# Interpretation of log(1 + x) transformed predictor

Interpretation of log transformed predictor neatly explains how to interpret a log transformed predictor in OLS. Does the interpretation change if there are 0s in the data and the transformation becomes log(1 + x) instead?

Some authors (e.g. Fox and Weisberg 2011) recommend adding a start (i.e. a positive constant) if a log transformation is necessary to correct skewness and improve symmetry, but the data contains zeros.

Consider a variation of the Ornstein example in CAR (p. 303):

require(car)
data(Ornstein)
boxplot(Ornstein$interlocks, horizontal = T)  The data is clearly right skewed, and contains 0s. summary(powerTransform(1 + Ornstein$interlocks))
## bcPower Transformation to Normality
##
##                         Est.Power Std.Err. Wald Lower Bound Wald Upper Bound
## 1 + Ornstein$interlocks 0.1248 0.053 0.0209 0.2287 ## ## Likelihood ratio tests about transformation parameters ## LRT df pval ## LR test, lambda = (0) 5.502335 1 0.0189911 ## LR test, lambda = (1) 262.431991 1 0.0000000  The powerTransform() function suggests that a log(1 + x) transformation here could be useful. boxplot(log(1 + Ornstein$interlocks), horizontal = T)


As you can see, symmetry is indeed improved.

Question: If this transformed variable were to be included in an OLS regression as an IV, would the coefficient estimates still have the usual interpretation of log transformed variables?

It depends, according to Wooldridge (2012) the percentage change interpretations are often closely preserved, except for changes beginning at $y = 0$ (where the percentage change is not defined). Strickly speaking, using $log(1+y)$ and then interpreting the estimates as if the variable were $log(y)$ is acceptable only if the data on y contain relatively few zeros.