The following is a step by step guide for calculating Wins Produced. Wins Produced is a model for estimating individual player contribution to winning. This model is based on the Wins Produced formula shown in the Wages of Wins and Stumbling on Wins (written by our esteemed General Manager David Berri, Martin Schmidt with Stacey Brook) To help we have provided a concrete set of data for our steps. This article calculates Wins Produced on the 2010-2011 Chicago Bulls.
Background: Wins Produced is grounded on strong principles
Offensive efficiency (points scored per offensive possession) and defensive efficiency (points given up per defensive possession) are good for modeling a team’s winning percentage. This simple model was noted by both Dean Oliver (2004) and John Hollinger (2002). In Berri (2008) this model is developed mathematically. For here, though, we are going to simply take the link between wins and the efficiency measures as given (one can read Berri (2008) for the math).
Here is the specific model linking winning percentage to offensive and defensive efficiency.
- The model was estimated with data from 1987-88 to 2010-11
- Data taken from Basketball Reference
- Dependent Variable is Winning Percentage
Adjusted R2 = 0.94
- Offensive Efficiency = Points Scored divided by Possessions Employed (PE)
- Defensive Efficiency = Points Surrendered divided by Possessions Acquired (PA)
- PE = FGA + 0.45*FTA + TO – REBO
- PA = DFGM + 0.45*DFTM + REBD + DTO + REBTM
|FGA = Field Goal Attempts||FTA = Free Throw Attempts|
|TO = Turnovers||REBO = Offensive Rebounds|
|DFGM = Opponent’s Field Goals Made||DFTM = Opponent’s Free Throws Made|
|REBD = Defensive Rebounds||DTO = Opponent’s Turnovers|
|REBTM = Team Rebounds|
The formulation for PE and PA is explained in Berri (2008).
The value for FTA and DFTM is explained in Berri (2008)
REBTM refers to Team Rebounds that change possession. This calculation is detailed in Berri (2008)
Determine the value, in terms of wins, of points and possessions.
This is done by differentiating the above wins model with respect to Points, Points Surrendered, PE, and PA.
Table one: The Value of Points and Possessions
Linking box score stats to individual players
With the value of PTS, DPTS, PE, and PA determined, we can now ascertain the value of all the individual elements of offensive and defensive efficiency (i.e. PTS, FGA, ORB, etc…). The model estimated for this example employed data from 1987-88 through the 2010-11 season. One should note that one can also estimate the same model with ABA and NBA data from 1973-74 to 1986-87 and see essentially the same values.
Of the statistics tabulated in the box score, only the values of blocked shots, personal fouls, and assists cannot be derived directly from the values of PTS, PE, DPTS, and PA listed above.
To determine the value of blocked shots, a simple regression was estimated that connected the opponent’s made field goals to opponent’s field goal attempts, blocked shots, and dummy variables for teams, years (1973-74 to 2010-11), and leagues (data from both the ABA and NBA was employed). This model explains 93% of the variation in opponent’s made shots. Furthermore, it indicates that each blocked shot reduces the opponent’s made shots by 0.637. Since each made two-point field goal by an opponent – as noted below – costs a team 0.031 wins, each blocked shot is worth 0.020 wins (i.e. 0.637*-0.031).
The value of personal fouls is calculated from the value of the opponent’s free throws made. Specifically, we first determine the percentage of personal fouls a player committed on a team. We then multiply this percentage by the number of free throws the opponent of a team made. For example, Derrick Rose committed 8.2% of Chicago’s personal fouls in 2010-11. Chicago’s opponents made 1,450 free throws, so Rose is charged with 119.5FTM(opp.).
The impact of assists will be detailed below. Before we get to that, here are the values of each of the box score statistics we can ascertain from the above steps.
Table Two: Value of Player and Team Statistics
|Player Variables||Marginal Value|
|Three Point Field Goals Made (3FGM)||0.064|
|Two Point Field Goals Made (2FGM)||0.032|
|Free Throws Made (FTM)||0.017|
|Missed Field Goals (FGMS)||-0.034|
|Missed Free Throws (FTMS)||-0.015|
|Offensive Rebounds (REBO)||0.034|
|Defensive Rebounds (REBD)||0.034|
|Opponent’s Free Throws Made [FTM(opp.)]||-0.017|
|Blocked Shots (BLK)||0.020|
|Team Variables||Marginal Value|
|Opponent’s Three Point Field Goals Made [3FGM(opp.)]||-0.064|
|Opponent’s Two Point Field Goals Made [2FGM(opp.)]||-0.031|
|Opponent’s Turnovers [TO(opp.)]||0.033|
|Team Turnovers (TOTM)||-0.034|
|Team Rebounds (REBTM)||0.033|
CALCULATING WINS PRODUCED
With these values, we can now ascertain the value of an individual player. Here are the steps one follows:
Step One: Calculate the value of a player’s production (PROD) and production per 48 minutes (P48)
The first step simply involves multiplying each player’s statistics by the corresponding value listed above.
PROD = 3FGM*0.064 + 2FGM*0.032 + FTM*0.017 + FGMS*-0.034 + FTMS*-0.015 + REBO*0.034 + REBD*0.034 + TO*-0.034 + STL*0.033 + FTM(opp.)*-0.017 + BLK*0.020
NOTE: One should note that the above values have been rounded off to three decimals. The results reported below for each player were ascertained without any rounding.
For Derrick Rose in 2010-11 the calculation would be as follows:
Rose PROD= 128*0.064 + 583*0.032 + 476*0.017 + 886*-0.034 + 79*-0.015 + 81*0.034 + 249*0.034 + 278*-0.034 + 85*0.033 + 119.5*-0.017 + 51*0.020 = 7.47
Rose P48 = (PROD / Minutes Played) * 48 = (7.47 / 3,026) * 48 = 0.119
Step Two: Adjust for teammate’s production of defensive rebounds
For the most part, player performance in the NBA does not depend on the player’s teammates. This can be easily seen when we consider how consistent a player’s P48 is across time. From 1973-74 to 2010-11, about 80% of a player’s P48 in the current season is explained by the player’s P48 in the past season (from a sample of nearly 5,000 player observations; players needed to play in at least 20 games and have at least 12 minutes per game in season to qualify for sample). And as noted in Stumbling on Wins, players in the NBA are much more consistent than what we generally see in the NFL or Major League Baseball.
Although players are consistent, it is the case that we do see some interaction effects in the NBA (as noted in The Wages of Wins and academic articles that were written well before The Wages of Wins appeared). To ascertain these effects, the following model was estimated:
player per minute statistical production (i.e. defensive rebounds per minute) = f(per minute performance in stat the previous season, age, age squared, percentage of games played last two seasons, dummy variable for position played, dummy variable for new coach, dummy variable for new team, dummy variable for year, stability of roster, and the teammates’ per-minute production of statistic).
This model allows us to see how a teammate’s production of a given statistic impacts a player’s per minute performance (after controlling for other factors that explain performance).
For example, consider field goal attempts. Estimating the above model for field goal attempts indicates a one-unit increase in the teammate’s field goal attempts per minute will reduce a player’s field goal attempts per-minute by 0.833. In other words, most of a player’s shot attempts come at the expense of his teammates. And this is why players should not be given credit for taking shots (see Player Efficiency Rating and NBA Efficiency for two metrics where such credit is given). In other words, shooting efficiency – not total points scored — is the primary determinant of a player’s offensive effectiveness.
For the other statistics, the results tend to be quite small and/or insignificant. The results also seem to depend upon the sample one considers and the specific independent variables one employs. In other words, the results are not robust. The one exception is seen with respect to defensive rebounds. The impact of the teammate’s defensive rebounds tends to similar regardless of sample considered and specific formulation of the model. And that suggests something is going on with respect to player’s rebounding on the defensive glass.
More specifically, for the above specification we see that each one-unit increase the teammate’s defensive rebounds reduces a player’s defensiverebounds per-minute by 0.504. In other words, some of a player’s defensive rebounds are taken from the player’s teammates. In contrast, on the offensive glass the teammate’s offensive rebounds per-minute do not have a statistically significant effect, suggesting that a player’s offensive rebounds do not come from the player’s teammates. These results make sense when we consider how teams go for rebounds on each end of the court. On the defensive end one suspects teams will send more players to the boards then they need because the team cannot begin its offensive possession if it doesn’t have the ball. Consequently, as already noted, one suspects that some defensive rebounds are just taken from a player’s teammates. On the offensive end, though, it’s a different story. Teams have to have some players back to play defense. So the offensive glass sees little competition among teammates, and therefore, offensive rebounds are primarily taken from a player’s opponent.
Given this result, here is how one incorporates the impact of a teammate’s defensive rebounds in the calculation of a player’s Wins Produced .
- Calculate for each player his teammates’ defensive rebounds per minute (TDRBPM). This is calculated as follows: TDRBPM = (Team DRB – Player DRB) / (Team Minutes Played – Player Minutes Played)
- For each player, multiply the TDRBPM by -0.504. This gives us how many defensive rebounds a player lost to his teammates.
- Multiple the result of step (2) by the value of a defensive rebound (i.e. 0.034) and then multiply by a player’s minutes.
- Sum step (3) across all members on a team.
- For each player, the value determined in step (4) is multiplied by the percentage of a player’s defensive rebounds a player captured. This gives us the value of the defensive rebounds an individual player captured from his teammates.
- Subtract step (5) from step (4), and add that value to PROD. Then divide this number – we can call this PRODDRBADJ — by minutes played and multiply by 48.
- Now here is what is done intuitively. Essentially we are determining the number of defensive rebounds a player’s teammates are taking from the player. We are also determining the number of defensive rebounds a player is taking from his teammates. The difference – or step (5) – decreases the P48 value for players who take a large number of defensive rebounds and increases the P48 value for players who get relatively few defensive rebounds.
Perhaps surprisingly, this adjustment does not have a very large impact on the evaluation of the players. There is a 0.98 correlation between P48 and P48DRBADJ. So this adjustment doesn’t dramatically alter the evaluation of most players.
For Derrick Rose, the new P48 number is 0.167. In other words, his value – once we adjust for teammates’ defensive rebounds – increases.
Step Three: Adjust for Assists
Assists are not part of the construction of offensive or defensive efficiency. But assists do impact outcomes. Specifically, a player’s shooting efficiency is related to the number of assists his teammates accumulate. To see this, the following model was estimated.
player’s adjusted field goal percentge = f(player’s adjusted field goal percentage last season, age, age squared, percentage of games played last two seasons, dummy variable for position played, dummy variable for new coach, dummy variable for new team, dummy variable for year, stability of roster, the teammates’ per-minute production of assists, the teammates’ adjusted field goal percentage).
The results from this model were incorporated into Wins Produced as follows.
- Calculate for each player his Teammates’ Assists per Minute (TAPM). This is calculated as follows: TAPM = (Team Assists – Player Assists) / (Team Minutes – Player Minutes)
- Multiply TAPM for each player by the coefficient on TAPM from the above model, or 0.725.
- Multiply step (2) by 2. This step allows us to see how TAPM impact a player’s points-per-field goal attempts (which is simply ADJ FG * 2).
- Multiply step (3) by field goal attempts taken. This allows us to see how many points a player scored should be credited to his teammates.
- Multiply step (4) by the impact points have on wins 0.032586. This allows us to see how much of a player’s production of wins should be credited to his teammates.
- Sum step v across all players on a team.
- Allocate the outcome of step #6 across all players on a team by the percentage of assists on the team that are credited to each player.
This approach will increase the value of a player who gets many assists (i.e. who helps his teammates shoot better). And decrease the value of a player who doesn’t get many assists. For Derrick Rose – a player who led the Chicago Bulls in assists in 2010-11 – his P48 value now rises to 0.231.
Note: In the above formulation, a teammate’s adjusted field goal percentage does have a small and positive impact on a player’s shooting efficiency. One should note that the effect is small, and the statistical significance of the effect does depend upon the specific formulation of the model. In other words, this result is not as robust as the result we see for defensive rebounds and assists. So it is ignored in the calculation of Wins Produced.
Step Four: Incorporate team defense and calculate adjusted P48.
From Table Two we see that there are five factors tracked for the team that are not tracked for individual players. These include 3FGM(opp.), 2FGM(opp.), TO(opp.), TOTM, and REBTM. Each of these statistics are tracked for the team, but not assigned to individual players. In addition, blocked shots are valued in terms of eliminated made shots by the opponent. The value of the erased made shots also has to be included in the team defensive adjustment. This is done by noting the blocked shots by the team (BLKTM).
Having identified the team defensive factors, these are allocated across the players according to the minutes the player plays. In other words, we treat defense as a team activity, not an individual action. This approach allows us to differentiate players who play on good and bad defensive teams. But the data limitations prevent us from differentiating between players who are relatively better or worse on an individual team.
The calculation of DEFTM48 begins with the Team Defense Adjustment.
Team Defense Adjustment = [(3FGM(opp.)*-0.064 + (2FGM(opp.)*-0.031 + TO(opp.)*0.033 + TOTM*-0.034 + REBTM*0.033 - BLKTM*0.200)/Minutes Played]*48
Bulls Team Defensive Adjustment = [(427*-0.064 + 2378*-0.032 + 567*0.033 + 57*-0.034 + 415.0*0.033 – 468*0.200)/19,830]*48 = -0.195
To calculate DEFTM48 we compare each team’s defensive adjustment to the league average.
DEFTM48 = League Average Team Defensive Adjustment – Team Defensive Adjustment
Bulls DEFTM48 = (-0.195)–(-0.217) = 0.022
DEFTM48 is incorporated into each player’s value by adding DEFTM48 to each player’s P48. The outcome of this calculation is called Adj. P48.
Rose Adj. P48 = 0.231 + (0.022) = 0.253
Table Three: Value of DEFTM48 in 2010-11
As one can see, the best defensive teams were Boston, Milwaukee, and Chicago. The worst defensive teams were Cleveland, Minnesota, and New York.
The average value, in absolute terms, of DEFTM48 is 0.010, so incorporting team defense results in a very small adjustment to an individual player’s per 48 minute performance. Furthermore, the correlation coefficient between P48 (adjusted for defensive rebounds and assists) and Adj. P48 in 2010-11 was 0.996.
Step Five: Adjusting for position played.
The average value for Adj. P48 is 0.217. But this value is not the same across all positions. As noted in The Wages of Wins (and in other writings before that book appeared), centers and power forwards get rebounds and tend not to commit turnovers. Guards are the opposite. The nature of basketball is that teams need guards and big men. Given nature of the game, players should be evaluated relative to their position averages. These are reported in Table Five.
Table Four: Value of Adj. P48 Across Positions
|Position||Average Adj. P48|
To incorporate the position averages we need to identify the position each player plays. For most players this is easy. For a few, though, it can be more challenging. It is important to note that positions in basketball are not like baseball or football. In baseball and football we can tell position by where a player appears on the field. In basketball, though, position designations are more arbitrary. Consequently, two analysts looking at the same team may designate positions differently.
Here is the process I follow:
1. Minutes are equal at each position
2. In general, players are allocated across the center and forwards position according to designations found at places like Yahoo.com, ESPN.com, Basketball-Reference.com, etc… and then by height and weight.
3. At the guard positions again I look at position designation, height, and weight. But I also consider number of assists per minute. The players who get more assists are generally considered point guards.
Table Five: Allocating Chicago Bulls across Positions in 2010-11
|Point Guards||Minutes at Position|
With positions ascertained, we can now calculate a player’s performance relative to the position average. For Rose the calculation would be as follows:
Rose Relative Adj. P48 = Adj. P48 – League Average Adj. P48 = 0.253 – 0.191 = 0.062
So per 48 minutes, Rose produced 0.062 more wins than an average point guard. Given that he played 3,026 minutes, we can now see that Rose produced 3.9 wins more than the average point guard.
Before moving on, what about a player like Carlos Boozer? Boozer is listed at power forward and center. To assess his productivity, we need to compare Boozer to an average power forward and an average center. This is done as follows:
Boozer’s Relative Adj. P48 at power forward = 0.264 – 0.256 = 0.009
Boozer’s Relative Adj. P48 at center = 0.264 – 0.296 = -0.032
One then weights these two calculations according to the time Boozer spent at each position. After this calculation we see that Boozer, per 48 minutes, produced 0.004 more than an average player at the position he played (so his Relative Adj. P48 was 0.004).
Step Six: Calculating WP48 and Wins Produced
If we stop after Step Five we will have a player’s production relative to the position average. What we want is a player’s Wins Produced per 48 minutes (WP48) and his Wins Produced.
As noted in The Wages of Wins, to move from relative wins to absolute wins you need to note the average number of wins produced by a player per 48 minutes. This is quite easy to calculate.
The average team will win 0.500 games. Since a team employs five players per 48 minutes, the average player must produce per 48 minutes 0.100 wins. Because teams do play overtime games once in awhile, the actual average production of wins per 48 minutes is 0.099 (and one should note, all this is true regardless of how you calculate Wins Produced).
Given what we know about an average player, WP48 is calculated as follows:
WP48 = Relative Adj. P48 + 0.099
For Rose the calculation is as follows:
Rose WP48 = 0.062 + 0.099 = 0.161
Again, Rose played 3,026 minutes. If he produced 0.161 wins per 48 minutes he then produced 10.2 wins for the season.
Rose Wins Produced = WP48 / 48 * Minutes Played = 0.161/48 * 3,026 = 10.2
Or you can think of it this way. An average player would have produced 6.3 wins in Rose’s minutes. We saw in Step Five that Rose produced 3.9 wins more than the average point guard. Therefore Rose’s Wins Produced must be 10.2.
Table Six reports the Wins Produced each player the Bulls employed in 2010-11. Again, an average team would win 41 games, and an average position would produce 8.2 victories. Looking over the roster, it appears the Bulls were above average at every single position. But although Derrick Rose led the team in Wins Produced, it was the shooting guard position – led by Ronnie Brewer – that led the team.
Table Six: Calculating Wins Produced for the Bulls in 2010-11
|Minutes Played||ADJP48||Relative ADJP48||WP48||Wins Produced|
Again Rose was the most productive player on this team when we consider Wins Produced. Table Seven, which just reports the overall production of each player, indicates that Ronnie Brewer and Joakim Noah were more productive than Rose when we consider WP48.
Table Seven: Wins Produced for the Bulls in 2010-11
|Player||Minutes Played||Position Number||ADJ P48||WP48||Wins Produced|
The summation of Wins Produced for this team was 60.4. And the Bulls actually did win 62 games in 2010-11. Table Eight reports for each team the summation of Wins Produced and actual wins. As one can see, the average difference – in absolute terms – is 2.6 wins. Again, Wins Produced is based on a model connecting wins to offensive and defensive efficiency. So the small difference between actual wins and the Summation of Wins Produced simply reflects the fact that the efficiency metrics do indeed explain team wins in the NBA.
Table Eight : Reviewing the Accuracy of Wins Produced in 2010-11
|Team||Actual Wins||Summation of
Here are the top 50 players in Wins Produced in 2010-11. Although Derrick Rose was the most productive player on the team with the most wins, he was only the 21st most productive player in the NBA. One should note that these 50 players produced about 530 of the league’s 1230 wins. There were 452 players in the NBA in 2010-10. If we go past these 50 players we see about 80% of the league’s wins were produced by 132 of the league’s top players. In other words, about 29% of the league’s players produced 80% of the league’s wins. In sum, contrary to what we saw with respect to the Bulls above, most wins in the NBA are produced by a minority of the league’s players.
Table Nine: The Top 50 Players in 2010-11
|Rank||Player||Team||Minutes||Position||ADJ P48||WP48||Wins Produced|
|1||Chris Paul||New Orleans||2,865||1.00||0.401||0.309||18.45|
|6||Pau Gasol||LA Lakers||3,037||4.79||0.422||0.234||14.81|
|8||Landry Fields||New York||2,541||2.00||0.295||0.237||12.52|
|13||Lamar Odom||LA Lakers||2,639||4.00||0.368||0.212||11.63|
|17||Kris Humphries||New Jersey||2,061||4.00||0.411||0.254||10.92|
|20||Manu Ginobili||San Antonio||2,426||2.62||0.280||0.204||10.32|
|24||Kevin Durant||Oklahoma City||3,038||3.08||0.247||0.155||9.80|
|25||Blake Griffin||LA Clippers||3,112||4.30||0.317||0.148||9.62|
|27||Serge Ibaka||Oklahoma City||2,216||4.17||0.369||0.206||9.49|
|32||Russell Westbrook||Oklahoma City||2,847||1.00||0.245||0.153||9.10|
|38||Andrew Bynum||LA Lakers||1,500||5.00||0.478||0.281||8.78|
|40||Tim Duncan||San Antonio||2,156||5.00||0.389||0.192||8.61|
|42||Emeka Okafor||New Orleans||2,273||5.00||0.373||0.176||8.35|
|43||DeAndre Jordan||LA Clippers||2,047||5.00||0.391||0.194||8.26|
|44||Thabo Sefolosha||Oklahoma City||2,049||2.52||0.265||0.192||8.19|
|45||Deron Williams||New Jersey-Utah||2,465||1.00||0.251||0.159||8.17|
|48||James Harden||Oklahoma City||2,189||2.00||0.229||0.171||7.78|
Berri, David J. and Martin B. Schmidt. 2010 Stumbling on Wins: Two Economists Explore the Pitfalls on the Road to Victory in Professional Sports. Financial Times Press (Princeton, N.J.)
Berri, David J. 2008. “A Simple Measure of Worker Productivity in the National Basketball Association.” In The Business of Sport, eds. Brad Humphreys and Dennis Howard, editors, 3 volumes, Westport, Conn.: Praeger.
Berri, David J., Martin B. Schmidt, and Stacey L. Brook. 2006. The Wages of Wins: Taking Measure of the Many Myths in Modern Sport. Stanford University Press. Paperback edition released in 2007.
Hollinger, John. 2002. Pro Basketball Prospectus 2002. Washington D.C.: Brassey’s Sports.
Oliver, Dean. 2004. Basketball on Paper. Washington D.C.: Brassey’s.