Analyze model diagnostic plots (e.g., residual, marginal model, QQ, and added variable plots) to assess quality of fit.

A regression that produces a high $R^2$ can still fit badly. The coefficients summarize averages, but diagnostic plots reveal what those averages hide: curvature, heteroskedasticity, fat tails, and a handful of observations doing the heavy lifting.

What residuals are doing. Ordinary residuals are $e_i=y_i-{\hat{y}}_i$. Standardized residuals divide by an estimate of their standard deviation, so they should be roughly $N(0,1)$ under the linear-model assumptions. Studentized (or deleted) residuals refit the model without observation $i$ before computing the residual, which is what you want when one point is suspected of distorting the fit.

$$r_i=\frac{e_i}{s\sqrt{1-h_{ii}}}$$

The $h_{ii}$ is the $i$-th diagonal of the hat matrix $H=X(X^{T}X{)}^{-1}{X}^T$. Points with high leverage shrink the raw residual mechanically; standardization corrects for that.

Residuals vs fitted values. Plot $e_i$ on the vertical axis and ${\hat{y}}_i$ on the horizontal. You want a structureless horizontal band centered at zero. Three failure modes appear repeatedly.

HIGH-FREQUENCY: A U-shape or inverted U means the response is non-linear in at least one predictor. A funnel (variance growing with $\hat{y}$) means heteroskedasticity.

Common mistakes

• Reading variance off a QQ plot. The QQ plot tells you about distributional shape, not spread across $\hat{y}$. Use the scale-location plot for variance.
• Treating high leverage as a defect. A point with $h_{ii}=0.30$ but a small residual is fine; it sits at an unusual $X$ but agrees with the fit. The combination of high leverage and large $|r_i|$ is what matters.
• Confusing the added variable plot with the residual-vs-predictor plot. The added variable plot partials out other predictors on both axes. Plotting raw residuals against $X_j$ does not, and gives the wrong diagnostic for functional form in the multiple-regression sense.

Bottom line

• Residuals vs fitted checks the mean structure: a curved smoother flags non-linearity, a funnel flags heteroskedasticity, and a flat band near zero is the target.
• QQ plot of standardized residuals flags non-normal errors: S-shape is heavy tails (kurtosis), banana is skew, off-line endpoints are individual outliers.
• Scale-location plot isolates non-constant variance using $\sqrt{|r_i|}$; a flat red line means $\text{Var}(\epsilon )$ is stable across $\hat{y}$.
• Added variable plot shows the slope of $X_j$ (equal to ${\hat{\beta }}_j$) after partialling out other predictors; flat means $X_j$ adds nothing, curvature means wrong functional form.

Exam shortcut

When you see a funnel in residuals-vs-fitted plus heavy tails on the QQ plot, jump straight to a log transformation of $y$ or a gamma GLM; the same fix kills both problems. For binary or count GLMs, skip residuals-vs-fitted as your first diagnostic and go to marginal model plots, since the binary residual band tells you nothing.

The full lesson (about 2,388 words, 16 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• C8

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