Package 'AID' reference manual

Title:	Box-Cox Power Transformation
Description:	Performs Box-Cox power transformation for different purposes, graphical approaches, assesses the success of the transformation via tests and plots, computes mean and confidence interval for back transformed data.
Authors:	Osman Dag [aut, cre], Muhammed Ali Yilmaz [aut], Ozgur Asar [ctb], Ozlem Ilk [ctb]
Maintainer:	Osman Dag <[email protected]>
License:	GPL (>= 2)
Version:	3.0
Built:	2024-11-23 16:30:05 UTC
Source:	CRAN

Box-Cox Power Transformation

Description

Performs Box-Cox power transformation for different purposes, graphical approaches, assesses the success of the transformation via tests and plots, computes mean and confidence interval for back transformed data.

Details

Package:	AID
Type:	Package
License:	GPL (>=2)

Average Annual Daily Traffic Data

Description

Average annual daily traffic data collected from the Minnesota Department of Transportation data base.

Usage

data(AADT)data(AADT)

Format

A data frame with 121 observations on the following 8 variables.

aadt: average annual daily traffic for a section of road
ctypop: population of county
lanes: number of lanes in the section of road
width: width of the section of road (in feet)
control: a factor with levels: access control; no access control
class: a factor with levels: rural interstate; rural noninterstate; urban interstate; urban noninterstate
truck: availability situation of road section to trucks
locale: a factor with levels: rural; urban, population <= 50,000; urban, population > 50,000

References

Cheng, C. (1992). Optimal Sampling for Traffic Volume Estimation, Unpublished Ph.D. dissertation, University of Minnesota, Carlson School of Management.

Neter, J., Kutner, M.H., Nachtsheim, C.J.,Wasserman, W. (1996). Applied Linear Statistical Models (4th ed.), Irwin, page 483.

Examples


library(AID)

data(AADT)
attach(AADT)
hist(aadt)
out <- boxcoxfr(aadt, class)
confInt(out)
library(AID)

data(AADT)
attach(AADT)
hist(aadt)
out <- boxcoxfr(aadt, class)
confInt(out)

Box-Cox Transformation for One-Way ANOVA

Description

boxcoxfr performs Box-Cox transformation for one-way ANOVA. It is useful to use if the normality or/and the homogenity of variance is/are not satisfied while comparing two or more groups.

Usage

boxcoxfr(y, x, option = "both", lambda = seq(-3, 3, 0.01), lambda2 = NULL, 
  tau = 0.05, alpha = 0.05, verbose = TRUE)boxcoxfr(y, x, option = "both", lambda = seq(-3, 3, 0.01), lambda2 = NULL, 
  tau = 0.05, alpha = 0.05, verbose = TRUE)

Arguments

`y`	a numeric vector of data values.
`x`	a vector or factor object which gives the group for the corresponding elements of y.
`option`	a character string to select the desired option for the objective of transformation. "nor" and "var" are the options which search for a transformation to satisfy the normality of groups and the homogenity of variances, respectively. "both" is the option which searches for a transformation to satisfy both the normality of groups and the homogenity of variances. Default is set to "both".
`lambda`	a vector which includes the sequence of feasible lambda values. Default is set to (-3, 3) with increment 0.01.
`lambda2`	a numeric for an additional shifting parameter. Default is set to lambda2 = 0.
`tau`	the feasible region parameter for the construction of feasible region. Default is set to 0.05. If tau = 0, it returns the MLE of transformation parameter.
`alpha`	the level of significance to check the normality and variance homogenity after transformation. Default is set to alpha = 0.05.
`verbose`	a logical for printing output to R console.

Details

Denote $y$ the variable at the original scale and $y'$ the transformed variable. The Box-Cox power transformation is defined by:

$y' = \left\{ \begin{array}{ll} \frac{y^\lambda - 1}{\lambda} \mbox{ , if $\lambda \neq 0$} \cr log(y) \mbox{ , if $\lambda = 0$} \end{array} \right.$

If the data include any nonpositive observations, a shifting parameter $\lambda_2$ can be included in the transformation given by:

$y' = \left\{ \begin{array}{ll} \frac{(y + \lambda_2)^\lambda - 1}{\lambda} \mbox{ , if $\lambda \neq 0$} \cr log(y + \lambda_2) \mbox{ , if $\lambda = 0$} \end{array} \right.$

Maximum likelihood estimation in feasible region (MLEFR) is used while estimating transformation parameter. MLEFR maximizes the likehood function in feasible region constructed by Shapiro-Wilk test and Bartlett's test. After transformation, normality of the data in each group and homogeneity of variance are assessed by Shapiro-Wilk test and Bartlett's test, respectively.

Value

A list with class "boxcoxfr" containing the following elements:

`method`	method applied in the algorithm
`lambda.hat`	the estimated lambda
`lambda2`	additional shifting parameter
`shapiro`	a data frame which gives the test results for the normality of groups via Shapiro-Wilk test
`bartlett`	a matrix which returns the test result for the homogenity of variance via Bartlett's test
`alpha`	the level of significance to assess the assumptions.
`tf.data`	transformed data set
`x`	a factor object which gives the group for the corresponding elements of y
`y.name`	variable name of y
`x.name`	variable name of x

Author(s)

Osman Dag, Ozlem Ilk

References

Dag, O., Ilk, O. (2017). An Algorithm for Estimating Box-Cox Transformation Parameter in ANOVA. Communications in Statistics - Simulation and Computation, 46:8, 6424–6435.

Examples


######

# Communication between AID and onewaytests packages

library(AID)
library(onewaytests)

# Average Annual Daily Traffic Data (AID)
data(AADT)

# to obtain descriptive statistics by groups (onewaytests)
describe(aadt ~ class, data = AADT)

# to check normality of data in each group (onewaytests)
nor.test(aadt ~ class, data = AADT)

# to check variance homogeneity (onewaytests)
homog.test(aadt ~ class, data = AADT, method = "Bartlett")


# to apply Box-Cox transformation (AID)
out <- boxcoxfr(AADT$aadt, AADT$class)

# to obtain transformed data
AADT$tf.aadt <- out$tf.data

# to conduct one-way ANOVA with transformed data (onewaytests)
result<-aov.test(tf.aadt ~ class, data = AADT)

# to make pairwise comparison (onewaytests)
paircomp(result)

# to convert the statistics into the original scale (AID)
confInt(out, level = 0.95)

######

library(AID)

data <- rnorm(120, 10, 1)
factor <- rep(c("X", "Y", "Z"), each = 40)
out <- boxcoxfr(data, factor, lambda = seq(-5, 5, 0.01), tau = 0.01, alpha = 0.01)
confInt(out, level = 0.95)

######




######

# Communication between AID and onewaytests packages

library(AID)
library(onewaytests)

# Average Annual Daily Traffic Data (AID)
data(AADT)

# to obtain descriptive statistics by groups (onewaytests)
describe(aadt ~ class, data = AADT)

# to check normality of data in each group (onewaytests)
nor.test(aadt ~ class, data = AADT)

# to check variance homogeneity (onewaytests)
homog.test(aadt ~ class, data = AADT, method = "Bartlett")


# to apply Box-Cox transformation (AID)
out <- boxcoxfr(AADT$aadt, AADT$class)

# to obtain transformed data
AADT$tf.aadt <- out$tf.data

# to conduct one-way ANOVA with transformed data (onewaytests)
result<-aov.test(tf.aadt ~ class, data = AADT)

# to make pairwise comparison (onewaytests)
paircomp(result)

# to convert the statistics into the original scale (AID)
confInt(out, level = 0.95)

######

library(AID)

data <- rnorm(120, 10, 1)
factor <- rep(c("X", "Y", "Z"), each = 40)
out <- boxcoxfr(data, factor, lambda = seq(-5, 5, 0.01), tau = 0.01, alpha = 0.01)
confInt(out, level = 0.95)

######

Box-Cox Transformation for Linear Models

Description

boxcoxlm performs Box-Cox transformation for linear models and provides graphical analysis of residuals after transformation.

Usage

boxcoxlm(x, y, method = "lse", lambda = seq(-3,3,0.01), lambda2 = NULL, plot = TRUE, 
  alpha = 0.05, verbose = TRUE)boxcoxlm(x, y, method = "lse", lambda = seq(-3,3,0.01), lambda2 = NULL, plot = TRUE, 
  alpha = 0.05, verbose = TRUE)

Arguments

`x`	a nxp matrix, n is the number of observations and p is the number of variables.
`y`	a vector of response variable.
`method`	a character string to select the desired method to be used to estimate Box-Cox transformation parameter. To use Shapiro-Wilk test method should be set to "sw". For method = "ad", boxcoxnc function uses Anderson-Darling test to estimate Box-Cox transformation parameter. Similarly, method should be set to "cvm", "pt", "sf", "lt", "jb", "mle", "lse" to use Cramer-von Mises, Pearson Chi-square, Shapiro-Francia, Lilliefors and Jarque-Bera tests, maximum likelihood estimation and least square estimation, respectively. Default is set to method = "lse".
`lambda`	a vector which includes the sequence of candidate lambda values. Default is set to (-3,3) with increment 0.01.
`lambda2`	a numeric for an additional shifting parameter. Default is set to lambda2 = 0.
`plot`	a logical to plot histogram with its density line and qqplot of residuals before and after transformation. Defaults plot = TRUE.
`alpha`	the level of significance to assess the normality of residuals after transformation. Default is set to alpha = 0.05.
`verbose`	a logical for printing output to R console.

Details

Denote $y$ the variable at the original scale and $y'$ the transformed variable. The Box-Cox power transformation is defined by:

$y' = \left\{ \begin{array}{ll} \frac{y^\lambda - 1}{\lambda} = \beta_0 + \beta_1x_1 + ... + \epsilon \mbox{ , if $\lambda \neq 0$} \cr log(y) = \beta_0 + \beta_1x_1 + ... + \epsilon \mbox{ , if $\lambda = 0$} \end{array} \right.$

If the data include any nonpositive observations, a shifting parameter $\lambda_2$ can be included in the transformation given by:

$y' = \left\{ \begin{array}{ll} \frac{(y + \lambda_2)^\lambda - 1}{\lambda} = \beta_0 + \beta_1x_1 + ... + \epsilon \mbox{ , if $\lambda \neq 0$} \cr log(y + \lambda_2) = \beta_0 + \beta_1x_1 + ... + \epsilon \mbox{ , if $\lambda = 0$} \end{array} \right.$

Maximum likelihood estimation and least square estimation are equivalent while estimating Box-Cox power transformation parameter (Kutner et al., 2005). Therefore, these two methods return the same result.

Value

A list with class "boxcoxlm" containing the following elements:

`method`	method preferred to estimate Box-Cox transformation parameter
`lambda.hat`	estimate of Box-Cox Power transformation parameter based on corresponding method
`lambda2`	additional shifting parameter
`statistic`	statistic of normality test for residuals after transformation based on specified normality test in method. For mle and lse, statistic is obtained by Shapiro-Wilk test for residuals after transformation
`p.value`	p.value of normality test for residuals after transformation based on specified normality test in method. For mle and lse, p.value is obtained by Shapiro-Wilk test for residuals after transformation
`alpha`	the level of significance to assess normality of residuals
`tf.y`	transformed response variable
`tf.residuals`	residuals after transformation
`y.name`	response name
`x.name`	x matrix name

Author(s)

Osman Dag, Ozlem Ilk

References

Asar, O., Ilk, O., Dag, O. (2017). Estimating Box-Cox Power Transformation Parameter via Goodness of Fit Tests. Communications in Statistics - Simulation and Computation, 46:1, 91–105.

Kutner, M. H., Nachtsheim, C., Neter, J., Li, W. (2005). Applied Linear Statistical Models. (5th ed.). New York: McGraw-Hill Irwin.

Examples


library(AID)

trees=as.matrix(trees)
boxcoxlm(x = trees[,1:2], y = trees[,3])

library(AID)

trees=as.matrix(trees)
boxcoxlm(x = trees[,1:2], y = trees[,3])

Ensemble Based Box-Cox Transformation via Meta Analysis for Normality of a Variable

Description

boxcoxmeta performs ensemble based Box-Cox transformation via meta analysis for normality of a variable and provides graphical analysis.

Usage

boxcoxmeta(data, lambda = seq(-3,3,0.01), nboot = 100, lambda2 = NULL, plot = TRUE, 
  alpha = 0.05, verbose = TRUE)boxcoxmeta(data, lambda = seq(-3,3,0.01), nboot = 100, lambda2 = NULL, plot = TRUE, 
  alpha = 0.05, verbose = TRUE)

Arguments

`data`	a numeric vector of data values.
`lambda`	a vector which includes the sequence of candidate lambda values. Default is set to (-3,3) with increment 0.01.
`nboot`	a number of Bootstrap samples to estimate standard errors of lambda estimates.
`lambda2`	a numeric for an additional shifting parameter. Default is set to lambda2 = 0.
`plot`	a logical to plot histogram with its density line and qqplot of raw and transformed data. Defaults plot = TRUE.
`alpha`	the level of significance to check the normality after transformation. Default is set to alpha = 0.05.
`verbose`	a logical for printing output to R console.

Details

Denote $y$ the variable at the original scale and $y'$ the transformed variable. The Box-Cox power transformation is defined by:

$y' = \left\{ \begin{array}{ll} \frac{y^\lambda - 1}{\lambda} \mbox{ , if $\lambda \neq 0$} \cr log(y) \mbox{ , if $\lambda = 0$} \end{array} \right.$

If the data include any nonpositive observations, a shifting parameter $\lambda_2$ can be included in the transformation given by:

$y' = \left\{ \begin{array}{ll} \frac{(y + \lambda_2)^\lambda - 1}{\lambda} \mbox{ , if $\lambda \neq 0$} \cr log(y + \lambda_2) \mbox{ , if $\lambda = 0$} \end{array} \right.$

Value

A list with class "boxcoxmeta" containing the following elements:

`method`	name of method
`lambda.hat`	estimate of Box-Cox Power transformation parameter
`lambda2`	additional shifting parameter
`result`	a data frame containing the result
`alpha`	the level of significance to assess normality.
`tf.data`	transformed data set
`var.name`	variable name

Author(s)

Muhammed Ali Yilmaz, Osman Dag

References

Yilmaz, M.A., Dag, O. (2022). Ensemble Based Box-Cox Transformation via Meta Analysis. Journal of Advanced Research in Natural and Applied Sciences, 8:3, 463–471.

Examples

library(AID)
data(textile)

out <- boxcoxmeta(textile[,1])
out$lambda.hat # the estimate of Box-Cox parameter 
out$tf.data # transformed data set

library(AID)
data(textile)

out <- boxcoxmeta(textile[,1])
out$lambda.hat # the estimate of Box-Cox parameter 
out$tf.data # transformed data set

Box-Cox Transformation for Normality of a Variable

Description

boxcoxnc performs Box-Cox transformation for normality of a variable and provides graphical analysis.

Usage

boxcoxnc(data, method = "sw", lambda = seq(-3,3,0.01), lambda2 = NULL, plot = TRUE, 
  alpha = 0.05, verbose = TRUE)boxcoxnc(data, method = "sw", lambda = seq(-3,3,0.01), lambda2 = NULL, plot = TRUE, 
  alpha = 0.05, verbose = TRUE)

Arguments

`data`	a numeric vector of data values.
`method`	a character string to select the desired method to be used to estimate Box-Cox transformation parameter. To use Shapiro-Wilk test method should be set to "sw". For method = "ad", boxcoxnc function uses Anderson-Darling test to estimate Box-Cox transformation parameter. Similarly, method should be set to "cvm", "pt", "sf", "lt", "jb", "ac", "mle" to use Cramer-von Mises, Pearson Chi-square, Shapiro-Francia, Lilliefors, Jarque-Bera tests, artificial covariate method and maximum likelihood estimation, respectively. Default is set to method = "sw".
`lambda`	a vector which includes the sequence of candidate lambda values. Default is set to (-3,3) with increment 0.01.
`lambda2`	a numeric for an additional shifting parameter. Default is set to lambda2 = 0.
`plot`	a logical to plot histogram with its density line and qqplot of raw and transformed data. Defaults plot = TRUE.
`alpha`	the level of significance to check the normality after transformation. Default is set to alpha = 0.05.
`verbose`	a logical for printing output to R console.

Details

Denote $y$ the variable at the original scale and $y'$ the transformed variable. The Box-Cox power transformation is defined by:

$y' = \left\{ \begin{array}{ll} \frac{y^\lambda - 1}{\lambda} \mbox{ , if $\lambda \neq 0$} \cr log(y) \mbox{ , if $\lambda = 0$} \end{array} \right.$

If the data include any nonpositive observations, a shifting parameter $\lambda_2$ can be included in the transformation given by:

$y' = \left\{ \begin{array}{ll} \frac{(y + \lambda_2)^\lambda - 1}{\lambda} \mbox{ , if $\lambda \neq 0$} \cr log(y + \lambda_2) \mbox{ , if $\lambda = 0$} \end{array} \right.$

Value

A list with class "boxcoxnc" containing the following elements:

`method`	method preferred to estimate Box-Cox transformation parameter
`lambda.hat`	estimate of Box-Cox Power transformation parameter based on corresponding method
`lambda2`	additional shifting parameter
`statistic`	statistic of normality test for transformed data based on specified normality test in method. For artificial covariate method, statistic is obtained by Shapiro-Wilk test for transformed data
`p.value`	p.value of normality test for transformed data based on specified normality test in method. For artificial covariate method, p.value is obtained by Shapiro-Wilk test for transformed data
`alpha`	the level of significance to assess normality.
`tf.data`	transformed data set
`var.name`	variable name

Author(s)

Osman Dag, Ozgur Asar, Ozlem Ilk

References

Asar, O., Ilk, O., Dag, O. (2017). Estimating Box-Cox Power Transformation Parameter via Goodness of Fit Tests. Communications in Statistics - Simulation and Computation, 46:1, 91–105.

Dag, O., Asar, O., Ilk, O. (2014). A Methodology to Implement Box-Cox Transformation When No Covariate is Available. Communications in Statistics - Simulation and Computation, 43:7, 1740–1759.

Examples


library(AID)

data(textile)

out <- boxcoxnc(textile[,1], method = "sw")
out$lambda.hat # the estimate of Box-Cox parameter based on Shapiro-Wilk test statistic 
out$p.value # p.value of Shapiro-Wilk test for transformed data 
out$tf.data # transformed data set
confInt(out) # mean and confidence interval for back transformed data


out2 <- boxcoxnc(textile[,1], method = "sf")
out2$lambda.hat # the estimate of Box-Cox parameter based on Shapiro-Francia test statistic
out2$p.value # p.value of Shapiro-Francia test for transformed data 
out2$tf.data 
confInt(out2) 
library(AID)

data(textile)

out <- boxcoxnc(textile[,1], method = "sw")
out$lambda.hat # the estimate of Box-Cox parameter based on Shapiro-Wilk test statistic 
out$p.value # p.value of Shapiro-Wilk test for transformed data 
out$tf.data # transformed data set
confInt(out) # mean and confidence interval for back transformed data


out2 <- boxcoxnc(textile[,1], method = "sf")
out2$lambda.hat # the estimate of Box-Cox parameter based on Shapiro-Francia test statistic
out2$p.value # p.value of Shapiro-Francia test for transformed data 
out2$tf.data 
confInt(out2)

Mean and Asymmetric Confidence Interval for Back Transformed Data

Description

confInt.boxcoxfr calculates mean and asymmetric confidence interval for back transformed data in each group and plots their error bars with confidence intervals.

Usage

## S3 method for class 'boxcoxfr'
confInt(x, level = 0.95, plot = TRUE, xlab = NULL, ylab = NULL, title = NULL, 
  width = NULL, verbose = TRUE, ...)## S3 method for class 'boxcoxfr'
confInt(x, level = 0.95, plot = TRUE, xlab = NULL, ylab = NULL, title = NULL, 
  width = NULL, verbose = TRUE, ...)

Arguments

`x`	a `boxcoxfr` object.
`level`	the confidence level.
`plot`	a logical to plot error bars with confidence intervals.
`xlab`	a label for the x axis, defaults to a description of x.
`ylab`	a label for the y axis, defaults to a description of y.
`title`	a main title for the plot.
`width`	a numeric giving the width of the little lines at the tops and bottoms of the error bars (defaults to 0.15).
`verbose`	a logical for printing output to R console.
`...`	additional argument(s) for methods.

Details

Confidence interval in each group is constructed separately.

Value

A matrix with columns giving mean, lower and upper confidence limits for back transformed data. These will be labelled as (1 - level)/2 and 1 - (1 - level)/2 in % (by default 2.5% and 97.5%).

Author(s)

Osman Dag

Examples


library(AID)

data(AADT)
attach(AADT)
out <- boxcoxfr(aadt, class)
confInt(out, level = 0.95)
library(AID)

data(AADT)
attach(AADT)
out <- boxcoxfr(aadt, class)
confInt(out, level = 0.95)

Mean and Asymmetric Confidence Interval for Back Transformed Data

Description

confInt.boxcoxmeta calculates mean and asymmetric confidence interval for back transformed data.

Usage

## S3 method for class 'boxcoxmeta'
confInt(x, level = 0.95, verbose = TRUE, ...)## S3 method for class 'boxcoxmeta'
confInt(x, level = 0.95, verbose = TRUE, ...)

Arguments

`x`	a `boxcoxmeta` object.
`level`	the confidence level.
`verbose`	a logical for printing output to R console.
`...`	additional argument(s) for methods.

Value

A matrix with columns giving mean, lower and upper confidence limits for back transformed data. These will be labelled as (1 - level)/2 and 1 - (1 - level)/2 in % (by default 2.5% and 97.5%).

Author(s)

Osman Dag, Muhammed Ali Yilmaz

Examples

library(AID)
data(textile)

out <- boxcoxmeta(textile[,1])
confInt(out) # mean and confidence interval for back transformed data

library(AID)
data(textile)

out <- boxcoxmeta(textile[,1])
confInt(out) # mean and confidence interval for back transformed data

Mean and Asymmetric Confidence Interval for Back Transformed Data

Description

confInt is a generic function to calculate mean and asymmetric confidence interval for back transformed data.

Usage

## S3 method for class 'boxcoxnc'
confInt(x, level = 0.95, verbose = TRUE, ...)## S3 method for class 'boxcoxnc'
confInt(x, level = 0.95, verbose = TRUE, ...)

Arguments

`x`	a `boxcoxnc` object.
`level`	the confidence level.
`verbose`	a logical for printing output to R console.
`...`	additional argument(s) for methods.

Value

A matrix with columns giving mean, lower and upper confidence limits for back transformed data. These will be labelled as (1 - level)/2 and 1 - (1 - level)/2 in % (by default 2.5% and 97.5%).

Author(s)

Osman Dag

Examples


library(AID)

data(textile)
out <- boxcoxnc(textile[,1])
confInt(out) # mean and confidence interval for back transformed data

library(AID)

data(textile)
out <- boxcoxnc(textile[,1])
confInt(out) # mean and confidence interval for back transformed data

Student Grades Data

Description

Overall student grades for a class thaught by Dr. Ozlem Ilk

Usage

data(grades)data(grades)

Format

A data frame with 42 observations on the following variable.

grades: a numeric vector for the student grades

Examples


library(AID)

data(grades)
hist(grades[,1])
out <- boxcoxnc(grades[,1])
confInt(out, level = 0.95)

library(AID)

data(grades)
hist(grades[,1])
out <- boxcoxnc(grades[,1])
confInt(out, level = 0.95)

Textile Data

Description

Number of Cycles to Failure of Worsted Yarn

Usage

data(textile)data(textile)

Format

A data frame with 27 observations on the following variable.

textile: a numeric vector for the number of cycles

References

Box, G. E. P., Cox, D. R. (1964). An Analysis of Transformations (with discussion). Journal of the Royal Statistical Society, Series B (Methodological), 26, 211–252.

Examples


library(AID)

data(textile)
hist(textile[,1])
out <- boxcoxnc(textile[,1])
confInt(out)
library(AID)

data(textile)
hist(textile[,1])
out <- boxcoxnc(textile[,1])
confInt(out)

Package 'AID'

Help Index

Box-Cox Power Transformation

Description

Details

Average Annual Daily Traffic Data

Description

Usage

Format

References

Examples

Box-Cox Transformation for One-Way ANOVA

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Box-Cox Transformation for Linear Models

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Ensemble Based Box-Cox Transformation via Meta Analysis for Normality of a Variable

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Box-Cox Transformation for Normality of a Variable

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Mean and Asymmetric Confidence Interval for Back Transformed Data

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Mean and Asymmetric Confidence Interval for Back Transformed Data

Description

Usage

Arguments

Value

Author(s)

Examples

Mean and Asymmetric Confidence Interval for Back Transformed Data

Description

Usage

Arguments

Value

Author(s)

Examples

Student Grades Data

Description

Usage

Format

Examples

Textile Data

Description

Usage

Format

References

Examples