Pattern to interpretation
| Pattern | Possible explanation | What to check next |
|---|---|---|
| Random band around zero | Mean structure and variance look broadly compatible | Check scale, dependence, and other diagnostics. |
| Smooth curve | Missing nonlinear term, transformation, or interaction | Plot partial relationships; revise the mean structure. |
| Funnel | Nonconstant residual variance | Compare variance-aware uncertainty or modeling; see homoscedasticity and heteroscedasticity. |
| Separate clusters | Omitted group, batch, hierarchy, or dependence | Identify design structure before adding ad hoc terms. |
| Vertical stripes | Discrete fitted values or grouped predictors | Decide whether the pattern follows the design. |
| One remote point | Large residual, leverage, influence, or data problem | Verify the record and compare fitted results with transparent sensitivity checks. |
Do not treat the plot as a personality test for models. A visible pattern is evidence about a specific assumption, not an automatic repair instruction.
Outlier, leverage, and influence
- Residual outlier: the observed response is far from its fitted value.
- High leverage: the predictor combination is far from the center of the design space.
- Influential observation: deleting or downweighting the case would materially change fitted results.
A point can have high leverage but sit close to the fitted line, producing a small residual. A point with a large residual near the center of the predictor space may have limited leverage. Influence combines residual size with leverage and the model’s geometry.
Cook’s distance summarizes how much the fitted values change when an observation is omitted.1 It is a screening diagnostic, not evidence that a case is erroneous. Verify data entry, sampling, model form, and substantive legitimacy before changing the analysis.
Cook’s distance has competing rules
The primary Cook paper defines the diagnostic but does not create one universal deletion threshold.1 Four verifiable operational treatments illustrate why the category matters:
| Value or rule | Source/context | Category |
|---|---|---|
| D > .5 deserves attention; D > 1 may be influential | Penn State regression diagnostics guidance2 | RULE OF THUMB |
| 4/n | UCLA OARC calls this “the convention cut-off point”4 | RULE OF THUMB |
| 4/(n−k−1) | Threshold Type 2 in the olsrr Cook’s D plot5 | SOFTWARE DEFAULT |
| 3 × mean(D) reference line | MATLAB display and olsrr Threshold Type 53 5 | SOFTWARE DEFAULT |
The two fraction rules are not interchangeable. 4/n depends only on sample size; 4/(n−k−1) uses residual degrees of freedom and rises as predictors are added. At n = 50, the rules are 0.080 and 0.083 with one predictor, but 0.080 and 0.103 with ten predictors. The chosen screen can therefore change which observations are flagged.
UCLA documents 4/n as an instructional heuristic, and olsrr documents the behavior of one software implementation. Neither source verifies the original methodological attribution commonly attached to the formula. ResultAtlas therefore prints the operational rules and their categories without claiming that Bollen and Jackman proposed 4/n or that Fox proposed 4/(n−k−1). A software line describes what a tool draws; it is not a deletion law.
Use sensitivity analysis, not automatic deletion
When a case is influential:
- verify the observation and units;
- identify why it has leverage or a large residual;
- check whether the model form is defensible in that region;
- compare estimates and conclusions with and without the case transparently;
- consider a method suited to the data-generating process;
- report the reason for any exclusion independently of the diagnostic value.
A high R-squared does not erase residual structure. VIF and Durbin–Watson diagnostics address different problems, while the QQ plot focuses on residual distribution shape.
A defensible reporting sentence
“Residuals showed a fan-shaped increase in spread and one high-leverage case. Cook’s distance was used as a screening diagnostic under the stated rule; estimates were reported with a sensitivity analysis rather than deleting the case automatically.”