APBRmetrics

Brian M · Joined: 25 Nov 2006 Posts: 40

I've been thinking about adjusted plus/minus lately. The big appeal to adjusted +/- is that it gives you an idea of how effective a player has been once you control for outside influences (a player's sub, the lineups he typically plays in, etc) that impinge on the unadjusted +/-.

However, it occurs to me that there is a limit to the extent to which adjusted +/- is really context free. Whenever we are able to control for factor X in a player's raw +/-, we are able to control for it because we can estimate the effect of X by comparing data from when X is present to data when X is absent. From this it follows that we can only control for those factors that are alternatively present and absent over the course of a season.

Now, there may be factors that have non-trivial effects on a player's effectiveness that are never really "off" over the course of a season. The most obvious example is the effect of a coach (assuming he is not fired midseason) and whatever offensive and defensive systems he favors.

There is also the possibility that roster effects like this may exist. For instance, suppose for the sake of argument that Steve Nash's impact on the effectiveness of his team's play is greater when he happens to be playing with lithe, athletic frontcourt players rather than lumbering, bruising frontcourt players. (One could also imagine guards that might be more effective with the bruisers.) If Nash were on a team where some of the regular frontcourt players were lumbering and others were lithe, we would be able to control for this effect to some extent. But if Nash were on a team where all his frontcourt players were of one type or the other, we would not.

More generally, whenever a player's effectiveness might be affected by the nature of the players around him, our ability to control for these effects are limited by the range of players that are actually on his team. So for instance, although we can control for the effects of the other Suns players on Nash's +/-, there is still a sense in which Nash's adjusted +/- is specific to the roster he happens to be a part of. Again, for the sake of argument, assume Nash is better off playing with lithe, athletic frontcourt players. Then put Nash on a team of slow bruisers and, all else being equal, his adjusted +/- will be worse than it is on the Suns.

So, it seems that adjusted +/- can never really give us a completely context-free sense of how effective a player is. Rather, they can give us a sense of how effective he is in a context where some set of factors (e.g. coaching, perhaps type of teammate, etc.) is held constant.

Perhaps that's all old news, in which case I would appreciate references to older discussions on the topic. Or perhaps my arguments are flawed or need qualifications in which case comments are welcome.

Dan Rosenbaum · Posted: Wed Apr 04, 2007 12:24 pm Post subject:

I certainly don't disagree with these observations, but we could say the same things about box score stats. We likely also could say the same thing of subjective evaluations. What I think this says more than anything is that extrapolating from the past to predict the future get more difficult as the context changes more signficantly.

DLew · Joined: 13 Nov 2006 Posts: 224

I view adjusted plus-minus statistics as the best measure we have of how good a player was, adjusted for the factors that we can adjust for (teammates, opponents, homecourt advantage). Now, sometime you don't want to know how good a player was, but instead how good he will be. If you want to know how good someone will be it makes sense to me to use how good they were as a starting point and then make adjustments to that based on what factors changed (age, teammates, coach, etc.).

Adjusted plus-minus tells you some things about a player, and it can serve as a useful starting point for determining other stuff that you want to know.

Mark · Joined: 20 Aug 2005 Posts: 807

Adjusted +/- does seem a valuable companion to the set of boxscore based information. Adjusted 4 factor analysis would seem to extend the usefulness. Is there any reason to believe or suspect that Dallas and Winston/Sagarin take this path and consider it a key part of their self-perceived edge on the competition?

Has anyone taken league distributions of pts, minutes, salaries and compared (or plotted) them with counterpart PER and adjusted +/- distributions?

If salary were based heavily or soley on adjusted +/- would they become more unequal than today? Looking beyond rookie contracts, on average do stars return surplus value (according to adjusted +/-) over cost or proportional? For what it is worth the top 25 highest paid guys playing average +5 to 6.

After the top 1/3rd of the talent pool does the question shift from returning positive value to minimizing any harm? Does such a change make a difference or is it just word substitution?

After your top 4 differencemakers should guys close to neutral on 4 factors for position be bid up instead of shopping for rotation guys with 1 strength (and a salary based on the strength) and then trying to offset any weaknesses? Do the better teams do the former more often than others do? Does Dallas? I guess it depends on how balanced and strong your top 4 are on the 4 factors. I can see adding a specialist to nudge team stats on a 4 factor needing assistance or to maintain a style of play that depends on a 4 factor strength from the bench like with the starter. But if the top 4 is strong and balanced, maybe neutral is pretty good from there (and possibly cheaper).

Brian M · Joined: 25 Nov 2006 Posts: 40

Thanks for the responses. I have another question: from a purely mathematical standpoint, is it possible that all the players on a given team might have a negative (or positive) adjusted plus/minus? Or is it the case that even with adjustments from the regression, some constraints necessitate that at least some players will show up with a positive (or negative) adjusted +/- to counterbalance that of their teammates? (I would imagine the answer is that it is indeed possible for an entire roster to be positive or negative on adjusted +/-.)

DLew · Joined: 13 Nov 2006 Posts: 224

It is definitely possible for an entire team to be negative or positive if the team was that bad/good and all the players on the team were of similar quality. It doesn't happen in real life though, because every team bad team has an above average player and every good team has a below average one.

gabefarkas · Joined: 31 Dec 2004 Posts: 1313 Location: Durham, NC

HoopStudies · Posted: Fri Apr 13, 2007 12:51 pm Post subject:

Mark · Joined: 20 Aug 2005 Posts: 807

Gabe, what I was thinking in general when I said "adjusted 4 factor analysis" was to use exactly the same methodology used for adjusted team +/- as a whole- the main adjustment being the presence of the other 9 players on the court who might be above average, near average or below average on each of the 4 factors on offensive side of the ball or defensive which affects the floor context and should affect the evaluation of the players 4 factors "results".

Unless anyone sees a reason that it is not appropriate or feasible. I would think it is, as team +/- is essentially the weighted product of the 4 factors after all. Turnovers and offensive rebounding seem straightforward. FT/FG might seem mainly individual but probably is affected by good passers, outsider shooters and maybe post scorers. If there are postive and negative impacts of teammates on others FG%s then the standard adjusted method should find it just as it finds impacts at the team +/- rollup level.

As for impact, the top 25 paid players averaging a +5-6 per 100 possessions impact on adjusted team +/- does seem pretty big and differencemaking. I asked a question about value and later tacked on a finding about impact but rather than saying I underestimated impact (though I started with a somewhat skeptical tone) I think it would be more accurate to say that I havent fully processed the surplus or proportionate value to cost argument. But it could be done if you found the average salary for a neutral adjusted +/- player (something that seems very worth doiing and different than "replacement level player" or average or median paid player) and plotted average adjusted + /- vs average salary.

In relation to the scale issue Dean is right that the top players will look more impressive measuring from the very top to very bottom rather than just to the neutral on team +/- player. who (just guessing here) may indeed be an above the median performer.

Mark · Joined: 20 Aug 2005 Posts: 807

In the frontiers of apbrmetrics thread I sketched out that you could try to split and find adjusted scores for players on the 4 factors offensive and defensive for them personally as well as other team plays while they were on the court where they werent not directly the leading actor but may have had a secondary role and deserve some of that credit.

But there would be methodological hurdles to overcome on the local splits.Scoring local results 100% based on boxscore matchup might be a simplistic starting point but you could do better. Ideally you'd want to go off videotape. Short of that you'd estimate off an intelligent matchup system from play by play logs as 82games does or could estimate using Mike G's starter/sub ratio based on minutes played method. And instead of assuming the local result is 100% based on the counterpart stat matchup you could blend in team offensive and defensive strength as part of weight/ real world influence for each 4 factor activity as you estimate is appropriate.

It may have been dense to read and too far flung but I went on to suggest that ultimately you might try to know not only average rates and adjusted impacts for these 16 splits of the 4 factors but also to try to find marginal 4 factor rates and adjusted marginal impacts perhaps off of game by game logs from where they exceed their average minutes or in the especially important case of shooting when they exceeded their average shots (though you could measure marginal rates at any point of their total history curve)

If you got thru these first 2 levels of analysis then I speculated you could try to build demographic based 4 factor production functions.If you did all that then it would seem you would have a pretty complete multi-level 4 factor model and database that you could continue to refine and query and project or simulate off of. Perhaps directly using calculus to move between levels of functions (or implictly) and game theory to try to address team and player matchups and so on. On a professional basis (for team, academia or as a book writer) or perhaps on a community project basis this seems doable with current methods and alot of effort. I don't know how far it will be pursued (or has already been privately) but I welcome any feedback on the sketch of this framework. At a minimum when I think about the frontier that is what came to mind.

gabefarkas · Joined: 31 Dec 2004 Posts: 1313 Location: Durham, NC

Mark · Joined: 20 Aug 2005 Posts: 807

I think you could use adjusted 4 factor analysis to say such things as Player X is a differencemaking sharpshooter if his adjusted FG% scored showed that or a differencemaking offensive rebounder if the adjusted scored showed that. Sticking with team 4 factor level data and impacts. If you did the personal / non-personal team split I've mentioned you coulod further identify the direct portion of the total effect and the indirect portion and the split may vary by player and situation and it may be worth knowing this split detail.

With regard to this question:
[Mark] "In relation to the scale issue Dean is right that the top players will look more impressive measuring from the very top to very bottom rather than just to the neutral on team +/- player. who (just guessing here) may indeed be an above the median performer."

[Gabe] "I would think that the exact average player's +/- would invariably be 0, no?"

I dont think it is invariable that average (mean or median) would be O on the adjusted scale, just that the score would be the weighted average adjusted score or the 50% percentile adjusted score. The adjusted scale is, I believe, court impact based not zero based on average player. I mistated that two posts above and may have mislead you into this and corected it so as to not misguide others. I do not know if the "average" player is neutral, positive or negative at 4 factor scores. What would be invariable, if I am thinking about it correctly, is that the sum of all the player's adjusted scores minute weighted for any of these 4 factors would balance back to zero but the distribution is unknown, could be heavily affected by top players offsetting the majority.

Mark · Joined: 20 Aug 2005 Posts: 807

Further brainstorming on the kinds of questions that adjusted 4 factor analysis could pursue:

1. Aggregrating data across the league do 40% 3pt shooters have a bigger impact than 35% 3 pt shooters? How much more? Is it sensible? How should it affect market salaries or coaching strategy?

2. Do 20 pt a game players who have some minimum level of "post play" have the large impact on team FG% that conventional wisdom credits them with?

3. How does the average benefit of above average penetrators compare to strong 3 pt shooters and post players? Does position appear to matter with this comparison?

4. If you had adjusted offensive rebounding impacts scores could you then look at breakouts of 5 man lineups and look at average scores for various C/PF combos of stout 7 footers, skinny 7 footers, quick hop sub 7 footers and burly sub 7 footers and decide how you want to run your rotation or shop in the summer?

5. Can you use the adjusted team (nonpersonal) split of adjusted offensive rebounding to identify or confirm the presence and impact of the guy who boxes out really well, beyond what the eye or unadjusted team on/off data shows?

6. Can you use the adjusted team (nonpersonal) split of adjusted turnovers rate to identify or confirm the presence and impact of the guy who contributes to teammates making fewer or more turnovers (apart from his own personal turnover rate and an improvement over the information contained in unadjusted team turnover rate on court that doesnt apportion out shares of responsibility) ?

7. Is the impact of a player that draws foulshots frequently better understood as the sum of his own activity and perhaps a positive impact on foulshots earned by others? In another case, alternatively, you could find a guy that draws a lot of personal fouls but when he passes leaves his teammates heaving last second outside shots and not getting as many foulshots as they normally would playing with others and find that split information diminishes the value of the initial player's activity and somewhat gives the other players an excuse for their shortfall.

Mark · Joined: 20 Aug 2005 Posts: 807

gabefarkas · Joined: 31 Dec 2004 Posts: 1313 Location: Durham, NC