Reading Regression Diagnostics: R-squared, VIF and Cook's Distance

A regression's coefficients are only as trustworthy as its diagnostics. Here's how to read the handful of numbers and plots that tell you whether to believe the model.

It's easy to fit a regression and read off the coefficients. The harder, more valuable habit is checking whether the model is sound. Four diagnostics carry most of the load.

R-squared (and adjusted)

R-squared is the share of the response's variation the model explains. Higher looks better, but it always rises when you add predictors — even useless ones. Adjusted R-squared penalizes that, so it's the one to compare models with. And a high R-squared never rescues a model whose assumptions are violated.

VIF: multicollinearity

When predictors are themselves correlated, the model can't tell their effects apart — coefficients become unstable and hard to interpret. The variance inflation factor measures this. A common rule of thumb: VIF above ~5 is worth a look, above ~10 is a problem. The fix is usually to drop or combine the redundant predictors.

Linear regression report in Stratum: coefficients table with the VIF column and R-squared — The coefficients table’s VIF column — on the diamonds model, carat and the size columns x, y, z all run past 20 (near-collinear); depth and table stay near 1.

Leverage and Cook's distance: influential points

Leverage flags observations with unusual predictor values; Cook's distance measures how much the whole fit would change if you dropped a point. A single high-Cook's-distance observation can swing your coefficients, so it deserves a look — is it an error, or a real but extreme case?

Influence (leverage) plot in Stratum, markers sized by Cook's distance — The Influence plot: leverage vs. studentized residual, marker size = Cook’s distance. One diamond — its width mis-entered as 59 mm — dwarfs every other point.

Residual plots

Finally, plot the residuals. A random, patternless cloud is good. A curve means the relationship isn't linear; a fan shape means the variance isn't constant; a drifting Q-Q plot of the residuals means non-normal errors. The plots catch what the summary numbers can't.

Residuals versus fitted plot in Stratum showing a fan shape — Residuals vs. fitted — the fan (variance grows with price) flags non-constant variance; the fix is to model log-price.

The order that works. Read adjusted R-squared for fit, VIF for tangled predictors, Cook's distance for points that dominate, then the residual plots for assumption checks. If all four look reasonable, trust the coefficients.

Put it into practice with How to Run a Linear Regression.

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Frequently asked questions

What is a good R-squared value?

It depends on the field — physical processes can hit 0.9+, while human-behavior data may be useful at 0.3. Compare models with adjusted R-squared, and never trust R-squared alone if the diagnostics fail.

What does VIF tell me?

The variance inflation factor measures multicollinearity — how much a predictor overlaps with the others. Above ~5 warrants a look; above ~10 usually means you should drop or combine predictors.

What is Cook's distance used for?

It measures how much the regression would change if a single observation were removed — a way to find influential points that may be dominating your coefficients.

Why do I need to look at residual plots?

Because the summary numbers can look fine while assumptions fail. Residual plots reveal non-linearity, non-constant variance, and non-normal errors that R-squared hides.