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Normality test and Power transformation

Some examples of Normality test and power transformation

#Normality test and transformation to normality

#generate uniform distribution which is not normal
normalNo1 <- runif(1000, min=2, max =4)
hist(normalNo1, breaks = 50)

#null statement: Distribution is normal
shapiro.test(normalNo1)

## 
##  Shapiro-Wilk normality test
## 
## data:  normalNo1
## W = 0.95354, p-value < 2.2e-16

#generate a normal distribution and add skewness
normalNo2<- rnorm(1000, mean=0, sd=1)
hist(normalNo2, breaks = 50)

normalNo2 <- exp(normalNo2)
hist(normalNo2, breaks = 50)

#null statement: Distribution is normal
shapiro.test(normalNo2)

## 
##  Shapiro-Wilk normality test
## 
## data:  normalNo2
## W = 0.66337, p-value < 2.2e-16

##Power Transformation based on Box and Cox (1964)
#lambda = -1. is a reciprocal transform.
#lambda = -0.5 is a reciprocal square root transform.
#lambda = 0.0 is a log transform.
#lambda = 0.5 is a square root transform.
#lambda = 1.0 is no transform.

x2 <- powerTransform(normalNo2 ~ 1, family="bcPower")
summary(x2)

## bcPower Transformation to Normality 
##    Est Power Rounded Pwr Wald Lwr Bnd Wald Upr Bnd
## Y1   -0.0193           0      -0.0745       0.0359
## 
## Likelihood ratio test that transformation parameter is equal to 0
##  (log transformation)
##                             LRT df    pval
## LR test, lambda = (0) 0.4694733  1 0.49323
## 
## Likelihood ratio test that no transformation is needed
##                            LRT df       pval
## LR test, lambda = (1) 1310.454  1 < 2.22e-16

lamb <- x2$lambda #lambda is very close to zero suggesting a log transform
lamb

##          Y1 
## -0.01930521

xNorm2 <- normalNo2^x2$lamb
hist(xNorm2, breaks = 50)

shapiro.test(xNorm2) ##the null statement that the distribution is normal could not be rejected.

## 
##  Shapiro-Wilk normality test
## 
## data:  xNorm2
## W = 0.99829, p-value = 0.4244