Nothing
R/103.ConfidenceIntervals_BASE_n_x.R
In proportion: Inference on Single Binomial Proportion and Bayesian Computations
Defines functions
ciWDx
ciSCx
ciASx
ciLRx
ciEXx
lufn103
exlim103l
exlim103u
ciTWx
ciLTx
ciBAx
ciBADx
ciAllx
Documented in
ciAllx
ciASx
ciBAx
ciEXx
ciLRx
ciLTx
ciSCx
ciTWx
ciWDx
#' Wald method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details Wald-type interval that results from inverting
#' large-sample test and evaluates standard errors at maximum likelihood estimates
#' for the given \code{x} and \code{n}
#' @return A dataframe with
#' \item{x}{ Number of successes (positive samples)}
#' \item{LWDx }{ Wald Lower limit}
#' \item{UWDx }{ Wald Upper Limit}
#' \item{LABB }{ Wald Lower Abberation}
#' \item{UABB }{ Wald Upper Abberation}
#' \item{ZWI }{ Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05;
#' ciWDx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#1.WALD
ciWDx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
###CRITICAL VALUES
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
#WALD METHOD
pW=x/n
qW=1-(x/n)
seW=sqrt(pW*qW/n)
LWDx=pW-(cv*seW)
UWDx=pW+(cv*seW)
if(LWDx<0) LABB="YES" else LABB="NO"
if(LWDx<0) LWDx=0
if(UWDx>1) UABB="YES" else UABB="NO"
if(UWDx>1) UWDx=1
if(UWDx-LWDx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LWDx,UWDx,LABB,UABB,ZWI))
}
##############################################################################
#' Base Score method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details A score test approach based on inverting the test
#' with standard error evaluated at the null hypothesis is due to
#' Wilson for the given \code{x} and \code{n}
#' @return A dataframe with
#' \item{x}{ Number of successes (positive samples)}
#' \item{LSCx }{ Score Lower limit}
#' \item{USCx }{ Score Upper Limit}
#' \item{LABB }{ Score Lower Abberation}
#' \item{UABB }{ Score Upper Abberation}
#' \item{ZWI }{ Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciSCx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#2.SCORE
ciSCx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
###CRITICAL VALUES
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
cv1=(cv^2)/(2*n)
cv2=(cv/(2*n))^2
#SCORE (WILSON) METHOD
pS=x/n
qS=1-(x/n)
seS=sqrt((pS*qS/n)+cv2)
LSCx=(n/(n+(cv)^2))*((pS+cv1)-(cv*seS))
USCx=(n/(n+(cv)^2))*((pS+cv1)+(cv*seS))
if(LSCx<0) LABB="YES" else LABB="NO"
if(LSCx<0) LSCx=0
if(USCx>1) UABB="YES" else UABB="NO"
if(USCx>1) USCx=1
if(USCx-LSCx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LSCx,USCx,LABB,UABB,ZWI))
}
##############################################################################
#' Base ArcSine method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details Wald-type interval for the given x and n using the arcsine
#' transformation of the parameter \code{p}; that is based on the normal approximation
#' for \eqn{sin^{-1}(p) }
#' @return A dataframe with
#' \item{x}{ Number of successes (positive samples)}
#' \item{LASx }{ ArcSine Lower limit}
#' \item{UASx }{ ArcSine Upper Limit}
#' \item{LABB }{ ArcSine Lower Abberation}
#' \item{UABB }{ ArcSine Upper Abberation}
#' \item{ZWI }{ Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciASx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#3.ARCSINE
ciASx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
#ARC-SINE METHOD
pA=x/n
#qA=1-pA
seA=cv/sqrt(4*n)
LASx=(sin(asin(sqrt(pA))-seA))^2
UASx=(sin(asin(sqrt(pA))+seA))^2
if(LASx<0) LABB="YES" else LABB="NO"
if(LASx<0) LASx=0
if(UASx>1) UABB="YES" else UABB="NO"
if(UASx>1) UASx=1
if(UASx-LASx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LASx,UASx,LABB,UABB,ZWI))
}
##############################################################################
#' Likelihood Ratio method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details Likelihood ratio limits for the given \code{x} and \code{n}
#' obtained as the solution to the equation in \code{p} formed as logarithm of
#' ratio between binomial likelihood at sample proportion and that of over
#' all possible parameters
#' @return A dataframe with
##' \item{x }{- Number of successes (positive samples)}
##' \item{LLRx }{ - Likelihood Ratio Lower limit}
##' \item{ULRx }{ - Likelihood Ratio Upper Limit}
##' \item{LABB }{ - Likelihood Ratio Lower Abberation}
##' \item{UABB }{ - Likelihood Ratio Upper Abberation}
##' \item{ZWI }{ - Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciLRx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#4.LIKELIHOOD RATIO
ciLRx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
####INPUT
y=x
###CRITICAL VALUES
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
#LIKELIHOOD-RATIO METHOD
likelhd = function(p) dbinom(y,n,p)
loglik = function(p) dbinom(y,n,p,log=TRUE)
mle=optimize(likelhd,c(0,1),maximum=TRUE)$maximum
cutoff=loglik(mle)-(cv^2/2)
loglik.optim=function(p){abs(cutoff-loglik(p))}
LLRx=optimize(loglik.optim, c(0,mle))$minimum
ULRx=optimize(loglik.optim, c(mle,1))$minimum
if(LLRx<0) LABB="YES" else LABB="NO"
if(LLRx<0) LLRx=0
if(ULRx>1) UABB="YES" else UABB="NO"
if(ULRx>1) ULRx=1
if(ULRx-LLRx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LLRx,ULRx,LABB,UABB,ZWI))
}
##############################################################################
#' Exact method of CI estimation
#' @param n - Number of trials
#' @param x - Number of sucess
#' @param alp - Alpha value (significance level required)
#' @param e - Exact method indicator in [0, 1] {1: Clopper Pearson, 0.5: Mid P}
#' @details Confidence interval for \code{p} (for the given \code{x} and \code{n}),
#' based on inverting equal-tailed binomial tests with null hypothesis \eqn{H0: p = p0 }
#' and calculated from the cumulative binomial distribution.
#' Exact two sided P-value is usually calculated as \deqn{P= 2[e*Pr(X = x) + min({Pr(X < x), Pr(X > x)})]}
#' where probabilities are found at null value of \code{p} and \eqn{0 <= e <= 1.}
#' @return A dataframe with
##' \item{x }{- Number of successes (positive samples)}
##' \item{LEXx }{ - Exact Lower limit}
##' \item{UEXx }{ - Exact Upper Limit}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05;e=0.5
#' ciEXx(x,n,alp,e) #Mid-p
#' x=5; n=5; alp=0.05;e=1 #Clopper Pearson
#' ciEXx(x,n,alp,e)
#' x=5; n=5; alp=0.05;e=c(0.1,0.5,0.95,1) #Range including Mid-p and Clopper-Pearson
#' ciEXx(x,n,alp,e)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#5.EXACT METHOD
ciEXx=function(x,n,alp,e)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (missing(e)) stop("'e' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp) >1) stop("'alpha' has to be between 0 and 1")
if ((class(e) != "integer") & (class(e) != "numeric") || any(e>1) || any(e<0)) stop("'e' has to be between 0 and 1")
if (length(e)>10) stop("'e' can have only 10 intervals")
nvar=length(e)
res <- data.frame()
for(i in 1:nvar)
{
lu=lufn103(x,n,alp,e[i])
res <- rbind(res,lu)
}
return(res)
}
lufn103<-function(x,n,alp,e)
{
LEXx=0
UEXx=0
LABB=0
UABB=0
ZWI=0
LEXx=exlim103l(x,n,alp,e)
UEXx=exlim103u(x,n,alp,e)
if(LEXx<0) LABB="YES" else LABB="NO"
if(LEXx<0) LEXx=0
if(UEXx>1) UABB="YES" else UABB="NO"
if(UEXx>1) UEXx=1
if(UEXx-LEXx==0) ZWI="YES" else ZWI="NO"
resx=data.frame(x,LEXx,UEXx,LABB, UABB, ZWI, e)
return(resx)
}
exlim103l=function(x,n,alp,e)
{
if(x==0)
{
LEX = 0
} else if(x==n){
LEX= (alp/(2*e))^(1/n)
}else
{
z=x-1
y=0:z
f1=function(p) (1-e)*dbinom(x,n,p)+sum(dbinom(y,n,p))-(1-(alp/2))
LEX= uniroot(f1,c(0,1))$root
}
return(LEX)
}
exlim103u=function(x,n,alp,e)
{
if(x==0)
{
UEX = 1-((alp/(2*e))^(1/n))
} else if(x==n){
UEX = 1
}else
{
z=x-1
y=0:z
f2= function(p) e*dbinom(x,n,p)+sum(dbinom(y,n,p))-(alp/2)
UEX =uniroot(f2,c(0,1))$root
}
return(UEX)
}
######################################################################
#' Wald-T method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details An approximate method based on a t_approximation of the
#' standardized point estimator for the given \code{x} and \code{n}; that is the point estimator
#' divided by its estimated standard error. Essential boundary modification is when \eqn{x = 0 or n},
#' \deqn{\hat{p}=\frac{(x+2)}{(n+4)}{phat=(x+2)/(n+4)}}
#' @return A dataframe with
#' \item{x}{ Number of successes (positive samples)}
#' \item{LTWx }{ Wald-T Lower limit}
#' \item{UTWx }{ Wald-T Upper Limit}
#' \item{LABB }{ Wald-T Lower Abberation}
#' \item{UABB }{ Wald-T Upper Abberation}
#' \item{ZWI }{ Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciTWx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#6.WALD-T
ciTWx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
#MODIFIED_t-WALD METHOD
if(x==0||x==n)
{
pTWx=(x+2)/(n+4)
#qTWx=1-pTWx
}else
{
pTWx=x/n
#qTWx=1-pTWx
}
f1=function(p,n) p*(1-p)/n
f2=function(p,n) (p*(1-p)/(n^3))+(p+((6*n)-7)*(p^2)+(4*(n-1)*(n-3)*(p^3))-(2*(n-1)*((2*n)-3)*(p^4)))/(n^5)-(2*(p+((2*n)-3)*(p^2)-2*(n-1)*(p^3)))/(n^4)
DOFx=2*((f1(pTWx,n))^2)/f2(pTWx,n)
cvx=qt(1-(alp/2), df=DOFx)
seTWx=cvx*sqrt(f1(pTWx,n))
LTWx=pTWx-(seTWx)
UTWx=pTWx+(seTWx)
if(LTWx<0) LABB="YES" else LABB="NO"
if(LTWx<0) LTWx=0
if(UTWx>1) UABB="YES" else UABB="NO"
if(UTWx>1) UTWx=1
if(UTWx-LTWx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LTWx,UTWx,LABB,UABB,ZWI))
}
#####################################################################
#' Logit Wald method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details Wald-type interval for all the given \code{x} and \code{n}
#' based on the logit transformation of \code{p};
#' that is normal approximation for \eqn{log(p/1-p)}
#' @return A dataframe with
#' \item{x }{- Number of successes (positive samples)}
#' \item{LLTx }{ - Logit Wald Lower limit}
#' \item{ULTx }{ - Logit Wald Upper Limit}
#' \item{LABB }{ - Logit Wald Lower Abberation}
#' \item{UABB }{ - Logit Wald Upper Abberation}
#' \item{ZWI }{ - Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciLTx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#7.LOGIT-WALD
ciLTx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
####INITIALIZATIONS
pLTx=0
qLTx=0
seLTx=0
lgitx=0
LLTx=0
ULTx=0
LABB=0
UABB=0
ZWI=0
###CRITICAL VALUES
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
#LOGIT-WALD METHOD
if(x==0)
{
pLTx=0
qLTx=1
LLTx = 0
ULTx = 1-((alp/2)^(1/n))
}
if(x==n)
{
pLTx=1
qLTx=0
LLTx= (alp/2)^(1/n)
ULTx=1
}
if(0<x&&x<n)
{
pLTx=x/n
qLTx=1-pLTx
lgitx=log(pLTx/qLTx)
seLTx=sqrt(pLTx*qLTx*n)
LLTx=1/(1+exp(-lgitx+(cv/seLTx)))
ULTx=1/(1+exp(-lgitx-(cv/seLTx)))
}
if(LLTx<0) LABB="YES" else LABB="NO"
if(LLTx<0) LLTx=0
if(ULTx>1) UABB="YES" else UABB="NO"
if(ULTx>1) ULTx=1
if(ULTx-LLTx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LLTx,ULTx,LABB,UABB,ZWI))
}
#####################################################################################
#' Bayesian method of CI estimation with Beta prior distribution
#' @param x - Number of sucess
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Shape parameter 1 for prior Beta distribution in Bayesian model. Can also be a vector of length n+1 priors.
#' @param b - Shape parameter 2 for prior Beta distribution in Bayesian model. Can also be a vector of length n+1 priors.
#' @details Highest Probability Density (HPD) and two tailed intervals are
#' provided for the given \code{x} and \code{n}. based on the conjugate prior \eqn{\beta(a, b)}
#' for the probability of success \code{p} of the binomial distribution so that the
#' posterior is \eqn{\beta(x + a, n - x + b)}.
#' @return A dataframe with
##' \item{x }{- Number of successes (positive samples)}
##' \item{LBAQx }{ - Lower limits of Quantile based intervals}
##' \item{UBAQx }{ - Upper limits of Quantile based intervals}
##' \item{LBAHx }{ - Lower limits of HPD intervals}
##' \item{UBAHx }{ - Upper limits of HPD intervals}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05; a=0.5;b=0.5;
#' ciBAx(x,n,alp,a,b)
#' x= 5; n=5; alp=0.05; a=c(0.5,2,1,1,2,0.5);b=c(0.5,2,1,1,2,0.5)
#' ciBAx(x,n,alp,a,b)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#8. BAYESIAN
ciBAx<-function(x,n,alp,a,b)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (missing(a)) stop("'a' is missing")
if (missing(b)) stop("'b' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
if(length(a)>1 || length(b)>1) {
if ((class(a) != "integer") & (class(a) != "numeric") || any(a < 0)) stop("'a' has to be a set of positive numeric vectors")
if ((class(b) != "integer") & (class(b) != "numeric") || any(b < 0)) stop("'b' has to be a set of positive numeric vectors")
if (length(a) <= n || length(b) <= n ) stop("'a' and 'b' vectors have to be equal to or greater than length n+1")
BAdfx=ciBADx(x,n,alp,a,b)
}
else
{
if (a<0 ) stop("'a' has to be greater than or equal to 0")
if (b<0 ) stop("'b' has to be greater than or equal to 0")
##############
#library(TeachingDemos) #To get HPDs
#Quantile Based Intervals
LBAQx=qbeta(alp/2,x+a,n-x+b)
UBAQx=qbeta(1-(alp/2),x+a,n-x+b)
LBAHx=TeachingDemos::hpd(qbeta,shape1=x+a,shape2=n-x+b,conf=1-alp)[1]
UBAHx=TeachingDemos::hpd(qbeta,shape1=x+a,shape2=n-x+b,conf=1-alp)[2]
BAdfx=data.frame(x,LBAQx,UBAQx,LBAHx,UBAHx)
}
return(BAdfx)
}
#######################################################################
ciBADx<-function(x,n,alp,a,b)
{
k=n+1
####INITIALIZATIONS
LBAQx=0
UBAQx=0
LBAHx=0
UBAHx=0
##############
for(i in 1:k)
{
#Quantile Based Intervals
LBAQx[i]=qbeta(alp/2,x+a[i],n-x+b[i])
UBAQx[i]=qbeta(1-(alp/2),x+a[i],n-x+b[i])
LBAHx[i]=TeachingDemos::hpd(qbeta,shape1=x+a[i],shape2=n-x+b[i],conf=1-alp)[1]
UBAHx[i]=TeachingDemos::hpd(qbeta,shape1=x+a[i],shape2=n-x+b[i],conf=1-alp)[2]
}
return(data.frame(x,LBAQx,UBAQx,LBAHx,UBAHx))
}
#####################################################################################
#' Specific CI estimation of 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine)
#' @param x - Number of sucess
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details The Confidence Interval of using 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine) for \code{n} given \code{alp} and \code{x}
#' @return A dataframe with
##' \item{name }{- Name of the method}
##' \item{x }{- Number of successes (positive samples)}
##' \item{LLT }{ - Lower limit}
##' \item{ULT }{ - Upper Limit}
##' \item{LABB }{ - Lower Abberation}
##' \item{UABB }{ - Upper Abberation}
##' \item{ZWI }{ - Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x= 5; n=5; alp=0.05;
#' ciAllx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#9.All methods
ciAllx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
#### Calling functions and creating df
WaldCI.df = ciWDx(x,n,alp)
ArcSineCI.df = ciASx(x,n,alp)
LRCI.df = ciLRx(x,n,alp)
ScoreCI.df = ciSCx(x,n,alp)
WaldLCI.df = ciLTx(x,n,alp)
AdWaldCI.df = ciTWx(x,n,alp)
WaldCI.df$method = as.factor("Wald")
ArcSineCI.df$method = as.factor("ArcSine")
LRCI.df$method = as.factor("Likelihood")
WaldLCI.df$method = as.factor("Logit-Wald")
ScoreCI.df$method = as.factor("Score")
AdWaldCI.df$method = as.factor("Wald-T")
Generic.1 = data.frame(method = WaldCI.df$method, x=WaldCI.df$x, LowerLimit = WaldCI.df$LWDx, UpperLimit = WaldCI.df$UWDx, LowerAbb = WaldCI.df$LABB, UpperAbb = WaldCI.df$UABB, ZWI = WaldCI.df$ZWI)
Generic.2 = data.frame(method = ArcSineCI.df$method, x=ArcSineCI.df$x, LowerLimit = ArcSineCI.df$LASx, UpperLimit = ArcSineCI.df$UASx, LowerAbb = ArcSineCI.df$LABB, UpperAbb = ArcSineCI.df$UABB, ZWI = ArcSineCI.df$ZWI)
Generic.3 = data.frame(method = LRCI.df$method, x=LRCI.df$x, LowerLimit = LRCI.df$LLRx, UpperLimit = LRCI.df$ULRx, LowerAbb = LRCI.df$LABB, UpperAbb = LRCI.df$UABB, ZWI = LRCI.df$ZWI)
Generic.4 = data.frame(method = ScoreCI.df$method, x=ScoreCI.df$x, LowerLimit = ScoreCI.df$LSCx, UpperLimit = ScoreCI.df$USCx, LowerAbb = ScoreCI.df$LABB, UpperAbb = ScoreCI.df$UABB, ZWI = ScoreCI.df$ZWI)
Generic.5 = data.frame(method = WaldLCI.df$method, x=WaldLCI.df$x, LowerLimit = WaldLCI.df$LLTx, UpperLimit = WaldLCI.df$ULTx, LowerAbb = WaldLCI.df$LABB, UpperAbb = WaldLCI.df$UABB, ZWI = WaldLCI.df$ZWI)
Generic.6 = data.frame(method = AdWaldCI.df$method, x=AdWaldCI.df$x, LowerLimit = AdWaldCI.df$LTWx, UpperLimit = AdWaldCI.df$UTWx, LowerAbb = AdWaldCI.df$LABB, UpperAbb = AdWaldCI.df$UABB, ZWI = AdWaldCI.df$ZWI)
Final.df= rbind(Generic.1,Generic.2,Generic.3,Generic.4,Generic.5, Generic.6)
return(Final.df)
}
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Defines functions ciWDx ciSCx ciASx ciLRx ciEXx lufn103 exlim103l exlim103u ciTWx ciLTx ciBAx ciBADx ciAllx
Documented in ciAllx ciASx ciBAx ciEXx ciLRx ciLTx ciSCx ciTWx ciWDx
#' Wald method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details Wald-type interval that results from inverting
#' large-sample test and evaluates standard errors at maximum likelihood estimates
#' for the given \code{x} and \code{n}
#' @return A dataframe with
#' \item{x}{ Number of successes (positive samples)}
#' \item{LWDx }{ Wald Lower limit}
#' \item{UWDx }{ Wald Upper Limit}
#' \item{LABB }{ Wald Lower Abberation}
#' \item{UABB }{ Wald Upper Abberation}
#' \item{ZWI }{ Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05;
#' ciWDx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#1.WALD
ciWDx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
###CRITICAL VALUES
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
#WALD METHOD
pW=x/n
qW=1-(x/n)
seW=sqrt(pW*qW/n)
LWDx=pW-(cv*seW)
UWDx=pW+(cv*seW)
if(LWDx<0) LABB="YES" else LABB="NO"
if(LWDx<0) LWDx=0
if(UWDx>1) UABB="YES" else UABB="NO"
if(UWDx>1) UWDx=1
if(UWDx-LWDx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LWDx,UWDx,LABB,UABB,ZWI))
}
##############################################################################
#' Base Score method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details A score test approach based on inverting the test
#' with standard error evaluated at the null hypothesis is due to
#' Wilson for the given \code{x} and \code{n}
#' @return A dataframe with
#' \item{x}{ Number of successes (positive samples)}
#' \item{LSCx }{ Score Lower limit}
#' \item{USCx }{ Score Upper Limit}
#' \item{LABB }{ Score Lower Abberation}
#' \item{UABB }{ Score Upper Abberation}
#' \item{ZWI }{ Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciSCx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#2.SCORE
ciSCx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
###CRITICAL VALUES
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
cv1=(cv^2)/(2*n)
cv2=(cv/(2*n))^2
#SCORE (WILSON) METHOD
pS=x/n
qS=1-(x/n)
seS=sqrt((pS*qS/n)+cv2)
LSCx=(n/(n+(cv)^2))*((pS+cv1)-(cv*seS))
USCx=(n/(n+(cv)^2))*((pS+cv1)+(cv*seS))
if(LSCx<0) LABB="YES" else LABB="NO"
if(LSCx<0) LSCx=0
if(USCx>1) UABB="YES" else UABB="NO"
if(USCx>1) USCx=1
if(USCx-LSCx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LSCx,USCx,LABB,UABB,ZWI))
}
##############################################################################
#' Base ArcSine method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details Wald-type interval for the given x and n using the arcsine
#' transformation of the parameter \code{p}; that is based on the normal approximation
#' for \eqn{sin^{-1}(p) }
#' @return A dataframe with
#' \item{x}{ Number of successes (positive samples)}
#' \item{LASx }{ ArcSine Lower limit}
#' \item{UASx }{ ArcSine Upper Limit}
#' \item{LABB }{ ArcSine Lower Abberation}
#' \item{UABB }{ ArcSine Upper Abberation}
#' \item{ZWI }{ Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciASx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#3.ARCSINE
ciASx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
#ARC-SINE METHOD
pA=x/n
#qA=1-pA
seA=cv/sqrt(4*n)
LASx=(sin(asin(sqrt(pA))-seA))^2
UASx=(sin(asin(sqrt(pA))+seA))^2
if(LASx<0) LABB="YES" else LABB="NO"
if(LASx<0) LASx=0
if(UASx>1) UABB="YES" else UABB="NO"
if(UASx>1) UASx=1
if(UASx-LASx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LASx,UASx,LABB,UABB,ZWI))
}
##############################################################################
#' Likelihood Ratio method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details Likelihood ratio limits for the given \code{x} and \code{n}
#' obtained as the solution to the equation in \code{p} formed as logarithm of
#' ratio between binomial likelihood at sample proportion and that of over
#' all possible parameters
#' @return A dataframe with
##' \item{x }{- Number of successes (positive samples)}
##' \item{LLRx }{ - Likelihood Ratio Lower limit}
##' \item{ULRx }{ - Likelihood Ratio Upper Limit}
##' \item{LABB }{ - Likelihood Ratio Lower Abberation}
##' \item{UABB }{ - Likelihood Ratio Upper Abberation}
##' \item{ZWI }{ - Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciLRx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#4.LIKELIHOOD RATIO
ciLRx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
####INPUT
y=x
###CRITICAL VALUES
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
#LIKELIHOOD-RATIO METHOD
likelhd = function(p) dbinom(y,n,p)
loglik = function(p) dbinom(y,n,p,log=TRUE)
mle=optimize(likelhd,c(0,1),maximum=TRUE)$maximum
cutoff=loglik(mle)-(cv^2/2)
loglik.optim=function(p){abs(cutoff-loglik(p))}
LLRx=optimize(loglik.optim, c(0,mle))$minimum
ULRx=optimize(loglik.optim, c(mle,1))$minimum
if(LLRx<0) LABB="YES" else LABB="NO"
if(LLRx<0) LLRx=0
if(ULRx>1) UABB="YES" else UABB="NO"
if(ULRx>1) ULRx=1
if(ULRx-LLRx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LLRx,ULRx,LABB,UABB,ZWI))
}
##############################################################################
#' Exact method of CI estimation
#' @param n - Number of trials
#' @param x - Number of sucess
#' @param alp - Alpha value (significance level required)
#' @param e - Exact method indicator in [0, 1] {1: Clopper Pearson, 0.5: Mid P}
#' @details Confidence interval for \code{p} (for the given \code{x} and \code{n}),
#' based on inverting equal-tailed binomial tests with null hypothesis \eqn{H0: p = p0 }
#' and calculated from the cumulative binomial distribution.
#' Exact two sided P-value is usually calculated as \deqn{P= 2[e*Pr(X = x) + min({Pr(X < x), Pr(X > x)})]}
#' where probabilities are found at null value of \code{p} and \eqn{0 <= e <= 1.}
#' @return A dataframe with
##' \item{x }{- Number of successes (positive samples)}
##' \item{LEXx }{ - Exact Lower limit}
##' \item{UEXx }{ - Exact Upper Limit}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05;e=0.5
#' ciEXx(x,n,alp,e) #Mid-p
#' x=5; n=5; alp=0.05;e=1 #Clopper Pearson
#' ciEXx(x,n,alp,e)
#' x=5; n=5; alp=0.05;e=c(0.1,0.5,0.95,1) #Range including Mid-p and Clopper-Pearson
#' ciEXx(x,n,alp,e)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#5.EXACT METHOD
ciEXx=function(x,n,alp,e)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (missing(e)) stop("'e' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp) >1) stop("'alpha' has to be between 0 and 1")
if ((class(e) != "integer") & (class(e) != "numeric") || any(e>1) || any(e<0)) stop("'e' has to be between 0 and 1")
if (length(e)>10) stop("'e' can have only 10 intervals")
nvar=length(e)
res <- data.frame()
for(i in 1:nvar)
{
lu=lufn103(x,n,alp,e[i])
res <- rbind(res,lu)
}
return(res)
}
lufn103<-function(x,n,alp,e)
{
LEXx=0
UEXx=0
LABB=0
UABB=0
ZWI=0
LEXx=exlim103l(x,n,alp,e)
UEXx=exlim103u(x,n,alp,e)
if(LEXx<0) LABB="YES" else LABB="NO"
if(LEXx<0) LEXx=0
if(UEXx>1) UABB="YES" else UABB="NO"
if(UEXx>1) UEXx=1
if(UEXx-LEXx==0) ZWI="YES" else ZWI="NO"
resx=data.frame(x,LEXx,UEXx,LABB, UABB, ZWI, e)
return(resx)
}
exlim103l=function(x,n,alp,e)
{
if(x==0)
{
LEX = 0
} else if(x==n){
LEX= (alp/(2*e))^(1/n)
}else
{
z=x-1
y=0:z
f1=function(p) (1-e)*dbinom(x,n,p)+sum(dbinom(y,n,p))-(1-(alp/2))
LEX= uniroot(f1,c(0,1))$root
}
return(LEX)
}
exlim103u=function(x,n,alp,e)
{
if(x==0)
{
UEX = 1-((alp/(2*e))^(1/n))
} else if(x==n){
UEX = 1
}else
{
z=x-1
y=0:z
f2= function(p) e*dbinom(x,n,p)+sum(dbinom(y,n,p))-(alp/2)
UEX =uniroot(f2,c(0,1))$root
}
return(UEX)
}
######################################################################
#' Wald-T method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details An approximate method based on a t_approximation of the
#' standardized point estimator for the given \code{x} and \code{n}; that is the point estimator
#' divided by its estimated standard error. Essential boundary modification is when \eqn{x = 0 or n},
#' \deqn{\hat{p}=\frac{(x+2)}{(n+4)}{phat=(x+2)/(n+4)}}
#' @return A dataframe with
#' \item{x}{ Number of successes (positive samples)}
#' \item{LTWx }{ Wald-T Lower limit}
#' \item{UTWx }{ Wald-T Upper Limit}
#' \item{LABB }{ Wald-T Lower Abberation}
#' \item{UABB }{ Wald-T Upper Abberation}
#' \item{ZWI }{ Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciTWx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#6.WALD-T
ciTWx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
#MODIFIED_t-WALD METHOD
if(x==0||x==n)
{
pTWx=(x+2)/(n+4)
#qTWx=1-pTWx
}else
{
pTWx=x/n
#qTWx=1-pTWx
}
f1=function(p,n) p*(1-p)/n
f2=function(p,n) (p*(1-p)/(n^3))+(p+((6*n)-7)*(p^2)+(4*(n-1)*(n-3)*(p^3))-(2*(n-1)*((2*n)-3)*(p^4)))/(n^5)-(2*(p+((2*n)-3)*(p^2)-2*(n-1)*(p^3)))/(n^4)
DOFx=2*((f1(pTWx,n))^2)/f2(pTWx,n)
cvx=qt(1-(alp/2), df=DOFx)
seTWx=cvx*sqrt(f1(pTWx,n))
LTWx=pTWx-(seTWx)
UTWx=pTWx+(seTWx)
if(LTWx<0) LABB="YES" else LABB="NO"
if(LTWx<0) LTWx=0
if(UTWx>1) UABB="YES" else UABB="NO"
if(UTWx>1) UTWx=1
if(UTWx-LTWx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LTWx,UTWx,LABB,UABB,ZWI))
}
#####################################################################
#' Logit Wald method of CI estimation
#' @param x - Number of successes
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details Wald-type interval for all the given \code{x} and \code{n}
#' based on the logit transformation of \code{p};
#' that is normal approximation for \eqn{log(p/1-p)}
#' @return A dataframe with
#' \item{x }{- Number of successes (positive samples)}
#' \item{LLTx }{ - Logit Wald Lower limit}
#' \item{ULTx }{ - Logit Wald Upper Limit}
#' \item{LABB }{ - Logit Wald Lower Abberation}
#' \item{UABB }{ - Logit Wald Upper Abberation}
#' \item{ZWI }{ - Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05
#' ciLTx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#7.LOGIT-WALD
ciLTx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
####INITIALIZATIONS
pLTx=0
qLTx=0
seLTx=0
lgitx=0
LLTx=0
ULTx=0
LABB=0
UABB=0
ZWI=0
###CRITICAL VALUES
cv=qnorm(1-(alp/2), mean = 0, sd = 1)
#LOGIT-WALD METHOD
if(x==0)
{
pLTx=0
qLTx=1
LLTx = 0
ULTx = 1-((alp/2)^(1/n))
}
if(x==n)
{
pLTx=1
qLTx=0
LLTx= (alp/2)^(1/n)
ULTx=1
}
if(0<x&&x<n)
{
pLTx=x/n
qLTx=1-pLTx
lgitx=log(pLTx/qLTx)
seLTx=sqrt(pLTx*qLTx*n)
LLTx=1/(1+exp(-lgitx+(cv/seLTx)))
ULTx=1/(1+exp(-lgitx-(cv/seLTx)))
}
if(LLTx<0) LABB="YES" else LABB="NO"
if(LLTx<0) LLTx=0
if(ULTx>1) UABB="YES" else UABB="NO"
if(ULTx>1) ULTx=1
if(ULTx-LLTx==0)ZWI="YES" else ZWI="NO"
return(data.frame(x,LLTx,ULTx,LABB,UABB,ZWI))
}
#####################################################################################
#' Bayesian method of CI estimation with Beta prior distribution
#' @param x - Number of sucess
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Shape parameter 1 for prior Beta distribution in Bayesian model. Can also be a vector of length n+1 priors.
#' @param b - Shape parameter 2 for prior Beta distribution in Bayesian model. Can also be a vector of length n+1 priors.
#' @details Highest Probability Density (HPD) and two tailed intervals are
#' provided for the given \code{x} and \code{n}. based on the conjugate prior \eqn{\beta(a, b)}
#' for the probability of success \code{p} of the binomial distribution so that the
#' posterior is \eqn{\beta(x + a, n - x + b)}.
#' @return A dataframe with
##' \item{x }{- Number of successes (positive samples)}
##' \item{LBAQx }{ - Lower limits of Quantile based intervals}
##' \item{UBAQx }{ - Upper limits of Quantile based intervals}
##' \item{LBAHx }{ - Lower limits of HPD intervals}
##' \item{UBAHx }{ - Upper limits of HPD intervals}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x=5; n=5; alp=0.05; a=0.5;b=0.5;
#' ciBAx(x,n,alp,a,b)
#' x= 5; n=5; alp=0.05; a=c(0.5,2,1,1,2,0.5);b=c(0.5,2,1,1,2,0.5)
#' ciBAx(x,n,alp,a,b)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#8. BAYESIAN
ciBAx<-function(x,n,alp,a,b)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (missing(a)) stop("'a' is missing")
if (missing(b)) stop("'b' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
if(length(a)>1 || length(b)>1) {
if ((class(a) != "integer") & (class(a) != "numeric") || any(a < 0)) stop("'a' has to be a set of positive numeric vectors")
if ((class(b) != "integer") & (class(b) != "numeric") || any(b < 0)) stop("'b' has to be a set of positive numeric vectors")
if (length(a) <= n || length(b) <= n ) stop("'a' and 'b' vectors have to be equal to or greater than length n+1")
BAdfx=ciBADx(x,n,alp,a,b)
}
else
{
if (a<0 ) stop("'a' has to be greater than or equal to 0")
if (b<0 ) stop("'b' has to be greater than or equal to 0")
##############
#library(TeachingDemos) #To get HPDs
#Quantile Based Intervals
LBAQx=qbeta(alp/2,x+a,n-x+b)
UBAQx=qbeta(1-(alp/2),x+a,n-x+b)
LBAHx=TeachingDemos::hpd(qbeta,shape1=x+a,shape2=n-x+b,conf=1-alp)[1]
UBAHx=TeachingDemos::hpd(qbeta,shape1=x+a,shape2=n-x+b,conf=1-alp)[2]
BAdfx=data.frame(x,LBAQx,UBAQx,LBAHx,UBAHx)
}
return(BAdfx)
}
#######################################################################
ciBADx<-function(x,n,alp,a,b)
{
k=n+1
####INITIALIZATIONS
LBAQx=0
UBAQx=0
LBAHx=0
UBAHx=0
##############
for(i in 1:k)
{
#Quantile Based Intervals
LBAQx[i]=qbeta(alp/2,x+a[i],n-x+b[i])
UBAQx[i]=qbeta(1-(alp/2),x+a[i],n-x+b[i])
LBAHx[i]=TeachingDemos::hpd(qbeta,shape1=x+a[i],shape2=n-x+b[i],conf=1-alp)[1]
UBAHx[i]=TeachingDemos::hpd(qbeta,shape1=x+a[i],shape2=n-x+b[i],conf=1-alp)[2]
}
return(data.frame(x,LBAQx,UBAQx,LBAHx,UBAHx))
}
#####################################################################################
#' Specific CI estimation of 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine)
#' @param x - Number of sucess
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @details The Confidence Interval of using 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine) for \code{n} given \code{alp} and \code{x}
#' @return A dataframe with
##' \item{name }{- Name of the method}
##' \item{x }{- Number of successes (positive samples)}
##' \item{LLT }{ - Lower limit}
##' \item{ULT }{ - Upper Limit}
##' \item{LABB }{ - Lower Abberation}
##' \item{UABB }{ - Upper Abberation}
##' \item{ZWI }{ - Zero Width Interval}
#' @family Base methods of CI estimation given x & n
#' @seealso \code{\link{prop.test} and \link{binom.test}} for equivalent base Stats R functionality,
#' \code{\link[binom]{binom.confint}} provides similar functionality for 11 methods,
#' \code{\link[PropCIs]{wald2ci}} which provides multiple functions for CI calculation ,
#' \code{\link[BlakerCI]{binom.blaker.limits}} which calculates Blaker CI which is not covered here and
#' \code{\link[prevalence]{propCI}} which provides similar functionality.
#' @examples
#' x= 5; n=5; alp=0.05;
#' ciAllx(x,n,alp)
#' @references
#' [1] 1993 Vollset SE.
#' Confidence intervals for a binomial proportion.
#' Statistics in Medicine: 12; 809 - 824.
#'
#' [2] 1998 Agresti A and Coull BA.
#' Approximate is better than "Exact" for interval estimation of binomial proportions.
#' The American Statistician: 52; 119 - 126.
#'
#' [3] 1998 Newcombe RG.
#' Two-sided confidence intervals for the single proportion: Comparison of seven methods.
#' Statistics in Medicine: 17; 857 - 872.
#'
#' [4] 2001 Brown LD, Cai TT and DasGupta A.
#' Interval estimation for a binomial proportion.
#' Statistical Science: 16; 101 - 133.
#'
#' [5] 2002 Pan W.
#' Approximate confidence intervals for one proportion and difference of two proportions
#' Computational Statistics and Data Analysis 40, 128, 143-157.
#'
#' [6] 2008 Pires, A.M., Amado, C.
#' Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods.
#' REVSTAT - Statistical Journal, 6, 165-197.
#'
#' [7] 2014 Martin Andres, A. and Alvarez Hernandez, M.
#' Two-tailed asymptotic inferences for a proportion.
#' Journal of Applied Statistics, 41, 7, 1516-1529
#' @export
#9.All methods
ciAllx<-function(x,n,alp)
{
if (missing(x)) stop("'x' is missing")
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (((class(x) != "integer") & (class(x) != "numeric")) || (x<0) || x>n || length(x)>1) stop("'x' has to be a positive integer between 0 and n")
if ((class(n) != "integer") & (class(n) != "numeric") || length(n) >1|| n<=0 ) stop("'n' has to be greater than 0")
if (alp>1 || alp<0 || length(alp)>1) stop("'alpha' has to be between 0 and 1")
#### Calling functions and creating df
WaldCI.df = ciWDx(x,n,alp)
ArcSineCI.df = ciASx(x,n,alp)
LRCI.df = ciLRx(x,n,alp)
ScoreCI.df = ciSCx(x,n,alp)
WaldLCI.df = ciLTx(x,n,alp)
AdWaldCI.df = ciTWx(x,n,alp)
WaldCI.df$method = as.factor("Wald")
ArcSineCI.df$method = as.factor("ArcSine")
LRCI.df$method = as.factor("Likelihood")
WaldLCI.df$method = as.factor("Logit-Wald")
ScoreCI.df$method = as.factor("Score")
AdWaldCI.df$method = as.factor("Wald-T")
Generic.1 = data.frame(method = WaldCI.df$method, x=WaldCI.df$x, LowerLimit = WaldCI.df$LWDx, UpperLimit = WaldCI.df$UWDx, LowerAbb = WaldCI.df$LABB, UpperAbb = WaldCI.df$UABB, ZWI = WaldCI.df$ZWI)
Generic.2 = data.frame(method = ArcSineCI.df$method, x=ArcSineCI.df$x, LowerLimit = ArcSineCI.df$LASx, UpperLimit = ArcSineCI.df$UASx, LowerAbb = ArcSineCI.df$LABB, UpperAbb = ArcSineCI.df$UABB, ZWI = ArcSineCI.df$ZWI)
Generic.3 = data.frame(method = LRCI.df$method, x=LRCI.df$x, LowerLimit = LRCI.df$LLRx, UpperLimit = LRCI.df$ULRx, LowerAbb = LRCI.df$LABB, UpperAbb = LRCI.df$UABB, ZWI = LRCI.df$ZWI)
Generic.4 = data.frame(method = ScoreCI.df$method, x=ScoreCI.df$x, LowerLimit = ScoreCI.df$LSCx, UpperLimit = ScoreCI.df$USCx, LowerAbb = ScoreCI.df$LABB, UpperAbb = ScoreCI.df$UABB, ZWI = ScoreCI.df$ZWI)
Generic.5 = data.frame(method = WaldLCI.df$method, x=WaldLCI.df$x, LowerLimit = WaldLCI.df$LLTx, UpperLimit = WaldLCI.df$ULTx, LowerAbb = WaldLCI.df$LABB, UpperAbb = WaldLCI.df$UABB, ZWI = WaldLCI.df$ZWI)
Generic.6 = data.frame(method = AdWaldCI.df$method, x=AdWaldCI.df$x, LowerLimit = AdWaldCI.df$LTWx, UpperLimit = AdWaldCI.df$UTWx, LowerAbb = AdWaldCI.df$LABB, UpperAbb = AdWaldCI.df$UABB, ZWI = AdWaldCI.df$ZWI)
Final.df= rbind(Generic.1,Generic.2,Generic.3,Generic.4,Generic.5, Generic.6)
return(Final.df)
}
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