1. Straight-line model
The red line is easy to summarize, but its residuals still carry a visible story.
Residuals vs fitted
Normal Q-Q
Scale-location
STA 235 · Week 6 · non-linear relationships
A curve can be a better business model than a line
A straight line says each additional customer contact changes sales by the same amount. That is sometimes too simple. A polynomial model lets the relationship bend—and the residual plots help us decide whether the bend is useful.
Example: coaching meetings and weekly sales
These simulated observations represent sales representatives. Extra coaching is helpful at first, but after a busy schedule is packed with meetings, time away from selling begins to matter.
Observed weekly sales versus coaching meetings
observed representativestraight-line fitquadratic fit
2. Quadratic model
The green curve accounts for the systematic bend. Now check whether the diagnostic pattern is calmer.
Residuals vs fitted
Normal Q-Q
Scale-location
How much curve is too much?
This second data set was generated from a quadratic relationship. Use the arrows to raise the polynomial order. Watch the blue fit chase small sample quirks while its score on new cases begins to suffer.
training observationsunderlying quadratic patternselected polynomial fit
Selected modelDegree 1
How to read this result
1 · Diagnose first
A smooth U-shape or upside-down U in residuals versus fitted values is evidence that a straight line misses structure.
2 · Add a purposeful term
Fit x and x² together. Keep both terms even if the individual linear coefficient is not impressive by itself.
3 · Re-check, then explain
The improved residual plot matters more than a slightly larger R². Describe how the slope changes, and stay within the observed x-range.