Nylon Calculus: Introducing Player Impact Plus-Minus

This probably went in. (Photo by Tim Warner/Getty Images)
This probably went in. (Photo by Tim Warner/Getty Images) /

Player Impact Plus-Minus (PIPM for short) is a plus-minus metric that I have been building out for nearly six months. It has finally reached a point where I am happy with the output along with the infrastructure of tools utilizing the metric.

The goal of any good metric is not just to measure what a player has done, but to be able to predict how they will perform in the future. This is where older metrics, like PER and Win Shares, struggle to maintain relevance. Statistical plus-minus metrics, such as the popular Box Plus-Minus, which only use box-score information improve in this regard, but can struggle mightily on the defensive end. Newer plus-minus metrics, such as Real Plus-Minus, and player-tracking metrics, such as Player Tracking Plus-Minus, are the current benchmarks for publicly available metrics. They both can describe what a player has done and are predictive of future results.

Without further ado, I present Player Impact Plus-Minus. There are three components of Player Impact Plus-Minus: a box-score prior, luck-adjusted on-off data, and luck-adjusted net rating. Together these three components are able to provide a descriptive function of what has happened as well as add insight into future results.

The box-score prior

The box-score prior is nothing out of the ordinary. I use pace adjusted per 36 minute statistics to calculate an initial estimate of offensive and defensive production. The equation for the offensive and defensive components is as follows:

These numbers were calculated via a weighted regression against a 15-year RAPM (Regularized Adjusted Plus-Minus) sample provided by Jeremias Engelmann, the creator of RAPM and co-creator of ESPN’s RPM.

Additionally, the above is used by itself for players who played prior to the 2000-01 season, the first in which plus-minus data is publically available. Similar to the calculation for Box Plus-Minus, there is a team adjustment done to the data for seasons before 2000-01 to sum to the schedule and opponent adjusted net rating for a team.

In statistics, R is known as the correlation coefficient where a value of 1 is perfect correlation and 0 is no correlation. In regression, the term is squared to create the coefficient of determination. Now while R^2, or the coefficient of determination, sounds nice and fancy, all it really does it explain the percent of variance in the target of a regression that the output explains.

In this case, Box Plus-Minus has an R^2 value of 0.661 with RAPM. This means that BPM is able to explain 66.1 percent of variance seen in RAPM samples. Just using my box-score estimate, I was able to achieve an R^2 of 0.732 mean an increase of 7.1 percent in the explanation of the variance of RAPM. Below is the R^2 value of by box-score estimate alone with other well-known public metrics:

As you can see, the main increase in accuracy of my box-score estimate comes on the offensive side of the court. Offense has long been much easier to track from just the box-score due to an increased availability of counting statistics.

Creating Player Impact Plus-Minus

The box-score prior from above is coupled with my luck-adjusted on-off ratings for each player using the following equation:

It is a relatively straight forward and simple formula, which was something important to me when designing the metric. DPI Box and OPI Box are calculated using the formula displayed earlier. D-On/Off and O-On/Off are the player’s luck-adjusted on-off net ratings. D-AVG and O-AVG are the player’s luck-adjusted on-court ORTG and DRTG relative to league average.

There are two regressions within the metric. All four plus-minus components multiplied by the square root of a player’s minutes played divided by 3936, the theoretical maximum minutes available if a player played 48 minutes a game for all 82 games. This first regression pulls all the plus-minus data towards zero accounting for variance in small samples even when using luck-adjusted data.

The second regression is to regress the overall offensive and defensive components to “replacement level” to again account for variance in small samples. Offense is regressed by 350 minutes of -1.7 level offensive play and defense is regressed by 450 minutes of -0.3 level defensive play. This is done to best limit error in small samples.

The last aspect of PIPM is a team adjustment in the same way of Box Plus-Minus. These adjustments are usually less than 0.5 points overall because of the accuracy of the components of the metric.

Overall, Player Impact Plus-Minus has an R^2 value of 0.875 with the 15-year sample of RAPM. The most impressive aspect to me is that Defensive Player Impact Plus-Minus has a massively improved R^2 of 0.843 with DRAPM compared to the 0.620 of DBPM and the 0.634 of my box-score prior. That means that the introduction of plus-minus data is massively able to improve the accuracy of this metric.

Wins Added

PIPM is designed to measure impact on a per 100 possessions basis, as most plus-minus metrics are. To convert to a cumulative impact, I created Wins Added. The formula for Wins Added is as follows:

While this may seem like just a big messy equation, and honestly it is, the results are a cumulative value of wins a player adds over a replacement level player. The first big mess is converting Player Impact Plus-Minus into a per 48-minute rate of Wins Added. The second component is the subtraction of “replacement level”, 0.426 WA per 48-minutes. The last component converts to a total Wins Added above a “replacement level” player.

Replacement level for this formula was determined with the idea that a replacement level team would win 10 games. Using a simple solver, the value of 0.426 was selected as the per 48-minute replacement level.

Historical Player Impact Plus-Minus and Wins Added for the NBA

Since 2000-01, when plus-minus data became available, Stephen Curry has the three of the top nine season in my Player Impact Plus-Minus database including the top ever during his 2016-17 campaign at +12.1 points per 100 possessions.

Including every season in the database (which goes back to 1973-74, the first in which starts were tracked), the top 15 seasons are as follows:

The list may appear somewhat prejudiced against older players, but that is simply because the lack of plus-minus information underrates what they were able to accomplish on the court. Looking at the top-15 seasons by Wins Added helps to even the playing field because older players played far more minutes.

Jordan’s 1987-88 season, the fifth best ever by Player Impact Plus-Minus, takes the cake as the most overall impactful season since 1973-74.

Player Impact Plus-Minus and Wins Added this season

This season, Stephen Curry once again sits on top with a PIPM of +7.61 points per 100 possessions. Due to missing time with injuries, he is not Wins Added king. That title belongs to James Harden as of today, at 6.6 Wins Added. Currently the worst PIPM score is held by Josh Jackson, with an almost impressive -4.93.

Looking at just the offensive side of the court, Curry once again holds the top spot with an Offensive Player Impact Plus-Minus of +6.74. The lowest O-PIPM in the league belongs to Orlando Magic big man Bismack Biyombo who provides -3.74 points per 100 possessions on offense.

On defense, the current top spot is held by Andre Roberson. Roberson’s D-PIPM has been +3.09 points per 100 possessions and the Thunder have fallen apart on defense during his recent injury. The worst D-PIPM score is a close race between Lou Williams and Jamal Crawford. Both known for their instant offense, they are separated by six ten-thousandths of a point and virtually tied at -3.58 points per 100 possessions.

Next: Nylon Calculus -- Defining and calculating luck-adjusted ratings for the NBA

Player Impact Plus-Minus tools

There are numerous tools I have created to help people view and use Player Impact Plus-Minus data.

The first tool is a league wide database that updates daily to view where each player ranks by PIPM and Wins Added.

There is a game-adjustable PIPM tool that allows you to look at a specific team over any range of games, along with luck-adjusted on-off data and complete player statistics.

Building off that, there is a similar game-adjustable PIPM tool that allows you to compare any two teams over any stretch of games, or compare the same team over different periods.

The final tool is a Trade Machine powered by PIPM. ESPN’s version is great

for making sure a trade is legal under the CBA, but the PER powered projections are virtually useless for evaluating a trade. This tool allows you to trade players and get an actual idea of how it will impact a team’s projected wins for the remainder of the season.

A tip when using any of these tools: just make a copy. Especially with the Trade Machine, there have been issues with multiple people trying to work on it at once. I will tweet out if I ever make any changes to any of the above tools. I personally make my own copies. It’s easier. Trust me and enjoy!