Outliers, Influential Points and Transforming for Linearity

Three kinds of unusual point

An outlier in regression is a point with a large residual — it lies far from the line vertically. A high-leverage point is one whose $x$-value is far from $\bar x$; it may sit right on the line and still have leverage. A point is influential if removing it changes the slope, the intercept or $r$ substantially. High-leverage points are the usual culprits: a single distant $x$ can drag the whole line toward itself and inflate $r$. The AP test asks you to describe the effect of removing such a point, not to delete it — you may only remove data when you know it is an error.

When the scatter is curved

A curved residual plot means the linear model is wrong for the raw data. The fix is to transform a variable and refit:

• Exponential model $y=ab^x$: take $\ln y$. Then $\ln y=\ln a+(\ln b)\,x$ is linear in $x$.
• Power model $y=ax^k$: take logs of both: $\ln y=\ln a+k\ln x$.
• A square-root or reciprocal transformation also appears, usually given by the problem.

Example: if the fitted line is $\widehat{\ln y}=1{,}10+0{,}18x$, then $\hat y=e^{1{,}10}\cdot e^{0{,}18x}\approx3{,}0\cdot(1{,}20)^x$, so $y$ grows by about $20\%$ per unit of $x$.

Judging the transformed model

The transformation is good if the new residual plot is patternless; a higher $r^2$ is supporting evidence but not the deciding one. Predictions must be back-transformed: predict $\ln y$, then exponentiate. And $r^2$ of the transformed fit describes variation in $\ln y$, not in $y$.