Position- and Shot-Location-Adjusted Marginal Scoring (Part I): Inside Scoring

This is my first post in a while obviously. The lockout is really draining my enthusiasm. Every once in a while, though, an idea hits you, and you have to run with it, lockout be darned.

I happened to be looking at Hoopdata's shot location stats, and was trying to think about how to compare the efficiency of players at different positions. For example, how do we compare the inside efficiency of a point guard and a center? On average, Hoopdata says that centers shoot 66% at the rim compared to 59% for point guards. In absolute terms, then, centers are more efficient on average. However, in basketball as in other sports, measuring efficiency in "absolute" terms may not be the most valuable thing to do. To paraphrase Albert Einstein via James Carville, Everything is relative, stupid.

Here's what I came up with after a little bit of thought. I'm going to illustrate the calculation with a couple of examples. First up is Tony Parker. Tony Parker averaged 7.3 shot attempts (per 40 minutes) at the rim in 2011 according to Hoopdata. His FG% at the rim was 65.4%. The average number of shot attempts at the rim for point guards is 4.03. The average FG% at the rim for point guards is 60%. So, not only did Parker take more shot attempts than the average point guard, he shot them at a considerably better percentage. I propose that his "adjusted marginal scoring" adjusted for position and shot location (PSAMS) is as follows:

PSAMS = 2*(7.30-4.03)*(0.654-0.50)+2*4.03*(0.654-0.60) = 1.18

Let me explain this formula in steps. The first term on the right side accounts for the additional number of shots that Parker took compared to the average point guard at the rim and the marginal increase in efficiency that those shots had over the average efficiency of what those shots would have had if they were taken by an average player taking an average shot (i.e. league average efficiency). In other words, the assumption is that if Parker didn't take those 3.3 "extra" shots, an average player shooting with average efficiency would have. Of course, that isn't always or even mostly the case, but it's a common reference point that enables us to compare all players fairly. Ok, still with me? The second part of the equation represents the marginal value of the shots that Parker was "supposed" to take relative to other point guards. In the case of Parker, both terms are positive, and his total of 1.18 ranks him very high (#6) on the list of 152 players that I looked at (who qualified by playing 40+ games of 25+ minutes per game).

The second example is Steve Nash. Nash only attempted 2 shots at the rim per 40 minutes. His FG% was 72.8%, which is very high. The calculation for Nash goes as follows:

PSAMS = 2*(2-4.03)*(0.60-0.50)+2*2.0*(0.728-0.60) = 0.27

In Nash's case, the first term is actually negative, because Nash took fewer shots at the rim than the average PG, and those shots are normally more efficient than an average shot, thus costing his team a little bit (compared to average!). Now, the good news is that Nash's marginal value on the two shots he did take was positive, because his efficiency on those shots was above what an average point guard would do. In the end, Nash ranked #73 on the list.

The values 4.03 and 0.60, which represent the shot attempts and FG% for PG at the rim, are replaced by their respective values at each of the other positions. In this post, I am just looking at shots at the rim (< 3ft from the basket according to Hoopdata). Here are the top 20 overall:

Top 20

Thaddeus Young comes out on top, which is not surprising, given his 7.9 attempts per 40 at 73.4% efficiency. Also, for Warriors fans reading this, note that Monta Ellis comes in at #11 overall and #2 among shooting guards.

RANK POS RANK NAME TEAM POS PSAMS
1 1 Thaddeus Young PHI 3 2.92
2 2 LeBron James MIA 3 1.92
3 1 Dwight Howard ORL 5 1.74
4 1 Dwyane Wade MIA 2 1.73
5 3 Shawn Marion DAL 3 1.60
6 1 Tony Parker SAS 1 1.44
7 4 Wilson Chandler NYK 3 1.34
8 2 Nene Hilario DEN 5 1.33
9 5 Kevin Durant OKC 3 1.24
10 6 Tayshaun Prince DET 3 1.15
11 2 Monta Ellis GSW 2 1.15
12 7 Jeff Green OKC 3 1.08
13 8 Paul Pierce BOS 3 1.06
14 1 Blake Griffin LAC 4 1.04
15 3 Marcin Gortat PHO 5 0.85
16 2 Carlos Boozer CHI 4 0.83
17 2 Russell Westbrook OKC 1 0.82
18 3 Eric Gordon LAC 2 0.77
19 4 Kevin Martin HOU 2 0.76
20 5 Landry Fields NYK 2 0.73

Here are the bottom 20 on the list:

Bottom 20  

RANK POS RANK NAME TEAM POS PSAMS
152 34 Marcus Camby POR 4 -1.56
151 19 Andrea Bargnani TOR 5 -1.42
150 33 Channing Frye PHO 4 -1.35
149 33 Daniel Gibson CLE 1 -1.28
148 32 Ersan Ilyasova MIL 4 -1.08
147 32 Carlos Delfino MIL 3 -0.90
146 34 Anthony Morrow NJN 2 -0.87
145 33 Sasha Vujacic NJN 2 -0.85
144 32 Derek Fisher LAL 1 -0.82
143 18 Chuck Hayes HOU 5 -0.80
142 31 Luke Ridnour MIN 1 -0.75
141 32 Anthony Parker CLE 2 -0.69
140 17 Roy Hibbert IND 5 -0.68
139 30 Brandon Jennings MIL 1 -0.67
138 16 Kwame Brown CHA 5 -0.65
137 29 Jason Kidd DAL 1 -0.65
136 31 Elton Brand PHI 4 -0.64
135 15 Brook Lopez NJN 5 -0.63
134 30 Antawn Jamison CLE 4 -0.60
133 31 John Salmons MIL 2 -0.60

Here is the full list of players by position, with their overall rank and positional rank:

Point Guards 

RANK POS RANK NAME TEAM PSAMS
6 1 Tony Parker SAS 1.44
17 2 Russell Westbrook OKC 0.82
22 3 Ty Lawson DEN 0.73
32 4 Derrick Rose CHI 0.55
38 5 Andre Miller POR 0.51
39 6 Tyreke Evans SAC 0.51
44 7 Rajon Rondo BOS 0.43
46 8 Beno Udrih SAC 0.42
55 9 John Wall WAS 0.30
56 10 Deron Williams UTH 0.30
58 11 Darren Collison IND 0.27
59 12 Ramon Sessions CLE 0.27
73 13 Steve Nash PHO 0.11
76 14 Kyle Lowry HOU 0.08
77 15 Jrue Holiday PHI 0.07
79 16 Mike Conley MEM 0.06
81 17 Baron Davis LAC 0.01
84 18 Rodney Stuckey DET -0.04
90 19 Stephen Curry GSW -0.12
93 20 Jameer Nelson ORL -0.16
96 21 George Hill SAS -0.18
99 22 Raymond Felton NYK -0.24
103 23 Chris Paul NOR -0.27
110 24 Jose Calderon TOR -0.32
115 25 Chauncey Billups DEN -0.39
119 26 Devin Harris NJN -0.44
121 27 D.J. Augustin CHA -0.46
130 28 Mike Bibby ATL -0.56
137 29 Jason Kidd DAL -0.65
139 30 Brandon Jennings MIL -0.67
142 31 Luke Ridnour MIN -0.75
144 32 Derek Fisher LAL -0.82
149 33 Daniel Gibson CLE -1.28

Shooting Guards

RANK POS RANK NAME TEAM PSAMS
4 1 Dwyane Wade MIA 1.73
11 2 Monta Ellis GSW 1.15
18 3 Eric Gordon LAC 0.77
19 4 Kevin Martin HOU 0.76
20 5 Landry Fields NYK 0.73
24 6 Andre Iguodala PHI 0.71
43 7 Kobe Bryant LAL 0.43
50 8 Wesley Matthews POR 0.37
53 9 Manu Ginobili SAS 0.31
54 10 Joe Johnson ATL 0.30
66 11 Vince Carter PHO 0.22
67 12 DeMar DeRozan TOR 0.20
72 13 Jason Richardson ORL 0.11
74 14 Arron Afflalo DEN 0.10
75 15 James Harden OKC 0.09
82 16 Thabo Sefolosha OKC -0.00
83 17 Jason Terry DAL -0.02
86 18 Richard Hamilton DET -0.07
87 19 Ray Allen BOS -0.08
95 20 Brandon Roy POR -0.18
100 21 Kirk Hinrich WAS -0.24
101 22 Brandon Rush IND -0.25
104 23 Stephen Jackson CHA -0.29
106 24 Nick Young WAS -0.30
113 25 O.J. Mayo MEM -0.37
114 26 Jamal Crawford ATL -0.38
116 27 Raja Bell UTH -0.39
117 28 Jodie Meeks PHI -0.41
120 29 Ben Gordon DET -0.46
124 30 J.J. Redick ORL -0.49
133 31 John Salmons MIL -0.60
141 32 Anthony Parker CLE -0.69
145 33 Sasha Vujacic NJN -0.85
146 34 Anthony Morrow NJN -0.87

Small Forwards

RANK POS RANK NAME TEAM PSAMS
1 1 Thaddeus Young PHI 2.92
2 2 LeBron James MIA 1.92
5 3 Shawn Marion DAL 1.60
7 4 Wilson Chandler NYK 1.34
9 5 Kevin Durant OKC 1.24
10 6 Tayshaun Prince DET 1.15
12 7 Jeff Green OKC 1.08
13 8 Paul Pierce BOS 1.06
23 9 Grant Hill PHO 0.72
27 10 Nicolas Batum POR 0.62
29 11 C.J. Miles UTH 0.59
31 12 Luol Deng CHI 0.59
35 13 Rudy Gay MEM 0.54
47 14 Andrei Kirilenko UTH 0.42
49 15 Trevor Ariza NOR 0.38
51 16 Danilo Gallinari NYK 0.35
57 17 Dorell Wright GSW 0.28
63 18 Jared Dudley PHO 0.24
68 19 Marvin Williams ATL 0.18
69 20 Danny Granger IND 0.18
70 21 Carmelo Anthony DEN 0.14
78 22 Mike Dunleavy IND 0.07
85 23 Richard Jefferson SAS -0.04
88 24 Gerald Wallace CHA -0.08
89 25 Luc Mbah a Moute MIL -0.11
105 26 Wesley Johnson MIN -0.30
107 27 Ron Artest LAL -0.31
108 28 Shane Battier HOU -0.31
123 29 Ryan Gomes LAC -0.48
128 30 Travis Outlaw NJN -0.52
129 31 Hedo Turkoglu ORL -0.53
147 32 Carlos Delfino MIL -0.90

Power Forwards

RANK POS RANK NAME TEAM PSAMS
14 1 Blake Griffin LAC 1.04
16 2 Carlos Boozer CHI 0.83
25 3 Lamar Odom LAL 0.65
28 4 Kenyon Martin DEN 0.61
30 5 Greg Monroe DET 0.59
33 6 Andray Blatche WAS 0.55
36 7 Amir Johnson TOR 0.53
37 8 Carl Landry SAC 0.53
40 9 LaMarcus Aldridge POR 0.51
41 10 Serge Ibaka OKC 0.48
42 11 Paul Millsap UTH 0.48
48 12 Kevin Garnett BOS 0.40
52 13 Zach Randolph MEM 0.34
60 14 Kris Humphries NJN 0.26
61 15 Josh Smith ATL 0.24
64 16 David Lee GSW 0.23
65 17 Pau Gasol LAL 0.22
71 18 Amare Stoudemire NYK 0.13
80 19 J.J. Hickson CLE 0.04
94 20 Chris Bosh MIA -0.18
97 21 Luis Scola HOU -0.21
98 22 Michael Beasley MIN -0.23
102 23 DeMarcus Cousins SAC -0.25
109 24 David West NOR -0.32
112 25 Brandon Bass ORL -0.34
126 26 Dirk Nowitzki DAL -0.50
127 27 Kevin Love MIN -0.52
131 28 Glen Davis BOS -0.59
132 29 Boris Diaw CHA -0.59
134 30 Antawn Jamison CLE -0.60
136 31 Elton Brand PHI -0.64
148 32 Ersan Ilyasova MIL -1.08
150 33 Channing Frye PHO -1.35
152 34 Marcus Camby POR -1.56

Centers

RANK POS RANK NAME TEAM PSAMS
3 1 Dwight Howard ORL 1.74
8 2 Nene Hilario DEN 1.33
15 3 Marcin Gortat PHO 0.85
21 4 Andrew Bynum LAL 0.73
26 5 Tyson Chandler DAL 0.62
34 6 DeAndre Jordan LAC 0.54
45 7 JaVale McGee WAS 0.42
62 8 Emeka Okafor NOR 0.24
91 9 Al Jefferson UTH -0.12
92 10 Tim Duncan SAS -0.15
111 11 Al Horford ATL -0.33
118 12 Marc Gasol MEM -0.42
122 13 Andrew Bogut MIL -0.46
125 14 Joakim Noah CHI -0.50
135 15 Brook Lopez NJN -0.63
138 16 Kwame Brown CHA -0.65
140 17 Roy Hibbert IND -0.68
143 18 Chuck Hayes HOU -0.80
151 19 Andrea Bargnani TOR -1.42

Well, that's all I got for now. As always, feedback is welcome.

Edit: As Alex pointed out in the comments, the league average eFG% is 50%, not 54%. The ratings/rankings have been changed to reflect that.

Edit #2: You might disagree with the notion of adjusting for position. For those who do, I've calculated SAMS, which simply uses the league average shot attempts per 40 (3.4) and FG% at the rim (65%), thus, ignoring any position dependency. 

SAMS

As you can see, Dwight Howard is now on top, but Thaddeus Young is (perhaps, surprisingly) still #2 and very close behind. For Warriors fans, note that Monta Ellis still ranks ahead of David Lee even though position is not taken into account.

RANK NAME TEAM POS SAMS
1 Dwight Howard ORL 5 2.74
2 Thaddeus Young PHI 3 2.68
3 Nene Hilario DEN 5 2.33
4 Marcin Gortat PHO 5 1.85
5 Andrew Bynum LAL 5 1.73
6 Blake Griffin LAC 4 1.71
7 LeBron James MIA 3 1.68
8 Tyson Chandler DAL 5 1.62
9 DeAndre Jordan LAC 5 1.54
10 Dwyane Wade MIA 2 1.54
11 Carlos Boozer CHI 4 1.50
12 JaVale McGee WAS 5 1.42
13 Shawn Marion DAL 3 1.35
14 Lamar Odom LAL 4 1.32
15 Kenyon Martin DEN 4 1.28
16 Greg Monroe DET 4 1.26
17 Emeka Okafor NOR 5 1.24
18 Tony Parker SAS 1 1.23
19 Andray Blatche WAS 4 1.22
20 Amir Johnson TOR 4 1.20
21 Carl Landry SAC 4 1.20
22 LaMarcus Aldridge POR 4 1.17
23 Serge Ibaka OKC 4 1.15
24 Paul Millsap UTH 4 1.15
25 Wilson Chandler NYK 3 1.09
26 Kevin Garnett BOS 4 1.07
27 Zach Randolph MEM 4 1.01
28 Kevin Durant OKC 3 0.99
29 Monta Ellis GSW 2 0.96
30 Kris Humphries NJN 4 0.93
31 Josh Smith ATL 4 0.91
32 Tayshaun Prince DET 3 0.91
33 David Lee GSW 4 0.90
34 Pau Gasol LAL 4 0.89
35 Al Jefferson UTH 5 0.88
36 Tim Duncan SAS 5 0.85
37 Jeff Green OKC 3 0.83
38 Paul Pierce BOS 3 0.82
39 Amare Stoudemire NYK 4 0.80
40 J.J. Hickson CLE 4 0.70
41 Al Horford ATL 5 0.67
42 Russell Westbrook OKC 1 0.60
43 Marc Gasol MEM 5 0.58
44 Eric Gordon LAC 2 0.58
45 Kevin Martin HOU 2 0.57
46 Andrew Bogut MIL 5 0.54
47 Landry Fields NYK 2 0.54
48 Andre Iguodala PHI 2 0.52
49 Ty Lawson DEN 1 0.51
50 Joakim Noah CHI 5 0.50
51 Chris Bosh MIA 4 0.49
52 Grant Hill PHO 3 0.47
53 Luis Scola HOU 4 0.45
54 Michael Beasley MIN 4 0.44
55 DeMarcus Cousins SAC 4 0.42
56 Nicolas Batum POR 3 0.38
57 Brook Lopez NJN 5 0.37
58 Kwame Brown CHA 5 0.35
59 C.J. Miles UTH 3 0.35
60 David West NOR 4 0.35
61 Luol Deng CHI 3 0.34
62 Derrick Rose CHI 1 0.34
63 Brandon Bass ORL 4 0.33
64 Roy Hibbert IND 5 0.32
65 Andre Miller POR 1 0.30
66 Tyreke Evans SAC 1 0.29
67 Rudy Gay MEM 3 0.29
68 Kobe Bryant LAL 2 0.23
69 Rajon Rondo BOS 1 0.21
70 Beno Udrih SAC 1 0.21
71 Chuck Hayes HOU 5 0.20
72 Wesley Matthews POR 2 0.18
73 Andrei Kirilenko UTH 3 0.17
74 Dirk Nowitzki DAL 4 0.17
75 Kevin Love MIN 4 0.15
76 Trevor Ariza NOR 3 0.13
77 Manu Ginobili SAS 2 0.12
78 Joe Johnson ATL 2 0.11
79 Danilo Gallinari NYK 3 0.11
80 John Wall WAS 1 0.09
81 Deron Williams UTH 1 0.09
82 Glen Davis BOS 4 0.08
83 Boris Diaw CHA 4 0.07
84 Antawn Jamison CLE 4 0.07
85 Darren Collison IND 1 0.05
86 Ramon Sessions CLE 1 0.05
87 Dorell Wright GSW 3 0.04
88 Elton Brand PHI 4 0.03
89 Vince Carter PHO 2 0.03
90 DeMar DeRozan TOR 2 0.01
91 Jared Dudley PHO 3 -0.01
92 Marvin Williams ATL 3 -0.06
93 Danny Granger IND 3 -0.07
94 Jason Richardson ORL 2 -0.08
95 Arron Afflalo DEN 2 -0.10
96 James Harden OKC 2 -0.10
97 Carmelo Anthony DEN 3 -0.10
98 Steve Nash PHO 1 -0.11
99 Kyle Lowry HOU 1 -0.14
100 Jrue Holiday PHI 1 -0.14
101 Mike Conley MEM 1 -0.15
102 Mike Dunleavy IND 3 -0.18
103 Thabo Sefolosha OKC 2 -0.19
104 Baron Davis LAC 1 -0.20
105 Jason Terry DAL 2 -0.21
106 Rodney Stuckey DET 1 -0.25
107 Richard Hamilton DET 2 -0.26
108 Ray Allen BOS 2 -0.27
109 Richard Jefferson SAS 3 -0.28
110 Gerald Wallace CHA 3 -0.33
111 Stephen Curry GSW 1 -0.34
112 Luc Mbah a Moute MIL 3 -0.36
113 Brandon Roy POR 2 -0.37
114 Jameer Nelson ORL 1 -0.38
115 George Hill SAS 1 -0.40
116 Ersan Ilyasova MIL 4 -0.42
117 Andrea Bargnani TOR 5 -0.42
118 Kirk Hinrich WAS 2 -0.43
119 Brandon Rush IND 2 -0.44
120 Raymond Felton NYK 1 -0.45
121 Stephen Jackson CHA 2 -0.48
122 Chris Paul NOR 1 -0.49
123 Nick Young WAS 2 -0.50
124 Jose Calderon TOR 1 -0.54
125 Wesley Johnson MIN 3 -0.54
126 Ron Artest LAL 3 -0.55
127 Shane Battier HOU 3 -0.56
128 O.J. Mayo MEM 2 -0.57
129 Jamal Crawford ATL 2 -0.57
130 Raja Bell UTH 2 -0.58
131 Chauncey Billups DEN 1 -0.60
132 Jodie Meeks PHI 2 -0.60
133 Ben Gordon DET 2 -0.65
134 Devin Harris NJN 1 -0.65
135 D.J. Augustin CHA 1 -0.67
136 J.J. Redick ORL 2 -0.68
137 Channing Frye PHO 4 -0.68
138 Ryan Gomes LAC 3 -0.72
139 Travis Outlaw NJN 3 -0.76
140 Mike Bibby ATL 1 -0.77
141 Hedo Turkoglu ORL 3 -0.77
142 John Salmons MIL 2 -0.79
143 Jason Kidd DAL 1 -0.86
144 Anthony Parker CLE 2 -0.89
145 Brandon Jennings MIL 1 -0.89
146 Marcus Camby POR 4 -0.89
147 Luke Ridnour MIN 1 -0.96
148 Derek Fisher LAL 1 -1.04
149 Sasha Vujacic NJN 2 -1.05
150 Anthony Morrow NJN 2 -1.06
151 Carlos Delfino MIL 3 -1.15
152 Daniel Gibson CLE 1 -1.50

 

28 thoughts on “Position- and Shot-Location-Adjusted Marginal Scoring (Part I): Inside Scoring”

  1. Evan, where does the .54 come from? That doesn't match up with league FG% or eFG% that I can find.

    Also, are the numbers in Nash's equation correct? If the first term is supposed to mark what he cost his team by not shooting enough lay-ups, shouldn't his lay-up percentage be there instead of the average point guards'? Presumably the Suns 'lost' two Nash lay-ups for two average shots, not two point guard lay-ups.

  2. Whoops. You're right about the 54%. I got eFG% and TS% mixed up in my memory. I'll need to fix that. The correct value should be 50%.

    As to your second point, I disagree. The fact is we don't actually know what % he would shoot if he took those extra shots. Most likely, those shots wouldn't be nearly as efficient, otherwise, he'd take many more. Also, in general, it doesn't make sense to me to penalize a guy for being more efficient, which is what you would be doing in that case. Just give him credit for what he actually did or did not do relative to a replacement.

  3. This cut of data is useful but I also have some caution about using it by itself. Typically heightened inside shots, especially from perimeter players, are accompanied by heightened mid-range shots (with a lower efficiency at least somewhat offsetting the greater efficiency of inside shots) from drives that get shut down short of an inside shot. On the other hand drives tend to be associated with more free throw shots as well. To get a fuller sense of the efficiency impact of a player I'd mainly look at overall TS% or TS% apart from 3 point shooting. I calculated what I called non-outside scoring index about 2 years ago at APBRmetrics. I didn't include free throw activity then but I should have. Ideally catch and shoot mid-range shots probably should be removed from the clump of inside activity and mid-range shots associated with drives / created shots. 3 points, catch and shot mid-range and everything else seems like a useful way to view use of player possessions to me. The detail of the last clump is important to look at too of course, but I think it is also important to view it together.

    1. Thanks for the comment. I plan to extend this analysis to mid-range and 3-pt shots, and then add up all the points and see what it gives. I agree that free throws are important, but I don't have free throw rates categorized by shot location (that would be nice). Probably one could assume fouls are mostly drawn on these close attempts, though. Another thing that I neglected here is the % assisted rate. In general, guards have a lower percentage of assisted shots, which is something to consider.

        1. It looks like a similar idea, obviously, but I don't think he is taking into account the shot frequency either. Where are the formulas? I didn't see any on that page.

        2. He says: Expected Points uses a player’s FGA from each area of the floor and multiplies it by the average number of points scored on that type of shot to come up with an Expected Point total from that area. The Expected Point total can than be compared to the actual number of points a player scored from that area to arrive at a Point Differential. This point differential is an expression of how a player shot compared to the league average, but I like that the comparison is drawn with actual point totals.

          http://www.hickory-high.com/?p=542

          The formula would apparently just be the efficiency differential between player and league average * FGA for shot location.

          "The average values of shots by location that I use (At Rim – 1.208, <10ft. – 0.856, 10-15ft. – 0.783, 16-23ft. – 0.801, 3PT – 1.081, FT – 0.759) were calculated by Albert Lyu of ThinkBlueCrew."

      1. Right, so it looks like the major distinction is that I'm accounting for the "extra" inside shots compared to average shot attempts (in the case of Parker), or in the case of players taking fewer shots than average (like Nash) the "missing" shots that would be expected.

  4. Looking back at the "non-outside scoring index" (without FTAs) it appears that league-wide "the mid-range component pulled the average yield of non 3 point shots down at team level... to essentially the average of 3 point shots. There is equilibrium at that level. Nearly exact." Getting high efficiency yield FTAs is the main net payoff on average from not settling for the 3 pointer. Backing out catch n shoot mid-rangers would increase the payoff, but I guess one could perhaps argue against backing them out since players usually had the choice to put the ball on the floor.

  5. In the case of Nash at least the usage argument is moot. His percentage is so high that he could take more lay-ups, even at the cost of dropping his percentage to 55%, and it would be completely worth trading any other two-pointer he takes for those lay-ups (in previous years he has taken more, and his percentage has only been as low as 66%. His lowest ever was 60% when he took as many as in your data set). He doesn't take more lay-ups because it either isn't in the team's game plan or his game plan. So it does hurt his team for him to not take those shots, unless you really believe that his third lay-up is going to destroy his percentage (which is empirically untrue) or that it would knock it below some other shot that a teammate is getting (because his next most efficient shot is a three, at 60% eFG).

    Nash shoots a very good percentage, but the same argument applies in degrees to everyone else. If a player shoots at an above-average rate, even if you believe in the usage trade-off he should in fact go ahead and trade a little efficiency to get more still-more-efficient-than-the-alternative shots. If he doesn't, he's hurting his team.

    1. Derek Fisher takes about the same number of inside shots as Nash. He is much more inefficient on those shots. Does he actually deserve more credit than Nash for those shots not taken? Does he get positive points because he is helping his team by not taking those shots? That doesn't make sense to me, but it's where your logic is headed.

      1. Why doesn't that make sense? Do you want players to take shots they aren't good at? If I'm a bad three point shooter, of course I hurt my team every time I take a three; by definition there are better shots that could be taken, either by me or a teammate. I guess I should make it a question for you; A player who shoots a lot and poorly would be penalized by your measure; why shouldn't poor shooters get credit for not shooting much?

        1. "Why shouldn't poor shooters get credit for not shooting much?"

          Let's take it to the logical next step. A player who completely "opts out" of shooting ends up with a rating of 0 by your logic. That means he is by definition an average shooter. Or do you actually go a step further and award him points for passing up shots that would be bad for him?

          That may make sense using David Berri logic, but not basketball logic. I realize it's a big divide between us, but that's the way I see it. I don't expect everyone to agree.

  6. I am a little confused. If the first part of the equation measures marginal efficiency relative to an average player shouldn't the marginal shots attempted be relative to the the number of shots taken by an average player (position independent) I was thinking

    (shots - shots_avg)*(percentage - percentage_avg) + (shots - shots_pos-avg)(percentage - percentage_pos-avg)

    Has there been any thoughts about using the median player for each shot, since the distributions may be skewed by a relatively few number of players taking a lot of a certain type of shot at a high efficiency. I was thinking of the relatively small number of three point specialists that may take an unusually high volume of shots (maybe at a high percentage). It probably isn't so skewed, but I haven't looked at the data. Thanks for doing this, this is super interesting and it gives me an excuse to think about statistics.

    Is the data that you are using freely available?

    1. Good question. I actually am comparing to the average shots by position in the PSAMS metric and average shots across all positions in the SAMS (the "P" stands for the position adjustment).

      Your idea of using the median is interesting. I think I'll look into it. Another way to really get a better handle on the mean is to use bootstrapping. I'm just not sure it's worth the effort.

  7. Evan, Dave Berri has nothing to do with it. Think of it this way; if you were Ben Wallace's coach, when and how much would you want him to shoot? I would say only dunks, and even then I'm nervous. He makes the team better by not shooting.

    It also seems reasonable that you would have a disclaimer for someone who never did something. As it stands you're only looking at people with more than a certain number of minutes, right? Never shooting shouldn't be an issue.

  8. Ben Wallace not shooting is more valuable then Ben Wallace shooting. I agree with you there. That doesn't mean his not shooting is valuable in an absolute sense (and judging by his -2.4 multi-year offensive RAPM, he is not valuable at all in an any sense offensively, relative or absolutely). Someone else has to take his shots for him, and that player gets penalized. In effect, Ben Wallace is "stealing" value from his teammates. To account for this, Wallace has to be penalized for his undershooting. The average player would not be penalized because his teammates wouldn't have to take those bad shots for him. The way I see it, this is really just basic accounting 101 for basketball.

    The reason I mentioned Berri is because he has explicitly argued against this kind of logic. He thinks usage is irrelevant to valuing players.

  9. I enjoyed reading this. Thanks for putting in the work on it.

    The PSAMS for PF's being the lowest confuses me. Does it have to do with less deviation from the league averages even among the relative "elite" in inside shooting percentages among PF's (where would I find that data)?

  10. Yeah, variance is what I meant. I'm not necessarily asking what specifically is controlling it, just that the numbers indicate that the variance in PF's is lower then in the other positions.

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