Terms
Player Value – What the player contributes toward improving or hurting his team’s chances of winning. More valuable players will have relatively more positive plays (made shots, rebounds, assists, etc.) than negative plays (missed shots, turnovers, etc.).
Player Value Score – The numeric representation of the player’s value. Like WAR in baseball, it is a one-number metric that attempts to combine all relevant statistical information in a way that accurately captures the relative value of each play. For instance, a made 3-point shot is more valuable than a made 2-point shot. In fact, it is 1 full point more valuable.
PPG +/- or Adj PPG +/- – The specific Player value score used by Charting the Hawks. It stands for points per game plus minus, and is the number of points per game a player provides relative to his hypothetical opponent. The Adj PPG +/- is schedule adjusted. For instance, a player with a PPG +/- of +2.50 is estimated to be 2.5 points per game better than the men he is matched up against. Another way to understand this number is to see this player as adding 2.5 points of value per game to his team’s chances of winning.
Introduction
The traditional way basketball player value is calculated is through the use of box score and play-by-play data, utilizing regression analysis and Team +/- scores to estimate the relative value of a player. While there are some advantages to these player value systems, many of the creators of these systems admit that a lack of data and questions of statistical accuracy mar the results.
Charting the Hawks is an innovative system that vastly improves upon the results of old player value metrics. It provides a more accurate player value score, and much more accurate defensive player value score than BPM, PER, WS, PRPG!, VORP, RAPM, EPM, or RAPTOR.
Viewing Basketball as a Possession Game
Just like baseball is a game of innings, basketball is a game of possessions. Teams play an equal number of possessions on offense as they do on defense (give or take a possession depending on who wins the jump ball and/or who gets the ball at the end of each half). Each possession either ends as a win or loss…if a team scores (doesn’t score) on offense it wins (loses) that offensive possession, if a team gets scored on (doesn’t get scored on) it loses (wins) that defensive possession.
A baseball game line score might look as follows:
010 020 200 5 runs
002 110 011 6 runs
(Hits and errors are left off).
We could do the same thing for basketball as well. Perhaps the first 20 possessions of a game would look as follows (separated into groups of 5 to make possessions easier to count):
02203 02002 02000 23002 = 20
02000 00000 20300 02030 = 12
In fact, this was the first 20 possessions of the recent UTEP vs. KU game. After a Jalen Wilson missed layup with 7:14, UTEP rebounded the ball and now was on its 21st offensive possession, leading 20-12.
Zooming in on the Possessions
The next thing we will do is look at each possession not from the offense’s perspective, but from Kansas’ perspective. Kansas has played 20 total possessions on both offense and defense, for a total of 40. Starting with defense, each time UTEP was held to 0 points, the possession clearly ended in Kansas’ favor, either as a blocked shot, rebound, steal, or forced turnover.
KU’s 1st 20 defensive possessions: R,-2,-2,R,-3,T,-2,R,R,-2,T,-2,B,R,R,-2,-3,R,R,-2
We’ve replaced the 0’s with letters…either B (block), R (rebound), S (steal) or T (turnover) to indicate how Kansas ended possessions it won on defense. We’ve also replaced the numbers with their negative counterparts to indicate that we’re viewing this from Kansas’s perspective. We’re looking at this from the perspective of KU giving up 20 points on defense. Commas are added to separate possessions.
Counting these up, Kansas gave up 20 points on 9 scoring possessions, ended 8 defensive possessions with a rebound, forced 2 non-steal turnovers, had 1 blocked shot, and had 0 steals in its first 20 defensive possessions.
KU’s 1st 20 offensive possessions: MRT,MR2a,T,MRM,M,M,M,M,M,M,2,T,3a,T,M,M,2,MRM,3a,M
We’ve now replaced all 0’s with letters…either M (miss) or T (turnover) to indicate how KU ended the possession. We’ve also added the letter R to indicate offensive rebounds. Offensive rebounds are not seen as creating a new possession but rather continuing the current possession. They are essentially the contra-entry to a missed shot. So, for instance, in KU’s first possession they started with M (missed shot) then followed with an R (offensive rebound) yet ended the possession with a T (turnover). 3 separate chartable plays were all made in the same possession.
There are also subscript a’s following certain numbers…these indicate that the play was assisted. Some of the points KU scored were assisted, others were unassisted. Counting up the possessions, KU scored 12 points on 5 scoring possessions, leaving 15 possessions without a score. Adding together missed shots with turnovers, and subtracting offensive possessions, we arrive at the following math: 15 M + 4 T – 4 R = 15 net non-scoring possessions used.
First 20 Possessions of UTEP game – Team Value
In this illustration we’re using the first 20 possessions as it keeps the possession size small enough to reference. Normally we would apply this process to all possessions of a game, and in fact all possessions of a season. But taking the first 20 possessions of the game, KU is down 12-20 to UTEP. We can state that KU’s team value is -8. That’s basically it. A team’s value is just the difference in points of the current result. From here, we can calculate that KU’s offensive value is 0.60 points per possession (12 points in 20 possessions), and its defensive value is 1.00 ppp. We can also calculate team FG%, Reb%, etc.
First 20 Possessions – Player Value
So far, all of this should be pretty straight-forward. But it is here where we begin to look at the logic behind breaking a basketball game down into a series of offensive and defensive possessions. We’ve established that, down 8 points through 20 offensive and defensive possessions each, KU as a team has a value of -8. Another way of looking at this is to we took each player’s individual contribution; it would sum to -8. Adding the player values of Marcus Garrett, Christian Braun, Ochai Agbaji, Jalen Wilson, David McCormack, Bryce Thompson, Mitch Lightfoot, and Dejuan Harris will necessarily equal the team’s score of -8. This system assumes the team just is the sum of its parts1.
For instance, it could be the case that each of the 8 KU players who logged minutes had a player value score of -1.0. (8 * -1.0 = -8). Although this is not very likely of course. There is likely some distribution, where there are a few players adding some value, but more doing poorly. Perhaps the distribution looks like this: +4, +3, +1, 0, -2, -3, -5, -6.
One basic way to calculate a player value score is to do so purely on points since the total team value score is equivalent to the point difference in the game. At this point in the game, KU’s 12 points were scored by the following players: Agbaji 6, Braun 2, Harris 2, Thompson 2. We are able to account for these points at least. But what about KU’s defense? How can we account for the 20 points KU has given up?
Unfortunately a box score won’t do us a lot of help, and a play-by-play analysis will only help a little. Any KU foul which results in FT’s for UTEP might be the fault of the fouler, or perhaps some other defender (sometimes a help-side defender, through no fault of his own, will foul to prevent an easy basket). Looking through the line score, we might try and see who scored and match that to a defender. But with the amount of switching KU does, combined with ball-screens and defensive rotation, we just don’t really know.
What we need is a new statistic, defensive points allowed. Just like we break down offensive possessions, giving value (in points) by who scored the ball; we will want to break down the defensive possessions and remove value (in points) by who got scored on. We’ll need video to help us know what happened. See this YouTube link.
(This is not the full game, but it does have most of the scoring plays. The game was first charted with the full game available, but for illustration purposes we’ll use the highlights).
Starting at about 1:38 on the video, we see UTEP guard Souley Boum drive around Christian Braun to the rim, scoring over the help from David McCormack. We see that Braun was a step slow, unable to keep his man in front. This possession is fairly clear. Braun was responsible for the team giving up 2 points. Therefore this possession would be charted on the ledger under his name, with -2 of value to Braun.
Charting the rest of the scoring plays for UTEP over the first 20 possessions, we arrive at the following defensive point values for Kansas defenders: McCormack -6, Braun -5, Harris -5, Lightfoot -2, Thompson -2. No one else got scored on. For more info on charting defensive possessions, see note below2.
From here we can attribute total player values in this format (Points Scored – Pts allowed = Net Value).
- Garrett (0 – 0) = 0
- Braun (2 – 5) = -3
- Agbaji (6 – 0) = +6
- Wilson (0 – 0) = 0
- McCormack (0 – 6) = -6
- Thompson (2 – 2) = 0
- Lightfoot (0 – 2) = -2
- Harris (2 – 5) = -3
If the math is done correctly, we will get -8 for a total (0-3+6+0-6+0-2-3), matching the score difference.
Charting defensive points allowed lets us in on a few things we might have otherwise missed. McCormack, Braun, and Harris are the three primarily responsible for UTEP’s (relatively) good offensive start. And while Garrett and Wilson have had slow offensive starts, they’ve defended their men well-enough to keep their value scores at neutral.
Value is More than Points
But a player can add value in ways other than producing points. He can rebound, steal the ball, block shots, or assist among other things. He can also lose value in other ways than getting scored on. He can turn it over or he can miss a shot. So we’ll want to track “possession” stats as well explore the difference between an assisted and unassisted basket.
At 3:54 in the video, Christian Braun takes a pass on the wing. He drives it middle with his left hand, collapsing the defense, and makes a fine chest-pass to an open Ochai Agbaji in the far corner. Agbaji hits the 3, cutting the deficit to 6. This is a clear-cut assist. Agbaji’s open 3 was set up by Braun’s penetration and skilled pass. It was an easier shot opportunity than if Agbaji had to generate the shot on his own with a set defender guarding him. Because of this, the 3 points of value will need to be split between the assist-man (Braun) and the scorer (Agbaji) to some degree, with the scorer receiving more of the value.
Possession stats include rebounds, blocked shots, steals, turnovers, missed shots, and forced turnovers. Fortunately for us, most of these stats are kept in the play-by-play data. The only stats missing a link to an individual player are forced turnovers as well as “team” rebounds. Using video evidence, these possessions are assigned to the individual who won these defensive possessions.
Once the possession stats are accounted for, we can devise offensive and defensive points per possession rate stats for each player. The formula for this is to take total points accounted for over total possessions used. We want to know usage, and reward efficient play with the value it deserves.
Without getting deep into the weeds, we see that Ochai Agbaji missed 2 shots over the first 20 possessions. These shots have an opportunity cost, so while he scored 6 points by making 2 3-point FG’s his offensive value was actually less than +6 due to him shooting four times. (Agbaji’s offensive value was also lower than +6 because both of his 3-point makes were assisted. Besides scoring and assisting, a player can also add offensive value through offensive rebounds, which serve to offset missed shots by creating another scoring chance in a possession).
On the other side, Ochai Agbaji snagged 3 defensive rebounds during this span, adding value on the defensive side of the ball by winning possessions back for Kansas. To calculate Agbaji’s entire value, we’d want to consider all relevant plays. He made 2 3-point FG’s that were assisted, missed 2 FG’s, and had 3 defensive rebounds.
The math at this stage can get complicated, but each statistical category is being awarded point value relative to the value it adds. Steals are more valuable than rebounds as they occur before a shot is attempted, and misses are more valuable than turnovers as missed shots can be rebounded whereas turnovers are an automatic loss of possession. Points scored also come with an opportunity cost of one possession. Once all points scored, possessions used, assists, misses, turnovers, offensive rebounds, points allowed, possessions scored on, defensive rebounds, blocks, steals, and turnovers forced are added up, the formula kicks in and provides the following value scores for the first 20 possessions:
- Garrett -0.93
- Braun -3.33
- Agbaji +3.87
- Wilson +0.01
- McCormack -4.86
- Thompson +0.45
- Lightfoot -1.52
- Harris -1.69
Adding these numbers up, we arrive at -8.00.
These numbers changed quite a bit by the time the game was over. Marcus Garrett continued to play elite defense (only gave up 2 points for the entire game), and his offense picked up in the second half to generate a +7.30 game value. This was more than the 5-point margin of the game, so he played a pivotal role in the final result of a KU victory and was the team’s MVP for that game.
Adj PPG +/-
The final calculation to make is an opponent adjustment, using publicly-available advanced rating systems. Given where UTEP is in the computer ratings (136th on KP, 147th on TR, 128th on BT, 167th on BPI), Kansas’s 5-point home margin of victory is reduced to a game score of -4.49. What this indicates is that the team played 4.49 points worse than that of an average game of a bubble-team.
Using this same concept, each player’s value score for the game is adjusted to account for opponent. After adjustment, Marcus Garrett’s value score is only +5.49. However, this still increases his Adjusted PPG +/- to +3.80 points per game for the season.
Offense and Defense PPG +/-
Additional, complicated steps are needed to divide between offense and defense. Basically what we want to know is what rate does a typical replacement player allow points, and then calculate from there. Using Garrett as an example, we know that since his Adj PPG +/- is +3.80, his Off +/- and Def +/- components sum to +3.80 in some fashion. In this instance, after doing the complicated math, Garrett’s defense (unsurprisingly) accounts for almost the entirety of the value as his Off +/- is +0.12 and Def +/- is +3.68. Garrett’s defensive score for this season (2021) is currently the highest defensive score CtH has calculated for any Kansas Jayhawk, slightly ahead of last season’s…you guessed it…Marcus Garrett.
Comparing Adj PPG +/- to Other Systems
Garrett’s Adj PPG +/- of +3.80 is the team’s highest, although only slightly higher than David McCormack’s +3.69. Either Garrett or McCormack will end the season as KU’s most valuable player. But if we were to take other value systems at face value, we wouldn’t know that.
College Basketball Reference currently has Marcus Garrett with a 3.0 WS (win share), which falls to fifth on the team. Likewise, his BPM (box plus-minus) is fifth on the team at 5.5. Garrett does have the team’s highest DBPM (defensive box plus-minus), so this stat is able capture some of his defensive value.
Bart Torvik’s value metric PRPG! has Garrett as 2.6 points per game better than a replacement player, once again below the other four Kansas starters and in fifth place for the team. It has him as 0.4 ppg worse than Jalen Wilson. This quite different from Adj PPG+/-, which is over 3.0 ppg in favor of Garrett.
Value Add Basketball has Garrett as the fourth-best Jayhawk, adding 5.79 pts/game. I’m not sure what the replacement level is in this system, but it nevertheless obviously does not account for his defensive prowess with the information it receives.
This isn’t to disparage these other systems, as they are doing the best with what they have, namely limited statistical information. But it does highlight why charting defense gives us a more-complete picture. What these other systems are able to do is give us national-level player data, whereas CtH only charts Kansas basketball. There simply isn’t enough time for one man to chart other teams.
Summary
Other systems of player value fail to fully take into account defense due to the lack of defensive statistical information. To account for this, Charting the Hawks created its own defensive charting system in order to accurately capture defensive value. In utilizing a possession-based system, it also accounts for “team rebounds” or non-steal turnovers that don’t normally show up in the stat sheet, such as drawn charges. Assists are also carefully sorted out, and rewarded based on passing merit instead of official scorer bias. Additionally, box score errors do occur with some frequency so these errors are corrected using game replays and the ability to rewind.
Footnotes
1 This doesn’t mean coaches don’t add value, rather the system incorporates the value added by coaches into the players’ value scores. Coach Self designing a play that allows McCormack to get an uncontested layup benefits McCormack’s value score. Players would have different value scores were they to play in a different system. This is true for all statistical measures. It would seem clear, based on his historic record and history of player development, that Self does a good job of getting players in the best position they can be in to succeed. A player value score simply tells us what value the player did add, not what he could have added under a different set of circumstances (whether that be more playing time, a different offensive system, etc.).
2 Defensive points allowed is a more subjective statistic than points scored. It is fairly obvious who scores the ball. For defense it isn’t always clear who the main defensive culprit is. Perhaps there are some reading this who attempted to chart the defensive possessions and got different results than I did. That’s ok! A group of 20 possessions is a small sample size. Over the course of the season, these interpretative differences will iron themselves out. (Value metrics aren’t meant to be exact, even the best sabermetricians calculate different WAR values for MLB players each season). What will remain is clear defensive differences between players of varying defensive skill. Regarding the 9 possessions where UTEP scored, I noted that points were scored due to a variety of ways including: giving up a layup off the dribble, giving up a step-back jumper, giving up an offensive rebound/stick-back, giving up penetration leading to an open jumper, and not recovering a catchable ball leading to a layup. More of the philosophy behind charting defensive points allowed will be given in later posts.
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