Box-Cox Normality Plot - Free Statistics and Forecasting Software (Calculators) v.1.2.1

:: Box-Cox Normality Plot - Free Statistics Software (Calculator) ::

All rights reserved. The non-commercial (academic) use of this software is free of charge. The only thing that is asked in return is to cite this software when results are used in publications.

This free online software (calculator) computes the Box-Cox Normality Plot. This analysis identifies the lambda (Box-Cox parameter) value that results in the (quasi-)optimal fit against the normal distribution. The software uses two computational algorithms to find the value for lambda. The first method maximizes the correlation from the normal probability plot (for all values between a user-specified minimum and maximum). The second method uses Maximum Likelihood Estimation (without limitation) and a Likelihood Ratio test against the nulhypotheses: lambda = 0 and lambda = 1.

Enter (or paste) your data delimited by hard returns.

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Data [reset data]
112 118 132 129 121 135 148 148 136 119 104 118 115 126 141 135 125 149 170 170 158 133 114 140 145 150 178 163 172 178 199 199 184 162 146 166 171 180 193 181 183 218 230 242 209 191 172 194 196 196 236 235 229 243 264 272 237 211 180 201 204 188 235 227 234 264 302 293 259 229 203 229 242 233 267 269 270 315 364 347 312 274 237 278 284 277 317 313 318 374 413 405 355 306 271 306 315 301 356 348 355 422 465 467 404 347 305 336 340 318 362 348 363 435 491 505 404 359 310 337 360 342 406 396 420 472 548 559 463 407 362 405 417 391 419 461 472 535 622 606 508 461 390 432

Type of transformation


Minimum lambda


Maximum lambda


Constant term to be added before analysis is performed (?)


Display table with original and transformed data?


Chart options
Width:
Height:

Source code of R module

library(car)
par2 <- abs(as.numeric(par2)*100)
par3 <- as.numeric(par3)*100
if(par4=="") par4 <- 0
par4 <- as.numeric(par4)
numlam <- par2 + par3 + 1
x <- x + par4
n <- length(x)
c <- array(NA,dim=c(numlam))
l <- array(NA,dim=c(numlam))
mx <- -1
mxli <- -999
for (i in 1:numlam)
{
l[i] <- (i-par2-1)/100
if (l[i] != 0)
{
if (par1 == "Full Box-Cox transform") x1 <- (x^l[i] - 1) / l[i]
if (par1 == "Simple Box-Cox transform") x1 <- x^l[i]
} else {
x1 <- log(x)
}
c[i] <- cor(qnorm(ppoints(x), mean=0, sd=1),sort(x1))
if (mx < c[i])
{
mx <- c[i]
mxli <- l[i]
x1.best <- x1
}
}
print(c)
print(mx)
print(mxli)
print(x1.best)
if (mxli != 0)
{
if (par1 == "Full Box-Cox transform") x1 <- (x^mxli - 1) / mxli
if (par1 == "Simple Box-Cox transform") x1 <- x^mxli
} else {
x1 <- log(x)
}
mypT <- powerTransform(x)
summary(mypT)
bitmap(file="test1.png")
plot(l,c,main="Box-Cox Normality Plot", xlab="Lambda",ylab="correlation")
mtext(paste("Optimal Lambda =",mxli))
grid()
dev.off()
bitmap(file="test2.png")
hist(x,main="Histogram of Original Data",xlab="X",ylab="frequency")
grid()
dev.off()
bitmap(file="test3.png")
hist(x1,main="Histogram of Transformed Data", xlab="X",ylab="frequency")
grid()
dev.off()
bitmap(file="test4.png")
qqPlot(x)
grid()
mtext("Original Data")
dev.off()
bitmap(file="test5.png")
qqPlot(x1)
grid()
mtext("Transformed Data")
dev.off()
load(file="createtable")
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,"Box-Cox Normality Plot",2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,"# observations x",header=TRUE)
a<-table.element(a,n)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,"maximum correlation",header=TRUE)
a<-table.element(a,mx)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,"optimal lambda",header=TRUE)
a<-table.element(a,mxli)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,"transformation formula",header=TRUE)
if (par1 == "Full Box-Cox transform") {
a<-table.element(a,"for all lambda <> 0 : T(Y) = (Y^lambda - 1) / lambda")
} else {
a<-table.element(a,"for all lambda <> 0 : T(Y) = Y^lambda")
}
a<-table.row.end(a)
if(mx<0) {
a<-table.row.start(a)
a<-table.element(a,"Warning: maximum correlation is negative! The Box-Cox transformation must not be used.",2)
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file="mytable.tab")
if(par5=="Yes") {
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,"Obs.",header=T)
a<-table.element(a,"Original",header=T)
a<-table.element(a,"Transformed",header=T)
a<-table.row.end(a)
for (i in 1:n) {
a<-table.row.start(a)
a<-table.element(a,i)
a<-table.element(a,x[i])
a<-table.element(a,x1.best[i])
a<-table.row.end(a)
}
a<-table.end(a)
table.save(a,file="mytable1.tab")
}
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,"Maximum Likelihood Estimation of Lambda",1,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,paste("<pre>",RC.texteval("summary(mypT)"),"</pre>",sep=""))
a<-table.row.end(a)
a<-table.end(a)
table.save(a,file="mytable3.tab")

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Cite this software as:

Wessa P., (2021), Box-Cox Normality Plot (v1.1.13) in Free Statistics Software (v1.2.1), Office for Research Development and Education, URL http://www.wessa.net/rwasp_boxcoxnorm.wasp/

The R code is based on :

NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, 2006-10-03.

Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations. Journal of the Royal Statisistical Society, Series B. 26 211-46.

Cook, R. D. and Weisberg, S. (1999) Applied Regression Including Computing and Graphics. Wiley

Fox, J. and Weisberg, S. (2011) An R Companion to Applied Regression, Second Edition, Sage.

Velilla, S. (1993) A note on the multivariate Box-Cox transformation to normality. Statistics and Probability Letters, 17, 259-263.

Weisberg, S. (2014) Applied Linear Regression, Fourth Edition, Wiley.

Yeo, I. and Johnson, R. (2000) A new family of power transformations to improve normality or symmetry. Biometrika, 87, 954-959.

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To cite Wessa.net in publications use:
Wessa, P. (2024), Free Statistics Software, Office for Research Development and Education,
version 1.2.1, URL https://www.wessa.net/

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Software Version : 1.2.1
Algorithms & Software : Patrick Wessa, PhD
Server : www.wessa.net

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:: Box-Cox Normality Plot - Free Statistics Software (Calculator) ::