APBRmetrics

We generate point estimates for each player. BPM, RAPM, Prior-informed RAPM, all generate a single value for NBA players.

How well do those predict lineup performance? Specifically, is there a bias when the lineup prediction is far away from 0? If the estimates predict a lineup to be +12 or -12, how well do those lineups, as a whole, match up with the observations? I believe I read somewhere there is bias, but I cannot recall details. I'm interested in whether the various types of metrics show a consistent bias one way or the other.

Relatedly--if there is a bias, how should we account for it? A player's point estimate should be based on the mean lineup of which that player is a part. If there is a downward bias (whole less than sum of parts) and a player's mean lineup has a +8 projection, how should that be accounted for?

Mean lineup by minutes will be a super-dink. Mean lineups by performance is likely to be a super-dink.


Directional bias on offensive efficiency is likely guided by sum of usage, for at least for the tail not summing to 100. Opposite tail for high sum of usage might have an upward bias on average but maybe not in cases of poor fit / shot competition.

In-sample, it looks like this

Seems fairly linear, with some noise. If I restrict to points that have larger sample size, R^2 increases. For the picture you see, it's 0.63
Could be a different story for out-of-sample, of course
My hunch is that's it's still linear, as long as you account for rubber band. If you don't, then it'll look like great lineups are underperforming expectations

Can you plot the one-to-one line on that chart? It appears the positive relationship is a lot less than one to one. Which is kind of what I was alluding to... If you add one point in expected you get 0.5 points in actual observed?

Which flavor of RAPM did you plot?

Note that the shallow slope may just be because of the RAPM shrinkage, and this is the best fit out of sample that can be achieved.

Also, are the estimates centered here? Looks like they may not be.

To clarify: my null hypothesis would be that there are diminishing returns at the lineup level. Certain skills are not additive, offensive creation on-ball being most notable. LeBron + Wade is less than the sum of their parts.

Note that this could complicate the elastic band effect being observed... If the lineup that is ahead by 10 points is disproportionately a sample of lineups that are also dealing with diminishing returns, that could overstate the rubber band effect.

The lower 17 in 'expected' are above that in 'actual'.
Only one of the top 7 'expected' exceed that value in 'actual'.

I'd prefer to focus on net points added to capture offense and defense. Then you could compare to cumulative lineup salary and dig for most efficient salary design for getting net results and look at how much, if any, was from synergy.

I made the last image in a bit of a rush
Here's a better picture that's not just better centered, but also shows sample size

These are low end efficiencies, barely a point per 100

Also previous graph the x axis was off by .05, low for all. Hence the apparent overachievement.

Interesting, Jerry! Informative chart.

Could you post predicted margin vs. actual margin, rather than just offensive ratings?

Do you happen to have BPM sums (predicted margin from BPM) for the same lineups available? BPM has no shrinkage/regression to the mean so I wonder how that would look.

@v-zero @gnomp

I wanted to refresh this question once again.

Conceptually, I continue to feel that diminishing returns from having a high "expected margin" because of the sum of RAPM point estimates for players on one team vs. the other could very much be collinear/conflated with the measured rubber band effect for having a significant lead. I believe that this "diminishing returns" effect is such that, in general, a player will have more apparent value on average teams/lineups than on elite teams/lineups. Relatedly, elite teams/lineups are ahead most of the time.

I suspect the measured rubber band effect is likely measuring 2 effects, added together, rather than 1.

I don't observe diminishing returns when I project lineup performance from my player impact model, but then my model has been built with accurate and well-calibrated prediction as its central goal, whereas RAPM has not.

Rubber banding exists, absolutely. I observe it in my in-play model. I don't account for it in my player impact metric because, well because I want it to be there. In 95% of cases a player's role, when they play, who they play with, the context in which they are on the floor, changes little from game to game to game to game. That being the case I just don't find any edge in trying to account for something, only to need to bake it back into the numbers later with a kludgy approximation. I also don't think that the rubber band is the same for every guy in the league, some players will relax with a lead more than others.

A point of theory regarding the lack of apparent diminishing returns. One might expect an above average player moving from an average to an above-average lineup to suffer statistically, from an individual perspective, as a consequence of there being but one ball to share. However, such effects will be offset by the fact that the underlying points per possession function for a lineup exhibits increasing returns. (This is to say that if you increase the propensities of all its arguments, i.e. to not turn the ball over, score, and gather offensive rebounds, by X%, then the points per possession increases by more than X%.)

I suspect that the apparent linearity of the above plot reflects these offsetting effects.

Celtics '24-25 biggest lineup, 356 minute sample:

Actual -0.4 / 100p
"Expected" by BPM +14.1
"Expected" by XRAPM +11.1

Celtics 2nd biggest lineup, 268 minute sample:

Actual +18.2 / 100p
"Expected" by BPM +10.6
"Expected" by XRAPM +9.8



Thunder biggest lineup, 316 minute sample:

Actual +16.4 / 100p
"Expected" by BPM +19.9
"Expected" by XRAPM +12.5

Sum of individual metrics off by huge amounts and moderate amounts here.


Cavs biggest lineup, 243 minute sample:

Actual +11.7 / 100p
"Expected" by BPM +16.9
"Expected" by XRAPM +9.1


Rockets biggest lineup, 451 minute sample:

Actual +4.1 / 100p
"Expected" by BPM +3.7
"Expected" by XRAPM +4.2

Rockets 2nd biggest lineup, 322 minute sample:

Actual -10.8 / 100p
"Expected" by BPM +8.5
"Expected" by XRAPM +11.6

6 cases, metrics predicted closely once, moderately off 2-3 times, way off 2-3 times.

Argument for linear relationship not supported at the apex this season.


Above graphs with 24-27 data points. Was it the biggest or something else? A 250 - 280 minute cutoff? Still a small sample. 100 minute minimum would give 84 data points in '23-24. 150 minutes, 47.

Universe of lineups is probably 15-18,000.
Unless there is data below or way below 250 minutes, the sample is small and does not provide a broader / general answer. Lineups on average may have a linear relationship (as expected) but we want reliability for specific lineups.

Net Points remains preferable to me.


Is lineup performance in last 20-40 games for bigger or biggest minute lineups a better predictor of future performance in next 20-40 games than sum of individual metrics? That would be an important test.

For these 6 lineups, lineup performance thru 1/31 predicted general performance direction from 2/1 in 5 cases. The miss had a quite small sample after 1/31.

I don't have the metric splits by date to see and directly compare their late season predictive power but they had 2.5 bad misses for the season.

Prediction is usually not near exact by either approach, though better by prior performance. An indication of good, mid or bad is useful though. Maybe some blend of metrics and actual to date could be somewhat better in a larger sample. In these 6 cases, a blend would have helped modestly in 2 cases, not in 4 cases.

Working with +/- adjusted for lineup opponent strength and sample size would be a better way to go than raw +/- of course. Likewise fully adjusted opponent lineup BPM is likely to be better than raw.

They were pretty close in these cases but RAPM was slightly closer to actual net points than BPM.


A sim using season average data is starting with a bag of stats that essentially
are a point estimate of a player when taken whole. With adjustments hopefully closer prediction of actual in lineup.


Pair RAPM might help. Someone with that level of data could check.

The metrics got basic direction right 4 times.