# Month: December 2021

• ## Introduction to Cryptbeam Plus/Minus (CrPM)

I talk about NBA impact metrics a lot; and as most familiar with my content know, I’ve even created a few. Now, I’m ready to add another player to the mix: “Cryptbeam Plus/Minus” (CrPM). With the metric’s namesake being the title of this website, CrPM joins a long line of composite metrics that aim to identify the difference-makers of the sport through quantitative analysis. Why? As the creator of Player-Tracking Plus/Minus (PTPM), Andrew Johnson, once said: “Because what the world needs now is another all in one basketball player metric.” [1]

Overview

CrPM draws most of its inspiration from two established impact metrics in Jacob Goldstein’s Player Impact Plus/Minus (PIPM) and Ben Taylor’s Augmented Plus/Minus (AuPM). The commonality of these three metrics is the shared “branch” of metric: I tend to differentiate between impact metrics by one of three types:

• Box: a composite metric that is calculated using the box score only
• Hybrid: a composite metric that is calculated using counting statistics outside the box score (e.g. player tracking and on-off data) — but may also include the box score
• APMs: a composite metric that uses ridge-regressed lineup data as the basis of its calculation

As for where CrPM falls under these categories, it’s formally a hybrid metric. The metric can be broken into two major component: a box score term and a plus-minus term. The box score “version” of CrPM can function as an impact metric on its own as an estimator of impact via traditionally-recorded counting stats: points, rebounds, assists, etc. Using Regularized Adjusted Plus/Minus (RAPM) as a basis, the box score is regressed onto this target to estimate a player’s per-possession impact on his team’s point differential.

Two plus-minus statistics are then added to the box-score estimate to create CrPM: on-court Plus/Minus, which is a team’s per-100 point differential (Net Rating) with a player on the floor, and on-off Plus/Minus, which subtracts the team’s Net Rating with a given player off the floor from his on-court rating. The goal of the plus-minus component is to fill in some of the gaps left in the immeasurable, e.g. what the box score can’t capture. Testing of the model did later reinforce this idea.

Regression Details

The regressions for both the box score and plus-minus variants of the metric were based on fourteen years of player-seasons from Jeremias Engelmann’s xRAPM model. (This means that, in the RAPM calculation, a player’s score is pushed toward the previous season rather than zero.) This provided a more stable base for the regression to capture a wider variety of player efficacy in shorter RAPM stints. Additionally, the penalization term for each season was homogenized to improve season-to-season interpretability.

The box score is not manipulated in the CrPM calculation outside of setting the stats as relative to league averages. The raw plus-minus terms, however, use minutes played as a stabilizer to draw results closer towards zero as playing time decreases. While some incorrectly label a larger sample size as more accurate, this adjustment serves to reduce variance in smaller samples to decrease the odds for larger error in these spots. These two branches of NBA statistics combine to create CrPM. [2]

The regressions, especially the box score ones, were all able to explain the variability in the target RAPM with a surprising degree of accuracy. The R^2 values for combined in and out-of-sample RAPM ranged from 0.725 to 0.750 for the box-score metrics and bumped up to 0.825 with plus-minus included. A large concern for most ordinary linear regressions is heteroskedasticity, which is when error rates become larger as the predicted variable increases. The breadth of performance captured in the RAPM target mitigated this, and CrPM serves as an accurate indicator of impact for average players and MVP players alike.

• Target response of RAPM was collected from Jeremias Engelmann’s website.
• Box scores and plus-minus data were collected from Basketball-Reference.

Because CrPM serves as an indicator of a player’s impact, it can do a solid job of identifying viable candidates for the league’s MVP award. Through the 18th of December, here are the top-10 players of the 2022 season with at least 540 minutes played per the metric:

1. Nikola Jokic (+11.8)

The reigning MVP has started his follow-up campaign with a bang, and looks to be the frontrunner to snag the award again. His placement in the box-score component of the metric (+12.4 per 100) would be first among all players since 1997 with at least 1,500 minutes played during the season. Jokic’s overall score of +11.8 in CrPM is tied with LeBron James’s legendary 2009 seasons for the best regular season on record.

2. Giannis Antetokounmpo (+9.4)

Voter fatigue in recent seasons has downplayed the regular-season greatness of the Greek Freak, but he seems as good as ever according to CrPM. Each of his last three seasons are top-45 seasons since 1997 per the metric, with his dominating MVP run in 2020 ranking fourth (+10.8) among all high-minute players on record.

3. Joel Embiid (+6.6)

The metric suggests Embiid provides massive two-way value, with his marks on offense and defense being nearly identical in both the box-score and plus-minus versions of the metric. Despite a shooting slump to start the season that followed an exceptional mid-range campaign for the big man, he still adds a ton of value in Philadelphia. So far, the 76ers outscore opponents by +4.3 per 100 when Embiid is on the floor and are +9.2 points better with him in the lineup.

4. Rudy Gobert (+6.5)

It’s not an NBA regular season nowadays without the analytics looking upon Rudy Gobert with perhaps a little too much enthusiasm. Regardless, he’s still a surefire candidate for the DPOY Award and a probable finalist, if not winner. Almost all of his impact comes from the defense end, adding only +0.4 points per 100 in the offensive component of CrPM, with the remaining +6.1 points coming from his all-time-level rim protection and paint anchoring.

5. Stephen Curry (+5.7)

Curry’s plus-minus portfolio isn’t as transcendent as it was during his all-time seasons in the mid-2010s, but a revamped Golden State roster that amplifies his strengths boosts those numbers into the upper echelon of NBA superstars once more. The Warriors outscore opponents by +14.1 points per 100 with Curry on the floor and are +12 points better with him on the court. While the box score can’t capture all of the value he brings to the table, Curry still looks like one of the best players in the league according to CrPM.

6. Kevin Durant (+5.5)

Curry’s former teammate is another shooting savant who continues to string together legendary offensive seasons with league-best marks in both scoring volume and efficiency. He’s arguably the greatest mid-range shooter in NBA history and makes Brooklyn’s offense great with these shots alone. Speaking of, the Nets look like a surefire title contender with Durant on the floor, when they outscore opponents by +7.3 points per 100 possessions.

7. Clint Capela (+4.6)

While I don’t think Clint Capela has been a top-10 player in the league this year, his placement illustrates arguably the biggest sign of caution I’d address when using CrPM. Because the box-score component receives a whole lot of the weight in the plus-minus-included formula, the metric is overly sensitive to defensive rebounds and stocks. Especially in the modern league, when spacing affects statistics like rebounds to a higher degree than ever, Capela is a case of, while being a valuable player nonetheless, a stylistic disadvantage of CrPM.

8. Jarrett Allen (+4.5)

I’m similarly not as fond with Allen’s actual ranking among the league’s best, but he’s been sneakily good this season. The Cavaliers have clearly stocked up on Michael’s Secret Stuff for 2022, because they’ve been better than the Nets with Durant when they put Jarrett Allen in the game. Cleveland outscores its opponents by +8.4 points per 100 with Allen in the lineup! One of the sport’s emerging two-way talents receives well-deserved credit in the analytics.

9. Karl-Anthony Towns (+4.4)

Towns has played like an all-league star for a while, and now that his longtime home of Minnesota has found its footing in 2022, his value is more evident than ever. CrPM views Towns as a clear value-add on offense and defense, and this is enough to make him look like a candidate for the All-NBA second team this year. He’s one of the On-Off kings so far, posting a +13.1 Net Plus/Minus in a similar vain to other top-player candidates.

10. Jimmy Butler (+4.4)

If this list were box-score only, Butler would be several spots higher with his +6.3 score in Box CrPM; however, plus-minus doesn’t look upon him like the other players on this list entering 2022. Miami plays like a surefire postseason team with him on the court, but they actually perform +2.4 points better with him off the floor! While this is very, very likely just noise that accompanies most plus-minus data like this, it doesn’t serve as a very good indicator of his impact, thus compressing his score in plus-minus-included CrPM.

PlayerTmGMPO CrPMD CrPMCrPM
Nikola JokićDEN247837.93.911.8
Giannis AntetokounmpoMIL268495.53.99.4
Joel EmbiidPHI196293.33.36.6
Rudy GobertUTA299200.46.16.5
Stephen CurryGSW289625.20.55.7
Kevin DurantBRK2710005.30.25.5
Clint CapelaATL298640.24.44.6
Jarrett AllenCLE289131.92.64.5
Karl-Anthony TownsMIN289633.01.44.4
Jimmy ButlerMIA186074.20.24.4
LaMelo BallCHO258283.21.24.4
DeMar DeRozanCHI248464.9-0.64.3
Trae YoungATL299906.2-2.04.2
Montrezl HarrellWAS317993.30.94.2
Myles TurnerIND30887-0.64.64.1
John CollinsATL299392.41.53.9
LeBron JamesLAL186673.10.73.8
Chris PaulPHO289063.60.33.8
Jusuf NurkićPOR307520.63.23.8
Dejounte MurraySAS289632.21.63.8
Donovan MitchellUTA289104.1-0.63.5
Jonas ValančiūnasNOP319801.52.03.5
Domantas SabonisIND3110571.91.53.4
Jrue HolidayMIL258193.00.43.3
Kristaps PorziņģisDAL216341.41.93.3
Jaren Jackson Jr.MEM297870.23.13.3
Jayson TatumBOS3010952.30.93.2
Andre DrummondPHI29571-2.86.03.2
Fred VanVleetTOR2810602.70.43.1
Anthony DavisLAL279551.02.23.1
LaMarcus AldridgeBRK255901.81.33.1
D’Angelo RussellMIN247812.90.13.0
Miles BridgesCHO3111351.91.02.9
James HardenBRK269421.90.92.8
Richaun HolmesSAC225961.51.32.8
Al HorfordBOS247110.22.42.6
Luka DončićDAL217352.30.22.5
Bobby PortisMIL257110.52.02.5
Alperen ŞengünHOU29538-0.12.62.5
Jarred VanderbiltMIN28682-1.64.02.4
Daniel GaffordWAS28601-1.13.42.3
Damian LillardPOR248733.9-1.62.3
Patrick BeverleyMIN215521.30.82.1
Ja MorantMEM196193.0-0.92.1
Wendell Carter Jr.ORL308650.21.82.0
Evan MobleyCLE25840-0.92.92.0
Mike ConleyUTA267273.2-1.21.9
Deandre AytonPHO206250.51.51.9
Malcolm BrogdonIND258832.7-0.81.9
Devin BookerPHO216763.1-1.21.9
Robert WilliamsBOS23638-0.42.31.9
Derrick RoseNYK266362.2-0.41.8
Paul GeorgeLAC248610.61.21.8
Darius GarlandCLE299913.0-1.31.8
Draymond GreenGSW28849-0.92.61.7
Jakob PoeltlSAS216060.51.11.6
Nikola VučevićCHI20663-0.92.51.6
De’Anthony MeltonMEM27654-0.62.21.6
Ricky RubioCLE318801.20.31.5
Brandon IngramNOP248552.3-0.81.5
Cole AnthonyORL237851.9-0.51.4
Alex CarusoCHI24685-0.11.51.4
Mo BambaORL27774-2.43.81.4
Monte MorrisDEN298752.4-1.11.3
Zach LaVineCHI279483.1-1.81.3
Shai Gilgeous-AlexanderOKC258731.9-0.61.2
Jalen BrunsonDAL277952.7-1.51.2
Pascal SiakamTOR175861.10.11.2
Mitchell RobinsonNYK27653-1.52.61.1
Aaron GordonDEN299451.2-0.21.1
Devin VassellSAS235590.11.01.1
Tyus JonesMEM306341.8-0.71.1
Cedi OsmanCLE255521.3-0.21.1
Kevon LooneyGSW30564-1.02.01.0
Deni AvdijaWAS31672-1.32.31.0
Christian WoodHOU28888-1.02.01.0
Desmond BaneMEM308631.3-0.31.0
Anthony EdwardsMIN2810060.50.51.0
Khris MiddletonMIL216501.00.01.0
Immanuel QuickleyNYK296331.8-0.90.9
Andrew WigginsGSW299011.7-0.80.9
Gary Trent Jr.TOR279330.60.20.8
Ivica ZubacLAC30742-1.21.90.7
Kyle AndersonMEM26565-1.31.90.6
Alec BurksNYK297740.40.30.6
Caris LeVertIND236681.3-0.70.6
Cody MartinCHO297990.10.40.6
Tyrese HaliburtonSAC289300.10.40.5
Larry Nance Jr.POR30650-0.91.40.5
Scottie BarnesTOR279730.10.40.5
Tyrese MaxeyPHI289691.9-1.50.5
Josh HartNOP247560.10.30.4
Gordon HaywardCHO3110521.4-1.00.4
CJ McCollumPOR248481.4-1.10.3
Mikal BridgesPHO289630.30.00.3
Tobias HarrisPHI217200.7-0.40.3
Lonzo BallCHI27958-0.91.20.3
Jordan ClarksonUTA297301.3-1.00.2
OG AnunobyTOR165880.5-0.30.2
Marcus SmartBOS29991-0.30.40.2
Kelly Oubre Jr.CHO319031.1-1.00.1
Lauri MarkkanenCLE226640.3-0.20.1
Kyle LowryMIA289620.9-0.90.1
Devonte’ GrahamNOP288761.3-1.20.1
Will BartonDEN258250.6-0.60.0
Grayson AllenMIL308750.7-0.70.0
Derrick WhiteSAS288750.3-0.4-0.1
T.J. McConnellIND245810.4-0.5-0.1
Franz WagnerORL319940.3-0.4-0.1
Jerami GrantDET24797-0.10.0-0.2
Danny GreenPHI23558-2.01.8-0.2
Russell WestbrookLAL3010780.2-0.4-0.2
Mason PlumleeCHO22562-1.61.5-0.2
Patty MillsBRK309051.6-1.8-0.2
Luke KennardLAC308700.8-1.1-0.2
Herb JonesNOP28766-1.71.4-0.3
Jae CrowderPHO28789-1.51.2-0.3
Nassir LittlePOR26603-1.61.3-0.3
Keldon JohnsonSAS278290.1-0.5-0.4
George HillMIL266860.1-0.5-0.4
Royce O’NealeUTA27833-1.40.9-0.5
Bojan BogdanovićUTA298641.8-2.2-0.5
Jordan PooleGSW288600.8-1.2-0.5
Reggie JacksonLAC309921.1-1.6-0.5
Gabe VincentMIA275320.3-0.8-0.5
Jae’Sean TateHOU30844-0.80.2-0.6
Lonnie WalkerSAS27611-0.1-0.5-0.6
Norman PowellPOR268251.0-1.6-0.6
Terry RozierCHO227080.5-1.1-0.7
Bogdan BogdanovićATL205640.5-1.2-0.7
Josh RichardsonBOS225550.1-0.8-0.7
Shake MiltonPHI256360.0-0.8-0.8
Carmelo AnthonyLAL30828-0.3-0.5-0.8
Georges NiangPHI286620.2-1.0-0.8
Anfernee SimonsPOR266181.3-2.2-0.8
Harrison BarnesSAC258360.2-1.1-0.9
De’Aaron FoxSAC299930.5-1.4-0.9
Matisse ThybullePHI23555-3.62.7-0.9
Danilo GallinariATL26568-0.1-0.8-0.9
Isaiah StewartDET26668-2.71.8-0.9
Cameron JohnsonPHO28682-0.6-0.3-0.9
Raul NetoWAS30607-0.2-0.8-0.9
Seth CurryPHI279291.1-2.0-1.0
Pat ConnaughtonMIL32929-0.4-0.6-1.0
Dennis SchröderBOS278940.8-1.9-1.0
Kevin HuerterATL287740.2-1.3-1.0
Malik MonkLAL28666-0.2-0.8-1.0
Tyler HerroMIA258220.5-1.5-1.1
Buddy HieldSAC308570.0-1.1-1.2
Julius RandleNYK301063-1.30.0-1.2
Joe InglesUTA297180.2-1.5-1.3
Spencer DinwiddieWAS267620.2-1.4-1.3
Grant WilliamsBOS28618-1.0-0.3-1.3
Tim Hardaway Jr.DAL28878-0.1-1.2-1.3
Precious AchiuwaTOR22579-2.81.3-1.4
Terance MannLAC29829-0.5-1.1-1.6
Josh GiddeyOKC26779-2.20.6-1.6
Chris DuarteIND29843-1.0-0.6-1.6
P.J. TuckerMIA30856-1.2-0.5-1.6
Bruce BrownBRK24539-2.50.9-1.7
Kyle KuzmaWAS29937-2.30.6-1.7
Dorian Finney-SmithDAL28899-1.80.0-1.8
Davion MitchellSAC29744-0.3-1.6-1.9
Robert CovingtonPOR30822-4.22.2-2.0
Luguentz DortOKC26841-0.3-1.7-2.0
Eric GordonHOU257450.0-2.0-2.0
Furkan KorkmazPHI27600-1.0-1.1-2.1
Nickeil Alexander-WalkerNOP31877-1.3-0.8-2.1
Kevin Porter Jr.HOU19574-2.20.1-2.2
Dwight PowellDAL28536-1.8-0.4-2.2
Duncan RobinsonMIA30849-1.3-0.9-2.2
Isaac OkoroCLE23663-1.3-1.0-2.3
Jeff GreenDEN29732-0.9-1.4-2.3
Eric BledsoeLAC30787-2.4-0.1-2.5
Chuma OkekeORL25562-3.51.1-2.5
Cam ReddishATL25558-1.5-1.0-2.5
Landry ShametPHO27562-0.6-2.0-2.5
RJ BarrettNYK25784-1.4-1.2-2.5
Kentavious Caldwell-PopeWAS31906-2.0-0.5-2.6
Justin HolidayIND25693-1.2-1.5-2.7
Darius BazleyOKC28765-4.51.9-2.7
Frank JacksonDET28620-0.6-2.1-2.7
Killian HayesDET23602-2.90.1-2.9
Facundo CampazzoDEN28567-1.9-1.0-2.9
Jeremiah Robinson-EarlOKC28608-2.90.0-3.0
Jalen SuggsORL21583-3.1-0.1-3.2
Evan FournierNYK30857-1.6-1.8-3.4
Malik BeasleyMIN29750-1.7-1.8-3.4
R.J. HamptonORL29536-2.4-1.1-3.5
Garrett TempleNOP29532-4.20.5-3.6