If you are reading this post most probably you know that a lot of work has been done in developing advanced metrics for evaluating players. The objective of all these metrics is to try an quantify the impact of each player in winning. There are metrics such as win shares, wins over replacement etc. that have been developed. These metrics are either not described in great detail or they are (unnecessarily in my opinion) complicated. In this post I will describe a simple metric – baptized “wins contributed” for lack of a better term – that aims at quantifying the number of wins that each player contributed to his team.

At a high level the idea behind “wins contributed” is to translate the adjusted plus-minus of a player to wins. To do this one needs (i) a mapping between points margin and wins, and (ii) information about the pace of a team (to adjust the plus-minus value that is computed per 100 possessions). Just a note before we get into the details, I am using data from the 2007-08 season since this is the season for which I managed to find all the data needed. I am working on data from the more recent seasons.

### Average win margin and number of wins

The first thing I set to examine was whether there is any relationships between the average win margin for a team and the number of wins they produce over the season. As we can see from the following figure there is a very strong linear relationship (R-squared = 0.96). The slope of the linear fit is 2.4, which means that an increase in the average win margin by 1 point translates to 2.4 additional wins expected. This is crucial for translating adjusted plus-minus to wins contributed.

### Adjusted Plus-Minus

The next element in our metric is the adjusted plus-minus. First introduced in hockey, a plus-minus metric tries to measure the actual impact of a player in a team. The main idea is to measure how many points a team scores and allows while a player is on the floor. This can be thought of as an all-encompassing metric that includes things such as a good screen, floor spacing etc. The raw plus-minus metric simply looks at the possessions a player was on the floor and calculates the differential between points scored and allowed during these possessions. If the differential is positive the player helps his team, otherwise not. The main problem with the raw plus-minus is that a player might usually be in a lineup with other players who are more influential in the game (think of James, Curry etc.) and hence, their own plus-minus is inflated. In order to control for that the adjusted plus-minus was introduced. The calculation of adjusted plus-minus is based on a linear regression where the dependent variable is the point differential observed during a *stint* (adjusted per 100 possessions). The independent variables are dummy variables that represent players. A player who is on the home team line up has a value of 1, while a player on the away lineup has a value of -1. All other players are 0. This is the central building block of the adjusted plus-minus calculation, with various details possibly differing from implementation to implementation (e.g., how you deal with players with few minutes played etc.). I have implemented my own version of adjusted plus-minus that you can find here. Using the data from the 2007-08 season following is the adjusted plus-minus calculated for each player (except the ones at the bottom 20-th percentile of play time).

The top-5 players for adjusted plus-minus were:

- Kevin Garnett (16.9) – Boston Celtics
- Allen Iverson (14.3) – Denver Nuggets
- Chauncey Billups (14.2) – Detroit Pistons
- Dwight Howard (13.9) – Orlando Magic
- Chris Paul (11.7) – New Orleans Hornets

The bottom-5 players for adjusted plus-minus were:

- Dominic McGuire (-13.0) – Washington Wizzards
- Eddy Curry (-11.2) – New York Knicks
- Deron Williams (-9.8) – Utah Jazz
- Kirk Snyder (-9.4) – Minnesota Timberwolves
- Josh Power (-9.2) – Los Angeles Clippers

### Mapping APM to Wins Contributed

The APM values computed above are the adjusted point differential per 100 possessions. However, depending on the pace of the team a game might be more or less than 100 possessions. In order to translate the point differential to wins contributed we need to adjust for the pace as well as the average time that the player spends on the court with the team. With p representing the pace of the team, m(i) the average minutes per game played by player i, and M(T) is the average minutes per game played by team T, the wins contributed from player i can be calculated as follows:

The 2.4 coefficient is the slope of the linear fit between average win margin and total wins described above. Using the above formula we get the following values for the top and bottom adjusted plus-minus players:

- Kevin Garnett: 25.6
- Allen Iverson: 30.7
- Chauncey Billups: 20.5
- Dwight Howard: 23.9
- Chris Paul: 20.2

- Dominic McGuire: -5.9
- Eddy Curry: -13.6
- Deron Williams: -17.5
- Kirk Snyder: -11.7
- Josh Power: -8.4

As we can see the rankings themselves have changed since the pace and the minutes played per game by the players are important factors as well. For example, Dominic McGuire had the lowest adjusted plus-minus but he only “cost” to his team 5.9 wins since he played less than 10 minutes per game.

**NBA Win Shares**

Basketball-reference reports a similar metric, namely, win shares, whose goal is “to divvy up credit for team success to the individuals on the team” – pretty much in line with what I have tried to develop in what discussed above. However, as you will see if you visit the link for the description of the metric it is not an easily digestable metric. Obviously there is not a way to obtain ground truth in order to see how good each metric does compared to the golden standard, but there is a specific check that we can do to get a feel for it. In particular, one would expect that a metric that tries to assign win credits among the players of a team, should provide a number approximately equal to the number of total wins for a team once the win shares/contributions of all of each players are summed up.

Max Horowitz was a great help in this part, since he provided me with the data that he had collected for the win shares and other advanced statistics (including plus-minus) for all the players over the past 17 seasons. I further collected the pace and the total wins for every team during these seasons and I was able to explore the following relationships:

- Total team wins-VS-total wins shares of a team (i.e., the sum of win shares for the team’s players)
- Total team wins-VS-total wins contributed of a team (i.e., the sum of the wins contributed defined for the team’s players above)

When using the total win shares from basketball reference we can see that it does pretty good job at capturing the total wins of a team (correlation 0.91). The linear regression between the total team wins and total win shares further explains 85% of the variability in the data.

On the other hand, the total wins contributed based on the plus-minus as described above, exhibits an even higher correlation (0.94) with the actual total wins, while, the linear fit shown below, captures 89% of the variance in the underlying data.

One interesting thing to note here is that the line fitted in the data above has an intercept approximately equal to 20 and a slope of 0.94. This means that when the total win contributions for the players of a team is 0, the actual wins for the team is 20. This might not look right but with a second thought it is not. In particular, a team with average players (i.e., PM = 0) will have a record of 40-40 (based on the relation between average win margin and total wins). A replacement level player has a PM = -2, so a team with only replacement players will be outscored by its opponents by a margin of -10 points. Based again on the relationship between average win margin and win totals this translates to 18 wins, that is, a team with replacement players will have 18 wins. This is very close to the intercept of 20 that we get. So if your PM win shares is 40, you should expect to have 58 wins in total, since the 40 are win shares above replacement.

So what is the big deal with the PM-based total win shares since other metrics (e.g., the basketball-reference one) performs almost equally good? The answer is its simplicitly and easy comprehension. Following, Occam’s razor way we prefer simplicitly!

Evaluating players and their impact on a team is always going to be an important problem but at the same time a challenging one. In this blog I have tried to provide a simple, yet illustrative measure of the contributions of a player to a team.