1. Model revenue in dollars
On the raw scale, the residual spread grows as fitted revenue grows. That fan shape is a warning.
Residuals vs fitted
Normal Q-Q
Scale-location
STA 235 · Week 6 · non-linear relationships
A log model describes percentage-style change
Some business outcomes grow by multipliers: 10% more, 1.25 times as much, doubled. When the spread grows with the outcome, modeling the log of the outcome can turn a curved, fanning relationship into a more regular straight-line pattern.
Example: product demos and weekly revenue
Businesses with more product demonstrations tend to have higher revenue, and the dollar-to-dollar spread is larger for the higher-revenue businesses. The simulated example has positive revenue, so logging the outcome is possible.
Weekly revenue versus product demonstrations
observed businessraw-dollar straight-line fitlog-model fit, returned to dollars
2. Model log(revenue)
On the log scale, the residual cloud is much more even. The green curve is the model’s prediction converted back to dollars.
Residuals vs fitted
Normal Q-Q
Scale-location
How to use a log transform responsibly
1 · Check the scale
Logging an outcome requires positive values. A log transform is useful when the relationship and residual spread suggest proportional, not constant-dollar, variation.
2 · Say “multiply” or “percent”
Interpret a one-unit x increase with eᵇ¹ or 100(eᵇ¹−1)%. Avoid describing a log coefficient as an added number of dollars.
3 · Back-transform with care
Exponentiating a fitted log value gives a typical predicted dollar amount. The arithmetic average can be a little higher when there is remaining variation.