I spent some time a while back discussing PER and its limitations. Today I'll take a similar look at adjusted plus-minus, or APM.

One of the weaknesses of PER is that it's a rather arbitrary linear combination of basketball statistics. As I pointed out, one can come up with alternate combinations that put any number of players on top of the PER list. In math nerd terms, any player on the convex hull of the statistics space can end up on top, given the right PER formula. With as many dimensions in that space as there are component statistics, that could end up being a lot of players.

And anyway, the bottom line of the game is winning, and there's no clear evidence that maximizing team PER (however you define that) maximizes your chances of winning. (It must be emphasized, by the way, that that's all any statistical approach can do: maximize chances. Basketball may be played on the floor, not on a piece of paper, but the small contingencies that lead to winning or losing are so complex and so numerous that the only thing we can do with them is treat them as essentially random events. Nothing is ever really certain in any practical sense.)

APM is a completely different approach to player assessment that attempts to remedy this weakness. Its purpose is to determine how much a player contributes to his team's scoring margin versus the opponents, which has been shown, to varying degrees of certainty, to be a good predictor of future winning percentage—better even than past winning percentage. It does this by calculating how much the team outscores its opponents with that player on the court. There's a few ways we could do this (just as there are multiple ways to define PER); I'll just be discussing one of them.

As its name implies, APM is an adjusted form of raw plus-minus, which we can call RPM for the moment. The difference between the two can best be illustrated using a simplified example. Suppose some Lakers players (Kobe, Pau, and Lamar) are participating in a two-on-two tournament, with substitutes allowed. Games are 48 minutes long. Let's say that in a particular game, Kobe and Pau open the game and play for 16 minutes, outscoring the opponents by 8. Pau and Lamar play the next 16 minutes, outscoring the opponents by just 2. Finally, Kobe and Lamar close the last 16 minutes, and outscore the opponents by 4. For the sake of simplicity, let's assume for now that the opponents have no sub and play the entire game with the same two players.

During the 32 minutes that Kobe's on the floor, his team outscores the opponents by a total of 12 points. Over a full 48-minute game, that would work out to a RPM of +18 (a 48-minute game is half again as long as Kobe's 32 minutes). Similarly, Pau's 48-minute RPM is +15, and Lamar's is +9.

However, you might ask, for instance, how much of Pau's RPM is due to his own contribution, and how much is due to sharing the court with Kobe? This is the question that APM seeks to answer. It attempts to account for the teammates one plays with, as well as the opponents one plays against (though we're keeping those constant for now).

One might compute the APMs of the three players as follows: Let Kobe's, Pau's, and Lamar's APM be represented by k, p, and l, respectively. From the first 16 minutes, we extrapolate that if Kobe and Pau played the entire game, they'd have outscored the opponents by 24 points. That could mean that both players have APMs of +24, or perhaps Kobe's is +28 and Pau's is +20, or maybe vice versa. There's not enough information to determine for sure. However, at any rate, they add up to 48:

k + p = 48

Similarly, we can write for the other two 16-minute segments

p + l = 12

k + l = 24

I'm not going to go through the gory algebra (I'm assuming you can do that yourself if you've read this far), but these three equations in three variables yield a unique solution: k = +30, p = +18, l = - 6. By way of interpretation, if you had two Kobes play against two average players for an entire game, the Kobes would win by 30 points. (Various versions of APM scale this so that you can just add up the APMs to determine the expected final winning margin. There's no significant difference between this and what we derived; they would just differ by a constant factor—the number of players—so that the scaled APMs would be +15, +9, and - 3, respectively.)

Note that nowhere in all of this computing did we say anything about scoring, rebounds, assists, steals, blocks, fouls, etc.—any of the statistics that make up aggregate parameters like PER. APM is entirely agnostic about what makes players valuable to their team; it simply measures that value. In a way, this is useful, because it completely short-circuits any assumptions about what makes players valuable in general; on the other hand, it sure would help if you knew why your player was valuable. APM can't really answer that. It is, in a very real sense, the holistic yin to PER's reductionistic yang.

Incidentally: What happens if the opponents do use different line-ups? Suppose the Lakers are playing the Magic, with Dwight Howard, Vince Carter, and Rashard Lewis. We'd use d, v, and r to represent their APMs, and assuming they played those line-ups in the same 16-minute segments as the Lakers did, we'd write out something like the following equations:

(k + p) - (d + v) = 48

(p + l) - (v + r) = 12

(k + l) - (d + r) = 24

Note that we now have three equations in six variables, which means that the scenario is said to be underdetermined: there won't be a unique solution to the equations, but multiple solutions (an infinite number, in fact). In general, there will be some kind of mathematical mismatch like this: There are as many variables as players, but as many equations as there are matchups, and those usually won't be equal. Since the number of matchups is larger than the number of players, though, you'll typically have overdetermined scenarios: there won't be any exact solutions at all; any combination of numbers will violate one equation or another.

That sounds bad, but in a sense, it's better than being underdetermined, because we can use statistical methods to determine the best near-solution to the equations—"best" in this case defined by how little the equations are violated as a whole. We can justify this by observing that players aren't robots—their performance varies up and down over the course of a game or a season—so some error in the equations is expected. Typically, the statistical method used is some form of linear regression, which is the same method used to identify likely correlations in all manner of scientific studies. In general, such methods work very well indeed.

I am, however, going to go off the reservation a little: I'm claiming that it might not work so well for basketball.

The key sticking point is hinted at by that name, linear regression, but it's present even in the deterministic case we worked out when Kobe, Pau, and Lamar were taking out their aggression on some hapless two-man team with a constant line-up. I said, for instance, that if Kobe and Pau both had APMs of +24, then they'd outscore the opponents, over an entire game, by those 24 points. Not so earthshattering; if they had in fact played the whole game, that's exactly the APM they'd have ended up with.

But then I also suggested that their APMs might be different: Kobe's could be higher and Pau's lower, or the other way around. And most crucially, I suggested that if one was higher, then the other must be lower by the same amount, so that they always add up to 48. In technical terms, we assume that APM combines linearly. That hidden assumption is part and parcel of the APM calculation; it is what allows us to make the determination that although Kobe's APM and Pau's could be any values individually, they must add up to 48. Without the linearity assumption, we can't write any equations at all; we can't compute APM, statistically or otherwise.

If you think about it, though, what justifies this addition of APMs? What makes us think that we can just add players willy-nilly, like numbers? I personally can't think of a thing that justifies that in anything close to a rigorous way. On the contrary, there's every possibility that they don't always add that way. If two players are both offensive powerhouses but defensive milquetoasts, they might both have good APMs because they spend all of their time playing with teammates that cover for their defensive weaknesses. Put them together, though, and since there's only one ball to score with, their collectively miserable defense might make them a net minus. (EDIT: Wayne Winston's version of APM, at the very least, tries to account for this. Look closely at Winston's answer to Question 5 here, and you'll see that his model includes an "interaction" factor that is a function of a pair of players. As a result, you have an affine relation instead of a linear one, and at least some of the first-order issues with linearity are taken care of.)

The linearity assumption is so seductive because it seems natural and jibes with lots of our experience. If I can grade 20 exams per hour, and you can too, then together we can grade 40 exams per hour. But in any endeavor that requires lots of teamwork and collaboration, the assumption becomes more tenuous. That doesn't unfortunately make it any less critical to the validity of things like APM. It simply has to be demonstrated for us to have any legitimate confidence in the value of APM; it isn't incumbent on anyone else to show that the linearity assumption doesn't hold, but for APM proponents to show that it does.

More insidiously, because linearity seems so natural, we are likely to miss its pivotal role in statistical measures like APM. Perhaps someone somewhere has done a study to validate the linearity assumption for APM. But if so, I haven't seen it, and I bet neither have most APM adherents. If you have, please share it!

One of the weaknesses of PER is that it's a rather arbitrary linear combination of basketball statistics. As I pointed out, one can come up with alternate combinations that put any number of players on top of the PER list. In math nerd terms, any player on the convex hull of the statistics space can end up on top, given the right PER formula. With as many dimensions in that space as there are component statistics, that could end up being a lot of players.

And anyway, the bottom line of the game is winning, and there's no clear evidence that maximizing team PER (however you define that) maximizes your chances of winning. (It must be emphasized, by the way, that that's all any statistical approach can do: maximize chances. Basketball may be played on the floor, not on a piece of paper, but the small contingencies that lead to winning or losing are so complex and so numerous that the only thing we can do with them is treat them as essentially random events. Nothing is ever really certain in any practical sense.)

APM is a completely different approach to player assessment that attempts to remedy this weakness. Its purpose is to determine how much a player contributes to his team's scoring margin versus the opponents, which has been shown, to varying degrees of certainty, to be a good predictor of future winning percentage—better even than past winning percentage. It does this by calculating how much the team outscores its opponents with that player on the court. There's a few ways we could do this (just as there are multiple ways to define PER); I'll just be discussing one of them.

As its name implies, APM is an adjusted form of raw plus-minus, which we can call RPM for the moment. The difference between the two can best be illustrated using a simplified example. Suppose some Lakers players (Kobe, Pau, and Lamar) are participating in a two-on-two tournament, with substitutes allowed. Games are 48 minutes long. Let's say that in a particular game, Kobe and Pau open the game and play for 16 minutes, outscoring the opponents by 8. Pau and Lamar play the next 16 minutes, outscoring the opponents by just 2. Finally, Kobe and Lamar close the last 16 minutes, and outscore the opponents by 4. For the sake of simplicity, let's assume for now that the opponents have no sub and play the entire game with the same two players.

During the 32 minutes that Kobe's on the floor, his team outscores the opponents by a total of 12 points. Over a full 48-minute game, that would work out to a RPM of +18 (a 48-minute game is half again as long as Kobe's 32 minutes). Similarly, Pau's 48-minute RPM is +15, and Lamar's is +9.

However, you might ask, for instance, how much of Pau's RPM is due to his own contribution, and how much is due to sharing the court with Kobe? This is the question that APM seeks to answer. It attempts to account for the teammates one plays with, as well as the opponents one plays against (though we're keeping those constant for now).

One might compute the APMs of the three players as follows: Let Kobe's, Pau's, and Lamar's APM be represented by k, p, and l, respectively. From the first 16 minutes, we extrapolate that if Kobe and Pau played the entire game, they'd have outscored the opponents by 24 points. That could mean that both players have APMs of +24, or perhaps Kobe's is +28 and Pau's is +20, or maybe vice versa. There's not enough information to determine for sure. However, at any rate, they add up to 48:

k + p = 48

Similarly, we can write for the other two 16-minute segments

p + l = 12

k + l = 24

I'm not going to go through the gory algebra (I'm assuming you can do that yourself if you've read this far), but these three equations in three variables yield a unique solution: k = +30, p = +18, l = - 6. By way of interpretation, if you had two Kobes play against two average players for an entire game, the Kobes would win by 30 points. (Various versions of APM scale this so that you can just add up the APMs to determine the expected final winning margin. There's no significant difference between this and what we derived; they would just differ by a constant factor—the number of players—so that the scaled APMs would be +15, +9, and - 3, respectively.)

Note that nowhere in all of this computing did we say anything about scoring, rebounds, assists, steals, blocks, fouls, etc.—any of the statistics that make up aggregate parameters like PER. APM is entirely agnostic about what makes players valuable to their team; it simply measures that value. In a way, this is useful, because it completely short-circuits any assumptions about what makes players valuable in general; on the other hand, it sure would help if you knew why your player was valuable. APM can't really answer that. It is, in a very real sense, the holistic yin to PER's reductionistic yang.

Incidentally: What happens if the opponents do use different line-ups? Suppose the Lakers are playing the Magic, with Dwight Howard, Vince Carter, and Rashard Lewis. We'd use d, v, and r to represent their APMs, and assuming they played those line-ups in the same 16-minute segments as the Lakers did, we'd write out something like the following equations:

(k + p) - (d + v) = 48

(p + l) - (v + r) = 12

(k + l) - (d + r) = 24

Note that we now have three equations in six variables, which means that the scenario is said to be underdetermined: there won't be a unique solution to the equations, but multiple solutions (an infinite number, in fact). In general, there will be some kind of mathematical mismatch like this: There are as many variables as players, but as many equations as there are matchups, and those usually won't be equal. Since the number of matchups is larger than the number of players, though, you'll typically have overdetermined scenarios: there won't be any exact solutions at all; any combination of numbers will violate one equation or another.

That sounds bad, but in a sense, it's better than being underdetermined, because we can use statistical methods to determine the best near-solution to the equations—"best" in this case defined by how little the equations are violated as a whole. We can justify this by observing that players aren't robots—their performance varies up and down over the course of a game or a season—so some error in the equations is expected. Typically, the statistical method used is some form of linear regression, which is the same method used to identify likely correlations in all manner of scientific studies. In general, such methods work very well indeed.

I am, however, going to go off the reservation a little: I'm claiming that it might not work so well for basketball.

The key sticking point is hinted at by that name, linear regression, but it's present even in the deterministic case we worked out when Kobe, Pau, and Lamar were taking out their aggression on some hapless two-man team with a constant line-up. I said, for instance, that if Kobe and Pau both had APMs of +24, then they'd outscore the opponents, over an entire game, by those 24 points. Not so earthshattering; if they had in fact played the whole game, that's exactly the APM they'd have ended up with.

But then I also suggested that their APMs might be different: Kobe's could be higher and Pau's lower, or the other way around. And most crucially, I suggested that if one was higher, then the other must be lower by the same amount, so that they always add up to 48. In technical terms, we assume that APM combines linearly. That hidden assumption is part and parcel of the APM calculation; it is what allows us to make the determination that although Kobe's APM and Pau's could be any values individually, they must add up to 48. Without the linearity assumption, we can't write any equations at all; we can't compute APM, statistically or otherwise.

If you think about it, though, what justifies this addition of APMs? What makes us think that we can just add players willy-nilly, like numbers? I personally can't think of a thing that justifies that in anything close to a rigorous way. On the contrary, there's every possibility that they don't always add that way. If two players are both offensive powerhouses but defensive milquetoasts, they might both have good APMs because they spend all of their time playing with teammates that cover for their defensive weaknesses. Put them together, though, and since there's only one ball to score with, their collectively miserable defense might make them a net minus. (EDIT: Wayne Winston's version of APM, at the very least, tries to account for this. Look closely at Winston's answer to Question 5 here, and you'll see that his model includes an "interaction" factor that is a function of a pair of players. As a result, you have an affine relation instead of a linear one, and at least some of the first-order issues with linearity are taken care of.)

The linearity assumption is so seductive because it seems natural and jibes with lots of our experience. If I can grade 20 exams per hour, and you can too, then together we can grade 40 exams per hour. But in any endeavor that requires lots of teamwork and collaboration, the assumption becomes more tenuous. That doesn't unfortunately make it any less critical to the validity of things like APM. It simply has to be demonstrated for us to have any legitimate confidence in the value of APM; it isn't incumbent on anyone else to show that the linearity assumption doesn't hold, but for APM proponents to show that it does.

More insidiously, because linearity seems so natural, we are likely to miss its pivotal role in statistical measures like APM. Perhaps someone somewhere has done a study to validate the linearity assumption for APM. But if so, I haven't seen it, and I bet neither have most APM adherents. If you have, please share it!

Bookmarked!

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