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eduKateSG · Why Mathematics? · Partial Residual Plots: Curvature After Adjustment
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 5–9
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 10–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–23
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when a straight-line regression may be hiding curvature? A partial residual plot keeps the original predictor on the horizontal axis and adds its fitted linear component back to the full-model residual. The result helps a modeller inspect the shape of one predictor's relationship with the response while the other predictors remain in the model.
This graph is a diagnostic, not a proof that a particular transformation is correct. It must be read with collinearity, leverage, residual and subject-matter checks.
Did You Know? The vertical coordinate can be written as residual + fitted coefficient × predictor. That small piece of algebra restores the target predictor's fitted contribution while leaving the other fitted contributions removed.
Full chapter index
Chapters 1–4 · Understand the question
Chapters 5–9 · Reproduce the example
Chapters 10–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–23 · Study and verify
A raw response-versus-predictor scatter plot mixes the target predictor with every measured variable correlated with it. A partial residual plot asks a narrower question: after the fitted effects of the other predictors have been accounted for, does the remaining relationship with the target predictor still look linear?
NIST defines the plotted vertical value as Res + beta-hat_i X_i and plots it against X_i. NIST also warns that the display can fail when X_i is highly correlated with other predictors. Read the NIST partial residual plot reference.
Start with the full-model residual e_i = y_i – y-hat_i. Add back only the fitted component beta-hat_j x_ij for the predictor under study. Plot c_i = e_i + beta-hat_j x_ij against x_ij. If the linear term is adequate, the cloud should be compatible with a straight component plus random scatter.
| Residual | Coefficient | Predictor |
|---|---|---|
| -1.2 | 0.8 | 4 |
Suppose a case has residual -1.2, target predictor x=4 and fitted coefficient 0.8. Its component is 0.8×4=3.2. Its partial residual is -1.2+3.2=2.0, so the plotted point is (4,2.0).
| Component | Partial residual | Plotted point |
|---|---|---|
| 3.2 | 2.0 | (4, 2.0) |
The horizontal axis remains the target predictor in its original units. That makes bends, thresholds and transformations easier to discuss than in a partial regression plot, whose horizontal axis is a residualised predictor.
A LOWESS or other restrained smoother can reveal systematic curvature, but its span controls how wiggly it appears. Compare the smoother with the fitted component line and with plausible scientific mechanisms rather than treating every bend as real.
When the target predictor overlaps strongly with other predictors, the partial residual plot can understate the true uncertainty and distort the apparent strength of the relationship. Pair it with variance inflation factors, condition indices and the observed predictor range.
A point far out on the x-axis may control the apparent bend or slope. Label influential cases and compare Cook's distance, DFFITS and DFBETAS before proposing a new functional form.
A fan-shaped vertical spread may indicate changing variance rather than mean curvature. Inspect ordinary residual-versus-fitted plots and consider a variance model or transformation supported by the measurement process.
If the effect of x differs by another predictor, one overall partial residual plot can average across distinct relationships. Use interaction-specific plots or conditional predictions when the model contains interactions.
First identify a reproducible shape. Then propose a defensible term such as log(x), x squared or a spline. Refit the model, compare diagnostics and out-of-sample performance, and preserve the original scale in the explanation.
CHAPTER 12 OF 23 · Check assumptions
12. Why visual evidence needs numerical checks
A plot is sensitive to axis limits, smoothing and sparse regions. Report the model formula, data range, smoother settings and uncertainty. A visual pattern becomes evidence only when it survives reasonable display and modelling choices.
Fit the full regression, save fitted values and residuals, copy the target predictor and coefficient, calculate e_i + beta-hat_j x_ij row by row, then create a scatter plot. Audit one row by hand before trusting the column.
Use a tested diagnostic routine or compute the coordinates explicitly. Preserve row IDs, preprocessing, missing-value rules and coefficient labels. Confirm that the coordinates match the model matrix actually fitted.
Partial residual plots support dose-response modelling, engineering calibration, environmental trend analysis, economics, education research and interpretable machine learning. The shared skill is diagnosing shape without confusing a model display with causation.
Model plant growth using light, water and temperature. Inspect the partial residual plot for light, then compare a linear term with a biologically plausible saturating or curved term. Explain which observations support the change.
It cannot establish causation, repair measurement error, guarantee a transformation, or substitute for external validation. It describes the fitted sample under the chosen adjustment set.
The vertical coordinate is not the raw response. A curved smoother is not automatically a better model. A visually straight display does not prove assumptions such as independence or constant variance.
CHAPTER 19 OF 23 · Explore applications
19. Connection to recent mathematics articles
The partial regression plots guide explains the different residual-on-residual construction. The condition indices guide helps diagnose the collinearity warning.
Students should show the coordinate calculation for at least one case and label both axes precisely. Parents can ask what other predictors were adjusted for, whether influential cases drive the shape and how a proposed curve was checked.
- Recalculate the worked point when the residual is 0.4.
- Explain why the horizontal axis helps with choosing log(x).
- Describe how high collinearity could make the display misleading.
### Is a partial residual plot the same as a partial regression plot? No. Their axes and main diagnostic purposes differ.
### Does curvature prove a quadratic term? No. It prompts a model comparison.
### Should every predictor receive a plot? Often yes, with priority given to terms central to the question.
Use the Mathematics Learning Hub to connect residual algebra with functions, graphs and statistical reasoning. Build both partial residual and partial regression plots for the same predictor and explain their different questions.
Extended practice laboratory
Practice case 1: Calculate one component
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 1. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Add one residual
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 2. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Change the coefficient
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 3. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Reverse the residual sign
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 4. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Move x within range
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 5. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Compare raw scatter
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 6. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Add a LOWESS curve
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 7. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Change smoothing span
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 8. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Inspect an edge point
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 9. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Label a leverage case
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 10. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Compare Cook distance
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 11. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Compare DFBETAS
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 12. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Increase collinearity
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 13. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Centre the predictor
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 14. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Apply log x
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 15. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Add x squared
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 16. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Add an interaction
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 17. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Inspect a fan shape
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 18. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Audit one row
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 19. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Check missing values
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 20. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Compare partial regression
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 21. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Restrict to observed range
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 22. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Write a transformation case
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 23. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Reject automatic deletion
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 24. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write a shape conclusion
Setup: Use the article’s partial-residual coordinate worked example as the starting record for case 25. Task: Change the input named in this case, recalculate every dependent quantity and state the comparison in one precise sentence.
Check: Check the defining identity, preserve the declared model and explain what additional graph, diagnostic or validation evidence is needed. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.