Model Fit and Diagnostic Tests

A linear model that fits well in-sample can still be wrong. Diagnostics are how you separate a usable fit from a regression that violates its own assumptions and lies about uncertainty.

Ordinary least squares (OLS) regression rests on four assumptions about the error term $\varepsilon_i$:

$$\varepsilon_i \stackrel{iid}{\sim} N(0, \sigma^2), \qquad E[Y \mid X] = X\beta$$

KEY: The mnemonic LINE ranks the assumptions by how badly violations hurt you. Linearity bias kills the coefficients themselves. Independence and Equal variance distort standard errors. Normality matters mainly for small-sample inference; the central limit theorem rescues large-sample $t$ and $F$ tests.

Four standard plots interrogate the four assumptions. Each one targets a specific failure mode.

A clean residuals vs. fitted plot shows a roughly horizontal band centered on zero with no funnel and no curvature. A funnel that opens to the right is the classic heteroscedasticity signature. A clear U-shape signals the linear functional form is wrong; consider a transformation or a polynomial term.

Common mistakes

• Reading a Q-Q plot at the tails only. Two or three points off the line at the extremes is expected. Look for systematic curvature across the bulk of the points before concluding non-normality.
• Confusing high leverage with influential. Leverage measures position in X-space only. An observation can sit far from the centroid yet land exactly on the regression line, contributing nothing influential. Influence requires both unusual X and a large residual.
• Using raw $R^2$ to compare models with different $p$. $R^2$ only goes up when predictors are added. Use adjusted $R^2$, AIC, BIC, or cross-validated error instead.

Bottom line

• OLS assumes LINE: Linearity, Independence, Normality of errors, Equal (constant) variance. Violations bias either the coefficients, the standard errors, or both.
• Residuals vs. fitted checks linearity and homoscedasticity; Q-Q plot checks normality; residuals vs. order or Durbin-Watson checks independence. One plot per assumption.
• Leverage $h_{ii} > 2(p+1)/n$ is high; Cook's distance $D_i > 1$ is influential; $|\text{studentized residual}| > 2$ is an outlier. Memorize these cutoffs.
• VIF > 10 signals serious multicollinearity; VIF between 5 and 10 is a warning. Inflates standard errors, not coefficients in expectation.

Exam shortcut

If the question gives you $n$, $p$, and a single leverage or Cook's value, plug straight into the cutoffs: $2(p+1)/n$ for leverage and 1 for Cook's. When a plot description includes the word "funnel," "fan," or "megaphone," the answer is heteroscedasticity.

The full lesson (about 4,463 words, 30 min read) adds 5 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 2d

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