @param theta single value for theta parameter.
#'
#' @details
#' \deqn{obs.freq \ge 0}
#' \deqn{x = 0,1,2,..}
#' \deqn{0 < p < 1}
#' \deqn{0 < theta }
#'
#' @return
#' The output of \code{fitMultiBin} gives the class format \code{fitMuB} 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{fitMuB} fitted probability values of \code{dMultiBin}.
#'
#' \code{NegLL} Negative Log Likelihood value.
#'
#' \code{p} estimated probability value.
#'
#' \code{theta} estimated theta parameter 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}.
#'
#' @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 <- EstMLEMultiBin(x=No.D.D,freq=Obs.fre.1,p=0.1,theta=.3)
#'
#' pMultiBin <- bbmle::coef(parameters)[1] #assigning the estimated probability value
#' thetaMultiBin <- bbmle::coef(parameters)[2] #assigning the estimated theta value
#'
#' #fitting when the random variable,frequencies,probability and theta are given
#' results <- fitMultiBin(No.D.D,Obs.fre.1,pMultiBin,thetaMultiBin)
#' results
#'
#' #extracting the AIC value
#' AIC(results)
#'
#' #extract fitted values
#' fitted(results)
#'
#' @export
fitMultiBin<-function(x,obs.freq,p,theta)
{
#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,theta))) | any(is.infinite(c(x,obs.freq,p,theta))) |
any(is.nan(c(x,obs.freq,p,theta))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dMultiBin(x,max(x),p,theta)
#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<-NegLLMultiBin(x,obs.freq,p,theta)
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),"fitMuB"=est,
"NegLL"=NegLL,"p"=p,"theta"=theta,"AIC"=2*2+2*NegLL,"call"=match.call())
class(final)<-c("fitMuB","fit")
return(final)
}
}
#' @method fitMultiBin default
#' @export
fitMultiBin.default<-function(x,obs.freq,p,theta)
{
return(fitMultiBin(x,obs.freq,p,theta))
}
#' @method print fitMuB
#' @export
print.fitMuB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Multiplicative Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated p value :",x$p," ,estimated theta parameter :",x$theta,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n")
}
#' @method summary fitMuB
#' @export
summary.fitMuB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Multiplicative Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated p value :",object$p," ,estimated theta parameter :",object$theta,"\n\t
X-squared :",object$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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