Direct comparison
Adjusted R² vs. R²: When It Diverges
R² always rises when you add predictors. Adjusted R² penalizes that and can fall. When the gap is trivial versus when it changes which model you report.
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How do R², Adjusted R² compare side by side?
The table below compares R², Adjusted R² across 13 procurement-relevant dimensions, from what it measures through same r² (0.50) and n=100, p=30.
Side-by-side comparison
| Dimension | R² | Adjusted R² |
|---|---|---|
| What it measures | Proportion of variance in the outcome explained by this model, in this sample | Same proportion, discounted for how many predictors it took relative to sample size |
| Formula | 1 − (SSE / SST) | 1 − [(1 − R²)(n − 1) / (n − p − 1)] |
| Effect of adding any predictor, useful or not | Always rises or stays exactly flat — mathematically cannot fall | Rises only if the predictor beats what chance alone would add; otherwise falls |
| Range (standard OLS with an intercept) | 0 to 1 | At or below R² — can go negative |
| Comparing models with different predictor counts | Always favors the model with more predictors, even noise ones | Built specifically for this comparison |
| Penalizes model complexity | No | Yes — that is its entire purpose |
| Worked example — base model: n=50, p=3 | 0.400 | 0.361 |
| Worked example — add one weak predictor: n=50, p=4 | 0.410 (rose 0.010) | 0.358 (fell 0.003) |
| Large sample, few predictors: n=600, p=6 | 0.350 | 0.343 — gap of 0.007, barely matters |
| Small sample, many predictors: n=30, p=10 | 0.550 | 0.313 — gap of 0.237, changes the conclusion |
| Same R² (0.50) and n=100, p=1 | 0.500 | 0.495 — gap 0.005 |
| Same R² (0.50) and n=100, p=10 | 0.500 | 0.444 — gap 0.056 |
| Same R² (0.50) and n=100, p=30 | 0.500 | 0.283 — gap 0.217 |
Common questions
Common questions about R² vs Adjusted R²
Should I report both R² and adjusted R² in a paper?
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Yes, once a model has more than a small handful of predictors relative to sample size. Report R² as the plain variance-explained figure and adjusted R² as the complexity-corrected one — a wide gap between them is itself a useful overfitting signal, not a discrepancy to hide.
Can adjusted R² actually be negative?
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Yes. If a model explains less variance than the number of predictors relative to sample size would predict by chance, the formula returns a negative value — it means the model fits worse than simply predicting the outcome’s mean for every observation.
Does a high adjusted R² mean the model is good?
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No — it means the model’s fit is not obviously inflated by predictor count. It says nothing about whether the model is correctly specified, causal, or predictive on new data; those are separate checks.
Is there a fixed rule of thumb for how many predictors is too many?
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No universal cutoff exists, but the gap between R² and adjusted R² widens quickly once predictor count approaches roughly 10–20% of the sample size. The size of that gap in your own output is a more direct signal than any fixed ratio.
