Estimating Player Contribution in Hockey with Regularized Logistic\n Regression
Bibliographic record
Abstract
We present a regularized logistic regression model for evaluating player\ncontributions in hockey. The traditional metric for this purpose is the\nplus-minus statistic, which allocates a single unit of credit (for or against)\nto each player on the ice for a goal. However, plus-minus scores measure only\nthe marginal effect of players, do not account for sample size, and provide a\nvery noisy estimate of performance. We investigate a related regression\nproblem: what does each player on the ice contribute, beyond aggregate team\nperformance and other factors, to the odds that a given goal was scored by\ntheir team? Due to the large-p (number of players) and imbalanced design\nsetting of hockey analysis, a major part of our contribution is a careful\ntreatment of prior shrinkage in model estimation. We showcase two recently\ndeveloped techniques -- for posterior maximization or simulation -- that make\nsuch analysis feasible. Each approach is accompanied with publicly available\nsoftware and we include the simple commands used in our analysis. Our results\nshow that most players do not stand out as measurably strong (positive or\nnegative) contributors. This allows the stars to really shine, reveals diamonds\nin the rough overlooked by earlier analyses, and argues that some of the\nhighest paid players in the league are not making contributions worth their\nexpense.\n
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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 | 0.006 | 0.020 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 0.003 |
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".