Extend the regression equation when the relationship is not linear or when an effect depends on context.
Revenue may rise quickly with advertising at first and then level off. A promotion may work well for new customers but have little effect for returning customers.
Transformations change the scale or shape of a relationship. Interaction terms allow the effect of one predictor to differ across the value of another.
Key question
How can a regression model represent curved relationships and effects that vary by group or condition?
What does a transformation change?
| Form | Typical purpose |
|---|---|
| log(x) | compress large values and represent diminishing returns |
| log(y) | model proportional rather than absolute changes |
| x² | represent curvature |
| standardization | place predictors on a comparable scale |
A transformation changes interpretation
After transforming a variable, the coefficient must be interpreted on the transformed scale rather than as a simple one-unit raw change.
Curved in the raw units
beforeA relationship that bends is hard for a straight line to follow.
Straight after a log
log(x)Compressing the large values turns the curve into a line.
An interaction
x₁ × x₂The slope for one variable depends on the level of the other.
How are log transformations interpreted?
| Model | Approximate interpretation of β |
|---|---|
| y = α + β log(x) | a 1% increase in x is associated with a β/100 unit change in y |
| log(y) = α + βx | a one-unit increase in x is associated with about 100β% change in y |
| log(y) = α + β log(x) | a 1% increase in x is associated with about β% change in y |
For larger coefficients, use the exact transformation rather than relying only on the small-change approximation.
What does an interaction mean?
Interaction model
ŷ = b₀ + b₁x + b₂z + b₃xz
Promotion by customer type
If z identifies returning customers, b₃ tells us how the promotion slope differs between new and returning customers.
How should main effects be read?
Do not drop main effects casually
When an interaction is included, its component main effects are usually retained so that the model remains interpretable and respects hierarchy.
Change transformations and interactions
Compare the raw and transformed relationship, then change the interaction coefficient and watch the two group-specific lines separate or converge.
With no interaction, the two lines have similar slopes. As the interaction moves away from zero, the slope for Group 1 changes relative to Group 0.
What to watch
A transformation changes the shape or scale of one relationship; an interaction changes which relationship applies under different conditions.