In the linear regression section of our book Practical Data Science in R, we use the example of predicting income from a number of demographic variables (age, sex, education and employment type). In the text, we choose to regress against
log10(income) rather than directly against income.
One obvious reason for not regressing directly against income is that (in our example) income is restricted to be non-negative, a restraint that linear regression can’t enforce. Other reasons include the wide distribution of values and the relative or multiplicative structure of errors on outcomes. A common practice in this situation is to use Poisson regression, or generalized linear regression with a log-link function. Like all generalized linear regressions, Poisson regression is unbiased and calibrated: it preserves the conditional expectations and rollups of the training data. A calibrated model is important in many applications, particularly when financial data is involved.
Regressing against the log of the outcome will not be calibrated; however it has the advantage that the resulting model will have lower relative error than a Poisson regression against income. Minimizing relative error is appropriate in situations when differences are naturally expressed in percentages rather than in absolute amounts. Again, this is common when financial data is involved: raises in salary tend to be in terms of percentage of income, not in absolute dollar increments.
Unfortunately, a full discussion of the differences between Poisson regression and regressing against log amounts was outside of the scope of our book, so we will discuss it in this note.