R-squared Measures for Multilevel Models with Three or More Levels
Bibliographic record
Abstract
Applications of multilevel models (MLMs) with three or more levels have increased alongside expanding software capability and dataset availability. Though researchers often express interest in R-squared measures as effect sizes for MLMs, R-squareds previously proposed for MLMs with three or more levels cover a limited subset of choices for how to quantify explained variance in these models. Additionally, analytic relationships between total and level-specific versions of MLM R-squared measures have not been clarified, despite such relationships becoming increasingly important to understand when there are more levels. Furthermore, the impact of predictor centering strategy on R-squared computation and interpretation has not been explicated for MLMs with any number of levels. To fill these gaps, we extend the Rights and Sterba two-level MLM R-squared framework to three or more levels, providing a general set of measures that includes preexisting three-level measures as special cases and yields additional results not obtainable from existing measures. We mathematically and pedagogically relate total and level-specific R-squareds, and show how all total and level-specific R-squared measures in our framework can be computed under any centering strategy. Finally, we provide and empirically demonstrate software (available in the r2mlm R package) to compute measures and graphically depict results.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.035 | 0.203 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.004 |
| Bibliometrics | 0.005 | 0.008 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.039 | 0.008 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".