R² and Adjusted R²: Interpreting Explanatory Power Without Confusing It with Prediction
R² describes variance explained in the estimation sample. Adjusted R² accounts for the number of predictors and sample size, but neither proves that a model predicts new observations well.
R² = 1 − SSE/SST and cannot decrease when predictors are added. Adjusted R² may decrease if a new predictor does not offset the complexity penalty. Use these measures for in-sample explanation and cross-validation/PLSpredict for out-of-sample prediction (SmartPLS, n.d.).
Explanation and prediction answer different questions
- R² / adjusted R²
Explained variance in the sample
- f²
The predictor’s unique contribution
- PLSpredict
Errors on held-out observations
- CVPAT
Compare predictive losses
Conceptual diagram. High R² alone does not establish out-of-sample prediction.
Source: Lê Minh · Conceptual illustration, not software output.1. How do R² and adjusted R² differ?
R² is the proportion of outcome variance explained within the sample. Adding a predictor cannot increase SSE in OLS, so R² cannot decrease even if the predictor contributes little (Kutner et al., 2005).
Adjusted R² accounts for n and the number of predictors k. It can decrease when the improvement is insufficient to offset added complexity, making it more suitable for comparing nested models with different numbers of predictors.
| Measure | Penalizes predictor count? | Can decrease when adding a predictor? | Purpose |
|---|---|---|---|
| R² | No | Not in OLS using the same data | In-sample explanation |
| Adjusted R² | Yes | Yes | Compare different predictor counts |
| Q²predict/RMSE | Through holdout evaluation | Yes | Out-of-sample prediction |
2. No R² threshold is appropriate in every field
An appropriate R² depends on noise, measurement and objectives. A moderate value may be useful in human behavior research but low in a controlled physical system.
Weak/moderate/substantial benchmarks are conventions from particular sources. Compare prior research, benchmarks, simpler models and relevance to decisions.
Overfitting, leakage, inappropriate controls or spurious relationships can produce high R². Inspect specification, residuals and out-of-sample prediction.
3. Read the results in SmartPLS
Record each endogenous variable
Each endogenous construct has its own R² and adjusted R².
Inspect the predictor structure
Check incoming predictors and collinearity; do not compare R² across outcomes with different scales.
Inspect f²
Identify each predictor’s contribution to the change in R².
Run PLSpredict
Examine holdout Q²predict and RMSE/MAE.
Compare plausible models
Prioritize theory, adjusted R² and stable prediction rather than the highest R².
4. Interpreting common patterns
| Change | Interpretation |
|---|---|
| R² and adjusted R² increase | The additional explanation offsets the complexity penalty |
| R² increases, adjusted R² decreases | The new predictor adds too little within the sample |
| Both high, prediction poor | Possible overfitting or differences in the new sample |
| Low R², useful prediction | A weak benchmark or hard-to-predict phenomenon; inspect actual errors |
5. Reporting template
Reporting example in English
The predictors explained R² = [ ] of the variance in [Y]; adjusted R² = [ ]. Adding [X] changed R² by [ ] and adjusted R² [increased/decreased], f² = [ ]. In PLSpredict, [RMSE/MAE] was [ ] relative to the benchmark, indicating [level of] out-of-sample predictive performance.
- Report R² separately for each outcome.
- Report adjusted R² when comparing different predictor counts.
- Do not call R² predictive accuracy.
- Avoid unsupported universal cutoffs.
- Also examine f² and PLSpredict.
Conclusion
R² and adjusted R² concern in-sample explanation. Assessing performance on new data requires out-of-sample predictive evaluation.
References
- Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2005). Applied linear statistical models (5th ed.). McGraw-Hill.https://www.mheducation.com/highered/product/applied-linear-statistical-models-kutner.html
- SmartPLS. (n.d.). PLS-SEM glossary.https://www.smartpls.com/documentation/literature/glossary/
- SmartPLS. (n.d.). PLSpredict.https://www.smartpls.com/documentation/algorithms-and-techniques/prediction-and-segmentation/predict/
Methodological note: Adjusted R² is an in-sample adjustment, not cross-validation. It reduces the preference for more predictors but does not eliminate overfitting.