OpenFloor #3: Introduction to +/- Models
An approach to measure player impact
Take a look at this video taken from Thinking Basketball’s video. Or take a look at the one in which I’m in it:
In the first video, LeBron James and Kyrie Irving opens up the whole corridor due to Stephen Curry’s gravity and Kevin Durant (KD) just drives for an easy dunk.
In the latter, I drive (number 10 in red) inside which forces a help from number 23 in greens. This opens up space for my big to get a better position near the basket which leads to an offensive rebound.
In the first example, Steph gets zero credit in the box score. In the second, I technically hurt my FG% by taking that layup. Yet both actions clearly impact the game and it’d be nice to capture that, wouldn’t be?
Measuring Player Impact
In the recent years, there has been a lot of new "advanced” metrics which kind of aims to capture the whole impact of a player. Some of them, like performance index rating (PIR), are product of bottom-up approach. What I mean by that is they are just linearly weighted equations where positive stuff like rebound, made field goal etc. have positive weights and negative stuff like turnovers have negative weights, then you just sum them to get the total. Well, there are some limitations to this:
We have to choose which metrics to include and then, we are limited by those metrics. Many of these bottom-up all-in-one metrics are giving more credit to players that have high usage since they include a lot of metrics related to usage while underestimating defense since, there’s no stat like “good defensive rotation”.
How do we decide on weights? We need to assign values to the actions on the court.
The alternative approach is called top-down approach. Plus-minus falls under this and is an attempt to capture player impact on the court. It’s very simple to calculate as well: It’s the score difference when a player is on the court. For example, if your team outscores opponents by 10 points while you’re playing, your plus-minus is +10. If you get outscored by 5, it’s -5. Simple as that.
It makes sense, right? If my team performs better when I’m on the court, I must be doing things in favor of my team. Also, it has its advantages. Conceptually, it captures everything. There’s no bias towards, for example, high usage players since you don’t put metrics into a mixer and come up with an end product.
However, there’s a problem. What if you are on the court with Jordan, Kobe, LeBron and Shaq, all the time? It’s almost impossible for you to end up with a bad plus-minus (+/-) although they are the ones carrying you to that +/- value.
Controlling For Other Players On The Court
Think about 2x2 scenario, where your team has 3 players: Player A, Player B, Player C. Player A and Player B starts the game and they outscore the opponent by 10 points. Then Player A and Player C outscore by 8 points. Then Player B and C by 4 points. So, what we have is:
Player A + Player B = 10
Player A + Player C = 8
Player B + Player C = 4
or:
Player A | Player B | Player C | Total
1 1 0 10
1 0 1 8
0 1 1 4
When you solve this system of equations, you end up with the following:
Player A = 7, Player B = 3, Player C = 1
This allows you to control (or adjust) for other players on the court, thus called Adjusted Plus-Minus (APM). In a real version, you’ll have bunch of rows like above and usually with both offensive and defensive variables for each player. Hence, you end up with values for offense and defense separately. Also, you were able to solve the system above but in the real version, you are not. Instead, we approximate the solution via good old linear regression. On top of that, there’s also the problem of multicollinearity: Some players always on the court (or off the court) together, or never play together etc. So, the trick is to use ridge regression instead of linear regression, shrinking parameter estimates. This version is called Regularized Adjusted Plus-Minus (RAPM).
EuroLeague Version
While most +/- models are NBA-focused, I built a RAPM version for the EuroLeague using 5 seasons of data. You can view all player estimates on my website, but here are the top six in offensive and defensive RAPM (filtered by possession count).
Usually, people just sum offensive and defensive parts to get the total. I include that but also provide weighted version, which weights the estimates proportional to the number of possessions the player played on each end.
More on +/- Framework
The +/- framework itself allows for many possibilities, that’s why there are so many versions of it. Some of them adjusting for “luck”, “aging” etc. Some are backward looking, meaning they measure impact in the past while some others are forward looking, better suited to understand the impact right now. Some others give more weight to recent games, called time-decaying RAPM. To sum up, you may come across bunch of different +/- models while surfing through the internet.
The +/- framework is highly flexible. For example, you could create a dummy variable for different roles a player takes or different portions of the game (such as garbage time) to contextualize a bit more. Additionally, you don’t necessarily have to run it for points. Same framework can be used to estimate who elevates teammates (beyond what assists capture), who forces more turnovers (beyond steal counts) etc. I may do and share them in the future.
That said, ridge regression complicates interpretation. You can’t simply subtract Player A’s RAPM from Player B’s and claim a concrete value of impact per 75 possessions. There’s shrinkage at play (the lambda term). If you’re interested in the details, you can check here.
While it gives information about “what”, it doesn’t provide an answer for “why”. There’s another framework called statistical plus-minus (SPM), which can help with why a player might have that +/-.