Departures from Linearity

Points that pull the line

• Not every point matters equally — some are influential 有影响的.
• An influential point noticeably changes the slope, intercept, or correlation if removed.
• The test is practical: fit the line with and without it and see if the answer moves.
• Two special kinds drive most of this: outliers and high-leverage points.

Outliers vs. high leverage

• An outlier 离群值 has a large residual — it's far from the line in the $y$-direction.
• A high-leverage point 高杠杆点 has an extreme $x$-value — far out along the horizontal.
• Leverage is about horizontal extremeness; an outlier is about vertical miss.
• A point that is both (extreme $x$ and off the trend) is the most influential of all.

A residual plot with no pattern supports a linear model; a curve or a fan does not

Anscombe's quartet: the same r and line with very different scatterplots

How one point distorts

• A high-leverage point acts like a lever: it can swing the slope toward itself.
• Removing an influential point can flip a slope's steepness — or even its sign.
• It can also inflate or deflate $r$, making a relationship look stronger or weaker than it is.
• Always ask whether your conclusion rests on one stubborn point.

Transformations for curves

• When the pattern is genuinely non-linear, a straight line is the wrong tool.
• A transformation 变换 (e.g. take logs, or square-root) can straighten a curved pattern.
• Fit the line to the transformed data, then transform predictions back.
• The limitation is real: forcing a line onto a curve gives biased, misleading predictions.

Write from a high-leverage point

• One x value is far to the right of the other x values.
• The point sits near the line but can still swing it.