Who actually moves the scoreboard? Player value with honest error bars

Every plus-minus argument has the same flaw: the players you share the floor with. A bench guard who rides shotgun with the MVP posts a glowing on/off number; a good starter marooned on a bad roster posts a grim one. This study builds the standard remedy, RAPM-family regularized regression over every five-man lineup. It builds it twice, in R and Python, for 2023-24 and the just-completed 2025-26 season. Then it does the two things public plus-minus work most often skips: it reports uncertainty honestly, and it rebuilds the whole thing from a second, independent unit of observation to see if the answer holds.

Terms used in this analysis

RAPM: Regularized adjusted plus-minus: a ridge regression crediting each player with net points per 100 possessions while adjusting for the other nine players on the floor.
stint: A stretch of game time with no substitution at either end - the unit of observation for the stint-level RAPM model.
plus-minus: Points scored minus points allowed while a player or lineup is on the floor.
ridge: Ridge regression: least squares with an L2 penalty that keeps many-parameter models stable at the cost of a little bias.
shrinkage weight: The fraction of an observed number that survives empirical-Bayes regression: 1 means fully trusted, 0 means replaced by the league mean.
CI: Confidence interval.
Spearman: Rank correlation: correlation of the orderings rather than the raw values.

The model's one free knob, how hard to shrink, is not a taste decision. Cross-validation chooses the ridge penalty (λ = 3200 in both seasons). The figure below shows why that choice is defensible rather than decorative: predictive error falls, bottoms out, and rises again as the penalty grows. The model simply takes the minimum.

The cross-validation curve that picks the ridge penalty. Necessary because shrinkage is the entire difference between this model and raw on/off. This figure is the evidence that the amount of shrinkage was chosen by held-out prediction, not by hand.

Raw on/off versus the regularized estimate for every qualifying player. The before/after of the method: the players pulled hardest toward zero are exactly the low-minute, star-adjacent profiles whose raw numbers are most contaminated: visible teammate-context correction, one dot per player.

What comes out the other side passes the eye test at the top: Jokić (+5.5 per 100), Anunoby, Brunson, George, Gilgeous-Alexander lead 2023-24; Wembanyama (+6.5) and Gilgeous-Alexander lead 2025-26. But the study's real headline is the width of the intervals: the median 95% bootstrap CI spans about ±2.5 points per 100 possessions. One season of lineup data genuinely cannot separate the 5th-best player from the 30th, and a model that pretends otherwise is selling something.

The 2023-24 player values with their bootstrap intervals: the study's most important figure precisely because of the error bars. The ordering is the headline everywhere else; here the honest message is how much the intervals overlap, which is what a front office actually needs to know before acting on a single-season number.

A ranking with intervals still deserves suspicion if it only exists in one model. So the study rebuilds valuation from scratch at a different grain: 69,767 substitution-to-substitution stints parsed from all 1,230 games of play-by-play, with an opponent-adjusted design. Its unpenalized intercept lands at +2.2 points per 100. That is the home-court advantage, estimated as a byproduct rather than assumed, and a built-in sanity check that the design matrix means what it claims. The two models share no code path and no data feed.

Stint-level RAPM against the aggregate lineup model, one dot per player (Spearman 0.96). The convergent-validity figure: two independently built models, different units of observation, different feeds, near-identical player ordering. Agreement this strong is the best public-data evidence that the values reflect the players rather than the plumbing.

Lineups themselves get the descriptive treatment, where the volume filter does the honest work: among five-man units with 1,000+ possessions, the best of 2023-24 was Milwaukee's Lopez–Lillard–Middleton–Antetokounmpo– Beasley at +15.3 per 100 over 1,302 possessions. Every lineup total is gated by exact reconstruction of official team totals. The model is not allowed to describe a season it cannot re-add.

The takeaway for a decision-maker is deliberately modest: with one public season, trust the tier, not the rank; trust a value more when two independent constructions agree on it; and treat any lineup-value claim without an error bar as marketing.

Additional figures

Player values (2025-26)

Shrinkage (2025-26)

Cv curve (2025-26)

Method, code, and verification

The full technical write-up (data provenance, model specification, validation gates, and stated limitations) is the README. The analysis is implemented independently in R and Python and reconciled to numeric tolerance; run_checks.sh replays every gate.