Load packages

library(tidyverse)
library(mosaic)

Motivation

What are data transformation good for? Why do we bother to transform variables for regression analysis? This post explores some nuances around these themes.

Simulate an exponentially distributed assocation

len <- 42  # 42 x values
x <- rep(runif(len), 30)  # each x value repeated 30 times
y <- dexp(x) + rnorm(length(x), mean = 0, sd = .01)  # add some noise

Plot it:

gf_point(y ~ x) %>% 
  gf_smooth()

If we now do a log transformation on \(y\), we get this:

gf_point(log(y) ~ x) %>% 
  gf_smooth()

A straigt line. That means, we have a linear trend, thus additivity holds - and additivity is a/the central assumption in lineare models.

For comparison, let’s log \(x\):

gf_point(y ~ log(x)) %>% 
  gf_smooth()

We get a different trend, but additivity does not hold anymore.

Gelman’s comment on the use of transformation (Chap. 5, p. 53ff)

Gelman and Hill (2007) point out three reasons for transformations:

1. Easier interpretation (more on that later)
2. Improved additivity (as seen above)
3. Improved normality of the residuls (see below)

Improved normality of the residuls

Consider the residulas of the nontransformed regression:

lm1 <- lm(y ~ x)
gf_histogram(~ resid(lm1), bins = 10)

Well, not toooo much of a normal distribution.

Compare the logged lm:

lm2 <- lm(log(y) ~ x)
gf_histogram(~ resid(lm2), bins = 10)

Much better.

Easier interpretation

Assume we have a regression model on log scale. As can be seen in the Figure below, small values of the exponential function (close to zero) are similar to the \(y=x+1\) straigt line. That means, we can interpret an logged coefficient of say 6% as a growth factor of 1.06. Note that this correspondence deteriorates for larger values.

gf_fun(exp(x) ~ x, xlim = c(-1,1)) %>% 
  gf_fun(x +1 ~ x , color = "red")