Track 9 · Correlation and Regression · Lesson 13 of 15
Transformations
Use log and polynomial transformations to model skewed responses and curved relationships, and interpret the coefficients on the transformed scale as percentages, elasticities or turning points.
House prices are right-skewed and an extra 10 m² is worth more in an expensive district than in a cheap one; a building uses energy for heating and cooling; extra advertising helps less once spending is already high. A straight line on the raw scale misdescribes all three. Transforming a variable keeps the convenience of linear regression while fitting the real shape.
Linear regression is linear in the coefficients, not necessarily in the original variables. You may replace by , or by or add , and still fit by least squares. Three signals suggest a transformation:
- Curvature in the scatter plot or in residuals against fitted values (see Residual analysis).
- A funnel: residuals spread out as fitted values grow — typical of prices, incomes and sales, where variation is proportional to size.
- Multiplicative thinking: effects that are naturally percentages (“10% more floor area, about 9% higher price”).
Included with a Formula Quest membership
It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Recognise curvature, funnels and multiplicative effects that call for a transformation
- Interpret log-response coefficients as percentage changes with 100(e^b − 1)
- Interpret a log–log slope as an elasticity
- Model a turning point with a squared term and locate it at −b1/(2b2)
- Back-transform log predictions and know they estimate a typical value