Topic 2.9 Analyzing Departures from Linearity: identify outliers, high-leverage, and influential points in regression, and use transformations to model a non-linear relationship.

What this topic is asking

The College Board (Topic 2.9) wants you to identify outliers, high-leverage points, and influential points in a regression, to understand how each affects the line, and to use transformations (such as logs or powers) to model a relationship that is not linear.

Outliers, leverage, and influence

These three ideas are related but distinct, and the exam tests whether you can keep them apart. Outlier is about a large residual (extreme in $y$ relative to the pattern). Leverage is about an extreme $x$ (far out along the horizontal axis). Influence is about effect: does removing the point change the line? A high-leverage point that lies on the trend has little influence (small residual, no change to the slope), whereas a high-leverage point off the trend is typically highly influential, because its distant $x$ lets it swing the line like a long lever arm. The cleanest test for influence is the thought experiment: imagine deleting the point; if the line moves a lot, the point was influential.

How influential points distort regression

Because the least-squares line minimizes squared vertical distances, an influential point can drag the slope and intercept toward itself and can inflate or deflate the correlation, giving a misleading summary of the bulk of the data. This is why Topic 2.5's warning that $r$ is not resistant matters here: a single influential point can make a weak relationship look strong, or hide a strong one. The practical advice the exam rewards is to identify such points (from the scatterplot and residual plot), to consider analyzing the data with and without them, and to report how the conclusions change, rather than silently letting one point dominate. You do not simply delete points, but you flag them and assess their effect, which is honest data analysis.

Transformations to achieve linearity

When the relationship itself is non-linear, the cure is not a different point but a transformation of a variable. If the scatterplot curves and the residual plot of a linear fit shows a systematic pattern (Topic 2.7), applying a function to $y$ or $x$ can straighten the relationship so a line fits the transformed data. Common transformations are the logarithm (taking $\ln(y)$ linearises exponential growth, where $y = a \cdot b^x$; taking $\ln$ of both variables linearises a power law $y = a x^b$) and powers or roots (such as $\sqrt{y}$). The workflow is: transform, refit the line to the transformed data, check that the new residual plot shows random scatter (confirming the transformation worked), and then back-transform to make predictions in the original units. For a log-$y$ model $\widehat{\ln(y)} = a + bx$, a prediction comes from computing $\widehat{\ln(y)}$ and then raising $e$ to that power: $\hat{y} = e^{\,a + bx}$. The back-transformation step is where marks are most often lost, because students stop at the transformed prediction; remembering that $\ln(y)$ must be undone with $e$ to recover $y$ is essential. Transformation is the topic's payoff: it extends the entire regression toolkit (line, $r$, $r^2$, residual analysis) to curved relationships, provided you transform first and interpret in the original units at the end.

Try this

Q1. A regression point lies far to the right (extreme $x$) but right on the trend line. Classify it (outlier, high leverage, influential?). [2 points]

• Cue. High leverage (extreme $x$) but not an outlier (small residual) and not necessarily influential (it lies on the pattern, so removing it changes the line little).

Q2. After fitting $\widehat{\ln(y)} = 1 + 0.4x$, you compute $\widehat{\ln(y)} = 3$ at some $x$. What is the predicted $y$? [1 point]

• Cue. Back-transform: $\hat{y} = e^{3} \approx 20.1$.