A simple demonstration of retransformation bias in log-linear models

retransformation-bias.R

#
# Simulate data
#
a <- 2
x <- seq(0, 1, length.out = 100)
n <- length(x)
sigma <- 0.5
set.seed(42)
lny <- rnorm(n, mean = a*x, sd = sigma)
y <- exp(lny)

#
# Expectations 
#
yexpect <- a*x #linear predictor  on log scale
expyexpect <- exp(yexpect) #incorrect expectation, just the exponent of the linear predictor

m <- exp(yexpect + (sigma^2)/2) #correct mean, accounts for errors
sd <- sqrt(exp(2 * yexpect + sigma^2)* (exp(sigma^2)-1)) #sd

plot(x, expyexpect, ylim = c(0, 20), type = 'l', col = "red") #incorrect
#Correct, with SD
lines(x, m)
lines(x, m-sd, lty = 2)
lines(x, m+sd, lty = 2)