Page 307 - Applied Statistics with R

14.1. RESPONSE TRANSFORMATION 307

where is a constant that does not depend on the mean, E[ ∣ = ]. A
transformation that accomplishes this is called a variance stabilizaing trans-
formation.

A common variance stabilizing transformation (VST) when we see increasing
variance in a fitted versus residuals plot is log( ). Also, if the values of a variable
range over more than one order of magnitude and the variable is strictly positive,
then replacing the variable by its logarithm is likely to be helpful.

A reminder, that for our purposes, log and ln are both the natural log. R uses
log to mean the natural log, unless a different base is specified.

We will now use a model with a log transformed response for the Initech data,

log( ) = + + .
1

0

Note, if we re-scale the model from a log scale back to the original scale of the
data, we now have

= exp( + ) ⋅ exp( )
0

1

which has the errors entering the model in a multiplicative fashion.

Fitting this model in R requires only a minor modification to our formula spec-
ification.

initech_fit_log = lm(log(salary) ~ years, data = initech)

Note that while log(y) is considered the new response variable, we do not
actually create a new variable in R, but simply transform the variable inside the
model formula.

plot(log(salary) ~ years, data = initech, col = "grey", pch = 20, cex = 1.5,
main = "Salaries at Initech, By Seniority")
abline(initech_fit_log, col = "darkorange", lwd = 2)