Introducing DRE: A (Hopefully) Better Simple Metric
Feb 14, 2015; New York, NY, USA; Golden State Warriors guard Klay Thompson (2) celebrates during the 2015 NBA All Star Three Point Contest competition at Barclays Center. Mandatory Credit: Bob Donnan-USA TODAY Sports
There are some really great, really sophisticated models for examining basketball in the public domain right now. ESPN’s Real Plus-Minus (“RPM”) being the most obvious one. The trouble with RPM is that it can be a bit of a black-box, even if you understand how it is calculated. It’s hard to know exactly which things a player is doing that are effecting his rating and to what degree. Additionally, RPM is calculated over season long data, so it is largely not useful for examining how a particular player played in a given game.
There are other ways of examining how a player played. There’s single game plus-minus, but that only really tells you how the score changed while the player was on the floor and it is subject to all the problems with raw, unadjusted plus-minus that you might have heard of by now: it is noisy, it is subject to teammate effects — both those a player plays with and his backups — and it also doesn’t tell you much of what the player actually did that was good or bad.
There are also simple, linear weights metrics. If you’re unfamiliar, all a linear weights metric does is add up the good things a player does and subtract the bad to come up with a one number measure of value. The “weights” portion of the name refers to the fact that different systems or metrics place different weights on how good or bad certain box-score actions are[1. For a really great comprehensive breakdown of the many linear weights metrics that have been developed for basketball analysis over the years, here‘s Five Thirty Eight’s Neil Paine, writing for Basketball Prospectus, with a great round-up.].
For instance, John Hollinger’s Game Score, a stripped-down version of his popular Player Efficiency Rating (PER) [2. PER itself is just a more convoluted linear weights metric.] sets its weights as follows:
"PTS + 0.4 * FG – 0.7 * FGA – 0.4*(FTA – FT) + 0.7 * ORB + 0.3 * DRB + STL + 0.7 * AST + 0.7 * BLK – 0.4 * PF – TOV."
You’ll note that the metric credits made baskets twice. First, the player gets credit for his points scored, but he also gets an addition .4 points in the metric for every shot made, while being debited .7 points for every shot attempt. This is almost certainly a means of crediting heavier-usage players for the fact that they are taking difficult shots. PER also makes this adjustment, though, in a much more complicated fashion. There’s nothing wrong with this adjustment, really. A tradeoff between usage and efficiency has been found in more than one study[3. Here’s one from Eli Witus, now the Vice President of Basketball Operations for the Houston Rockets. Here’s another from our own Justin Willard.]. There’s no empirical basis, though, for the specific values Hollinger set for Game Score.
Another popular linear weights metric is Alternate Win Score.[4. Much of the reason for AWS’s popularity is that Neil Paine study referenced in footnote 1. He found AWS to be the best of all the linear weights metrics at predicting results across low-continuity and high-continuity contexts – i.e. when a player changed teams and roles and also when a player stayed in the same team and role.] The weights for Alternate Win Score are as follows:
"PTS+ 0.7*ORB + 0.3*DRB + STL +0.5*BLK + 0.5*AST -0.7 *(FGA-FG)-FG-0.35*(FTA-FT)-0.5*FT-TOV-0.5*PF"
You can see a lot of overlap between the values selected between Game Score and AWS.
After spending some time thinking about the importance of being able to track game to game fluctuations in performance[5. Admittedly, much of this is due to my desire to keep track of how Derrick Rose is doing coming back from his knee injuries. So far, very up and down.], I decided to take a shot at creating my own very simple linear weights metric, but I wanted to be a little more discerning in how I arrived at the value of my weights[6. It’s more than a little unfair of me to say Hollinger and others who created linear weights metrics weren’t being discerning because they didn’t have access to the same data we have now, which is at the core of how I settled upon my weights. They did the best they could with the information they had and, as we’ll see, did a pretty good job of eye-balling things.]. So I did what made the most sense to me and ran a simple regression of per 100 possession basic box-score stats against Jerry Engelmann’s[7. The creator of ESPN’s Real Plus-Minus.] long-run regularized adjusted plus-minus data set[8. Regularized adjusted plus-minus is the basis of Real Plus-Minus. It essentially takes plus-minus data from every different lineup run out in an NBA season, adjusts for opponents and teammates using what amounts to algebra on steroids, to distill how much of an impact any given player had on the score, per 100 possessions. This data set includes 14 seasons worth of data, so one of the big problems with adjusted plus-minus models, noise as a result of too small a sample, is removed. This makes it great for my purpose, as the estimates of player value produced by long-run RAPM are very trustworthy.]. This allowed me to determine the relative value or “weight” of each stat typically included in linear-weights metrics. Here’s what the result of my regression looked like:
As you can see, all of the statistics I selected produced statistically significant results. (That’s what the little “***” annotation on the right is all about.) Tweaking the results such that all of the weights are set on a scale where points are weighted to 1, we get the following:
"PTS + .2*TRB + 1.7*STL + .535*BLK + .5*AST – .9*FGA – .35*FTA – 1.4*TOV"
One thing that should jump out right away is that I used total rebounds, rather than breaking them out by offense and defense. That’s because when I did break them out by offense and defense in my initial regression, offensive rebounds were not significant in predicting plus-minus impact (at least as measured by RAPM). I didn’t want to completely leave out offensive rebounds, because I do believe they are important and have value, so I included them by running all rebounds together as part of the regression. It should be noted, however, that my values suggest that previous linear weights metrics greatly overvalued rebounds.
Another big takeaway from these values is the importance of possession of the ball. A steal is worth roughly 1.7 points, presumably because it takes possession from the opposing team — resulting in zero points for them — and provides an easier scoring opportunity on the other end for the offense. Similarly, turnovers are worth -1.4 points, costing nearly as much as a steal helps. The reason these two numbers are probably not identical is that dead-ball turnovers don’t result in the same easy scoring opportunities for the team receiving possession that steals do.
The value of threes are implicitly included in these weights, as a made 2 point shot is worth 2 points – .9 for the FGA, so 1.1 points, whereas a made three is worth 2.1 points to the player’s impact. Similarly, the value of getting to the foul line should be obvious. A player who gets to the line and makes both of his attempts provides an estimated 1.3 points of value, .2 points more than a made 2 point shot. Much of this likely has to do with the extra value provided by getting to the line — i.e. getting the opposing team in foul trouble[9. Interestingly, personal fouls committed were not statistically significant as a predictor of plus-minus impact as measured by RAPM. As such, they were left out of my linear weights.].
I should mention briefly that I tried to include FG in the way that Hollinger did in his Game Score to credit high-usage players for their more difficult roles, but it didn’t help the fit with plus-minus impact[10. as measured by long-run RAPM].
The metric as described produces a rough measure of net points contributed per 100 possessions, but in order to turn the metric into something like the plus-minus derivative that it is, you need to subtract out league average performance. Thus far this year, league average performance has been roughly 4.9 points per 36 minutes played or .136 points per minute[11. This doesn’t account for pace differences between teams, but the aim here is relative simplicity.]. This makes sense. If you look at the regression I ran, -6.8 is the intercept on those per 100 possession stats. This means that 6.8 is the league average value produced per 100 possessions. Converting that to 96 possessions[12.because league average pace has been roughly 96 possessions per 48 minutes this season], you get 6.5 and if you divide that by the 48 minutes those 96 possessions occur in, you get that same .136 per minute value. So league average value per possession seems to be pretty consistent over time.
So, in order to determine an estimate of points contributed above average in a game, the calculation is as follows:
"PTS + .2*TRB + 1.7*STL + .535*BLK + .5*AST – .9*FGA – .35*FTA – 1.4*TOV – .136*Minutes"
Now, it’s time to name my creation. Thankfully, I was able to ask Twitter’s advice and our man, Layne Vashro came through with the following name, Daily RAPM Estimate or DRE.
Now that we’ve established what DRE is and how I arrived at the values, who’s had the best and worst games this season, as measured by DRE?
The Worst
Russell Westbrook, -17.3, January 7, 2015 vs. Kings (83-104 loss)
This was the classic Bad Russell game. Westbrook got up 19 shots in just 27 minutes, but made only 3 of them to score just 10 points. He also turned the ball over 7 times. That number of empty possessions – from missed shots and turnovers – unsurprisingly does a lot of damage to a team’s bottom line. This was an incredibly tough game for the Thunder (Kevin Durant’s DRE in this one was also below average -2.4 in 33 minutes) and with Westbrook playing so poorly it should come as no surprise that the Thunder lost in blowout fashion.
Derrick Rose, -18.5 DRE, January 27, 2015 vs. Golden State (113-111 win)
This was the much discussed overtime game in which Derrick Rose took 33 shots to score 30 points and turned the ball over a whopping 11 times in 43 minutes, while dishing just one assist. Of course, Rose finished the game with this:
So all was forgiven.
Michael Carter-Williams, -21.0 DRE, December 27, 2014 vs. Utah (71-88 loss)
This was easily the worst game by any individual player in terms of impact within a single game. MCW was an absolutely abysmal 2 for 20 from the field, scored just 8 points, and had 6 turnovers in 37 minutes of playing time. Taking that many shots to score just 8 points is obviously terrible and 6 turnovers is quite a few. MCW also didn’t lodge any defensive box-score numbers, producing 0 steals and blocks. It should come as no surprise then that, given how many minutes he played and how little he produced, that MCW’s post-Christmas stinker was the worst performance of the season.
The Best
Stephen Curry, +22.2, November 25, 2014 vs. Miami (114-97 win)
Steph Curry has, arguably, been the league’s best player this season, so his place in the top 3 performances on the season likely comes as a shock to no one. In Golden State’s late November beatdown of Miami, Curry went nova going off for 40 points on 19 shots in 37 minutes, nabbing 3 steals, dishing 7 assists to just 2 turnovers, pulling in 6 rebounds, and blocking a shot just for good measure. Stephen Curry is, in a word, ridiculous.
James Harden, +22.9, December 22, 2014 vs. Portland (110-95 win)
If Curry isn’t the league’s MVP this year, the man with the next best case is Houston’s James Harden, so, again, finding the Beard in this slot is hardly surprising. Just before Christmas, Harden dropped 44 points on 26 shots and 13 free throw attempts (typical Harden). He added 7 assists to just 1 turnover and tallied 5(!) steals. Harden managed to do all this in just 32 minutes of playing time. Just an incredible performance.
Klay Thompson, +26.7, January 23, 2015 vs. Sacramento (126-101 win)
You remember this one. This was easily the best performance of the year. Klay Thompson scored 37 points in an impossible, perfect 3rd quarter. For the night, he dropped 52 points on 25 shots (!!!!), 5 assists, 4 steals, and 2 blocks. Klay was such a towering inferno that even his 4 turnovers in just 33 minutes of playing time couldn’t put a dent in this game for the ages. I mean, just go ahead and relive this, it was awesome:
Thanks to the coding and data scraping wizardry of our own Darryl Blackport, you can find DRE for every game played by every player this season, in a searchable and sortable form, here. On that page, you’ll also find cumulative DRE over the course of the whole season put together by good dude and All-American basketball writing prospect, Seth Partnow. I hope you, the reader, will find this information useful. Enjoy responsibly!