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

Average Annual Daily Traffic Data

Description

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

Usage

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 Optimal Sampling for Traffic Volume Estimation, Unpublished Ph.D. dissertation, University of Minnesota, Carlson School of Management.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

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)

Arguments

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:

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)

######

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)

Arguments

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:

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])

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)

Arguments

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:

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

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)

Arguments

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:

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) 

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, ...)

Arguments

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)

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, ...)

Arguments

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

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, ...)

Arguments

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

Student Grades Data

Description

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

Usage

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)

Textile Data

Description

Number of Cycles to Failure of Worsted Yarn

Usage

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)