R/301.Expec_Leng_BASE_All.R
In RajeswaranV/proportion: Inference on Single Binomial Proportion and Bayesian Computations
Defines functions
explAll
lengthAll
lengthBA
exlim301u
exlim301l
glengthEX301
lengthEX
lengthLR
lengthTW
lengthLT
lengthAS
lengthSC
lengthWD
Documented in
lengthAll
lengthAS
lengthBA
lengthEX
lengthLR
lengthLT
lengthSC
lengthTW
lengthWD
#' Expected length and sum of length of Wald method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Wald-type intervals using sum of length of the \eqn{n + 1}
#' intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthWD(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
##### 1.WALD sum of length for a given n and alpha level
lengthWD<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pW=0
qW=0
seW=0
LW=0
UW=0
s=5000
LEW=0 #LENGTH OF INTERVAL
ewiW=matrix(0,k,s) #sum of length quantity in sum
ewW=0 #sum of length
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
#WALD METHOD
for(i in 1:k)
{
pW[i]=x[i]/n
qW[i]=1-(x[i]/n)
seW[i]=sqrt(pW[i]*qW[i]/n)
LW[i]=pW[i]-(cv*seW[i])
UW[i]=pW[i]+(cv*seW[i])
if(LW[i]<0) LW[i]=0
if(UW[i]>1) UW[i]=1
LEW[i]=UW[i]-LW[i]
}
#sumLEW=sum(LEW)
####sum of length
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiW[i,j]=LEW[i]*stats::dbinom(i-1, n,hp[j])
}
ewW[j]=sum(ewiW[,j]) #sum of length
}
#ELW=data.frame(hp,ewW)
sumLen=sum(LEW)
explMean=mean(ewW)
explSD=stats::sd(ewW)
explMax=max(ewW)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of Score method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of score test approach using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthSC(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
##### 2.SCORE - sum of length for a given n and alpha level
lengthSC<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pS=0
qS=0
seS=0
LS=0
US=0
s=5000
LES=0 #LENGTH OF INTERVAL
ewiS=matrix(0,k,s) #sum of length quantity in sum
ewS=0 #sum of length
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
cv1=(cv^2)/(2*n)
cv2=(cv/(2*n))^2
#SCORE (WILSON) METHOD
for(i in 1:k)
{
pS[i]=x[i]/n
qS[i]=1-(x[i]/n)
seS[i]=sqrt((pS[i]*qS[i]/n)+cv2)
LS[i]=(n/(n+(cv)^2))*((pS[i]+cv1)-(cv*seS[i]))
US[i]=(n/(n+(cv)^2))*((pS[i]+cv1)+(cv*seS[i]))
if(LS[i]<0) LS[i]=0
if(US[i]>1) US[i]=1
LES[i]=US[i]-LS[i]
}
#sumLES=sum(LES)
####sum of length
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiS[i,j]=LES[i]*stats::dbinom(i-1, n,hp[j])
}
ewS[j]=sum(ewiS[,j]) #sum of length
}
#ELS=data.frame(hp,ewS)
sumLen=sum(LES)
explMean=mean(ewS)
explSD=stats::sd(ewS)
explMax=max(ewS)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of ArcSine method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Wald-type interval for the arcsine transformation of the parameter
#' \code{p} using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthAS(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
##### 3. ARC SINE - sum of length for a given n and alpha level
lengthAS<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pA=0
qA=0
seA=0
LA=0
UA=0
s=5000
LEA=0 #LENGTH OF INTERVAL
ewiA=matrix(0,k,s) #sum of length quantity in sum
ewA=0 #sum of length
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
#WALD METHOD
#ARC-SINE METHOD
for(i in 1:k)
{
pA[i]=x[i]/n
qA[i]=1-pA[i]
seA[i]=cv/sqrt(4*n)
LA[i]=(sin(asin(sqrt(pA[i]))-seA[i]))^2
UA[i]=(sin(asin(sqrt(pA[i]))+seA[i]))^2
if(LA[i]<0) LA[i]=0
if(UA[i]>1) UA[i]=1
LEA[i]=UA[i]-LA[i]
}
#sumLEA=sum(LEA)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiA[i,j]=LEA[i]*stats::dbinom(i-1, n,hp[j])
}
ewA[j]=sum(ewiA[,j]) #Expected Length
}
sumLen=sum(LEA)
explMean=mean(ewA)
explSD=stats::sd(ewA)
explMax=max(ewA)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of Logit Wald method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Wald-type interval based on the logit
#' transformation of \code{p} using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthLT(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
##### 4.LOGIT-WALD - sum of length for a given n and alpha level
lengthLT<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pLT=0
qLT=0
seLT=0
lgit=0
LLT=0
ULT=0
s=5000
LELT=0 #LENGTH OF INTERVAL
ewiLT=matrix(0,k,s) #sum of length quantity in sum
ewLT=0 #sum of length
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
#LOGIT-WALD METHOD
pLT[1]=0
qLT[1]=1
LLT[1] = 0
ULT[1] = 1-((alp/2)^(1/n))
pLT[k]=1
qLT[k]=0
LLT[k]= (alp/2)^(1/n)
ULT[k]=1
for(j in 1:(k-2))
{
pLT[j+1]=x[j+1]/n
qLT[j+1]=1-pLT[j+1]
lgit[j+1]=log(pLT[j+1]/qLT[j+1])
seLT[j+1]=sqrt(pLT[j+1]*qLT[j+1]*n)
LLT[j+1]=1/(1+exp(-lgit[j+1]+(cv/seLT[j+1])))
ULT[j+1]=1/(1+exp(-lgit[j+1]-(cv/seLT[j+1])))
if(LLT[j+1]<0) LLT[j+1]=0
if(ULT[j+1]>1) ULT[j+1]=1
}
for(i in 1:k)
{
LELT[i]=ULT[i]-LLT[i]
}
#sumLET=sum(LELT)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiLT[i,j]=LELT[i]*stats::dbinom(i-1, n,hp[j])
}
ewLT[j]=sum(ewiLT[,j]) #Expected Length
}
sumLen=sum(LELT)
explMean=mean(ewLT)
explSD=stats::sd(ewLT)
explMax=max(ewLT)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of Wald-T method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of approximate method based on a t_approximation of the
#' standardized point estimator using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthTW(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
##### 5.t-WALD - sum of length for a given n and alpha level
lengthTW<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pTW=0
qTW=0
seTW=0
LTW=0
UTW=0
DOF=0
cv=0
s=5000
LETW=0 #LENGTH OF INTERVAL
ewiTW=matrix(0,k,s) #sum of length quantity in sum
ewTW=0 #sum of length
#MODIFIED_t-WALD METHOD
for(i in 1:k)
{
if(x[i]==0||x[i]==n)
{
pTW[i]=(x[i]+2)/(n+4)
qTW[i]=1-pTW[i]
}else
{
pTW[i]=x[i]/n
qTW[i]=1-pTW[i]
}
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)
DOF[i]=2*((f1(pTW[i],n))^2)/f2(pTW[i],n)
cv[i]=stats::qt(1-(alp/2), df=DOF[i])
seTW[i]=cv[i]*sqrt(f1(pTW[i],n))
LTW[i]=pTW[i]-(seTW[i])
UTW[i]=pTW[i]+(seTW[i])
if(LTW[i]<0) LTW[i]=0
if(UTW[i]>1) UTW[i]=1
LETW[i]=UTW[i]-LTW[i]
}
#sumLETW=sum(LETW)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiTW[i,j]=LETW[i]*stats::dbinom(i-1, n,hp[j])
}
ewTW[j]=sum(ewiTW[,j]) #Expected Length
}
sumLen=sum(LETW)
explMean=mean(ewTW)
explSD=stats::sd(ewTW)
explMax=max(ewTW)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of likelihood Ratio method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Likelihood ratio limits using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthLR(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
#####6.LIKELIHOOD RATIO - sum of length for a given n and alpha level
lengthLR<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
y=0:n
k=n+1
####INITIALIZATIONS
mle=0
cutoff=0
LL=0
UL=0
s=5000
LEL=0 #LENGTH OF INTERVAL
ewiL=matrix(0,k,s) #sum of length quantity in sum
ewL=0 #Simulation run to generate hypothetical p
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
#LIKELIHOOD-RATIO METHOD
for(i in 1:k)
{
likelhd = function(p) stats::dbinom(y[i],n,p)
loglik = function(p) stats::dbinom(y[i],n,p,log=TRUE)
mle[i]=stats::optimize(likelhd,c(0,1),maximum=TRUE)$maximum
cutoff[i]=loglik(mle[i])-(cv^2/2)
loglik.optim=function(p){abs(cutoff[i]-loglik(p))}
LL[i]=stats::optimize(loglik.optim, c(0,mle[i]))$minimum
UL[i]=stats::optimize(loglik.optim, c(mle[i],1))$minimum
LEL[i]=UL[i]-LL[i]
}
#sumLEL=sum(LEL)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiL[i,j]=LEL[i]*stats::dbinom(i-1, n,hp[j])
}
ewL[j]=sum(ewiL[,j]) #Expected Length
}
sumLen=sum(LEL)
explMean=mean(ewL)
explSD=stats::sd(ewL)
explMax=max(ewL)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###################################################################################################
#' Expected length and sum of length of Exact method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param e - Exact method indicator in [0, 1] {1: Clopper Pearson, 0.5: Mid P}
#' The input can also be a range of values between 0 and 1.
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Confidence interval for \code{p} based on inverting equal-tailed
#' binomial tests with null hypothesis \eqn{H0: p = p0} using sum of length of the \eqn{n + 1} intervals.
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' \dontrun{
#' n=5; alp=0.05;e=0.5;a=1;b=1
#' lengthEX(n,alp,e,a,b)
#' n=5; alp=0.05;e=1;a=1;b=1 #Clopper-Pearson
#' lengthEX(n,alp,e,a,b)
#' n=5; alp=0.05;e=c(0.1,0.5,0.95,1);a=1;b=1 #Range including Mid-p and Clopper-Pearson
#' lengthEX(n,alp,e,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
##### 7.EXACT EMTHOD sum of length for a given n and alpha level
lengthEX<-function(n,alp,e,a,b)
{
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (missing(e)) stop("'e' is missing")
if (missing(a)) stop("'a' is missing")
if (missing(b)) stop("'b' is missing")
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")
if ((class(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
nvar=length(e)
res <- data.frame()
for(i in 1:nvar)
{
lu=glengthEX301(n,alp,e[i],a,b)
res <- rbind(res,lu)
}
return(res)
}
glengthEX301<-function(n,alp,e,a,b)
{
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
LEX=0
UEX=0
s=5000
LEEX=0 #LENGTH OF INTERVAL
ewiEX=matrix(0,k,s) #sum of length quantity in sum
ewEX=0 #sum of length
#EXACT METHOD
LEX[1]=0
UEX[1]= 1-((alp/(2*e))^(1/n))
LEX[k]=(alp/(2*e))^(1/n)
UEX[k]=1
LEEX[1]=1-((alp/(2*e))^(1/n))
LEEX[k]=1-((alp/(2*e))^(1/n))
for(i in 1:(k-2))
{
LEX[i+1]=exlim301l(x[i+1],n,alp,e)
UEX[i+1]=exlim301u(x[i+1],n,alp,e)
LEEX[i+1]=UEX[i+1]-LEX[i+1]
}
#sumLEEX=sum(LEEX)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiEX[i,j]=LEEX[i]*stats::dbinom(i-1, n,hp[j])
}
ewEX[j]=sum(ewiEX[,j]) #Expected Length
}
sumLen=sum(LEEX)
explMean=mean(ewEX)
explSD=stats::sd(ewEX)
explMax=max(ewEX)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL,e)
return(df.Summary)
}
#####TO FIND LOWER LIMITS
exlim301l=function(x,n,alp,e)
{
z=x-1
y=0:z
f1=function(p) (1-e)*stats::dbinom(x,n,p)+sum(stats::dbinom(y,n,p))-(1-(alp/2))
LEX= stats::uniroot(f1,c(0,1))$root
return(LEX)
}
#####TO FIND UPPER LIMITS
exlim301u=function(x,n,alp,e)
{
z=x-1
y=0:z
f2 = function(p) e*stats::dbinom(x,n,p)+sum(stats::dbinom(y,n,p))-(alp/2)
UEX = stats::uniroot(f2,c(0,1))$root
return(UEX)
}
###############################################################################################################
#' Expected length and sum of length of Bayesian method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @param a1 - Beta Prior Parameters for Bayesian estimation
#' @param a2 - Beta Prior Parameters for Bayesian estimation
#' @details Evaluation of Bayesian Highest Probability Density (HPD) and two
#' tailed intervals using sum of length of the \eqn{n + 1}
#' intervals for the Beta - Binomial conjugate prior model for the probability of success \code{p}
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' \item{method}{ The method used - Quantile and HPD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1;a1=1;a2=1
#' lengthBA(n,alp,a,b,a1,a2)
#' @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 sum of length for a given n and alpha level
lengthBA<-function(n,alp,a,b,a1,a2)
{
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 (missing(a1)) stop("'a1' is missing")
if (missing(a2)) stop("'a2' is missing")
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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
if ((class(a1) != "integer") & (class(a1) != "numeric") || length(a1)>1 || a1<0 ) stop("'a1' has to be greater than or equal to 0")
if ((class(a2) != "integer") & (class(a2) != "numeric") || length(a2)>1 || a2<0 ) stop("'a2' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
LBAQ=0
UBAQ=0
LBAH=0
UBAH=0
s=5000
LEBAQ=0 #LENGTH OF INTERVAL
LEBAH=0
ewiBAQ=matrix(0,k,s) #sum of length quantity in sum
ewBAQ=0
ewiBAH=matrix(0,k,s) #sum of length quantity in sum
ewBAH=0 #sum of length
#library(TeachingDemos) #To get HPDs
for(i in 1:k)
{
#Quantile Based Intervals
LBAQ[i]=stats::qbeta(alp/2,x[i]+a1,n-x[i]+a2)
UBAQ[i]=stats::qbeta(1-(alp/2),x[i]+a1,n-x[i]+a2)
LBAH[i]=TeachingDemos::hpd(stats::qbeta,shape1=x[i]+a1,shape2=n-x[i]+a2,conf=1-alp)[1]
UBAH[i]=TeachingDemos::hpd(stats::qbeta,shape1=x[i]+a1,shape2=n-x[i]+a2,conf=1-alp)[2]
LEBAQ[i]=UBAQ[i]-LBAQ[i]
LEBAH[i]=UBAH[i]-LBAH[i]
}
# sumLEBAQ=sum(LEBAQ)
# sumLEBAH=sum(LEBAH)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiBAQ[i,j]=LEBAQ[i]*stats::dbinom(i-1, n,hp[j])
ewiBAH[i,j]=LEBAH[i]*stats::dbinom(i-1, n,hp[j])
}
ewBAQ[j]=sum(ewiBAQ[,j])
ewBAH[j]=sum(ewiBAH[,j]) #Expected Length
}
sumLenBAQ=sum(LEBAQ)
explMeanBAQ=mean(ewBAQ)
explSDBAQ=stats::sd(ewBAQ)
explMaxBAQ=max(ewBAQ)
explLLBAQ=explMeanBAQ-(explSDBAQ)
explULBAQ=explMeanBAQ+(explSDBAQ)
df.SummaryBAQ=data.frame(sumLen=sumLenBAQ,explMean=explMeanBAQ,
explSD=explSDBAQ,explMax=explMaxBAQ,
explLL=explLLBAQ,explUL=explULBAQ,method="Quantile")
sumLenBAH=sum(LEBAH)
explMeanBAH=mean(ewBAH)
explSDBAH=stats::sd(ewBAH)
explMaxBAH=max(ewBAH)
explLLBAH=explMeanBAH-(explSDBAH)
explULBAH=explMeanBAH+(explSDBAH)
df.SummaryBAH=data.frame(sumLen=sumLenBAH,explMean=explMeanBAH,
explSD=explSDBAH,explMax=explMaxBAH,
explLL=explLLBAH,explUL=explULBAH,method="HPD")
df.Summary=rbind(df.SummaryBAQ,df.SummaryBAH)
return(df.Summary)
}
########################################################################################
#' Expected Length summary calculation using 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine)
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details The sum of length for 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine) for \code{n} given alpha \code{alp} and Beta parameters \code{a}, \code{b}
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' \item{method}{ The name of the method}
#' @family Expected length of base methods
#' @examples
#' \dontrun{
#' n=5; alp=0.05;a=1;b=1
#' lengthAll(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
##### 9. sum of length for a given n and alpha level for all methods
lengthAll<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
#### Calling functions and creating df
df1 = lengthWD(n,alp,a,b)
df2 = lengthSC(n,alp,a,b)
df3 = lengthAS(n,alp,a,b)
df4 = lengthLT(n,alp,a,b)
df5 = lengthTW(n,alp,a,b)
df6 = lengthLR(n,alp,a,b)
df1$method = "Wald"
df2$method = "Score"
df3$method = "ArcSine"
df4$method = "Logit-Wald"
df5$method = "Wald-T"
df6$method = "Likelihood"
Final.df= rbind(df1,df2,df3,df4,df5,df6)
return(Final.df)
}
########################################################################################
# Expected length and sum of length using 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine)
#
##### 10. Expected length for a given n and alpha level for 6 base methods
explAll<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
#### Calling functions and creating df
df.1 = gexplWD(n,alp,a,b)
df.2 = gexplSC(n,alp,a,b)
df.3 = gexplAS(n,alp,a,b)
df.4 = gexplLT(n,alp,a,b)
df.5 = gexplTW(n,alp,a,b)
df.6 = gexplLR(n,alp,a,b)
df.new= rbind(df.1,df.2,df.3,df.4,df.5,df.6)
return(df.new)
}
RajeswaranV/proportion documentation built on June 17, 2022, 9:11 a.m.
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Defines functions explAll lengthAll lengthBA exlim301u exlim301l glengthEX301 lengthEX lengthLR lengthTW lengthLT lengthAS lengthSC lengthWD
Documented in lengthAll lengthAS lengthBA lengthEX lengthLR lengthLT lengthSC lengthTW lengthWD
#' Expected length and sum of length of Wald method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Wald-type intervals using sum of length of the \eqn{n + 1}
#' intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthWD(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
##### 1.WALD sum of length for a given n and alpha level
lengthWD<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pW=0
qW=0
seW=0
LW=0
UW=0
s=5000
LEW=0 #LENGTH OF INTERVAL
ewiW=matrix(0,k,s) #sum of length quantity in sum
ewW=0 #sum of length
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
#WALD METHOD
for(i in 1:k)
{
pW[i]=x[i]/n
qW[i]=1-(x[i]/n)
seW[i]=sqrt(pW[i]*qW[i]/n)
LW[i]=pW[i]-(cv*seW[i])
UW[i]=pW[i]+(cv*seW[i])
if(LW[i]<0) LW[i]=0
if(UW[i]>1) UW[i]=1
LEW[i]=UW[i]-LW[i]
}
#sumLEW=sum(LEW)
####sum of length
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiW[i,j]=LEW[i]*stats::dbinom(i-1, n,hp[j])
}
ewW[j]=sum(ewiW[,j]) #sum of length
}
#ELW=data.frame(hp,ewW)
sumLen=sum(LEW)
explMean=mean(ewW)
explSD=stats::sd(ewW)
explMax=max(ewW)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of Score method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of score test approach using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthSC(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
##### 2.SCORE - sum of length for a given n and alpha level
lengthSC<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pS=0
qS=0
seS=0
LS=0
US=0
s=5000
LES=0 #LENGTH OF INTERVAL
ewiS=matrix(0,k,s) #sum of length quantity in sum
ewS=0 #sum of length
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
cv1=(cv^2)/(2*n)
cv2=(cv/(2*n))^2
#SCORE (WILSON) METHOD
for(i in 1:k)
{
pS[i]=x[i]/n
qS[i]=1-(x[i]/n)
seS[i]=sqrt((pS[i]*qS[i]/n)+cv2)
LS[i]=(n/(n+(cv)^2))*((pS[i]+cv1)-(cv*seS[i]))
US[i]=(n/(n+(cv)^2))*((pS[i]+cv1)+(cv*seS[i]))
if(LS[i]<0) LS[i]=0
if(US[i]>1) US[i]=1
LES[i]=US[i]-LS[i]
}
#sumLES=sum(LES)
####sum of length
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiS[i,j]=LES[i]*stats::dbinom(i-1, n,hp[j])
}
ewS[j]=sum(ewiS[,j]) #sum of length
}
#ELS=data.frame(hp,ewS)
sumLen=sum(LES)
explMean=mean(ewS)
explSD=stats::sd(ewS)
explMax=max(ewS)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of ArcSine method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Wald-type interval for the arcsine transformation of the parameter
#' \code{p} using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthAS(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
##### 3. ARC SINE - sum of length for a given n and alpha level
lengthAS<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pA=0
qA=0
seA=0
LA=0
UA=0
s=5000
LEA=0 #LENGTH OF INTERVAL
ewiA=matrix(0,k,s) #sum of length quantity in sum
ewA=0 #sum of length
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
#WALD METHOD
#ARC-SINE METHOD
for(i in 1:k)
{
pA[i]=x[i]/n
qA[i]=1-pA[i]
seA[i]=cv/sqrt(4*n)
LA[i]=(sin(asin(sqrt(pA[i]))-seA[i]))^2
UA[i]=(sin(asin(sqrt(pA[i]))+seA[i]))^2
if(LA[i]<0) LA[i]=0
if(UA[i]>1) UA[i]=1
LEA[i]=UA[i]-LA[i]
}
#sumLEA=sum(LEA)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiA[i,j]=LEA[i]*stats::dbinom(i-1, n,hp[j])
}
ewA[j]=sum(ewiA[,j]) #Expected Length
}
sumLen=sum(LEA)
explMean=mean(ewA)
explSD=stats::sd(ewA)
explMax=max(ewA)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of Logit Wald method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Wald-type interval based on the logit
#' transformation of \code{p} using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthLT(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
##### 4.LOGIT-WALD - sum of length for a given n and alpha level
lengthLT<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pLT=0
qLT=0
seLT=0
lgit=0
LLT=0
ULT=0
s=5000
LELT=0 #LENGTH OF INTERVAL
ewiLT=matrix(0,k,s) #sum of length quantity in sum
ewLT=0 #sum of length
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
#LOGIT-WALD METHOD
pLT[1]=0
qLT[1]=1
LLT[1] = 0
ULT[1] = 1-((alp/2)^(1/n))
pLT[k]=1
qLT[k]=0
LLT[k]= (alp/2)^(1/n)
ULT[k]=1
for(j in 1:(k-2))
{
pLT[j+1]=x[j+1]/n
qLT[j+1]=1-pLT[j+1]
lgit[j+1]=log(pLT[j+1]/qLT[j+1])
seLT[j+1]=sqrt(pLT[j+1]*qLT[j+1]*n)
LLT[j+1]=1/(1+exp(-lgit[j+1]+(cv/seLT[j+1])))
ULT[j+1]=1/(1+exp(-lgit[j+1]-(cv/seLT[j+1])))
if(LLT[j+1]<0) LLT[j+1]=0
if(ULT[j+1]>1) ULT[j+1]=1
}
for(i in 1:k)
{
LELT[i]=ULT[i]-LLT[i]
}
#sumLET=sum(LELT)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiLT[i,j]=LELT[i]*stats::dbinom(i-1, n,hp[j])
}
ewLT[j]=sum(ewiLT[,j]) #Expected Length
}
sumLen=sum(LELT)
explMean=mean(ewLT)
explSD=stats::sd(ewLT)
explMax=max(ewLT)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of Wald-T method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of approximate method based on a t_approximation of the
#' standardized point estimator using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthTW(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
##### 5.t-WALD - sum of length for a given n and alpha level
lengthTW<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
pTW=0
qTW=0
seTW=0
LTW=0
UTW=0
DOF=0
cv=0
s=5000
LETW=0 #LENGTH OF INTERVAL
ewiTW=matrix(0,k,s) #sum of length quantity in sum
ewTW=0 #sum of length
#MODIFIED_t-WALD METHOD
for(i in 1:k)
{
if(x[i]==0||x[i]==n)
{
pTW[i]=(x[i]+2)/(n+4)
qTW[i]=1-pTW[i]
}else
{
pTW[i]=x[i]/n
qTW[i]=1-pTW[i]
}
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)
DOF[i]=2*((f1(pTW[i],n))^2)/f2(pTW[i],n)
cv[i]=stats::qt(1-(alp/2), df=DOF[i])
seTW[i]=cv[i]*sqrt(f1(pTW[i],n))
LTW[i]=pTW[i]-(seTW[i])
UTW[i]=pTW[i]+(seTW[i])
if(LTW[i]<0) LTW[i]=0
if(UTW[i]>1) UTW[i]=1
LETW[i]=UTW[i]-LTW[i]
}
#sumLETW=sum(LETW)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiTW[i,j]=LETW[i]*stats::dbinom(i-1, n,hp[j])
}
ewTW[j]=sum(ewiTW[,j]) #Expected Length
}
sumLen=sum(LETW)
explMean=mean(ewTW)
explSD=stats::sd(ewTW)
explMax=max(ewTW)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###############################################################################################################
#' Expected length and sum of length of likelihood Ratio method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Likelihood ratio limits using sum of length of the \eqn{n + 1} intervals
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1
#' lengthLR(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
#####6.LIKELIHOOD RATIO - sum of length for a given n and alpha level
lengthLR<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
####INPUT n
y=0:n
k=n+1
####INITIALIZATIONS
mle=0
cutoff=0
LL=0
UL=0
s=5000
LEL=0 #LENGTH OF INTERVAL
ewiL=matrix(0,k,s) #sum of length quantity in sum
ewL=0 #Simulation run to generate hypothetical p
###CRITICAL VALUES
cv=stats::qnorm(1-(alp/2), mean = 0, sd = 1)
#LIKELIHOOD-RATIO METHOD
for(i in 1:k)
{
likelhd = function(p) stats::dbinom(y[i],n,p)
loglik = function(p) stats::dbinom(y[i],n,p,log=TRUE)
mle[i]=stats::optimize(likelhd,c(0,1),maximum=TRUE)$maximum
cutoff[i]=loglik(mle[i])-(cv^2/2)
loglik.optim=function(p){abs(cutoff[i]-loglik(p))}
LL[i]=stats::optimize(loglik.optim, c(0,mle[i]))$minimum
UL[i]=stats::optimize(loglik.optim, c(mle[i],1))$minimum
LEL[i]=UL[i]-LL[i]
}
#sumLEL=sum(LEL)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiL[i,j]=LEL[i]*stats::dbinom(i-1, n,hp[j])
}
ewL[j]=sum(ewiL[,j]) #Expected Length
}
sumLen=sum(LEL)
explMean=mean(ewL)
explSD=stats::sd(ewL)
explMax=max(ewL)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL)
return(df.Summary)
}
###################################################################################################
#' Expected length and sum of length of Exact method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param e - Exact method indicator in [0, 1] {1: Clopper Pearson, 0.5: Mid P}
#' The input can also be a range of values between 0 and 1.
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details Evaluation of Confidence interval for \code{p} based on inverting equal-tailed
#' binomial tests with null hypothesis \eqn{H0: p = p0} using sum of length of the \eqn{n + 1} intervals.
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' @family Expected length of base methods
#' @examples
#' \dontrun{
#' n=5; alp=0.05;e=0.5;a=1;b=1
#' lengthEX(n,alp,e,a,b)
#' n=5; alp=0.05;e=1;a=1;b=1 #Clopper-Pearson
#' lengthEX(n,alp,e,a,b)
#' n=5; alp=0.05;e=c(0.1,0.5,0.95,1);a=1;b=1 #Range including Mid-p and Clopper-Pearson
#' lengthEX(n,alp,e,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
##### 7.EXACT EMTHOD sum of length for a given n and alpha level
lengthEX<-function(n,alp,e,a,b)
{
if (missing(n)) stop("'n' is missing")
if (missing(alp)) stop("'alpha' is missing")
if (missing(e)) stop("'e' is missing")
if (missing(a)) stop("'a' is missing")
if (missing(b)) stop("'b' is missing")
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")
if ((class(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
nvar=length(e)
res <- data.frame()
for(i in 1:nvar)
{
lu=glengthEX301(n,alp,e[i],a,b)
res <- rbind(res,lu)
}
return(res)
}
glengthEX301<-function(n,alp,e,a,b)
{
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
LEX=0
UEX=0
s=5000
LEEX=0 #LENGTH OF INTERVAL
ewiEX=matrix(0,k,s) #sum of length quantity in sum
ewEX=0 #sum of length
#EXACT METHOD
LEX[1]=0
UEX[1]= 1-((alp/(2*e))^(1/n))
LEX[k]=(alp/(2*e))^(1/n)
UEX[k]=1
LEEX[1]=1-((alp/(2*e))^(1/n))
LEEX[k]=1-((alp/(2*e))^(1/n))
for(i in 1:(k-2))
{
LEX[i+1]=exlim301l(x[i+1],n,alp,e)
UEX[i+1]=exlim301u(x[i+1],n,alp,e)
LEEX[i+1]=UEX[i+1]-LEX[i+1]
}
#sumLEEX=sum(LEEX)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiEX[i,j]=LEEX[i]*stats::dbinom(i-1, n,hp[j])
}
ewEX[j]=sum(ewiEX[,j]) #Expected Length
}
sumLen=sum(LEEX)
explMean=mean(ewEX)
explSD=stats::sd(ewEX)
explMax=max(ewEX)
explLL=explMean-(explSD)
explUL=explMean+(explSD)
df.Summary=data.frame(sumLen,explMean,explSD,explMax,explLL,explUL,e)
return(df.Summary)
}
#####TO FIND LOWER LIMITS
exlim301l=function(x,n,alp,e)
{
z=x-1
y=0:z
f1=function(p) (1-e)*stats::dbinom(x,n,p)+sum(stats::dbinom(y,n,p))-(1-(alp/2))
LEX= stats::uniroot(f1,c(0,1))$root
return(LEX)
}
#####TO FIND UPPER LIMITS
exlim301u=function(x,n,alp,e)
{
z=x-1
y=0:z
f2 = function(p) e*stats::dbinom(x,n,p)+sum(stats::dbinom(y,n,p))-(alp/2)
UEX = stats::uniroot(f2,c(0,1))$root
return(UEX)
}
###############################################################################################################
#' Expected length and sum of length of Bayesian method
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @param a1 - Beta Prior Parameters for Bayesian estimation
#' @param a2 - Beta Prior Parameters for Bayesian estimation
#' @details Evaluation of Bayesian Highest Probability Density (HPD) and two
#' tailed intervals using sum of length of the \eqn{n + 1}
#' intervals for the Beta - Binomial conjugate prior model for the probability of success \code{p}
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' \item{method}{ The method used - Quantile and HPD}
#' @family Expected length of base methods
#' @examples
#' n=5; alp=0.05;a=1;b=1;a1=1;a2=1
#' lengthBA(n,alp,a,b,a1,a2)
#' @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 sum of length for a given n and alpha level
lengthBA<-function(n,alp,a,b,a1,a2)
{
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 (missing(a1)) stop("'a1' is missing")
if (missing(a2)) stop("'a2' is missing")
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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
if ((class(a1) != "integer") & (class(a1) != "numeric") || length(a1)>1 || a1<0 ) stop("'a1' has to be greater than or equal to 0")
if ((class(a2) != "integer") & (class(a2) != "numeric") || length(a2)>1 || a2<0 ) stop("'a2' has to be greater than or equal to 0")
####INPUT n
x=0:n
k=n+1
####INITIALIZATIONS
LBAQ=0
UBAQ=0
LBAH=0
UBAH=0
s=5000
LEBAQ=0 #LENGTH OF INTERVAL
LEBAH=0
ewiBAQ=matrix(0,k,s) #sum of length quantity in sum
ewBAQ=0
ewiBAH=matrix(0,k,s) #sum of length quantity in sum
ewBAH=0 #sum of length
#library(TeachingDemos) #To get HPDs
for(i in 1:k)
{
#Quantile Based Intervals
LBAQ[i]=stats::qbeta(alp/2,x[i]+a1,n-x[i]+a2)
UBAQ[i]=stats::qbeta(1-(alp/2),x[i]+a1,n-x[i]+a2)
LBAH[i]=TeachingDemos::hpd(stats::qbeta,shape1=x[i]+a1,shape2=n-x[i]+a2,conf=1-alp)[1]
UBAH[i]=TeachingDemos::hpd(stats::qbeta,shape1=x[i]+a1,shape2=n-x[i]+a2,conf=1-alp)[2]
LEBAQ[i]=UBAQ[i]-LBAQ[i]
LEBAH[i]=UBAH[i]-LBAH[i]
}
# sumLEBAQ=sum(LEBAQ)
# sumLEBAH=sum(LEBAH)
hp=sort(stats::rbeta(s,a,b),decreasing = FALSE) #HYPOTHETICAL "p"
for (j in 1:s)
{
for(i in 1:k)
{
ewiBAQ[i,j]=LEBAQ[i]*stats::dbinom(i-1, n,hp[j])
ewiBAH[i,j]=LEBAH[i]*stats::dbinom(i-1, n,hp[j])
}
ewBAQ[j]=sum(ewiBAQ[,j])
ewBAH[j]=sum(ewiBAH[,j]) #Expected Length
}
sumLenBAQ=sum(LEBAQ)
explMeanBAQ=mean(ewBAQ)
explSDBAQ=stats::sd(ewBAQ)
explMaxBAQ=max(ewBAQ)
explLLBAQ=explMeanBAQ-(explSDBAQ)
explULBAQ=explMeanBAQ+(explSDBAQ)
df.SummaryBAQ=data.frame(sumLen=sumLenBAQ,explMean=explMeanBAQ,
explSD=explSDBAQ,explMax=explMaxBAQ,
explLL=explLLBAQ,explUL=explULBAQ,method="Quantile")
sumLenBAH=sum(LEBAH)
explMeanBAH=mean(ewBAH)
explSDBAH=stats::sd(ewBAH)
explMaxBAH=max(ewBAH)
explLLBAH=explMeanBAH-(explSDBAH)
explULBAH=explMeanBAH+(explSDBAH)
df.SummaryBAH=data.frame(sumLen=sumLenBAH,explMean=explMeanBAH,
explSD=explSDBAH,explMax=explMaxBAH,
explLL=explLLBAH,explUL=explULBAH,method="HPD")
df.Summary=rbind(df.SummaryBAQ,df.SummaryBAH)
return(df.Summary)
}
########################################################################################
#' Expected Length summary calculation using 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine)
#' @param n - Number of trials
#' @param alp - Alpha value (significance level required)
#' @param a - Beta parameters for hypo "p"
#' @param b - Beta parameters for hypo "p"
#' @details The sum of length for 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine) for \code{n} given alpha \code{alp} and Beta parameters \code{a}, \code{b}
#' @return A dataframe with
#' \item{sumLen}{ The sum of the expected length}
#' \item{explMean}{ The mean of the expected length}
#' \item{explSD}{ The Standard Deviation of the expected length}
#' \item{explMax}{ The max of the expected length}
#' \item{explLL}{ The Lower limit of the expected length calculated using mean - SD}
#' \item{explUL}{ The Upper limit of the expected length calculated using mean + SD}
#' \item{method}{ The name of the method}
#' @family Expected length of base methods
#' @examples
#' \dontrun{
#' n=5; alp=0.05;a=1;b=1
#' lengthAll(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
##### 9. sum of length for a given n and alpha level for all methods
lengthAll<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
#### Calling functions and creating df
df1 = lengthWD(n,alp,a,b)
df2 = lengthSC(n,alp,a,b)
df3 = lengthAS(n,alp,a,b)
df4 = lengthLT(n,alp,a,b)
df5 = lengthTW(n,alp,a,b)
df6 = lengthLR(n,alp,a,b)
df1$method = "Wald"
df2$method = "Score"
df3$method = "ArcSine"
df4$method = "Logit-Wald"
df5$method = "Wald-T"
df6$method = "Likelihood"
Final.df= rbind(df1,df2,df3,df4,df5,df6)
return(Final.df)
}
########################################################################################
# Expected length and sum of length using 6 base methods (Wald, Wald-T, Likelihood, Score, Logit-Wald, ArcSine)
#
##### 10. Expected length for a given n and alpha level for 6 base methods
explAll<-function(n,alp,a,b)
{
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(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(a) != "integer") & (class(a) != "numeric") || length(a)>1 || a<0 ) stop("'a' has to be greater than or equal to 0")
if ((class(b) != "integer") & (class(b) != "numeric") || length(b)>1 || b<0 ) stop("'b' has to be greater than or equal to 0")
#### Calling functions and creating df
df.1 = gexplWD(n,alp,a,b)
df.2 = gexplSC(n,alp,a,b)
df.3 = gexplAS(n,alp,a,b)
df.4 = gexplLT(n,alp,a,b)
df.5 = gexplTW(n,alp,a,b)
df.6 = gexplLR(n,alp,a,b)
df.new= rbind(df.1,df.2,df.3,df.4,df.5,df.6)
return(df.new)
}
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