Use residuals to decide whether your model is useful

Define what a successful model would do

A model can describe a pattern, estimate a quantity or support a decision. State which purpose matters in your investigation and how you will judge success. A small average error may be useful for one application and unacceptable for another. Define the variable range and conditions in which you intend to use the model.

Before fitting, inspect the data and document their source, units and exclusions. Model evaluation cannot repair a dataset whose rows refer to incompatible measurements or whose missing values were silently replaced with zeros.

Calculate residuals consistently

A residual is the observed response minus the response predicted by the fitted model. In an illustrative example, an observed value of 42 and a prediction of 39 produce a residual of +3. The model underestimates this observation by three units. A prediction of 45 would give -3 and an overestimate.

Show one representative calculation and label the residual axis with appropriate units. Keep the sign convention consistent. Residuals add information beyond the original scatter plot because they show the remaining discrepancy after the model's predicted pattern is removed.

Read patterns before selecting another equation

Plot residuals against the explanatory variable or fitted values. A curved pattern may indicate that a straight line misses the relationship's shape. Larger spread at one end suggests that prediction error varies across the range. Clusters may reflect a condition such as weekday versus weekend measurements.

Describe what the actual plot shows, then investigate the context. An unusual residual could reveal a transcription mistake, a rare but genuine observation or a missing explanatory factor. Do not remove it solely because the model looks better without it.

Compare models with a reason

Choose a competing model because it addresses a pattern or contextual issue you identified. For example, if residuals from a linear fit form a clear curve, a suitable curved model may be worth testing. Explain its form and parameters, and check whether its behavior makes sense in the original context.

A more flexible model can fit existing observations closely while making implausible predictions between or beyond them. Compare residual behavior, understandable mathematics and prediction usefulness together. An improved fit statistic alone does not settle the choice.

Test predictions where possible

If sufficient data are available, use a clearly defined portion for fitting and another portion to inspect predictions. Explain how you chose the split and whether observations are independent. For time-ordered data, a random split may conceal changing conditions; predicting a later period can test a different question.

• Keep test observations out of the fitting step.
• Compare errors using the same units and data for both models.
• Explain why the error size matters for the intended use.
• Restrict extrapolation unless the model and context justify it.

Answer at the strength of the evidence

State which model you prefer, the evidence for that choice and where it remains unreliable. Distinguish descriptive association from a causal explanation. If a missing condition explains large errors, propose a realistic refinement rather than claiming the model is universally accurate. Include enough calculation and software information for the reader to understand how you reached the judgment.