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Adjusted Plus-Minus Ratings:
New and Improved for 2007-2008

By Steve Ilardi, Ph.D. and Aaron Barzilai, Ph.D

Like most devoted NBA fans, we enjoy looking at boxscores. They’re often incredibly informative. And yet boxscores fail to capture many important elements of play ?lockdown defense, screens, rotations, hustle plays, and so on ?that can affect the game’s bottom line

That’s why we believe it’s necessary to “think outside the boxscore?in order to adequately measure each player’s true value to his team. The plus-minus statistic, which tracks all changes in scoring while each player is on the court, is one promising approach that’s gained traction in recent years. But this metric suffers from a key drawback: each player’s rating is heavily dependent upon the quality of his on-court teammates. Even a mere role player on a great team (e.g., Glen Davis) typically has a higher plus-minus rating than a superstar on a bad team (Dwyane Wade). Fortunately, this limitation can be overcome through the use of advanced mathematical techniques, which make it possible to isolate the unique effects of each player on the court. In other words, we can statistically adjust each player’s plus-minus rating to account for the simultaneous impact of all his teammates and opponents [1]. Hence the name: adjusted plus-minus [2].

At first blush, the metric might even seem like the “holy grail?of basketball statistics ?a single measure that captures the precise effect of each player on his team’s bottom-line scoring margin. But it, too, has a major drawback: as a mathematical estimate, each adjusted plus-minus rating contains measurement noise, i.e., a margin of error.

It’s important, therefore, to get this noise (error) level as low as possible, and we’ve taken an important step in that direction with the present set of ratings. Specifically, we’ve used five seasons?worth of data (provided by www.nbatop.com) ?weighted very heavily in favor of the 2007-2008 season ?to disentangle the individual effects of teammates who frequently appear on the court at the same time. As a result, we are able to present below the most accurate (low-noise) adjusted plus-minus ratings ever to appear in the public domain. In addition, we’ve modeled separately each player’s impact on offense and defense, treating these as completely independent variables. (Both innovations are explained in detail below the fold.)

Accordingly, the tables that follow include offensive, defensive, and total (net) adjusted plus-minus ratings for each player who logged at least 300 minutes in the 2007-2008 season, with high-minutes (> 2000 total regular-season minutes) and low-minutes players listed separately. These ratings reflect each player’s points contributed per 100 possessions, as compared with the league average. As an added bonus, both tables are sortable by adjusted plus-minus ratings, team, and position.

Players with 2000+ Minutes

Offense
Defense
Overall
 Team  Pos  Player
Adj +/-
Error
Adj +/-
Error
Adj +/-
Error
Minutes
BOS 4   Garnett, Kevin 6.88  1.06  7.59  1.06  14.47  1.41  2328 
CLE 3   James, LeBron 9.24  1.13  2.41  1.12  11.65  1.50  3027 
SAN 4   Duncan, Tim 4.39  1.27  4.46  1.26  8.85  1.69  2651 
SAN 2   Ginobili, Manu 4.93  0.97  3.66  0.97  8.59  1.29  2299 
LAL 2   Bryant, Kobe 8.96  1.11  -0.47  1.11  8.49  1.48  3192 
BOS 3   Pierce, Paul 7.33  1.02  0.48  1.02  7.81  1.35  2874 
LAC 1   Davis, Baron 5.96  1.10  1.46  1.10  7.42  1.47  3196 
TOR 3   Moon, Jamario 3.88  1.37  3.19  1.37  7.07  1.82  2166 
PHX 1   Nash, Steve 10.01  1.16  -3.24  1.16  6.77  1.55  2780 
DAL 1   Kidd, Jason 5.98  1.08  0.49  1.08  6.47  1.44  2906 
NJ 1   Harris, Devin 6.89  1.16  -0.46  1.16  6.43  1.54  2022 
HOU 3   Artest, Ron 1.79  1.09  4.52  1.09  6.31  1.46  2172 
UTA 3   Kirilenko, Andrei 4.28  1.21  1.94  1.21  6.22  1.62  2217 
SAC 5   Miller, Brad 2.08  1.21  4.03  1.21  6.11  1.61  2513 
TOR 4   Bosh, Chris 3.71  1.13  2.40  1.13  6.10  1.50  2425 
DAL 4   Nowitzki, Dirk 4.67  1.17  1.35  1.17  6.02  1.56  2769 
LAL 4   Gasol, Pau 2.50  0.96  3.12  0.96  5.62  1.28  2351 
POR 4   Aldridge, LaMarcus 2.21  1.51  3.06  1.51  5.27  2.02  2649 
DET 5   Wallace, Rasheed 1.61  1.17  3.66  1.17  5.26  1.56  2346 
NO 5   Chandler, Tyson 0.58  1.20  4.63  1.20  5.21  1.60  2783 
MIN 2   McCants, Rashad 3.43  1.30  1.56  1.30  4.99  1.73  2018 
CHI 3   Deng, Luol 4.52  1.24  0.43  1.24  4.95  1.65  2128 
NO 3   Stojakovic, Peja 5.05  1.15  -0.19  1.15  4.86  1.53  2711 
HOU 5   Ming, Yao 0.21  1.17  4.56  1.17  4.77  1.56  2044 
NO 1   Paul, Chris 9.24  1.52  -4.54  1.52  4.69  2.02  3006 
ORL 5   Howard, Dwight 2.78  1.45  1.71  1.45  4.49  1.93  3088 
DEN 4   Martin, Kenyon -1.86  1.14  6.05  1.14  4.19  1.52  2160 
NJ 2   Carter, Vince 1.89  1.09  2.27  1.09  4.17  1.46  2959 
MIL 5   Bogut, Andrew 0.39  1.50  3.71  1.50  4.10  1.99  2720 
IND 3   Dunleavy, Mike 2.29  1.09  1.74  1.09  4.04  1.45  2953 
MIA 3   Marion, Shawn -0.48  0.98  4.34  0.98  3.87  1.31  2315 
ORL 3   Turkoglu, Hedo 4.58  1.13  -0.74  1.13  3.83  1.50  3026 
DAL 3   Howard, Josh 0.52  1.13  3.27  1.13  3.79  1.50  2764 
DET 3   Prince, Tayshaun 0.33  1.34  3.34  1.33  3.67  1.78  2694 
ORL 3   Lewis, Rashard 3.10  1.01  0.42  1.01  3.52  1.35  3076 
DAL 1   Terry, Jason 6.01  1.07  -2.49  1.07  3.52  1.42  2579 
OK 4   Collison, Nick 1.92  1.18  1.57  1.18  3.49  1.57  2230 
UTA 2   Brewer, Ronnie 1.30  1.30  2.05  1.30  3.35  1.74  2088 
DEN 2   Iverson, Allen 4.97  1.15  -1.64  1.15  3.34  1.54  3424 
WAS 4   Jamison, Antawn 4.63  1.18  -1.45  1.18  3.19  1.57  3061 
GSW 3   Maggette, Corey 2.43  1.09  0.70  1.09  3.13  1.45  2502 
MIN 2   Miller, Mike 3.72  1.14  -0.63  1.14  3.10  1.52  2474 
LAC 5   Camby, Marcus -1.34  1.23  4.34  1.23  3.00  1.64  2758 
ATL 4   Smith, Josh -1.28  1.14  4.19  1.14  2.91  1.52  2873 
HOU 2   McGrady, Tracy 4.89  1.18  -2.05  1.18  2.85  1.57  2440 
MIN 3   Gomes, Ryan 3.18  1.30  -0.37  1.30  2.82  1.73  2434 
CHA 3   Wallace, Gerald -0.20  1.23  2.98  1.23  2.78  1.64  2376 
HOU 3   Battier, Shane 0.87  1.07  1.83  1.07  2.69  1.43  2907 
UTA 1   Williams, Deron 2.63  1.53  0.00  1.52  2.63  2.03  3059 
WAS 5   Haywood, Brendan -1.65  1.20  4.18  1.20  2.52  1.60  2228 
NY 2   Crawford, Jamal 3.40  1.18  -0.96  1.18  2.44  1.57  3190 
GSW 5   Biedrins, Andris 0.47  1.22  1.86  1.22  2.33  1.62  2079 
DET 1   Billups, Chauncey 4.80  1.30  -2.54  1.29  2.27  1.73  2522 
ORL 2   Bogans, Keith 0.70  1.03  1.57  1.03  2.27  1.37  2198 
CHA 1   Felton, Raymond 1.97  1.33  0.28  1.33  2.25  1.77  2972 
ATL 2   Johnson, Joe 1.90  1.12  0.13  1.12  2.04  1.49  3337 
PHX 4   Stoudemire, Amare 4.39  1.12  -2.63  1.11  1.76  1.48  2677 
WAS 3   Butler, Caron 2.10  1.07  -0.43  1.07  1.67  1.42  2314 
MEM 1   Lowry, Kyle 0.24  1.68  1.23  1.67  1.47  2.23  2089 
POR 3   Outlaw, Travis 2.52  1.36  -1.13  1.36  1.39  1.80  2186 
GSW 2   Jackson, Stephen 0.66  1.00  0.62  1.00  1.28  1.33  2855 
POR 2   Roy, Brandon 2.41  1.46  -1.23  1.47  1.18  1.95  2792 
CHA 3   Carroll, Matt 2.56  1.18  -1.40  1.18  1.16  1.57  2016 
DET 2   Hamilton, Richard 1.00  1.24  -0.08  1.24  0.93  1.66  2424 
BOS 2   Allen, Ray 2.73  1.00  -1.80  1.00  0.92  1.33  2624 
CHA 3   Richardson, Jason 1.46  1.04  -0.55  1.04  0.91  1.38  3149 
LAC 2   Mobley, Cuttino 0.77  1.01  0.12  1.01  0.89  1.34  2702 
PHX 3   Hill, Grant -1.73  1.00  2.51  1.00  0.78  1.33  2222 
PHI 2   Iguodala, Andre -1.56  1.29  2.29  1.29  0.73  1.71  3242 
PHI 5   Dalembert, Samuel -0.39  1.26  1.10  1.26  0.71  1.67  2724 
NO 4   West, David -3.14  1.31  3.83  1.31  0.69  1.74  2870 
DET 4   McDyess, Antonio -0.26  1.08  0.89  1.08  0.63  1.44  2285 
OKC 1   Watson, Earl 0.27  1.23  0.31  1.23  0.59  1.63  2273 
LAL 1   Fisher, Derek -1.09  1.06  1.55  1.06  0.46  1.41  2249 
PHO 2   Bell, Raja -1.97  1.05  2.15  1.05  0.18  1.39  2646 
CLE 5   Wallace, Ben -4.27  0.89  4.44  0.89  0.17  1.19  2205 
LAL 3   Odom, Lamar -3.53  1.00  3.64  1.00  0.11  1.33  2921 
MIL 2   Redd, Michael 4.55  1.23  -4.47  1.23  0.08  1.64  2702 
PHO 4   Diaw, Boris -1.82  1.02  1.84  1.02  0.03  1.36  2308 
UTA 4   Boozer, Carlos 1.97  1.26  -1.95  1.26  0.02  1.68  2827 
ATL 4   Horford, Al -1.46  1.39  1.47  1.39  0.01  1.85  2540 
TOR 2   Parker, Anthony 1.21  1.50  -1.36  1.50  -0.15  2.00  2634 
GSW 2   Ellis, Monta -0.14  1.27  -0.14  1.27  -0.27  1.69  3071 
UTA 5   Okur, Mehmet -0.37  1.17  0.05  1.17  -0.32  1.55  2390 
NYK 4   Randolph, Zach 0.87  1.04  -1.27  1.04  -0.40  1.38  2244 
SAC 4   Moore, Mikki -0.58  1.07  0.15  1.07  -0.42  1.42  2385 
3   Childress, Josh 2.34  1.10  -2.76  1.10  -0.42  1.47  2274 
PHI 1   Miller, Andre 2.56  1.20  -3.08  1.20  -0.52  1.60  3016 
TOR 1   Calderon, Jose 3.14  1.54  -3.69  1.54  -0.55  2.05  2484 
SA 1   Parker, Tony 0.67  1.29  -1.22  1.28  -0.56  1.71  2312 
CLE 5   Ilgauskas, Zydrunas -2.70  1.14  2.07  1.13  -0.62  1.51  2222 
CHI 1   Hinrich, Kirk -1.98  1.12  1.26  1.11  -0.72  1.48  2380 
WAS 2   Stevenson, DeShawn 0.93  1.11  -1.70  1.11  -0.77  1.48  2564 
WAS 2   Daniels, Antonio 0.88  1.12  -1.79  1.12  -0.91  1.50  2161 
SAC 3   Garcia, Francisco 1.82  1.24  -2.75  1.24  -0.93  1.65  2096 
HOU 1   Alston, Rafer -0.51  1.11  -0.46  1.11  -0.97  1.48  2526 
HOU 4   Scola, Luis -2.28  1.37  1.27  1.37  -1.02  1.83  2024 
CHI 3   Nocioni, Andres 1.02  1.17  -2.17  1.17  -1.15  1.56  2020 
SA 3   Bowen, Bruce -4.01  1.18  2.83  1.17  -1.18  1.56  2448 
MEM 3   Gay, Rudy 0.42  1.37  -1.74  1.37  -1.32  1.82  3000 
SAC 1   Udrih, Beno -0.97  1.26  -0.40  1.26  -1.37  1.68  2080 
NYK 4   Lee, David -1.35  1.28  -0.20  1.28  -1.55  1.70  2356 
CLE 1   Williams, Mo 0.86  1.27  -2.67  1.27  -1.81  1.69  2410 
DEN 3   Anthony, Carmelo 2.32  1.20  -4.15  1.20  -1.83  1.60  2807 
GSW 4   Harrington, Al -0.42  0.96  -1.45  0.96  -1.87  1.28  2190 
POR 1   Blake, Steve 0.03  1.19  -2.04  1.19  -2.01  1.59  2424 
SAC 4   Martin, Kevin 3.47  1.20  -5.63  1.20  -2.16  1.59  2216 
1   Navarro, Juan Carlos 1.55  1.36  -3.76  1.36  -2.21  1.81  2117 
SA 2   Finley, Michael -0.14  0.91  -2.13  0.91  -2.27  1.21  2204 
MIL 3   Jefferson, Richard 0.73  1.16  -3.18  1.16  -2.45  1.55  3201 
LAC 5   Kaman, Chris -4.86  1.19  1.97  1.19  -2.88  1.58  2083 
IND 4   Murphy, Troy -0.78  1.15  -2.11  1.15  -2.89  1.52  2106 
MEM 2   Jaric, Marko -2.97  1.10  -0.05  1.10  -3.02  1.46  2190 
CHI 2   Gordon, Ben 0.10  1.16  -3.47  1.16  -3.37  1.55  2291 
CHA 5   Okafor, Emeka -3.15  1.30  -0.26  1.30  -3.41  1.73  2718 
PHO 1   Barbosa, Leandro 1.85  1.02  -5.43  1.02  -3.58  1.36  2421 
CHI 4   Gooden, Drew -2.31  1.14  -1.36  1.14  -3.67  1.52  2122 
CHI 2   Hughes, Larry -1.06  1.04  -2.77  1.04  -3.83  1.38  2020 
POR 3   Webster, Martell -1.62  1.35  -2.29  1.35  -3.91  1.79  2132 
ATL 3   Williams, Marvin -1.03  1.16  -3.03  1.16  -4.07  1.55  2763 
IND 3   Granger, Danny -1.04  1.18  -3.28  1.18  -4.32  1.57  2876 
LAC 2   Davis, Ricky -2.17  1.03  -2.24  1.03  -4.41  1.37  2963 
MIN 4   Jefferson, Al -0.07  1.36  -4.54  1.36  -4.61  1.81  2919 
BOS 1   Rondo, Rajon -4.20  1.37  -0.48  1.37  -4.67  1.82  2306 
OKC 3   Green, Jeff -2.04  1.34  -3.03  1.34  -5.07  1.79  2253 
OKC 2   Durant, Kevin -1.36  1.43  -4.11  1.43  -5.47  1.90  2768 
SAC 2   Salmons, John -3.47  1.07  -2.08  1.07  -5.55  1.42  2517 
LAC 3   Thornton, Al -3.13  1.36  -4.55  1.36  -7.68  1.81  2158 
IND 1   Jack, Jarrett -5.48  1.33  -2.42  1.34  -7.89  1.78  2229 


Players with <2000 Minutes

Offense
Defense
Overall
 Team  Pos  Player
Adj +/-
Error
Adj +/-
Error
Adj +/-
Error
Minutes
UTA 1   Price, Ronnie 1.13  1.86  6.04  1.86  7.17  2.48  588 
DET 4   Johnson, Amir 2.65  2.05  3.97  2.05  6.62  2.73  764 
PHX 5   O'Neal, Shaquille 1.12  1.03  5.49  1.03  6.61  1.36  1748 
HOU 4   Landry, Carl 2.04  1.89  4.51  1.89  6.55  2.51  711 
HOU 5   Mutombo, Dikembe -1.26  1.49  7.17  1.49  5.91  1.98  619 
HOU 4   Hayes, Chuck -4.22  1.36  9.76  1.36  5.55  1.81  1575 
MIA 2   Wade, Dwyane 7.47  1.14  -2.06  1.15  5.41  1.52  1954 
PHI 3   Young, Thaddeus 4.11  1.38  1.21  1.38  5.32  1.84  1554 
NJ 4   Najera, Eduardo -0.18  1.18  5.47  1.18  5.29  1.57  1664 
MIA 5   Mourning, Alonzo -1.61  1.61  6.64  1.61  5.03  2.15  389 
CLE 1   Gibson, Daniel 2.59  1.19  2.37  1.19  4.96  1.59  1765 
MIN 4   Cardinal, Brian 2.71  1.62  2.17  1.63  4.88  2.16  440 
DET 4   Maxiell, Jason 3.27  1.24  1.46  1.24  4.73  1.65  1768 
POR 4   Frye, Channing 0.80  1.33  3.83  1.33  4.62  1.77  1342 
MIA 4   Lasme, Stephane -1.11  3.42  5.47  3.43  4.36  4.55  303 
BOS 2   Allen, Tony -0.30  1.27  4.45  1.27  4.15  1.70  1373 
LAL 3   Radmanovic, Vladimir 4.49  1.00  -0.57  1.00  3.93  1.33  1483 
SAC 4   Songaila, Darius 0.15  1.19  3.48  1.19  3.63  1.58  1554 
GSW 1   Dickau, Dan 3.57  1.49  -0.38  1.49  3.19  1.98  1040 
IND 5   Foster, Jeff -0.07  1.09  3.22  1.09  3.16  1.46  1884 
MIL 3   Griffin, Adrian -0.59  1.74  3.31  1.73  2.72  2.31  307 
SAN 3   Udoka, Ime -0.46  1.16  3.08  1.16  2.62  1.54  1316 
POR 5   Przybilla, Joel -2.39  1.31  4.99  1.31  2.60  1.75  1820 
MIA 1   Quinn, Chris 5.12  1.69  -2.75  1.69  2.38  2.25  1340 
MIA 3   Jones, James 3.31  1.23  -0.94  1.23  2.37  1.63  1276 
OKC 4   Wilcox, Chris 0.25  1.15  2.10  1.15  2.35  1.54  1739 
CHI 5   Noah, Joakim -0.60  1.43  2.71  1.43  2.10  1.90  1534 
CHA 3   Dudley, Jared 0.90  1.53  1.18  1.53  2.08  2.03  1384 
SAN 4   Thomas, Kurt -2.65  0.96  4.64  0.96  1.99  1.28  1584 
IND 2   Owens, Andre -1.37  2.43  3.29  2.43  1.92  3.24  392 
PHI 4   Evans, Reggie -1.73  1.10  3.57  1.09  1.85  1.46  1879 
SAC 5   Hawes, Spencer 1.32  1.88  0.51  1.87  1.84  2.50  931 
PHI 4   Brand, Elton 4.22  1.24  -2.39  1.24  1.83  1.66  274 
PHX 2   Giricek, Gordan -1.96  1.20  3.72  1.20  1.76  1.59  831 
UTA 3   Korver, Kyle 0.80  1.01  0.94  1.01  1.74  1.34  1733 
BOS 1   Cassell, Sam 1.02  1.09  0.61  1.09  1.62  1.45  1273 
DEN 3   Balkman, Renaldo -1.66  1.60  3.25  1.59  1.59  2.12  952 
IND 1   Tinsley, Jamaal 1.68  1.41  -0.17  1.41  1.51  1.88  1293 
DAL 1   Barea, Jose 0.44  2.26  1.06  2.27  1.50  3.01  460 
MEM 3   Jacobsen, Casey -1.61  1.63  3.06  1.63  1.45  2.16  547 
MIL 4   Gadzuric, Dan -1.56  1.62  2.97  1.62  1.41  2.16  534 
TOR 5   O'Neal, Jermaine -3.63  1.23  4.94  1.23  1.31  1.64  1206 
DEN 3   Patterson, Ruben 2.25  1.32  -0.93  1.32  1.31  1.76  327 
LAL 5   Bynum, Andrew 1.00  1.47  0.29  1.47  1.29  1.96  1008 
NY 3   Jeffries, Jared 0.47  1.12  0.81  1.12  1.28  1.49  1325 
ATL 5   Pachulia, Zaza -0.05  1.26  1.33  1.26  1.28  1.68  944 
CHI 4   Ruffin, Michael -0.56  1.60  1.79  1.60  1.23  2.14  632 
SAN 4   Horry, Robert -2.89  1.22  4.10  1.22  1.21  1.62  580 
PHI 5   Ratliff, Theo -2.68  1.41  3.85  1.41  1.17  1.87  437 
LAL 2   Vujacic, Sasha 5.92  1.22  -4.79  1.22  1.13  1.62  1279 
IND 2   Jones, Eddie 0.03  1.07  1.03  1.07  1.06  1.42  922 
DEN 5   Hilario, Nene -4.40  1.49  5.45  1.49  1.05  1.99  266 
SA 1   Mason, Roger 1.97  1.50  -0.97  1.51  1.00  2.00  1707 
NO 3   Bowen, Ryan -0.64  1.71  1.60  1.70  0.96  2.28  660 
ATL 5   Williams, Shelden -1.88  1.57  2.74  1.57  0.87  2.09  778 
CHI 4   Thomas, Tyrus -1.38  1.37  2.21  1.37  0.83  1.82  1330 
UTA 4   Millsap, Paul 0.03  1.44  0.75  1.43  0.77  1.91  1702 
NYK 1   Robinson, Nate 2.64  1.26  -1.91  1.26  0.72  1.67  1883 
BOS 5   Perkins, Kendrick -4.79  1.21  5.50  1.21  0.71  1.61  1913 
MIN 2   Snyder, Kirk 1.18  1.57  -0.49  1.57  0.69  2.10  761 
MIN 5   Harrison, David -1.78  1.64  2.39  1.64  0.61  2.19  702 
TOR 3   Kapono, Jason 0.63  1.17  -0.09  1.17  0.54  1.56  1530 
CLE 4   Varejao, Anderson -4.11  1.29  4.59  1.29  0.48  1.72  1320 
NJ 2   Dooling, Keyon -2.29  1.18  2.62  1.18  0.34  1.57  1334 
OKC 3   Gelabale, Mickael -0.39  1.91  0.64  1.91  0.26  2.55  465 
BOS 4   Powe, Leon -0.03  1.46  0.27  1.46  0.24  1.95  809 
HOU 2   Head, Luther -0.37  1.24  0.60  1.24  0.23  1.65  1379 
LAC 3   Thomas, Tim -1.06  1.03  1.25  1.03  0.19  1.37  1940 
SA 5   Oberto, Fabricio -1.57  1.14  1.76  1.14  0.19  1.52  1646 
CLE 2   Szczerbiak, Wally -0.27  0.96  0.39  0.97  0.12  1.28  1736 
UTA 2   Miles, C.J. 2.06  1.80  -1.99  1.79  0.07  2.39  689 
DAL 5   Diop, DeSagana -5.54  1.31  5.54  1.31  0.00  1.74  1296 
TOR 3   Graham, Joey 1.27  1.75  -1.31  1.75  -0.05  2.33  331 
LAL 3   Walton, Luke 0.57  1.06  -0.65  1.06  -0.08  1.41  1729 
ATL 1   Bibby, Mike 2.69  1.08  -2.78  1.08  -0.09  1.44  1574 
MIL 1   Jones, Damon 2.56  1.08  -2.67  1.08  -0.11  1.44  1336 
NO 3   Wright, Julian 2.01  1.85  -2.29  1.85  -0.28  2.46  640 
GSW 2   Watson, C.J. 0.45  2.61  -0.79  2.61  -0.33  3.47  368 
ATL 3   Evans, Maurice 0.63  1.06  -0.97  1.05  -0.34  1.40  1721 
HOU 2   Barry, Brent 2.11  1.12  -2.53  1.12  -0.41  1.49  554 
LAL 3   Ariza, Trevor -1.55  1.48  1.13  1.47  -0.42  1.97  546 
1   Pargo, Jannero -1.05  1.26  0.54  1.26  -0.51  1.67  1497 
IND 5   Nesterovic, Rasho -4.63  1.10  4.05  1.10  -0.58  1.47  1486 
IND 3   Daniels, Marquis -1.49  1.10  0.85  1.10  -0.64  1.46  1546 
WAS 1   Dixon, Juan 0.25  1.25  -0.91  1.25  -0.67  1.66  668 
CHI 3   Sefolosha, Thabo -3.24  1.41  2.54  1.40  -0.70  1.88  1436 
MIA 5   Magloire, Jamaal -3.18  1.40  2.43  1.40  -0.75  1.86  286 
NYK 1   Duhon, Chris -1.98  1.19  1.21  1.19  -0.77  1.59  1490 
DAL 3   George, Devean -2.25  1.21  1.45  1.21  -0.80  1.61  821 
TOR 2   Delfino, Carlos -0.92  1.20  0.11  1.19  -0.82  1.59  1928 
POR 5   LaFrentz, Raef 0.54  1.68  -1.41  1.68  -0.88  2.23  291 
SA 4   Bonner, Matt -0.66  1.38  -0.24  1.38  -0.91  1.84  852 
MIN 1   Ollie, Kevin -0.97  1.73  0.04  1.73  -0.93  2.31  300 
MIN 2   Carney, Rodney 0.33  1.50  -1.30  1.50  -0.96  2.00  1038 
CHI 5   Gray, Aaron 0.64  2.06  -1.61  2.06  -0.97  2.75  613 
WAS 1   Arenas, Gilbert 0.18  1.29  -1.19  1.29  -1.02  1.71  425 
DEN 3   Kleiza, Linas 1.41  1.20  -2.47  1.20  -1.05  1.59  1889 
OKC 4   Smith, Joe -0.78  0.96  -0.29  0.96  -1.06  1.28  1726 
UTA 5   Collins, Jarron -3.58  1.42  2.52  1.42  -1.06  1.89  702 
NJ 2   Simmons, Bobby -1.43  1.19  0.36  1.19  -1.07  1.58  1521 
OKC 2   Mason, Desmond -1.45  1.11  0.36  1.11  -1.09  1.47  1702 
MEM 2   Crittenton, Javaris -0.91  1.97  -0.18  1.97  -1.09  2.62  678 
CHA 2   Anderson, Derek -2.32  1.46  1.21  1.46  -1.11  1.95  396 
MIN 5   Collins, Jason -6.49  1.21  5.35  1.21  -1.15  1.62  1172 
NO 3   Butler, Rasual -2.57  1.31  1.36  1.31  -1.21  1.75  875 
NO 5   Ely, Melvin -2.66  1.49  1.44  1.49  -1.22  1.98  621 
NO 1   James, Mike -0.26  1.14  -0.99  1.14  -1.24  1.52  720 
CLE 1   Snow, Eric -3.69  1.31  2.44  1.30  -1.25  1.74  305 
NJ 1   Armstrong, Darrell -0.89  1.39  -0.41  1.39  -1.31  1.85  548 
DAL 4   Williams, Shawne -1.06  1.56  -0.33  1.56  -1.40  2.07  967 
UTA 1   Knight, Brevin -2.94  1.20  1.47  1.20  -1.46  1.60  1674 
CLE 2   West, Delonte -2.40  1.07  0.88  1.07  -1.52  1.43  1534 
NYK 1   Marbury, Stephon 2.62  1.20  -4.14  1.20  -1.52  1.60  805 
MIN 3   Brewer, Corey -2.66  1.48  1.14  1.48  -1.53  1.97  1802 
BOS 4   Scalabrine, Brian -2.81  1.50  1.20  1.49  -1.60  1.99  512 
NYK 2   Richardson, Quentin -2.46  1.07  0.83  1.07  -1.63  1.42  1841 
PHI 1   Williams, Louis -1.65  1.45  -0.09  1.45  -1.74  1.92  1862 
NO 2   Peterson, Morris -0.73  1.10  -1.02  1.10  -1.75  1.47  1796 
MIL 4   Croshere, Austin -1.47  1.46  -0.28  1.46  -1.75  1.94  457 
MIN 5   Richard, Chris 0.06  2.29  -1.82  2.29  -1.76  3.04  556 
SAC 1   Jackson, Bobby -3.06  1.16  1.25  1.16  -1.80  1.55  1389 
DEN 2   Smith, J.R. 1.98  1.18  -3.82  1.19  -1.83  1.58  1421 
MIL 4   Allen, Malik -0.91  1.29  -0.93  1.29  -1.84  1.72  1096 
CLE 2   Pavlovic, Sasha -4.56  1.19  2.70  1.19  -1.87  1.58  1188 
MEM 3   Walker, Antoine -2.53  1.05  0.47  1.05  -2.06  1.40  892 
LAL 5   Mbenga, DJ -1.74  2.28  -0.34  2.27  -2.08  3.03  326 
MIL 2   Bell, Charlie -0.86  1.27  -1.29  1.27  -2.14  1.69  1628 
NO 3   Posey, James -2.66  0.95  0.47  0.95  -2.19  1.26  1821 
MIN 5   Doleac, Michael -4.27  1.77  2.01  1.77  -2.25  2.36  257 
MIL 1   Ridnour, Luke 0.58  1.26  -2.90  1.26  -2.32  1.68  1219 
BOS 1   House, Eddie -1.79  1.23  -0.61  1.23  -2.40  1.63  1480 
IND 1   Ford, T.J. -0.23  1.43  -2.18  1.42  -2.41  1.90  1199 
NJ 5   Krstic, Nenad -4.04  1.41  1.63  1.41  -2.41  1.88  812 
ORL 4   Cook, Brian -1.40  1.42  -1.02  1.42  -2.42  1.88  629 
MIL 5   Voskuhl, Jake -3.60  1.67  1.15  1.67  -2.44  2.22  386 
MIA 5   Barron, Earl -6.36  1.86  3.88  1.86  -2.48  2.48  889 
MIA 3   Wright, Dorell -2.89  1.44  0.39  1.43  -2.50  1.91  1104 
MEM 1   Conley, Mike 3.64  1.83  -6.19  1.83  -2.55  2.44  1381 
MEM 5   Milicic, Darko -5.59  1.25  3.01  1.25  -2.58  1.67  1663 
IND 1   Diener, Travis 0.45  1.55  -3.03  1.55  -2.58  2.06  1356 
ORL 4   Garrity, Pat -4.85  2.03  2.24  2.04  -2.62  2.70  284 
DET 1   Stuckey, Rodney -2.23  1.56  -0.42  1.56  -2.65  2.08  1081 
GSW 4   Wright, Brandan -3.35  2.55  0.70  2.54  -2.66  3.39  376 
ORL 2   Pietrus, Mickael -1.41  1.14  -1.27  1.14  -2.68  1.51  1316 
NO 3   Wells, Bonzi -2.61  1.05  -0.08  1.05  -2.69  1.39  1561 
MEM 2   Ross, Quinton -3.47  1.20  0.79  1.20  -2.69  1.60  1505 
GSW 4   Turiaf, Ronny -2.93  1.30  0.22  1.30  -2.71  1.73  1458 
PHI 5   Smith, Jason -1.20  1.67  -1.56  1.67  -2.76  2.22  1106 
NYK 4   Rose, Malik -2.89  1.51  0.03  1.51  -2.86  2.01  494 
PHI 2   Rush, Kareem -0.80  1.18  -2.22  1.18  -3.03  1.58  1504 
GSW 2   Azubuike, Kelenna -1.59  1.25  -1.46  1.25  -3.04  1.66  1732 
MIL 1   Sessions, Ramon 2.51  2.52  -5.62  2.53  -3.12  3.36  450 
MIL 5   Elson, Francisco -6.37  1.27  3.21  1.27  -3.16  1.70  813 
ORL 1   Johnson, Anthony -2.83  1.15  -0.43  1.14  -3.26  1.52  1536 
DET 5   Brown, Kwame -4.38  1.31  1.11  1.31  -3.26  1.74  713 
PHO 3   Barnes, Matt 0.75  1.14  -4.08  1.14  -3.33  1.51  1414 
TOR 5   Bargnani, Andrea -2.04  1.28  -1.33  1.28  -3.37  1.71  1861 
MIA 5   Blount, Mark -5.30  1.17  1.91  1.17  -3.39  1.56  1542 
MIN 4   Smith, Craig -0.83  1.42  -2.62  1.42  -3.45  1.89  1546 
MIA 3   Diawara, Yakhouba -5.16  1.68  1.65  1.68  -3.51  2.24  542 
OKC 5   Petro, Johan -6.63  1.34  3.10  1.34  -3.53  1.78  1303 
2   Hardaway, Anfernee -4.03  1.86  0.42  1.87  -3.61  2.48  325 
NYK 2   Jones, Fred -1.82  1.05  -1.81  1.05  -3.62  1.40  1756 
PHI 2   Green, Willie -2.53  1.27  -1.10  1.27  -3.63  1.70  1970 
DAL 2   Green, Gerald -3.74  1.83  0.11  1.83  -3.63  2.43  361 
DET 2   Afflalo, Arron -5.69  1.86  2.02  1.84  -3.67  2.47  970 
UTA 3   Harpring, Matt 0.44  1.23  -4.13  1.23  -3.70  1.64  1374 
LAL 1   Farmar, Jordan -5.19  1.44  1.48  1.44  -3.71  1.92  1690 
BOS 4   Davis, Glen -2.84  1.62  -0.87  1.61  -3.71  2.15  940 
PHI 4   Marshall, Donyell -0.77  1.32  -2.95  1.32  -3.72  1.75  340 
NJ 5   Boone, Josh -3.13  1.42  -0.59  1.43  -3.73  1.90  1773 
ORL 1   Nelson, Jameer -2.80  1.41  -0.94  1.41  -3.74  1.88  1961 
MIA 4   Haslem, Udonis -3.78  1.25  0.04  1.25  -3.74  1.67  1806 
MIN 2   Foye, Randy -1.72  1.46  -2.20  1.46  -3.92  1.95  1259 
MIA 5   Johnson, Alexander 0.33  1.94  -4.31  1.94  -3.98  2.58  549 
NJ 4   Swift, Stromile -4.29  1.27  0.28  1.28  -4.01  1.70  845 
GSW 1   Williams, Marcus 2.35  1.63  -6.37  1.63  -4.02  2.17  854 
TOR 4   Humphries, Kris -5.95  1.52  1.89  1.52  -4.05  2.02  925 
MEM 3   Buckner, Greg -1.35  1.36  -2.73  1.36  -4.07  1.81  519 
DEN 1   Atkins, Chucky -0.40  1.29  -3.73  1.29  -4.13  1.72  352 
DEN 1   Parker, William (Smush) -3.35  1.44  -0.90  1.44  -4.25  1.92  592 
CHA 5   Hollins, Ryan -1.91  2.14  -2.37  2.14  -4.27  2.85  532 
DAL 2   Stackhouse, Jerry -1.29  1.10  -3.01  1.10  -4.31  1.47  1412 
NJ 3   Jianlian, Yi -5.62  1.59  1.29  1.60  -4.33  2.12  1647 
DAL 3   Wright, Antoine -6.84  1.33  2.28  1.33  -4.56  1.77  1231 
MIN 1   Telfair, Sebastian -3.45  1.26  -1.20  1.26  -4.64  1.67  1933 
DAL 5   Dampier, Erick -5.88  1.20  1.20  1.20  -4.68  1.60  1755 
MIA 5   Anthony, Joel -0.91  2.71  -3.86  2.71  -4.77  3.61  498 
ATL 2   Murray, Ronald (Flip) -1.39  1.19  -3.41  1.19  -4.80  1.58  873 
LAC 1   Williams, Jason -0.32  1.12  -4.52  1.12  -4.84  1.49  1886 
CHA 1   Boykins, Earl 0.30  1.26  -5.20  1.26  -4.90  1.68  577 
MIL 1   Lue, Tyronn -0.19  1.29  -4.98  1.29  -5.17  1.72  736 
WAS 5   Blatche, Andray -4.23  1.41  -0.96  1.41  -5.19  1.88  1675 
NYK 3   Chandler, Wilson -3.48  1.93  -1.76  1.93  -5.25  2.57  685 
LAC 1   Hart, Jason -5.12  1.50  -0.16  1.50  -5.28  2.00  606 
DET 3   Herrmann, Walter -3.90  2.03  -1.38  2.03  -5.28  2.70  373 
LAC 4   Skinner, Brian -4.72  1.27  -0.74  1.26  -5.46  1.69  844 
NJ 3   Hayes, Jarvis -2.72  1.22  -2.76  1.22  -5.49  1.62  1287 
DEN 1   Carter, Anthony -2.34  1.24  -3.16  1.24  -5.50  1.65  1960 
1   McInnis, Jeff -1.32  1.16  -4.20  1.16  -5.53  1.55  1410 
NJ 5   Williams, Sean -4.34  1.54  -1.20  1.54  -5.53  2.05  1278 
MIL 4   Villanueva, Charlie -1.22  1.30  -4.55  1.30  -5.78  1.73  1829 
PHI 2   Ivey, Royal -2.15  1.37  -3.65  1.36  -5.80  1.82  1437 
ORL 2   Redick, J.J. 2.71  2.35  -8.54  2.36  -5.83  3.13  276 
MIA 2   Cook, Daequan -3.73  1.57  -2.15  1.57  -5.88  2.09  1441 
ORL 1   Arroyo, Carlos -1.28  1.28  -4.63  1.28  -5.91  1.70  1269 
MIA 3   Powell, Kasib -8.55  3.30  2.64  3.31  -5.91  4.39  304 
1   Stoudamire, Damon -2.61  1.24  -3.35  1.24  -5.97  1.65  1037 
MIA 1   Banks, Marcus -2.20  1.48  -3.90  1.47  -6.11  1.96  569 
SAC 2   Douby, Quincy -5.33  1.79  -1.00  1.79  -6.33  2.38  876 
NJ 3   Nachbar, Bostjan -4.31  1.17  -2.10  1.17  -6.41  1.55  1659 
SAC 3   Jones, Bobby -9.50  1.99  3.04  1.99  -6.46  2.65  530 
DEN 4   Howard, Juwan -2.84  1.35  -3.62  1.35  -6.46  1.80  346 
NO 2   Brown, Devin -3.14  1.06  -3.34  1.06  -6.48  1.42  1762 
ORL 5   Jones, Dwayne -2.99  2.26  -3.52  2.26  -6.51  3.01  473 
2   Stoudamire, Salim -1.73  1.76  -4.80  1.77  -6.53  2.35  401 
SAC 4   Thomas, Kenny -5.90  1.34  -0.67  1.34  -6.57  1.79  281 
NJ 2   Hassell, Trenton -2.93  1.22  -3.76  1.21  -6.69  1.62  773 
NYK 2   Collins, Mardy -5.05  1.85  -1.78  1.85  -6.83  2.46  634 
WAS 2   Young, Nick -4.24  1.81  -2.81  1.82  -7.05  2.41  1158 
ORL 5   Foyle, Adonal -7.68  1.55  0.62  1.55  -7.05  2.07  774 
NO 5   Armstrong, Hilton -4.35  1.85  -2.79  1.84  -7.14  2.46  732 
TOR 5   Brezec, Primoz -1.34  1.51  -5.81  1.51  -7.15  2.01  475 
NJ 2   Storey, Awvee -2.82  3.04  -4.45  3.03  -7.27  4.05  258 
ATL 1   Law, Acie -5.12  1.74  -2.21  1.74  -7.33  2.32  865 
WAS 5   Pecherov, Oleksiy -0.62  2.91  -6.71  2.93  -7.33  3.88  320 
CHA 5   Mohammed, Nazr -4.77  1.06  -2.57  1.06  -7.34  1.41  1650 
LAL 5   Mihm, Chris -4.41  1.68  -3.06  1.68  -7.47  2.24  278 
SA 1   Vaughn, Jacque -5.91  1.30  -1.62  1.29  -7.53  1.73  1139 
HOU 1   Brooks, Aaron -1.34  2.21  -6.18  2.22  -7.53  2.94  608 
HOU 2   Strawberry, D. J. -4.84  2.89  -2.89  2.87  -7.73  3.85  270 
DAL 4   Bass, Brandon -6.78  1.38  -1.34  1.38  -8.12  1.84  1556 
OKC 2   Wilkins, Damien -3.15  1.18  -4.97  1.18  -8.13  1.57  1843 
POR 4   Diogu, Ike -5.31  1.99  -3.58  2.00  -8.89  2.65  305 
CHA 4   Davidson, Jermareo -7.18  2.77  -1.86  2.77  -9.04  3.69  322 
NYK 5   Curry, Eddy -5.80  1.18  -3.58  1.18  -9.37  1.57  1530 
LAC 4   Williams, Aaron -6.92  1.80  -2.77  1.80  -9.69  2.40  331 
MEM 4   Warrick, Hakim -6.03  1.28  -3.90  1.28  -9.93  1.70  1754 
LAL 3   Newble, Ira -6.60  1.49  -3.49  1.48  -10.08  1.98  698 
LAL 4   Powell, Josh -4.12  1.53  -6.54  1.53  -10.66  2.03  1227 
CHA 4   Brown, Andre -2.38  2.67  -8.66  2.68  -11.04  3.55  286 
POR 1   Rodriguez, Sergio -7.72  2.04  -3.80  2.04  -11.52  2.72  628 
WAS 4   McGuire, Dominic -9.06  2.16  -3.01  2.14  -12.07  2.87  695 
LAC 3   Novak, Steve -1.95  2.75  -10.18  2.76  -12.13  3.67  264 

Noise Reduction
As noted above, adjusted plus-minus ratings ?as mathematical estimates ?tend to be noisy. For example, Ilardi’s 2006-2007 ratings had a typical standard error of about 3.0, which implies a margin of error for each player estimate (at a 95% confidence level) of roughly +/- 5.9 points [3]. Obviously, the presence of so much noise greatly limits the usefulness of the metric. To take a telling example: Chris Paul had a 2006-2007 rating of + 4.1 pts (per 40 minutes), with a margin of error of roughly +/- 5.9; so, we can only say with 95% confidence that his performance was somewhere between that of a sub-mediocre player (-1.8) and one of the top players in the league (+10.0)!

But where does the noise come from, and how can it be eliminated? Mostly, it results from the fact that teams tend to put the same players on the court together at the same time. That is, many players?minutes are strongly inter-correlated, so the underlying adjusted plus-minus model has a hard time disentangling individual player effects at a high level of accuracy. In addition, the number of unique observations (i.e., lineups) of a given player in a single season is surprisingly small [4], typically under 1,000.

However, there exists a straightforward solution to both of these problems: use multiple seasons?worth of data. How many seasons of data are needed? As many as possible. With only one season, the standard errors are very high ?typically around 5.0 points per 100 possessions (with corresponding margins of error of roughly +/- 10.0 points). Such ratings, of course, are highly suspect. With two seasons of data, the noise level drops by about 40%, but it’s still uncomfortably high. The picture, however, gets appreciably clearer with each additional season that’s added to the model.

In this article, we present adjusted plus-minus ratings that utilize five seasons?worth of data ?ratings with an unprecedented lack of noise. As you’ll see, the average standard errors have now dropped down to about 1.1 points per 100 possessions [5], 80% lower than the errors based on only one season. This makes them dramatically more useful.

Of course, many readers will want to know how a given player performed last year (2007-2008), not how he performed on average over the past five seasons. Fortunately, the inputs to the adjusted plus-minus model can be weighted to emphasize data from the most recent season. This approach increases the noise level slightly in comparison with an equal weighting of all seasons, but it still reduces it substantially. Accordingly, the ratings herein can be regarded as estimates of 2007-2008 player performance, with data from the previous four seasons included in the model at fractional weightings [6] for the purpose of reducing estimation errors.

Separating Offense from Defense
We’ve also taken another important step toward enhancing the usefulness of the adjusted plus-minus metric by including a separate analysis of each player’s performance on the offensive and defensive sides of the ball. To do so, we’ve constructed a separate variable to model each player’s possessions on offense (Xi) and defense (Di), respectively. Thus, for each unique 10-player lineup that appears in each game, two separate lines of data are generated for use in the ensuing regression model: one line reflects scoring efficiency (points per 100 possessions) during those possessions when the home team is on offense and the away team on defense, and one represents the converse scenario (i.e., possessions in which the away team is on offense and the home team on defense). The model can be rendered as follows:

EFFICIENCY = b0 + b1X1 + b2X2 + . . . + bKXK + bK+1D1 + bK+2D2 + . . . + bK+KXK + homeflag + e, where

EFFICIENCY = 100 * (points per possession of home team [or away team])

X1 = 1 for possessions when player 1 is on offense, = 0 if player 1 is not playing (or playing on defense)
XK = 1 if player K is on offense, = 0 if player K is not playing (or on defense)
D1 = -1 for possessions when player 1 is on defense, = 0 if player 1 is not playing (or playing on offense)
DK = -1 if player K is on defense, = 0 if player K is not playing (or on offense)

homeflag = 1 for possessions when home team is on offense, = 0 when away team is on offense
e = error term

b0 (a constant) measures the average efficiency of all lineups
b1 measures the difference in offensive efficiency between player 1 and reference players
bK measures the difference in offensive efficiency between player K and reference players
bK+1 measures the difference in defensive efficiency between player 1 and reference players
bK+K measures the difference in defensive efficiency between player K and reference players
homeflag measures the efficiency advantage of the home team

It’s important to note at this juncture that Eli Witus, in an important article at his Count the Basket blog, has used a somewhat different, two-step estimation technique to derive a set of offensive and defensive adjusted plus-minus estimates for the 2007-2008 season. However, the offensive and defensive ratings presented herein have the virtue of possessing much lower standard errors. This is attributable to two factors: (a) we’ve used five seasons?worth of data, whereas Eli used just one; and (b) we have incorporated player offensive and defensive effects directly into our model, rather than utilizing a two-step (indirect) procedure to derive them from a net (offense + defense) estimate.

Observations
In reflecting on these ratings, we have hit upon a number of potentially provocative observations, among them (in no particular order):

  • Kevin Garnett was arguably the league’s most valuable player last year, with an astonishingly high net adjusted plus-minus rating of +14.47, nearly 3 points higher than that of the next highest player, Lebron James (+11.65). This difference was statistically significant (i.e., outside the margin of error).
  • Both Garnett and LeBron James had a more substantial impact than Kobe Bryant, a fact largely attributable to Bryant’s merely average defensive rating.
  • Although Chris Paul was one of the Top 3 offensive players in the league (+9.24), he was actually a liability on the defensive side of the ball (-4.54). As a result, his overall contribution (+4.69), while impressive, was not commensurate with that of a top MVP candidate.
  • Short point guards (Paul, Tony Parker, Mike Conley, T.J. Ford, Damon Stoudamire, etc.) pretty consistently show up in the model as defensive liabilities. We leave it to interested readers to provide a definitive explanation for this phenomenon. (We suspect it owes at least in part to their difficulty contending jump shots, especially when called upon in rotations to defend players that may be 6 or more inches taller.)
  • Because of his defensive prowess (+4.52), Ron Artest was rated as one of the top 20 players in the league last year. Historically, Houston has obtained many players whose impressive adjusted plus-minus ratings belie their less-heralded reputations: Artest (+6.31), Carl Landry (+6.55), Chuck Hayes (+5.55), and Shane Battier (+2.69). If Yao Ming (+4.77) and McGrady (+2.85) remain reasonably healthy this year, the Rockets may be regarded as the leading contenders to emerge from the West.
  • Due to their poor defense and (in many cases) inconsistent overall effort, several players with eye-popping boxscore stats appear to be much less valuable than widely believed, among them: Carmelo Anthony (-1.83), Al Jefferson (-4.61), Kevin Martin (-2.16), Richard Jefferson (-2.45), Ben Gordon (-3.37), Jose Calderon (-0.55), Michael Redd (+0.08), and Carlos Boozer (+0.02).
  • Jamario Moon, the 27-year-old rookie who cracked the Raptors?starting lineup last season with his relentless hustle and off-the-charts athleticism, emerged as one of the league’s better overall players, with a total adjusted plus-minus rating of + 7.07. Because there was only one season’s worth of data available to gauge his performance, his standard error was on the high side (1.82) [7]; still, we can be 95% confident that his true overall value last year was at least +3.6 points [8] per 100 possessions ?i.e., at worst, he was very good. It remains to be seen if this was something of a fluke, or if his performance this season will justify his elevation to the pantheon of elite NBA players.
Future Directions
We believe the offensive, defensive, and overall adjusted plus-minus ratings presented herein provide a useful set of metrics for evaluating the contributions of NBA players. In fact, the defensive ratings are particularly valuable because, unlike boxscore based ratings, they measure the total performance of a player on defense, rather than a small subset of activities.

Nevertheless, there are several ways in which the value of adjusted plus-minus analyses may be enhanced even further in the future:

  1. Further reductions in estimation errors. Although the present error levels are greatly improved, it’s always desirable to bring the “noise?level of estimates down as low as possible. This can be accomplished, in part, by adding even more seasons?worth of data to the model.

    Dan Rosenbaum has also demonstrated the possibility of reducing standard errors further by augmenting (blending) adjusted plus-minus estimates with regression-based estimates derived from other statistics (statistical plus-minus). Although this approach runs the risk of introducing some bias into the ensuing estimates, we believe this risk can be reduced by extending the approach to encompass: (a) the use of non-boxscore-based metrics, such as the many innovative stats available on nbatop.com (e.g., defensive eFG% allowed), and (b) separate statistical plus-minus estimates for the metric’s offensive and defensive components.

  2. Player interactions. Some players seem to be more effective ?or less effective ?when they are joined on the court by certain teammates. In statistical terms, this is referred to as a player-by-player interaction effect: the performance of Player A is contingent upon the presence/absence of Player B. Such effects are of great interest to coaches, GMs, and fans alike, and they can ?in principle ?be evaluated within the statistical framework of adjusted plus-minus models.
  3. Coaching effects. Does coaching really matter, above and beyond the talent a coach has available to put on the floor? It’s an empirical question, one which our adjusted plus-minus model ?with 5 seasons?worth of data to draw upon ?can now begin to answer. By entering each coach as a separate random variable, we will be able to derive a precise estimate of the added value of each team’s coach, above and beyond the effect of the team’s players.
  4. Time-trend analyses. What impact, from an adjusted plus-minus standpoint, is a given player likely to have in the 2008-2009 season? The answer is largely predictable, in principle, on the basis of his past performance, age, years in the league, position, and so on. But the precise relationship of these variables to adjusted plus-minus performance is presently unknown: a question for future study. It has been shown by Ed Kupfer, for example, that performance on several metrics (shooting percentage, passing, rebounding) generally peaks in the mid-late 20’s: it will be interesting to see if the same holds true for adjusted plus-minus ratings.

Footnotes

[1] This approach relies on the same mathematical tools employed by epidemiologists when they need to estimate, say, the harmful effects of a particular toxin in the environment while controlling for the effects of all other relevant hazards: asbestos, cigarette smoke, radon, smog, etc.

[2] The process was independently developed by at least three different sources ?Dan Rosenbaum, Sagarin and Winston (for the Dallas Mavericks), and Ilardi (for the Kansas Jayhawks) ?and fully described in a seminal article by Rosenbaum.

[3] It’s standard in statistics to derive the margin of error by constructing a 95% confidence interval around each estimate ?i.e., a range within which we can be 95% confident the true value of the variable actually falls. The standard error of each player’s estimate can be used to construct this interval, since the actual value of each player’s rating will fall in a roughly normal (bell-shaped) curve around the estimate, with a distribution defined in part by the magnitude of this standard error (se). Simply put: we can be 95% confident that the true value falls in a range defined by +/- 1.96(se). Since the typical standard error in the 2006-07 analyses was roughly 3.0, the margin of error equals +/- (1.96)(3.0).

[4] The basic unit of observation for the model is the lineup ?a set of 10 players that appear together on the court for a given number of possessions, until some further change in lineup (for either team) occurs.

[5] For the separate offensive and defensive ratings of high-minutes players; the standard errors are a bit higher for total (net) ratings.

[6] Specifically, the 2007-2008 season was weighted at 1.0, the 2006-2007 at 0.2, and the 2003-2006 seasons at 0.125. For all five seasons, playoffs were accorded double the weighting of the regular season ?a means of accounting for the heightened importance of the NBA’s “second season.?

[7] The standard error of the total score can be estimated using the variance sum law. It’s approximately equal to the square root of the following quantity: (seoffense)2 + (sedefense)2 ?2(.107) (seoffense) (sedefense)

[8] That is, 7.07 ?(1.96)(1.82)


About the Authors

Steve Ilardi is a professor of clinical psychology at the University of Kansas, and former statistical consultant to the KU men’s basketball team under Roy Williams. With the support of assistant coaches Jerod Haase and Ben Miller, Ilardi developed and implemented an adjusted plus-minus model of player evaluation at KU, one similar to the models independently developed by Dan Rosenbaum and Jeff Sagarin. In his ‘day job? Ilardi is a clinical researcher who has worked to develop a novel, lifestyle-based treatment for depressive illness.
- For more on his treatment of depression, see www.psych.ku.edu/TLC
- For his KU staff page see http://www.psych.ku.edu/psych_people/faculty_Stephen_Ilardi.shtml.


Aaron Barzilai “played?on the varsity basketball team at MIT as an undergraduate before earning his Ph.D. in Mechanical Engineering at Stanford University. He currently works as a consultant for a global consulting firm and has experience in the pharmaceutical, financial services, and online publishing industries. Aaron developed the website basketballvalue.com and would like to spend more time on basketball analytics. He can be contacted via email at [email protected]


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