Journal of Quantitative Analysis in
Sports
Volume 7, Issue 2
2011
Article 5
Stratified Odds Ratios for Evaluating NBA
Players Based on their Plus/Minus Statistics
Douglas M. Okamoto, Data to Information to Knowledge
Recommended Citation:
Okamoto, Douglas M. (2011) "Stratified Odds Ratios for Evaluating NBA Players Based on
their Plus/Minus Statistics," Journal of Quantitative Analysis in Sports: Vol. 7: Iss. 2, Article 5.
Available at: http://www.bepress.com/jqas/vol7/iss2/5
DOI: 10.2202/1559-0410.1320
©2011 American Statistical Association. All rights reserved.
Stratified Odds Ratios for Evaluating NBA
Players Based on their Plus/Minus Statistics
Douglas M. Okamoto
Abstract
In this paper, I estimate adjusted odds ratios by fitting stratified logistic regression models to
binary response variables, games won or lost, with plus/minus statistics as explanatory variables.
Adapted from ice hockey, the plus/minus statistic credits an NBA player one or more points
whenever his team scores while he is on the basketball court. Conversely, the player is debited
minus one or more points whenever the opposing team scores. Throughout the NBA season, the
league’s better players are likely to have positive plus/minus statistics as reported by
Yahoo!Sports and 82games.com. Crude or unadjusted odds ratios estimate the relative
probabilities of a player having a positive plus/minus in a win, versus a negative plus/minus in a
loss. Home and away games are twin strata with teams playing 41 home games and 41 road games
during an 82-game regular season. Stratum-specific odds ratios vary because some players
perform better at home than on the road and vice versa. In order to adjust for home court
advantage, stratified odds ratios and their 95 percent confidence intervals are estimated for each of
the Los Angeles Lakers during the 2009–2010 regular season.
KEYWORDS: plus/minus statistic, odds ratio, logistic regression model
Okamoto: Stratified Odds Ratios for Evaluating NBA Players
1. Introduction
Adapted from ice hockey, the plus/minus statistic credits an NBA player one or
more points whenever his team scores while he is on the basketball court.
Conversely, the player is debited one or more points whenever the opposing team
scores. At the end of the game, a player’s pluses and minuses are totaled to get his
plus/minus (+/-) statistic. The Lakers won the Pacific Division of the Western
Conference with a 57-25 win/loss record during the 2009-2010 regular season. In
Figure 1, blue circles indicate Laker wins and red circles indicate Laker losses.
Each dot corresponds to an ordered pair (x, y), with the x-coordinate equal to
Kobe Bryant’s +/- and the y-coordinate equal to the winning or losing margin.
When Kobe is a plus, {X > 0}, a minus, {X < 0}; when the Lakers win, {Y>0},
they lose, {Y<0}.
50
40
LA Laker Winning or Losing Margin
30
20
10
0
-30
-20
-10
0
10
20
30
40
50
-10
-20
-30
Kobe Bryant Plus/Minus
Win
Loss
Figure 1. Scatterplot of Los Angeles Laker Winning or Losing Margin vs.
Kobe Bryant Plus/Minus: 2009–2010 NBA Regular Season (73 Games)
Kobe was a plus in 48 of 51 Laker wins (NE quadrant) and 5 of 22 losses
(SE quadrant). He was a minus in 3 Laker wins (NW quadrant), and 17 losses
(SW quadrant). Not shown in the scatterplot are 9 Laker games (6 wins, 3 losses)
in which Kobe did not play.
Published by Berkeley Electronic Press, 2011
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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 2, Art. 5
2. Odds Ratios
The relative odds of the Lakers winning or losing when Kobe was a plus or minus
are defined in terms of the conditional probabilities of Y given X, where the
joint probability distribution of two binary random variables is as follows: a)
{X=1} if Kobe was a plus, {X=0} if Kobe was a minus; and b) {Y=1} if the
Lakers won, {Y=0} if the Lakers lost. The odds ratio (OR) is the relative
probability of a Laker win when Kobe is a plus divided by the relative probability
of a Laker win when he is a minus:
equals [(48)(17)]/[(5)(3)] = 54 or 54:1, calculated from the cross product of four
cell counts in the following two-by-two contingency table.
Table 1. Laker Wins/Losses vs. Kobe Bryant Plus/Minus
Win
Loss
Totals
Minus
3
17
20
Plus
48
5
53
Totals
51
22
73
The Los Angeles Lakers were 54 times more likely to have won when Kobe
was plus in a win. Conversely, the Lakers were 54 times more likely to have lost
when he was minus in a loss. Despite his 54:1 odds ratio, Kobe Bryant ranked
second to Laker forward Pau Gasol whose [(40)(14)]/[(4)(2)] = 70 or 70:1 odds
ratio calculated from Table 2 led the team in the 2009–2010 regular season.
Table 2. Laker Wins/Losses vs. Pau Gasol Plus/Minus
Win
Loss
Totals
Minus
4
14
18
Plus
40
2
42
Totals
44
16
60
Odds ratios and 95 percent confidence intervals for Kobe Bryant, Pau Gasol
and eight of their Laker teammates are shown in Figure 2.
http://www.bepress.com/jqas/vol7/iss2/5
DOI: 10.2202/1559-0410.1320
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Okamoto: Stratified Odds Ratios for Evaluating NBA Players
70
Pau Gasol
54
Kobe Bryant
31
Ron Artest
27
Derek Fisher
16
Lamar Odom
10
Andrew Bynum
7.8
Luke Walton
6.2
Jordan Farmar
Shannon Brown
Sasha Vujacic
1.0
3.4
3.1
10.0
100.0
1000.0
10000.0
Figure 2. Odds Ratio Chart for the Los Angeles Lakers: 2009–2010 NBA
Regular Season (82 Games)
3. Logistic Regression Models
Fitting a logistic regression model to the logit transform of relative probabilities
of the Los Angeles Lakers winning or losing,
l ogit  log  Pr Y  1 / Pr{Y  0}      X
where X is a Laker player’s plus/minus statistic and α an intercept term, yields a
maximum likelihood estimate of the logistic regression coefficient, , or a log
odds ratio that is the antilogarithm of his odds ratio as calculated in Section 2,
e.g., exp(3.996) = 54 or 54:1, for Kobe Bryant.
Fitting a second logistic multiple regression model to the logit transform
of relative probabilities of the Los Angeles Lakers winning or losing,
l ogit  log  Pr Y  1 / Pr{Y  0}     1  X 1   2  X 2
Published by Berkeley Electronic Press, 2011
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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 2, Art. 5
where X1 and X2 are plus/minus statistics for two Laker players, say Kobe Bryant
and Pau Gasol, yields maximum likelihood estimates of two logistic regression
coefficients, 1 and 2. Taking antilogarithms of these two log odds ratios,
exp(2.553) = 13 or 13:1, for Kobe Bryant and exp(3.733) = 42 or 42:1, for Pau
Gasol. During the 2009–2010 regular seasons, Bryant and Gasol only played
together in 56 of 82 Laker regular season games, winning 41 and losing 15 games.
Table 3a. Kobe Bryant Plus/Minus vs. Pau Gasol Plus/Minus – Laker Wins
GASOL +/Plus
Zero
Minus
Totals
BRYANT+/Plus Minus Totals
32
2
34
0
3
3
4
0
4
36
5
41
Table 3b. Kobe Bryant Plus/Minus vs. Pau Gasol Plus/Minus – Laker Losses
GASOL +/Plus
Zero
Minus
Totals
BRYANT+/Plus Minus Totals
0
1
1
1
1
2
2
10
12
3
12
15
In Table 3a, there are 5 Laker wins in which Bryant is a minus, but Gasol is not;
whereas, in 4 Laker wins Bryant is a plus and Gasol a minus. Similarly, in Table
3b there are 2 Laker losses in which Bryant is a plus and Gasol a minus; whereas,
in 1 Laker loss Bryant is a minus and Gasol a plus.
4. Stratified Odds Ratios
The relative odds of the Lakers winning a home game when Kobe was a plus
equals [(30)(5)]/[(1)(1)] = 150 or 150:1, calculated from Table 4a. The relative
odds of the Lakers winning an away game when Kobe was a plus equals
[(18)(12)]/[(4)(2)] = 27 or 27:1, calculated from Table 4b.
http://www.bepress.com/jqas/vol7/iss2/5
DOI: 10.2202/1559-0410.1320
4
Okamoto: Stratified Odds Ratios for Evaluating NBA Players
Table 4a. Laker Wins/Losses vs. Kobe Bryant Plus/Minus – Home Games
Minus
1
5
6
Win
Loss
Totals
Plus
30
1
31
Totals
31
6
37
Table 4b. Laker Wins/Losses vs. Kobe Bryant Plus/Minus – Away Games
Minus
2
12
14
Win
Loss
Totals
Plus
18
4
22
Totals
20
16
36
O d dRatio
s R a tio
(H o mand
e a nAway)
d Aw ay)
Odds
(Home
Dividing Kobe Bryant’s home odds ratio by his away odds ratio, the
Lakers were five-and-a-half times more likely to have won at home than on the
road when Kobe Bryant was a plus. Consequently, the estimation of stratified
odds ratios or adjusted odds ratios that take into account stratification is a
necessary refinement.
Figure 3 shows the odds ratios for Kobe Bryant and eight of his Laker
teammates, with home games represented in gold and away games represented in
purple.
1000.0
1000.0
100.0
100.0
10.0
10.0
1.0
Lamar
Derek
Ron
Kobe
Pau
Vujacic Shanno
Brow n Jordan
Farmar Andrew
By num Lamar
Odom
Sasha
Vujacic n Brown Farmar Bynum Odom
2.0
2.2
2.7
6.0
34
Home OR
Home OR 2.0
2.2
2.7
6.0
34
5.1
6.1
11
13
9.2
Aw ayOR
OR 5.1
Away
6.1
11
13
9.2
Fisher
Derek
Artest
Ron
Bry
ant
Kobe
Gasol
Pau
Fisher
62
62
Artest
999
999
Bryant
150
150
Gasol
18
18
14
14
17
17
27
27
999
999
1.0
Sasha Shannon Jordan
Andrew
Figure 3. Odds Ratio Chart for the Los Angeles Lakers: 2009–2010 NBA
Regular Season (41 Home Games, 41 Away Games)
Published by Berkeley Electronic Press, 2011
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Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 2, Art. 5
The logit transform of relative probabilities of the Lakers winning or losing
as a function of a player’s common log odds ratio  for both home and away
games is modeled as follows:
l ogit  log  Pr Y  1 / Pr{Y  0}   1    X or  2    X
where and  are stratum-specific, nuisance parameters. Fitting a stratified
logistic regression model yields a conditional likelihood estimate for the log odds
ratio, , conditioned on sufficient statistics for the two nuisance parameters as in
Mehta and Patel (1995). For example, Kobe Bryant’s stratified odds ratio equals
the antilogarithm of his log odds ratio, exp (3.695) = 40 or 40:1.
(
y
Pau Gasol
50
Kobe Bryant
40
Ron Artest
29
Derek Fisher
21
Lamar Odom
13
Andrew Bynum
8.6
Jordan Farmar
6.2
Luke Walton
3.7
Shannon Brown
3.6
Sasha Vujacic
0.1
)
5.6
1.0
10.0
100.0
1000.0
Figure 4. Stratified Odds Ratio Chart for the Los Angeles Lakers: 2009–2010
NBA Regular Season (41 Home Games, 41 Away Games)
Pau Gasol led the Lakers during the 2009–2010 regular season with a
stratified odds ratio, 50:1, which compared to Kobe Bryant’s 40:1 stratified odds
ratio means the Lakers were 25 percent more likely to win when he was plus (or
lose when he was minus) than they were when Kobe was plus (or minus).
http://www.bepress.com/jqas/vol7/iss2/5
DOI: 10.2202/1559-0410.1320
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Okamoto: Stratified Odds Ratios for Evaluating NBA Players
5. 2010 NBA Playoffs The Lakers won their second straight NBA Championship with an overall record
of 16 wins and 7 losses in the 2010 NBA Playoffs. In Figure 5, Kobe Bryant was
plus in 14 of 16 Laker playoff wins (blue circles) and minus in 6 of 7 Laker
playoff losses (red circles). He was neither plus nor minus in 1 playoff win (white
circle), minus in another playoff win, and plus in 1 playoff loss.
30
25
LA Laker Winning or Losing Margin
20
15
10
5
0
-20
-15
-10
-5
0
5
10
15
20
25
30
-5
-10
-15
-20
-25
Kobe Bryant Plus/Minus
Win
Loss
Figure 5. Scatterplot of Los Angeles Laker Winning or Losing Margin vs.
Kobe Bryant Plus/Minus: 2010 NBA Championship Playoffs (23 Games)
If the single playoff game in which Kobe Bryant was neither plus nor
minus is excluded, then his odds ratio was [(14)(6)/(1)(1)] = 84 or 84:1. The
Lakers were 84 times more likely to have won when Kobe was plus than they
were when he was minus in a win.
Table 5. Laker Wins/Losses vs. Kobe Bryant Plus/Minus
Win
Loss
Totals
Published by Berkeley Electronic Press, 2011
Minus
1
6
7
Plus
14
1
15
Totals
15
7
22
7
Journal of Quantitative Analysis in Sports, Vol. 7 [2011], Iss. 2, Art. 5
K. Bryant
84
P. Gasol
39
L. Odom
39
15
J. Farmar
10
L. Walton
D. Fisher
5.5
S. Brown
4.4
R. Artest
3.0
2.4
A. Bynum
0.3
J. Powell
0.0
0.1
1.0
10.0
100.0
1000.0
10000.0
Figure 6. Odds Ratio Chart for the Los Angeles Lakers: 2010 NBA
Championship Playoffs (23 Games)
With his 84:1 odds ratio more than twice the odds ratio of any of his
teammates, Kobe Bryant led the Los Angeles Lakers to their second consecutive
NBA Championship and earned his second straight Larry O’ Brien trophy as the
Finals MVP. His stratified odds ratio could not be calculated because he was
minus in the single Laker playoff loss at home against the Boston Celtics, and
plus in all 5 of their playoff wins away from home. Pau Gasol and Lamar Odom
whose 39:1 odds ratio during the playoffs tied them for second, had stratified odds
ratios of 23:1 and 29:1, respectively. Unlike Kobe Bryant, Pau Gasol and Lamar
Odom were minuses in one of the 5 playoff games the Lakers won on the road.
References
Mehta C.R. and Patel N.R. (1995). Exact logistic regression: theory and
examples, Statistics in Medicine, 14: pp. 2143-60.
Rosenbaum, Dan T. (2004). Measuring how NBA players help their teams win,
82games.com, http://www.82games.com/comm30.htm.
http://www.bepress.com/jqas/vol7/iss2/5
DOI: 10.2202/1559-0410.1320
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