Box Cox Transformation – Estimating Lambda for Box Cox Transformation for ANOVA

anovadata transformationmaximum likelihoodmodelingr

Assumptions:

In an ANOVA where the normality assumptions are violated, the Box-Cox transformation can be applied to the response variable. The lambda can be estimated by the using maximum likelihood to optimize the normality of the model residuals.

Question:

When the estimates for lambda in the null model and the full model differ, how should lambda be estimated?

My Data:

In my data the lambda estimate for the null model is -2.3 and the lambda estimate for the full model is -2.8. Transforming the response using these different parameters and preforming the ANOVA leads to different F-statistics.

I have produced below a simplified version of the analysis using beta distributions with different parameters to simulate non-normality. Unfortunately, in this example the results of the ANOVA are insensitive to the different estimates of lambda. So, it doesn't fully illustrate the problem.

library(ggplot2)
library(MASS)
library(car)


#Generating random beta-distributed data
n=200
df <- rbind(
  data.frame(x=factor(rep("a1",n)), y=rbeta(n,2,5)), # more left skewed
  data.frame(x=factor(rep("a2",n)), y=rbeta(n,2,2))) # less left skewed

print(qplot(data=df, color=x, x=y, geom="density"))

print("Untransformed Analaysis of Variance:")
m.null <- lm(y ~ 1, df)
m.full <- lm(y ~ x, df)
print(anova(m.null, m.full))

# Estimate Maximum Liklihood Box-Cox transform parameters for both models
bc.null <- boxcox(m.null); bc.null.opt <- bc.null$x[which.max(bc.null$y)]
bc.full <- boxcox(m.full); bc.full.opt <- bc.full$x[which.max(bc.full$y)]

print(paste("ML Box-Cox estimate for null model:",bc.null.opt))
print(paste("ML Box-Cox estimate for full model:",bc.full.opt))

df$y.bc.null <- bcPower(df$y, bc.null.opt)
df$y.bc.full <- bcPower(df$y, bc.full.opt)

print(qplot(data=df, x=x, y=y.bc.null, geom="boxplot"))
print(qplot(data=df, x=x, y=y.bc.full, geom="boxplot"))


print("Analysis of Variance with optimial Box-Cox transform for null model")
m.bc_null.null <- lm(y.bc.null ~ 1, data=df)
m.bc_null.full <- lm(y.bc.null ~ x, data=df)
print(anova(m.bc_null.null, m.bc_null.full))

print("Analysis of Variance with optimial Box-Cox transform for full model")
m.bc_full.null <- lm(y.bc.null ~ 1, data=df)
m.bc_full.full <- lm(y.bc.null ~ x, data=df)
print(anova(m.bc_full.null, m.bc_full.full))

Best Answer

It is not appropriate to do ordinary ANOVA after using the same dataset to fit lambda. The analysis should be unified, penalizing for uncertainty in lambda (a parameter to be estimated, and included in the covariance matrix).

Related Question