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eduKateSG · Why Mathematics? · Adjusted R-Squared: Fit per Degree of Freedom
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 21–26
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 regression model can improve its ordinary R-squared simply by adding predictors? Adjusted R-squared compares residual variation per remaining degree of freedom with total variation per total degree of freedom. It rewards a new term only when the reduction in residual error is large enough to offset the loss of freedom.
This makes adjusted R-squared useful for comparing candidate ordinary least-squares models fitted to the same response observations. It is still an in-sample criterion, not a guarantee of future prediction or a proof of causation.
Did You Know? Adjusted R-squared can fall when ordinary R-squared rises. The new predictor improved training fit, but not enough to justify its complexity under this criterion.
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 21–26 · Study and verify
In ordinary least squares with an intercept, adding a predictor cannot increase residual sum of squares. Ordinary R-squared therefore rises or stays unchanged even when the new term contributes little beyond sample noise.
Penn State gives adjusted R-squared as 1-[(n-1)/(n-p)](1-R-squared), where p counts fitted parameters under its notation. The same expression compares mean residual error with total mean-square variation. Read Penn State STAT 501.
Let n=50, p=4 including the intercept, and ordinary R-squared=0.70. Then adjusted R-squared=1-(49/46)(0.30)=1-0.319565=0.680435, or about 0.680.
| n | p including intercept | R-squared |
|---|---|---|
| 50 | 4 | 0.7 |
The residual denominator n-p shrinks when parameters are added. A more flexible model must reduce SSE enough to overcome that smaller denominator; otherwise residual mean square rises and adjusted R-squared falls.
| Multiplier | Adjusted R-squared |
|---|---|
| 49/46 | 0.68 |
Penn State shows adjusted R-squared equals 1-[(n-1)/SSTO]MSE. On the same response data, maximising adjusted R-squared is therefore equivalent to minimising the model's residual mean square.
Candidate models must use the same response values and observations. Changing missing-data rows, transformations or weights changes SSTO and the reference problem, so the numbers no longer support a simple ranking.
Count estimated coefficients according to the fitted design matrix, usually including the intercept in the n-p denominator. A categorical variable, spline or interaction can consume several parameters.
Adjusted R-squared can be negative when the fitted model's degree-of-freedom-adjusted error exceeds the total variance benchmark. Negative does not mean impossible; it means weak in-sample performance under this comparison.
Ordinary R-squared is a descriptive fraction of sample variation accounted for under the model. Adjusted R-squared modifies that fit measure for parameter count; it does not become an out-of-sample score.
PRESS evaluates leave-one-out prediction errors, whereas adjusted R-squared uses training residual mean square. A model can rank well on one and poorly on the other, especially with influential observations or overfitting.
AIC and BIC penalise maximised likelihood, not R-squared directly. Their penalties and theoretical aims differ, so agreement is reassuring but disagreement is a reason to inspect assumptions and purpose.
Adjusted R-squared can compare models with different term sets on the same data. It does not provide a hypothesis test, confidence level or automatic rule for accepting one model.
Do not drop required main effects merely because adjusted R-squared rises. Preserve scientifically sensible hierarchy and pre-specified controls before using the score to compare optional terms.
A high adjusted R-squared can coexist with unstable individual coefficients. Inspect VIFs, condition indices and coefficient uncertainty when explanation matters.
NIST warns that a high R-squared does not guarantee a good model. Check residual shape, variance, independence, unusual cases and functional form. Read the NIST model-validation guidance.
Record n, p, R-squared, n-p, the penalty multiplier and adjusted R-squared in labelled cells. Recompute from SSE, SSTO and MSE as an independent check.
Use software that states its intercept and uncentred-model conventions. Preserve the same analysis rows across candidates and return R-squared, adjusted R-squared, SSE, MSE and parameter count together.
Adjusted R-squared appears in engineering models, economics, environmental studies, education research and explanatory analytics. The transferable skill is judging whether extra structure earns its place.
Fit one-, two- and three-predictor models to the same dataset. Track ordinary and adjusted R-squared, then explain exactly why the rankings diverge.
Adjusted R-squared is not a probability, a causal score or a validation-set result. The highest value does not prove the model is true, fair, stable or useful outside the sample.
Suppose a three-parameter model has R-squared 0.695 and a four-parameter model has R-squared 0.700 on the same 50 observations. Recalculate both adjusted values rather than choosing the second model from ordinary R-squared alone. Then inspect which term was added, its uncertainty, the residual pattern and PRESS. The arithmetic disciplines the comparison, while the surrounding evidence determines whether the extra term improves the model for its real purpose.
The Mallows Cp guide gives another complexity-adjusted criterion. The PRESS guide adds leave-one-out prediction.
Students should state n and the exact parameter count before calculating. Parents can ask whether candidate models used the same rows and whether predictive validation agrees with the in-sample score.
- Recalculate for R-squared=0.72.
- Explain why adding a predictor can lower adjusted R-squared.
- Describe why changing missing-data rows invalidates a direct comparison.
### Can adjusted R-squared exceed ordinary R-squared? With the usual intercept formula it is no greater.
### Can it be negative? Yes.
### Does the largest value choose the final model? No. It identifies a candidate for further checks.
Use the Mathematics Learning Hub to connect degrees of freedom, variance and model comparison. Build a candidate table with R-squared, adjusted R-squared, PRESS and parameter count.
Extended practice laboratory
Practice case 1: Record n
Setup: Start from the article’s adjusted-R-squared worked example for practice case 1. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Count intercept
Setup: Start from the article’s adjusted-R-squared worked example for practice case 2. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Count slope parameters
Setup: Start from the article’s adjusted-R-squared worked example for practice case 3. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Calculate n minus p
Setup: Start from the article’s adjusted-R-squared worked example for practice case 4. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Record ordinary R squared
Setup: Start from the article’s adjusted-R-squared worked example for practice case 5. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Calculate one minus R squared
Setup: Start from the article’s adjusted-R-squared worked example for practice case 6. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Form penalty multiplier
Setup: Start from the article’s adjusted-R-squared worked example for practice case 7. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Calculate adjusted R squared
Setup: Start from the article’s adjusted-R-squared worked example for practice case 8. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Increase R squared
Setup: Start from the article’s adjusted-R-squared worked example for practice case 9. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Add one predictor
Setup: Start from the article’s adjusted-R-squared worked example for practice case 10. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Add a useless predictor
Setup: Start from the article’s adjusted-R-squared worked example for practice case 11. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Improve SSE enough
Setup: Start from the article’s adjusted-R-squared worked example for practice case 12. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Calculate from SSE
Setup: Start from the article’s adjusted-R-squared worked example for practice case 13. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Calculate from SSTO
Setup: Start from the article’s adjusted-R-squared worked example for practice case 14. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Calculate MSE
Setup: Start from the article’s adjusted-R-squared worked example for practice case 15. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Verify MSE equivalence
Setup: Start from the article’s adjusted-R-squared worked example for practice case 16. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Create a negative value
Setup: Start from the article’s adjusted-R-squared worked example for practice case 17. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Compare same rows
Setup: Start from the article’s adjusted-R-squared worked example for practice case 18. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Change missing-data rows
Setup: Start from the article’s adjusted-R-squared worked example for practice case 19. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Compare PRESS
Setup: Start from the article’s adjusted-R-squared worked example for practice case 20. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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 Mallows Cp
Setup: Start from the article’s adjusted-R-squared worked example for practice case 21. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Inspect VIF
Setup: Start from the article’s adjusted-R-squared worked example for practice case 22. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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: Inspect residuals
Setup: Start from the article’s adjusted-R-squared worked example for practice case 23. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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 causal claim
Setup: Start from the article’s adjusted-R-squared worked example for practice case 24. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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 model-comparison conclusion
Setup: Start from the article’s adjusted-R-squared worked example for practice case 25. Task: Change the named input, recompute all dependent quantities and compare the result with the original.
Check: Check the defining identity, state the data and model conditions, and name one diagnostic or validation step needed before a decision. 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.