Inspect Residuals Against Fitted Values Before Trusting a Regression

Define the residual being plotted

For ordinary least squares, a raw residual is the observed outcome minus its fitted value. Other models offer different residual definitions, so label the type rather than presenting every diagnostic as an ordinary difference. Use the observations actually included in the fitted model and retain their identifiers for follow-up.

NIST’s model-validation guidance emphasizes that a high R-squared is not enough to establish fit. A residual plot can reveal structure compressed away by a single summary measure.

Look for distinct patterns

Plot residuals against fitted values and relevant predictors. A curved average pattern suggests the mean structure may be inadequate. A widening vertical spread suggests changing residual variation. Separate bands may reflect groups, discrete outcomes or omitted structure rather than a single bad observation.

Use a horizontal zero reference and enough transparency to reveal overlapping points. Do not interpret the absence of an obvious pattern in a tiny sample as proof that assumptions hold. Where data are clustered or ordered in time, inspect that structure directly as well.

Read a hypothetical pattern

Imagine fitted values 1, 2, 3, 4 and 5 with illustrative residuals 2, −1, −2, −1 and 2. Their sum is zero, yet the pattern is curved. A zero average residual therefore does not establish that the model captures the relationship across the fitted range.

If instead residual magnitudes are usually around 1 for low fitted values and around 5 for high fitted values, the concern is changing spread. Adding a quadratic mean term may not address that variance pattern. Distinguish the symptom before choosing a remedy.

Investigate and document the response

Check source values, units, transformations and model coding before changing the scientific model. If a nonlinear term is justified, explain its form and avoid choosing it solely to remove every visible fluctuation. If uncertainty estimation changes, state whether the fitted mean itself changed.

Retain the original and revised diagnostic plots with the model versions. Report important remaining patterns rather than presenting only the most attractive figure. A residual audit is a structured investigation, not a rule that all points outside an arbitrary band must be deleted. The final model should remain connected to the research question, with diagnostics used to assess how well its assumptions approximate the observed data.

Source verification: 6 November 2026. Examples are hypothetical.