Diagnostics on the Least-Squares Regression Line
Learn how to evaluate whether a linear model is appropriate using R², residual plots, and influential observation analysis.
The Coefficient of Determination ()
Definition
is the percentage of total variation in the response variable that is explained by the least-squares regression line.
R² equals the correlation coefficient squared
LogicLens: Explained vs. Unexplained
Percentage of variation explained by the regression line
Percentage of variation unexplained (due to other factors)
Example
If , then:
= 72.25% explained
= 27.75% unexplained
Residual Analysis & Scatter Plots
The Residual Plot
Plot the residuals () on the y-axis against the explanatory variable () on the x-axis.
LogicLens: Three Things to Check
Good: No discernible pattern — linear model is appropriate
Bad: U-shape or curve — non-linear relationship
Good: Same spread throughout — homoscedasticity
Bad: Funnel/fan shape — heteroscedasticity
Good: All points near the e=0 line
Bad: Points far from zero line
✓ Good Residual Plot
Random scatter around zero — linear model appropriate
✗ U-Shaped Pattern
Systematic curve — relationship is non-linear
Influential Observations
Definition
An influential observation is a point that significantly affects the slope or intercept of the regression line when included or removed.
LogicLens: Leverage vs. Influence
High-Leverage Point
A point that is far from the mean of x (extreme in the x-direction).
Has the potential to influence the line, but doesn't necessarily do so.
Influential Point
A point that actually changes the regression line significantly.
All influential points have high leverage, but not all high-leverage points are influential.
Visual Comparison
High leverage but on the line — NOT influential
High leverage and off the pattern — INFLUENTIAL
Try It Yourself
Residual Plot Explorer
Residual Plot
Model Appears Valid
Residuals show random scatter around zero with no discernible pattern. The linear model is appropriate for this data.
Quick Reference: What to Look For
No pattern = linear model OK
No funnel = equal variance
Check points far from e=0
Adaptive Assessment
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