Score+: a simple metric to combine volume and efficiency

[Moonbeam]

Recently I've become interested in coming up with a way to jointly summarize scoring volume and efficiency. It seems that most people consider separate statistics for these (e.g. Points per 100 or Points per 36 for volume and TS% or Points per shot attempts for efficiency) and then list them separately, perhaps accounting for league averages. I think it's worthwhile to try to combine volume and efficiency, and I have thought of some very simple metrics to do so which I am tentatively calling Score+, PosScore+, and TeamScore+.

Essentially, Score+ is an estimate of the number of additional points per 100 possessions that a player produces relative to league average, given the same number of true shot attempts.

The formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(League Average TS%)

Players with higher usage (measured by true shot attempts per 100 possessions) will tend to be further away from 0:

So a player with a TS that is 3% above league average who averages 20 TSA per 100 possessions will have a Score+ of 0.03*2*20 = 1.2, meaning that replacing his true shot attempts with league average would reduce a team's scoring by 1.2 points per 100 possessions.

On the other hand, a player with a TS that is only 2.5% above league average but who averages 30 TSA per 100 possessions will have a Score+ of 0.025*2*30 = 1.5, meaning that replacing his true shot attempts with league average would reduce a team's scoring by 1.5 points per 100 possessions.

When looking at the data, I noticed that there were some important positional trends in TS that could be taken into account. Looking at post-merger seasons, it seems that bigs had a more noticeable advantage in TS immediately following the merger, but guards have tended to close the gap a bit:



Hence I also have a position-based version called PosScore+, for which the formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(League Position Average TS%)

The interpretation is similar to that of Score+, except PosScore+ represents the number of additional points per 100 possessions a player produces relative to league average at his position, given the same number of true shot attempts. This might be useful given the way rule changes have impacted scoring ability and efficiency at different positions over time.

Finally, I've included a team variation called TeamScore+, for which the formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(Team Average TS%)

Here is a spreadsheet with the top individual Score+, PosScore+, and TeamScore+ seasons since 1951-52 (when player minutes were first recorded). I've also included career leaders in these categories as well as other notable players (Hall of Famers and players with at least 10,000 career points scored).

These are very simple metrics, but I think they could be useful tools for summarizing the volume x efficiency combination.

What's next? The big question is how to account for increased usage. The theory seems to be that players suffer a drop in efficiency with increased usage, and it makes some sense conceptually - would someone like Tony Allen still be able to produce a TS of 0.541 in 2011-12 if we jacked up his true shot attempts to 25 per 100? This brilliant analysis seems to support the notion of a drop in efficiency with increased usage, but it considers overall offensive production (including turnovers), not just scoring efficiency.

I've tried to look at this effect by comparing each player's TS% relative to his career average against usage (again, measured by true shot attempts per 100 relative to career average), including all player seasons with at least 100 true shot attempts from 1951-52 to present. The scatterplot below seems to indicate no significant pattern.



Edit: I changed the x-axis from TSA per 100 to TSA per 100 relative to career average.

Fitting a linear regression model of relative TS% on relative true shot attempts per 100 does not indicate a significant effect of TSA per 100. I've also tried including a quadratic age effect, and the coefficient for relative TSA per 100 is significant but tiny:

Code: Select all

Coefficient  Estimate  Std. Error  t ratio  p-value
Intercept    -0.22679   0.02565     -22.11  <0.0001
TSA per 100  -0.00030   0.00009      -3.32   0.0009
Age           0.01681   0.00075      22.39  <0.0001
Age^2        -0.00031   0.00001     -22.54  <0.0001


This suggests that for every additional TSA per 100, a player's TS% is expected to decrease by 0.0003. It's a tiny, tiny effect but perhaps I'm thinking about it the wrong way. Once I have a better feel for it I may extend these metrics to try to measure "player scoring value".

For now, however, I thought I'd share these "naive" metrics.

[D Nice]

Spoiler:

Recently I've become interested in coming up with a way to jointly summarize scoring volume and efficiency. It seems that most people consider separate statistics for these (e.g. Points per 100 or Points per 36 for volume and TS% or Points per shot attempts for efficiency) and then list them separately, perhaps accounting for league averages. I think it's worthwhile to try to combine volume and efficiency, and I have thought of some very simple metrics to do so which I am tentatively calling Score+, PosScore+, and TeamScore+.

Essentially, Score+ is an estimate of the number of additional points per 100 possessions that a player produces relative to league average, given the same number of true shot attempts.

The formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(League Average TS%)

Players with higher usage (measured by true shot attempts per 100 possessions) will tend to be further away from 0:

So a player with a TS that is 3% above league average who averages 20 TSA per 100 possessions will have a Score+ of 0.03*2*20 = 1.2, meaning that replacing his true shot attempts with league average would reduce a team's scoring by 1.2 points per 100 possessions.

On the other hand, a player with a TS that is only 2.5% above league average but who averages 30 TSA per 100 possessions will have a Score+ of 0.025*2*30 = 1.5, meaning that replacing his true shot attempts with league average would reduce a team's scoring by 1.5 points per 100 possessions.

When looking at the data, I noticed that there were some important positional trends in TS that could be taken into account. Looking at post-merger seasons, it seems that bigs had a more noticeable advantage in TS immediately following the merger, but guards have tended to close the gap a bit:



Hence I also have a position-based version called PosScore+, for which the formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(League Position Average TS%)

The interpretation is similar to that of Score+, except PosScore+ represents the number of additional points per 100 possessions a player produces relative to league average at his position, given the same number of true shot attempts. This might be useful given the way rule changes have impacted scoring ability and efficiency at different positions over time.

Finally, I've included a team variation called TeamScore+, for which the formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(Team Average TS%)

Here is a spreadsheet with the top individual Score+, PosScore+, and TeamScore+ seasons since 1951-52 (when player minutes were first recorded). I've also included career leaders in these categories as well as other notable players (Hall of Famers and players with at least 10,000 career points scored).

These are very simple metrics, but I think they could be useful tools for summarizing the volume x efficiency combination.

What's next? The big question is how to account for increased usage. The theory seems to be that players suffer a drop in efficiency with increased usage, and it makes some sense conceptually - would someone like Tony Allen still be able to produce a TS of 0.541 in 2011-12 if we jacked up his true shot attempts to 25 per 100? This brilliant analysis seems to support the notion of a drop in efficiency with increased usage, but it considers overall offensive production (including turnovers), not just scoring efficiency.

I've tried to look at this effect by comparing each player's TS% relative to his career average against usage (again, measured by true shot attempts per 100), including all player seasons with at least 100 true shot attempts from 1951-52 to present. The scatterplot below seems to indicate no significant pattern.



Fitting a linear regression model of relative TS% on true shot attempts per 100 does not indicate a significant effect of TSA per 100. I've also tried including a quadratic age effect, but the coefficient for TSA per 100 is still insignificant. I'm still not entirely convinced that there is no efficiency dropoff, so I'll continue to investigate.

For now, however, I thought I'd share these "naive" metrics.

Good stuff. Your data essentially highlights the mis-application of "adjusted for league avg TS%" methodology I've literally been the only one to speak on in regards to '98-'04 perimeter players.

All positions were not impacted unilaterally by the rule change, it makes no sense to "credit" bigs who played in that era the same as wings for the TS% adjustment (in comparison to post '04 players) because bigs obviously aren't going to see a large change in TS% unless they are a Chris Bosh or David Robinson type of player. Meanwhile the ~2.5% people usually amp 98-04 wings by is probably a significant underestimation, and a figure closer to +4% would be a much more accurate adjustment (IMO).

Also, an interesting addition to the study (though I know it would be exhaustive) would be to measure the efficiency of these added shots not versus league averages, but team-by-team averages, as that is really what matters. For example if you area 53 TS% player and the league average is 54 TS%, if your teammates shoot a collective 51TS% you are still adding value by taking more "sub-league average" shot attempts.

Diminishing return data (aka player's %s after 15 shots, 18 shots, 20 shots, etc) would also be extremely helpful. For example if a guy plays on a team that (besides him) shoots 55TS% and he is a 60TS% player but only shoots 52 TS% once crossing the 18 FGA threshold he is actually hurting, not helping, his team by adding on these shots. This would be some invaluable data to add but Im not sure where you could get it (other than a granular analysis of play-by-play data).

Just some random thoughts really, but some cool ideas thrown around here.

[Moonbeam]

Spoiler:

Recently I've become interested in coming up with a way to jointly summarize scoring volume and efficiency. It seems that most people consider separate statistics for these (e.g. Points per 100 or Points per 36 for volume and TS% or Points per shot attempts for efficiency) and then list them separately, perhaps accounting for league averages. I think it's worthwhile to try to combine volume and efficiency, and I have thought of some very simple metrics to do so which I am tentatively calling Score+, PosScore+, and TeamScore+.

Essentially, Score+ is an estimate of the number of additional points per 100 possessions that a player produces relative to league average, given the same number of true shot attempts.

The formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(League Average TS%)

Players with higher usage (measured by true shot attempts per 100 possessions) will tend to be further away from 0:

So a player with a TS that is 3% above league average who averages 20 TSA per 100 possessions will have a Score+ of 0.03*2*20 = 1.2, meaning that replacing his true shot attempts with league average would reduce a team's scoring by 1.2 points per 100 possessions.

On the other hand, a player with a TS that is only 2.5% above league average but who averages 30 TSA per 100 possessions will have a Score+ of 0.025*2*30 = 1.5, meaning that replacing his true shot attempts with league average would reduce a team's scoring by 1.5 points per 100 possessions.

When looking at the data, I noticed that there were some important positional trends in TS that could be taken into account. Looking at post-merger seasons, it seems that bigs had a more noticeable advantage in TS immediately following the merger, but guards have tended to close the gap a bit:



Hence I also have a position-based version called PosScore+, for which the formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(League Position Average TS%)

The interpretation is similar to that of Score+, except PosScore+ represents the number of additional points per 100 possessions a player produces relative to league average at his position, given the same number of true shot attempts. This might be useful given the way rule changes have impacted scoring ability and efficiency at different positions over time.

Finally, I've included a team variation called TeamScore+, for which the formula is (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(Team Average TS%)

Here is a spreadsheet with the top individual Score+, PosScore+, and TeamScore+ seasons since 1951-52 (when player minutes were first recorded). I've also included career leaders in these categories as well as other notable players (Hall of Famers and players with at least 10,000 career points scored).

These are very simple metrics, but I think they could be useful tools for summarizing the volume x efficiency combination.

What's next? The big question is how to account for increased usage. The theory seems to be that players suffer a drop in efficiency with increased usage, and it makes some sense conceptually - would someone like Tony Allen still be able to produce a TS of 0.541 in 2011-12 if we jacked up his true shot attempts to 25 per 100? This brilliant analysis seems to support the notion of a drop in efficiency with increased usage, but it considers overall offensive production (including turnovers), not just scoring efficiency.

I've tried to look at this effect by comparing each player's TS% relative to his career average against usage (again, measured by true shot attempts per 100), including all player seasons with at least 100 true shot attempts from 1951-52 to present. The scatterplot below seems to indicate no significant pattern.



Fitting a linear regression model of relative TS% on true shot attempts per 100 does not indicate a significant effect of TSA per 100. I've also tried including a quadratic age effect, but the coefficient for TSA per 100 is still insignificant. I'm still not entirely convinced that there is no efficiency dropoff, so I'll continue to investigate.

For now, however, I thought I'd share these "naive" metrics.

Good stuff. Your data essentially highlights the mis-application of "adjusted for league avg TS%" methodology I've literally been the only one to speak on in regards to '98-'04 perimeter players. All positions were not impacted unilaterally by the rule change, it makes no sense to "credit" bigs who played in that era the same as wings for the TS% adjustment (in comparison to post '04 players) because bigs obviously aren't going to see a large change in TS% unless they are a Chris Bosh or David Robinson type of player. Meanwhile the ~2.5% people usually amp 98-04 wings by is probably a significant underestimation, and a figure closer to +4% would be a much more accurate adjustment (IMO).

Also, an interesting addition to the study (though I know it would be exhaustive) would be to measure the efficiency of these added shots not versus league averages, but team-by-team averages, as that is really what matters. For example if you area 53 TS% player and the league average is 54 TS%, if your teammates shoot a collective 51TS% you are still adding value by taking more "sub-league average" shot attempts.

Diminishing return data (aka player's %s after 15 shots, 18 shots, 20 shots, etc) would also be extremely helpful. For example if a guy plays on a team that (besides him) shoots 55TS% and he is a 60TS% player but only shoots 52 TS% once crossing the 18 FGA threshold he is actually hurting, not helping, his team by adding on these shots. This would be some invaluable data to add but Im not sure where you could get it (other than a granular analysis of play-by-play data).

Just some random thoughts really, but some cool ideas thrown around here.

Thanks for the feedback. I have added a TeamScore+ metric which does just as you say - measure a player's additional points per 100 possessions against his team's average TS%. Perhaps I should instead change the metric to compare to the average TS% of his teammates instead, as you have implied.

[D Nice]

Thanks for the feedback. I have added a TeamScore+ metric which does just as you say - measure a player's additional points per 100 possessions against his team's average TS%. Perhaps I should instead change the metric to compare to the average TS% of his teammates instead, as you have implied.

Ah yeah, missed that initially, and yeah, if you could relativize it by team (excluding the target player) I think that would breed for more useful data.

Another interesting take-off would be classifying players by "player type" and projecting what there efficiency "adjustment" should be based on their play-style. I think where your graph indicates bigs getting more efficient after the rule change (albeit to a lesser extent than smalls) it’s the bigs who did little more than catch and finish/OREB that see most of the benefit your graph captures (as well as the D-Rob/Bosh types I mentioned). This phenomena is DIRECTLY tied to the game becoming easier on penetrating guards for the simple rationale that more penetration doesn’t just mean more efficient shots for little guys, it’s also easier offense for the guys who don’t do much for themselves.

Let's take a look at 2 groups: Post Bigs (Shaq, Duncan, Pau, Sheed, Yao, Z-Bo) and what I like to call "pump-fake" bigs. The former is the type that really should see no change in efficiency because they're shot distribution won't change much. They're not the type to be spoon-fed most of their points and they aren't the type to draw fouls/free-throws based on hand-checking.The latter, while not the "spoon-fed" type, DOES generate a good deal of offense from hand-check type situations. The natural next step would be to examine the exclusively "catch-and-finish" roleplayer type bigs but I'm too lazy to compile any sort of list on those guys at the moment.

Post Bigs

Duncan 98-04: 56TS%
Duncan 05-10: 55TS%

Shaq 98-04: 58TS%
Shaq 05-06: 58TS%

Pau 02-04: 56TS%
Pau 05-07: 57TS%

Sheed 98-04: 55TS%
Sheed 05-08: 52TS%

Yao 04: 59TS%
Yao 05-09: 60TS%

Z-Bo 04: 53TS%
Z-Bo 05-13: 52TS%

Pump Fake Bigs

KG 99-04: 54TS%
KG 05-10: 57TS%

Brand 01-04: 55TS%
Brand 05-07: 57TS%

Dirk 01-04: 59TS%
Dirk 05-11: 59TS%

True to form the face-up players (Brand, Dirk, KG) saw collective efficiency rise while the true post players saw little fluctuation whatsoever (they experienced a moderate decline, actually).

Explanation for some of the windows chosen: With Pau his efficiency went way up when he became a #2 and had Kobe to spoon-feed so I only used his years as a #1, and with Shaq he was obviously done being a relevant #1 after ’06 so u cut it off there. Z-Bo and Yao only really had 1 "primeish" year under the former rule-set but I think a full season is a decent enough sample size to use as a baseline, particularly when these guys playstyles didn't really change much after the rule (and officiating "emphases") changes went into effect.

Iverson would make a perfect case-study for the opposite player type (penetrating guards). Unfortunately Kobe/T-Mac would be guys who could be included on the basis of having prime seasons in both windows but these guys "rim-running" primes pretty much ended the year the rules changed ('05), and if not, certainly did the year after ('06), and Lebron/Wade wouldn't be very useful since they were merely rookies in 2004.

Pierce, to me, would be like Dirk. Hand-checking would have little to no impact on his efficiency because the shots he generates don't really change under a different set of rules/officiating concentrations, so any efficiency change (or lackthereof) would follow suit. This doesn't mean when comparing him to another '02 wing you give one guy a boost while holding Pierce's constant, rather that when comparing him to a 2010 wing (for example) you should probably temper his adjustment (or really not make one at all) because an '02 Pierce ported into '10 would probably have the exact same efficiency as he did back in '02. Again, I think this is where a classification type of study could be majorly helpful to assessing the impact of environments on different sets of players.

[Moonbeam]

Thanks for the feedback. I have added a TeamScore+ metric which does just as you say - measure a player's additional points per 100 possessions against his team's average TS%. Perhaps I should instead change the metric to compare to the average TS% of his teammates instead, as you have implied.

Ah yeah, missed that initially, and yeah, if you could relativize it by team (excluding the target player) I think that would breed for more useful data.

Another interesting take-off would be classifying players by "player type" and projecting what there efficiency "adjustment" should be based on their play-style. I think where your graph indicates bigs getting more efficient after the rule change (albeit to a lesser extent than smalls) it’s the bigs who did little more than catch and finish/OREB that see most of the benefit your graph captures (as well as the D-Rob/Bosh types I mentioned). This phenomena is DIRECTLY tied to the game becoming easier on penetrating guards for the simple rationale that more penetration doesn’t just mean more efficient shots for little guys, it’s also easier offense for the guys who don’t do much for themselves.

Let's take a look at 2 groups: Post Bigs (Shaq, Duncan, Pau, Sheed, Yao, Z-Bo) and what I like to call "pump-fake" bigs. The former is the type that really should see no change in efficiency because they're shot distribution won't change much. They're not the type to be spoon-fed most of their points and they aren't the type to draw fouls/free-throws based on hand-checking.The latter, while not the "spoon-fed" type, DOES generate a good deal of offense from hand-check type situations. The natural next step would be to examine the exclusively "catch-and-finish" roleplayer type bigs but I'm too lazy to compile any sort of list on those guys at the moment.

Post Bigs

Duncan 98-04: 56TS%
Duncan 05-10: 55TS%

Shaq 98-04: 58TS%
Shaq 05-06: 58TS%

Pau 02-04: 56TS%
Pau 05-07: 57TS%

Sheed 98-04: 55TS%
Sheed 05-08: 52TS%

Yao 04: 59TS%
Yao 05-09: 60TS%

Z-Bo 04: 53TS%
Z-Bo 05-13: 52TS%

Pump Fake Bigs

KG 99-04: 54TS%
KG 05-10: 57TS%

Brand 01-04: 55TS%
Brand 05-07: 57TS%

Dirk 01-04: 59TS%
Dirk 05-11: 59TS%

True to form the face-up players (Brand, Dirk, KG) saw collective efficiency rise while the true post players saw little fluctuation whatsoever (they experienced a moderate decline, actually).

Explanation for some of the windows chosen: With Pau his efficiency went way up when he became a #2 and had Kobe to spoon-feed so I only used his years as a #1, and with Shaq he was obviously done being a relevant #1 after ’06 so u cut it off there. Z-Bo and Yao only really had 1 "primeish" year under the former rule-set but I think a full season is a decent enough sample size to use as a baseline, particularly when these guys playstyles didn't really change much after the rule (and officiating "emphases") changes went into effect.

Iverson would make a perfect case-study for the opposite player type (penetrating guards). Unfortunately Kobe/T-Mac would be guys who could be included on the basis of having prime seasons in both windows but these guys "rim-running" primes pretty much ended the year the rules changed ('05), and if not, certainly did the year after ('06), and Lebron/Wade wouldn't be very useful since they were merely rookies in 2004.

Pierce, to me, would be like Dirk. Hand-checking would have little to no impact on his efficiency because the shots he generates don't really change under a different set of rules/officiating concentrations, so any efficiency change (or lackthereof) would follow suit. This doesn't mean when comparing him to another '02 wing you give one guy a boost while holding Pierce's constant, rather that when comparing him to a 2010 wing (for example) you should probably temper his adjustment (or really not make one at all) because an '02 Pierce ported into '10 would probably have the exact same efficiency as he did back in '02. Again, I think this is where a classification type of study could be majorly helpful to assessing the impact of environments on different sets of players.

Oh, there definitely is room for a more nuanced look at player types when working on these types of statistics. The problem is that while I'm pretty familiar with the big-name players, to be as exhaustive as would be required for my data set, I'd have to be able to categorize everyone!

[Moonbeam]

I've taken a look at a version of Score+ for the postseason as well. Basically, the formula is:

Postseason Score+ = (Points per 100 possessions) - 2*(True Shot Attempts per 100 possessions)*(Opponent Defensive TS%)

I've also looked at offensive rating differential in the same way called O+, where

O+ = Player ORTG - Opponent DRTG

Here is a spreadsheet that I will update as I have time with individual series and postseasons.

[Nivek]

Does it make sense to use TS% for this? Free throw shooting is included in TS%, but FT% is unlikely to change much when a guy shoots more or fewer per 100 possessions. Might it make more sense to use efg to look at changes in efficiency with usage?

For overall offensive efficiency, I think Dean Oliver's offensive rating would work well with the formulation you've developed here.

Really interesting stuff.

EDIT -- Just saw in your last post that you've done the same analysis using ortg.

[Moonbeam]

Does it make sense to use TS% for this? Free throw shooting is included in TS%, but FT% is unlikely to change much when a guy shoots more or fewer per 100 possessions. Might it make more sense to use efg to look at changes in efficiency with usage?

For overall offensive efficiency, I think Dean Oliver's offensive rating would work well with the formulation you've developed here.

Really interesting stuff.

EDIT -- Just saw in your last post that you've done the same analysis using ortg.

I'm not sure why it doesn't make sense to use TS. I think it in terms of the direct scoreboard benefit you get by going to a given player. Inherent in that is the ability to make shots and draw fouls, and the proportion in which the player takes (and makes) free throws, two-point and three-point field goals. There's nothing in any of these formulas about the perceived tradeoff between usage and efficiency anyway (though it's something I'm keen to look at).

I haven't done O+ for the regular season, but it seems easy enough to do. Maybe I'll add that to the spreadsheet.

[Nivek]

Does it make sense to use TS% for this? Free throw shooting is included in TS%, but FT% is unlikely to change much when a guy shoots more or fewer per 100 possessions. Might it make more sense to use efg to look at changes in efficiency with usage?

For overall offensive efficiency, I think Dean Oliver's offensive rating would work well with the formulation you've developed here.

Really interesting stuff.

EDIT -- Just saw in your last post that you've done the same analysis using ortg.

I'm not sure why it doesn't make sense to use TS. I think it in terms of the direct scoreboard benefit you get by going to a given player. Inherent in that is the ability to make shots and draw fouls, and the proportion in which the player takes (and makes) free throws, two-point and three-point field goals. There's nothing in any of these formulas about the perceived tradeoff between usage and efficiency anyway (though it's something I'm keen to look at).

I haven't done O+ for the regular season, but it seems easy enough to do. Maybe I'll add that to the spreadsheet.

I think my thinking was that it would be interesting to see the results broken into segments and as well as rolled up into a big Score+ burrito.

Free throw shooting probably won't vary much whether a guy shoots more or fewer. Field goal shooting probably will change when a guy shoots more frequently because the defense will pay more attention to him. Or, maybe not.

I think this could be a good tool for looking at changes in efficiency based on possession usage. Dean Oliver had previously estimated that for every 1% increase in usage, a player's offensive rating would drop by 1 point per 100 possessions. Dean's ortg includes scoring, shooting efficiency, assists, offensive rebounds, and turnovers. The approach you've described here might be a way of looking at that issue in greater detail, and for breaking apart those aspects of efficiency to see what typically changes as players change usage rates and efficiency.