A residual is the observed response minus its fitted value: . Residuals are what the model leaves behind, so their structure is evidence about what the model missed.
Residuals versus fitted values
Plot against . Curvature suggests a missing nonlinear term. A funnel shape suggests changing error variance. Separate bands can reveal an omitted group variable.
Order and quantile plots
Plot residuals in time or collection order to find drift, cycles, or dependence. A normal quantile plot checks whether the residual distribution is compatible with normal-based intervals and tests; departures in the tails often matter most.
Unusual observations
Large residuals indicate responses poorly predicted by the model. High-leverage points have unusual predictor values. Influential points materially change the fit when included. These are different properties and should be investigated separately.
Connect each pattern to a violated claim
Residual analysis is strongest when each plot is tied to a model assumption:
| Pattern | Likely issue | Possible next step |
|---|---|---|
| U-shaped mean | missing curvature | add a justified nonlinear term |
| fan-shaped spread | nonconstant variance | transform, weight, or model variance |
| runs over time | dependent errors | model temporal structure |
| isolated large residual | unusual response | verify and assess influence |
Standardized residuals and leverage
Raw residuals have unequal variances when leverage differs. Internally studentized residuals divide by an estimated standard deviation involving . Cook’s distance combines residual size and leverage to approximate how much the fitted coefficients change when an observation is removed.
Check your understanding
Why might a point with a modest residual still be influential?
Show the reasoning
If its predictor values give it high leverage, it can pull the fitted model toward itself, making its own residual look modest while substantially changing coefficients and other predictions.