Background for what RAPM is: +/- was a revolution for the NBA because it allowed for a completely new method at evaluating players. You look at how a team scores and defends with you on the court and without you. When you set players as variables, you can use regression to calculate player impact. It's a full scope view of what matters in a game: outscoring your opponent. However, it's noisy for a number of reasons. One is that some player combinations are rare (this is known as collinearity.) Another is that the models don't deal well with low minute players, as they don't have enough of a sample for an accurate estimate and will often produce a ludicrous result just to "fit" the data better.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Using a prior set of values, Bayesian analysis, greatly improves the results. Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
As the league tried to recover from the lockout season, a new dynasty took form. Although it sounds odd now, the Shaq and Kobe duo were once disappointments, but with Phil Jackson on board as a coach and Shaq putting in more work than he ever had before the Lakers had an all-time great season -- 67 wins, a +8.6 point differential, and a championship. It was the dawn of a new era, and while it wasn't technically a new millennium (that's 2001), a number of new stars were surfacing. Garnett was second in MVP voting and did everything for his Minnesota team. Iverson took a step forward scoring 28 a game. Vince Carter reinvigorated the all-star weekend with his epic dunk contest (oh and he played basketball too.) And other young players emerged -- Dirk went from an unknown German into a intriguingly good player, Ray Allen topped 20 points per game for the first time, Kobe did so as well (the first of 14 such seasons), and Elton Brand was rookie of the year.
How does NPI RAPM view the players? Well, just as a reminder, non prior informed RAPM often has wonky results because there's not enough data -- Rodney Rogers tops the list. He was a big forward who had a career year, shooting efficiently and spacing the floor for the 53 win Phoenix Suns, who toppled the Duncan-less Spurs in the playoffs. Simply put, when Rogers was on the floor the Suns outscored their opponents by 9.6 points per 100 possessions, but it dropped to virtually 0 without him. He also won sixth man of the year; the fact that Phoenix plays with the best with him on the court is impressive. And the adjusted +/- rating, RAPM, obviously agrees that the Suns were better with him. Shaq, however, is third, curiously behind Terry Porter (Spurs that season.) Stockton and Vince Carter continue to be plus/minus stars even without the bias of a prior, and Payton looks great again on offense. As for a historically underrated player, Bo Outlaw was fourth overall -- and he was fifth in 1997 (NPI).
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
The preferred form of RAPM, however, is in the spreadsheet below. The top twenty consists of stars and highly respected players with unique skillsets in Sabonis, Rasheed Wallace, Divac, Mutombo, Eddie Jones, and Robert Horry. But Shaq destroys everyone. Playing 40 minutes a game, he would, going purely by the numbers, take an average team to 60 wins.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
RAPM and MVP voting agree on the first two names, and the rest of the top five in voting are rated well too. The divergence starts with big men who, mainly due to defense, are found to be more valuable from plus/minus, but there are only a couple guys in the top ten in MVP voting who aren't highly rated. Iverson came in 7th, winning the MVP next season, but he had one of the worst defensive plus/minus values. Webber also had fairly mediocre plus/minus stats, but was voted 9th. The all-NBA voting follows a similar path as 11 of the 15 players were in the top 25 in RAPM. The other four names are known for not being rated well in the stats community: the aforementioned Iverson and Webber, Stephon Marbury, and Kobe Bryant, who's actually a significant negative on defense. For the Rookie of the Year results, Elton Brand and Steve Francis shared the trophy, but according to this stat they were two of the worst rookies. Instead the guys who came in third and fourth in the voting, Odom and Andre Miller, should have won -- they were first and second, respectively, in RAPM, where Odom in particular was a very valuable player at +3.4. Rookies are rarely that valuable.
The best offensive player was Shaq -- not surprising because he averaged nearly 30 points a game on great efficiency with almost 4 assists. Karl Malone was second; apparently he slipped on defense but was still a scoring machine. Grant Hill, in his last great season, was third, as he was a point forward who scored 26 a game. One surprise is that Gary Payton's value appears to be more on offense because he's fourth here, matching previous results. Shooting legend Reggie Miller was fifth in offense in his last all-star season. Iverson, criticized for his shot selection, was actually eighth. Defense, of course, is dominated by big men -- Mutombo is first again, the giant Shawn Bradley second, and the underrated Bo Outlaw third (though he did pick one up vote for Defensive Player of the Year.) David Robinson was fourth, as RAPM finds him to be the basis for San Antonio's defense. Rasheed Wallace and Vlade Divac, not known for defense, followed closely. The Defensive Player of the Year was actually Mourning, who was tenth in defensive RAPM -- not a terrible choice according to plus/minus, but I suspect there was some voter fatigue with Mutombo. As a last note, this is believed to be Shaq's best season on defense, but he doesn't show up, and instead the highest rated Laker, and highest rated perimeter player in the league, was actually Derek Fisher.
RAPM, like any metric, isn't perfect, but it can perform as well or better than the popular box score metrics PER and Win Shares. For example, while Kobe's defensive +/- doesn't necessarily mean he was a "bad" defender, you can treat it like robust evidence about what effect his defense had in 2000. But every single metric agrees that Shaquille O'Neal stormed the league and was by far the best player. Plus/minus stats are often used to identify undervalued players, and in this context it's more of a historical retrospective for unheralded guys like Divac, but sometimes it's just fun seeing what legends have done in the past. Shaq owned the NBA that year.
Click here for the link to the spreadsheet.
Showing posts with label rapm. Show all posts
Showing posts with label rapm. Show all posts
Wednesday, March 26, 2014
Monday, March 3, 2014
1999 RAPM: non-prior and prior informed
Background for what RAPM is: +/- was a revolution for the NBA because it allowed a completely new method at evaluating players. You look at how a team scores and defends with you on the court and without you. When you set players as variables, you can use regression to calculate player impact. It's a full scope view of what matters in a game: outscoring your opponent. However, it's noisy for a number of reasons. One is that some player combinations are rare (this is known as collinearity.) Another is that the models don't deal well with players with low minutes, as they don't have enough of a sample for an accurate estimate and will often produce a ludicrous result just to "fit" the data better.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
The NBA lockout that delayed the 1999 season put a serious dent in the league and wasn't aided by the retirement of Michael Jordan and the dismantling of the Bulls. It was a transition year, with the old guard dropping off with the exception of iron-man Karl Malone, while the next generation was still developing and coming into their own. With a shortened season opening in February, games jammed together with too many back-to-back's, a team known for professionalism, led by a second year Duncan and a 33 year-old David Robinson, won the championship. This was Shaq pre-Phil Jackson but post-Jordan; it's a truly lost season.
Nevertheless, the games were played, and we can't ignore what happened. Surprisingly, by NPI RAPM, the best player in 1999 was ... David Robinson, who had a monster defensive impact. Just a season before Duncan and Robinson were neck-and-neck on a per possession basis, but the Admiral leaps ahead here. Though with a 24.9 PER and a league-leading 0.261 Win Shares per 48 minutes, and stats that did not drop in the postseason, perhaps it shouldn't be surprising. However, he only played 31.7 minutes in the regular season, lending credence to an argument for another candidate like Alonzo Mourning. Jaren Jackson is probably the oddest name near the top of the list. He was a Spurs teammate but only scored 6.4 points per game. And he's why we turn to prior-informed versions....
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
With two seasons as seed data behind it, RPI RAPM (an adjusted ridge-regression model given prior information) is a more powerful tool. After a dominating 1998, Shaq loses the top spot due to poorer defense. This was probably Alonzo's greatest season with award-caliber defense coupled with potent scoring (20 points in a slow-paced league and a 56 TS%.) He was second in MVP voting -- justified here. As for some surprising results, Blaylock continues to be a plus/minus star, and Rasheed Wallace rockets to the top on the deep but strong Blazers team that nearly defeated the Lakers in 2000. Charles Barkley also continues his one-sided results: his offensive RAPM rises to 6.16 but his defense slips to -2.64.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
Fourth in MVP voting was Iverson. He's 75th here with good offensive value despite his poor shooting efficiency, but that's submarined by his porous defense, according to RAPM. Duncan was third, and he's backed-up by a very strong +5.2 mark. Rookie of the year Vince Carter, however, climbs from the rookie prior of -2 (given to all rookies) to +3.21 overall. He heads a pretty strong rookie class including Pierce, Brad Miller, and Dirk Nowitzki. Jason Williams received some rookie love, but by this method he was a significantly negative player. For validation of the model, 16 of the top 24 players were on all-NBA teams -- or perhaps that's validation of the all-NBA teams. The lowest rated all-NBA player was the young Kobe Bryant at -1.2 and it wasn't close: he was ranked 267th and the next lowest all-NBA player was McDyess at 117.
The aforementioned Barkley was the best offensive player, according to RPI RAPM, and that's not just because of his past heroics: his non-prior informed RAPM was second. Shaq's stats were muted by fewer minutes and the slogging pace of the lockout season, but, with the first of his strong of 30+ PER seasons, he's second in RPI RAPM on offense, barely trailing Barkley. Malone rounds out the top three, and the next highest players are more intriguing: Reggie's three-point bombing is probably underrated by most box score metrics at fourth here, Grant Hill's fifth, and Jeff Hornacek, again, has a strong showing at sixth followed by Blaylock. Both Hornacek and Blaylock are players that appear to be underrated by a method that wasn't available in the 90's. Mutombo, David Robinson, and Mourning were top three by defense. Mourning won the Defensive Player of the Year award, which wasn't a bad choice, necessarily, but Mutombo has a huge lead in RAPM. Rasheed was fourth, and his value was demonstrated later when he was traded to the Pistons to complete one of the greatest defensive teams ever. Jaren Jackson appears to be an anomalous result, ranking fifth, but he was San Antonio's first "three-and-D" player. He concentrated on defense, and his value does not show up in box scores. His NPI RAPM wasn't entirely misleading. Trailing him were Shawn Bradley, Olajuwon, and Garnett. Gigantic size might actually be underrated because Yao Ming's defense looked better under the RAPM microscope and Bradley joins Gheorghe Muresan as another 7' 7" player with strong defensive results.
With three seasons of data, outliers can be weeded out and patterns emerge. We can reevaluate past legends like David Robinson and Stockton who rate well, but sub-stars should receive more discussion. Mookie Blaylock, by NPI RAPM, was 8th in 1997, 11th in 1998, and 36th in 1999. By prior-informed regression, he was fifth in 1999. Blaylock played in the shadows of point guards like Stockton and Kidd, but he may have been better than we thought. Defense is notoriously tricky to judge by basic stats, and this is where RAPM can be most illuminating. Bo Outlaw stands out as a defensive force who wasn't heralded. Divac fares well as a two-way center, often better than his famous teammate Chris Webber. And, again, we shouldn't completely toss aside Jaren Jackson's RAPM results. He was a forgotten role player and not highly regarded, but like Shane Batter we should look beyond the box score.
Click here for the link to the spreadsheet.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
The NBA lockout that delayed the 1999 season put a serious dent in the league and wasn't aided by the retirement of Michael Jordan and the dismantling of the Bulls. It was a transition year, with the old guard dropping off with the exception of iron-man Karl Malone, while the next generation was still developing and coming into their own. With a shortened season opening in February, games jammed together with too many back-to-back's, a team known for professionalism, led by a second year Duncan and a 33 year-old David Robinson, won the championship. This was Shaq pre-Phil Jackson but post-Jordan; it's a truly lost season.
Nevertheless, the games were played, and we can't ignore what happened. Surprisingly, by NPI RAPM, the best player in 1999 was ... David Robinson, who had a monster defensive impact. Just a season before Duncan and Robinson were neck-and-neck on a per possession basis, but the Admiral leaps ahead here. Though with a 24.9 PER and a league-leading 0.261 Win Shares per 48 minutes, and stats that did not drop in the postseason, perhaps it shouldn't be surprising. However, he only played 31.7 minutes in the regular season, lending credence to an argument for another candidate like Alonzo Mourning. Jaren Jackson is probably the oddest name near the top of the list. He was a Spurs teammate but only scored 6.4 points per game. And he's why we turn to prior-informed versions....
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
With two seasons as seed data behind it, RPI RAPM (an adjusted ridge-regression model given prior information) is a more powerful tool. After a dominating 1998, Shaq loses the top spot due to poorer defense. This was probably Alonzo's greatest season with award-caliber defense coupled with potent scoring (20 points in a slow-paced league and a 56 TS%.) He was second in MVP voting -- justified here. As for some surprising results, Blaylock continues to be a plus/minus star, and Rasheed Wallace rockets to the top on the deep but strong Blazers team that nearly defeated the Lakers in 2000. Charles Barkley also continues his one-sided results: his offensive RAPM rises to 6.16 but his defense slips to -2.64.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
Fourth in MVP voting was Iverson. He's 75th here with good offensive value despite his poor shooting efficiency, but that's submarined by his porous defense, according to RAPM. Duncan was third, and he's backed-up by a very strong +5.2 mark. Rookie of the year Vince Carter, however, climbs from the rookie prior of -2 (given to all rookies) to +3.21 overall. He heads a pretty strong rookie class including Pierce, Brad Miller, and Dirk Nowitzki. Jason Williams received some rookie love, but by this method he was a significantly negative player. For validation of the model, 16 of the top 24 players were on all-NBA teams -- or perhaps that's validation of the all-NBA teams. The lowest rated all-NBA player was the young Kobe Bryant at -1.2 and it wasn't close: he was ranked 267th and the next lowest all-NBA player was McDyess at 117.
The aforementioned Barkley was the best offensive player, according to RPI RAPM, and that's not just because of his past heroics: his non-prior informed RAPM was second. Shaq's stats were muted by fewer minutes and the slogging pace of the lockout season, but, with the first of his strong of 30+ PER seasons, he's second in RPI RAPM on offense, barely trailing Barkley. Malone rounds out the top three, and the next highest players are more intriguing: Reggie's three-point bombing is probably underrated by most box score metrics at fourth here, Grant Hill's fifth, and Jeff Hornacek, again, has a strong showing at sixth followed by Blaylock. Both Hornacek and Blaylock are players that appear to be underrated by a method that wasn't available in the 90's. Mutombo, David Robinson, and Mourning were top three by defense. Mourning won the Defensive Player of the Year award, which wasn't a bad choice, necessarily, but Mutombo has a huge lead in RAPM. Rasheed was fourth, and his value was demonstrated later when he was traded to the Pistons to complete one of the greatest defensive teams ever. Jaren Jackson appears to be an anomalous result, ranking fifth, but he was San Antonio's first "three-and-D" player. He concentrated on defense, and his value does not show up in box scores. His NPI RAPM wasn't entirely misleading. Trailing him were Shawn Bradley, Olajuwon, and Garnett. Gigantic size might actually be underrated because Yao Ming's defense looked better under the RAPM microscope and Bradley joins Gheorghe Muresan as another 7' 7" player with strong defensive results.
With three seasons of data, outliers can be weeded out and patterns emerge. We can reevaluate past legends like David Robinson and Stockton who rate well, but sub-stars should receive more discussion. Mookie Blaylock, by NPI RAPM, was 8th in 1997, 11th in 1998, and 36th in 1999. By prior-informed regression, he was fifth in 1999. Blaylock played in the shadows of point guards like Stockton and Kidd, but he may have been better than we thought. Defense is notoriously tricky to judge by basic stats, and this is where RAPM can be most illuminating. Bo Outlaw stands out as a defensive force who wasn't heralded. Divac fares well as a two-way center, often better than his famous teammate Chris Webber. And, again, we shouldn't completely toss aside Jaren Jackson's RAPM results. He was a forgotten role player and not highly regarded, but like Shane Batter we should look beyond the box score.
Click here for the link to the spreadsheet.
Saturday, December 21, 2013
1997-98 RAPM: prior Informed (RPI)
Background for what RAPM is: +/- was a revolution for the NBA because it allowed a completely new method at evaluating players. You look at how a team scores and defends with you on the court and without you. When you set players as variables, you can use regression to calculate player impact. It's a full scope view of what matters in a game: outscoring your opponent. However, it's noisy for a number of reasons. One is that some player combinations are rare (this is known as collinearity.) Another is that the models don't deal well with players with low minutes, as they don't have enough of a sample for an accurate estimate and will often produce a ludicrous result just to "fit" the data better.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
Single season advanced +/- models are interesting, but they're prone to fluky results and collinearity issues, even with RAPM. This is where prior-informed comes from: instead of assuming all the players have the same value, use a starting point based on the results of the previous season. This "daisy-chain" style produces some of the most reliable and usable +/- stats, especially after you get three or four seasons in a chain. There's no official name for this yes in basketball circles, an easy nomenclature, so I'm using something similar to NPI (non-prior informed): RPI, which means it's a "pure" RAPM model informed only by a previous RAPM model. I've put the results in the same spreadsheet but another tab for a quick comparison.
The names at the top are all great to elite players with guys who have MVPs and a few surprising results to keep things interesting. One of the purposes of +/- is to identify players who have a positive but hidden impact on the game like Shane Battier. This is like a retroactive spotlight on the underrated games of the late 90's. Mookie Blaylock is rated as the third best player thanks to fantastic defense for a point guard and plenty of offensive value, even though his TS% was 46 (he relied on the shortened line, which ended in '97.) There are a lot of debates about Nash versus Stockton, and here's fuel for the fire, as Stockton rates very well in advanced +/-. Divac wasn't invited to many all-star games, but it appears his impact is worthy of a selection. Robert Horry, the man with seven titles, also looks like more than a role player here.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
One new tweak was attempted for this model: using a different lambda for rookies. The common technique is to give all rookies the same prior, but in a prior-informed model this can be harsh for rookies. Obviously, a player with a stat from the previous season shouldn't be treated the same as a rookie. To deal with this problem, I used the same negative prior for rookies but cut the lambda in half (this is done with the penalty factor in R.) Consequently, Tim Duncan's great rookie season has been given freedom to shine. He's rated just a hair above David Robinson, which agrees with many subjective opinions on the value of the two big men, and 22nd overall. Only four other rookies were significantly above average: Brevin Knight at +3.6, who somehow didn't make the all-rookie team; Ron Mercer at +2.0; Keith Van Horn at +1.1; and Zydrunas Ilgauskas at +0.7.
The top offensive player was Karl Malone with Barkley and Shaq close behind. Tim Hardaway and Jordan round out the top five. This shows most of Shaq's impact was on offense even when he was younger. Barkley's an interesting result since this was past his prime; perhaps he really was an offensive savant. Sweet shooters Reggie Miller and Hornacek were a short distance away from Jordan on offense, suggesting that outside shooting was quite valuable even in the 90's. As for defense, Mutombo blows everyone away with a +5.7 rating and secures his Defensive Player of the Year trophy. Mourning, McKie, Ewing, and Tyrone Hill fill out the top five. David Robinson and Olajuwon weren't far behind, but this was past their respective primes. McKie's probably one of the most underrated role players of his era, and Hill is probably forgotten but he was part of those tough defensive Iverson/76ers teams.
As a last note, ten out of the top 23 players were on an all-NBA team, while the lowest rated player was Vin Baker at +0.2 and, perhaps not surprisingly, the lowest rated all-star was Antoine Walker at -2.5.
Click here for the link to the spreadsheet.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
Single season advanced +/- models are interesting, but they're prone to fluky results and collinearity issues, even with RAPM. This is where prior-informed comes from: instead of assuming all the players have the same value, use a starting point based on the results of the previous season. This "daisy-chain" style produces some of the most reliable and usable +/- stats, especially after you get three or four seasons in a chain. There's no official name for this yes in basketball circles, an easy nomenclature, so I'm using something similar to NPI (non-prior informed): RPI, which means it's a "pure" RAPM model informed only by a previous RAPM model. I've put the results in the same spreadsheet but another tab for a quick comparison.
The names at the top are all great to elite players with guys who have MVPs and a few surprising results to keep things interesting. One of the purposes of +/- is to identify players who have a positive but hidden impact on the game like Shane Battier. This is like a retroactive spotlight on the underrated games of the late 90's. Mookie Blaylock is rated as the third best player thanks to fantastic defense for a point guard and plenty of offensive value, even though his TS% was 46 (he relied on the shortened line, which ended in '97.) There are a lot of debates about Nash versus Stockton, and here's fuel for the fire, as Stockton rates very well in advanced +/-. Divac wasn't invited to many all-star games, but it appears his impact is worthy of a selection. Robert Horry, the man with seven titles, also looks like more than a role player here.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
One new tweak was attempted for this model: using a different lambda for rookies. The common technique is to give all rookies the same prior, but in a prior-informed model this can be harsh for rookies. Obviously, a player with a stat from the previous season shouldn't be treated the same as a rookie. To deal with this problem, I used the same negative prior for rookies but cut the lambda in half (this is done with the penalty factor in R.) Consequently, Tim Duncan's great rookie season has been given freedom to shine. He's rated just a hair above David Robinson, which agrees with many subjective opinions on the value of the two big men, and 22nd overall. Only four other rookies were significantly above average: Brevin Knight at +3.6, who somehow didn't make the all-rookie team; Ron Mercer at +2.0; Keith Van Horn at +1.1; and Zydrunas Ilgauskas at +0.7.
The top offensive player was Karl Malone with Barkley and Shaq close behind. Tim Hardaway and Jordan round out the top five. This shows most of Shaq's impact was on offense even when he was younger. Barkley's an interesting result since this was past his prime; perhaps he really was an offensive savant. Sweet shooters Reggie Miller and Hornacek were a short distance away from Jordan on offense, suggesting that outside shooting was quite valuable even in the 90's. As for defense, Mutombo blows everyone away with a +5.7 rating and secures his Defensive Player of the Year trophy. Mourning, McKie, Ewing, and Tyrone Hill fill out the top five. David Robinson and Olajuwon weren't far behind, but this was past their respective primes. McKie's probably one of the most underrated role players of his era, and Hill is probably forgotten but he was part of those tough defensive Iverson/76ers teams.
As a last note, ten out of the top 23 players were on an all-NBA team, while the lowest rated player was Vin Baker at +0.2 and, perhaps not surprisingly, the lowest rated all-star was Antoine Walker at -2.5.
Click here for the link to the spreadsheet.
Friday, December 6, 2013
1997-98 RAPM: Non-prior Informed
Background for what RAPM is: +/- was a revolution for the NBA because it allowed a completely new method at evaluating players. You look at how a team scores and defends with you on the court and without you. When you set players as variables, you can use regression to calculate player impact. It's a full scope view of what matters in a game: outscoring your opponent. However, it's noisy for a number of reasons. One is that some player combinations are rare (this is known as collinearity.) Another is that the models don't deal well with players with low minutes, as they don't have enough of a sample for an accurate estimate and will often produce a ludicrous result just to "fit" the data better.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
Continuing my work on breaking down the new play-by-play data from NBA.com, here's the non-prior informed (NPI) version of RAPM. This version is less predictive as an informed RAPM set, and it often has some wonky results, but it's another tool to use in evaluating NBA players. 1998 was an intriguing season historically as it was the last for the Jordan on the Bulls, preceded the destructive lockout that delayed the '99 season, and was perhaps the last hurrah for a generation of stars who gave way to Duncan, Shaq, Kobe, et al. Going by his stats, it wasn't Jordan's best season during their six-peat, but he still nailed a crucial shot over the Jazz for a championship. It was the second straight year they met in the finals with Jordan also taking back the MVP trophy after Malone stole it in 1997. However, by RAPM and considering minutes, Karl Malone was the most valuable player.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
Although Jordan and Malone got all the MVP attention, Shaq was an absolute monster -- a +6 RAPM on non-prior informed is enormous (all players, including Shaq, were regressed to 0 as a prior, so a high value is impressive.) Alas, he only played 2000 minutes as he missed many games with injuries. The iron-man Stockton was actually the same: he only played 1800+ minutes with nearly 20 missed games. If you scoff at these two leading the league, refer to NBA.com's raw +/- for 1998 where both players lead the league. Jordan is only 6th here and near teammate Kukoc. Strangely, Kukoc's defensive RAPM is one of the highest for the season; there's perhaps a weird interaction between those two and Pippen's missed games. This is where a prior-informed version would excel: assume Jordan has a superstar impact, Harper and Kukoc with more modest impacts, and crunch the numbers from there.
As for the leaders for each category, Barkley was nearly +5 just on offense, while interestingly Olajuwon was a paltry -0.8. This is likely a one-year hiccup due to limited data, but it's an intriguing note. Shaq was second, as that was one of his many seasons destroying the league: a 58.7 TS% with a 33 usage. Reggie Miller's shooting brings him in at number three, and MVP Malone is a close fourth. Detlef Schrempf was fifth, as the Sonics survived the loss of Kemp.
On defense, there were a few bizarre players in the top spots, but there were also the usual suspects. Mutombo won the DPOTY award, and RAPM agrees: he has the top spot. Yet Ben Wallace, and David Robinson were third, and fifth, respectively, while Kukoc and Jaren Jackson were second and fourth. As for an unheralded guy, Aaron McKie was a close sixth. He's known for being a tough defender, but not Defensive Player of the Year worthy. He was traded midseason (Pistons to the 76ers), but both teams were not enormously better with him; both actually only slightly improved on defense when he was on the team. Also, Duncan won Rookie of the Year in one of the best rookie seasons ever; RAPM agrees.
With two consecutive season files completed for RAPM, a prior-informed version can now be created for this season. Hopefully, some of the strange results like Kukoc a +4 on defense will be eliminated or reduced. Look for a prior-informed version to be posted soon along with a version of RAPM informed by a statistical plus/minus. The great MVP debates for Jordan and Malone now have more information, and it's exciting to have these advanced metrics for legends including those two players and young, explosive Shaq.
Click here for the link to the spreadsheet.
Edit: I fixed a problem in the matchup file producing better results for version 2.0.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
Continuing my work on breaking down the new play-by-play data from NBA.com, here's the non-prior informed (NPI) version of RAPM. This version is less predictive as an informed RAPM set, and it often has some wonky results, but it's another tool to use in evaluating NBA players. 1998 was an intriguing season historically as it was the last for the Jordan on the Bulls, preceded the destructive lockout that delayed the '99 season, and was perhaps the last hurrah for a generation of stars who gave way to Duncan, Shaq, Kobe, et al. Going by his stats, it wasn't Jordan's best season during their six-peat, but he still nailed a crucial shot over the Jazz for a championship. It was the second straight year they met in the finals with Jordan also taking back the MVP trophy after Malone stole it in 1997. However, by RAPM and considering minutes, Karl Malone was the most valuable player.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
Although Jordan and Malone got all the MVP attention, Shaq was an absolute monster -- a +6 RAPM on non-prior informed is enormous (all players, including Shaq, were regressed to 0 as a prior, so a high value is impressive.) Alas, he only played 2000 minutes as he missed many games with injuries. The iron-man Stockton was actually the same: he only played 1800+ minutes with nearly 20 missed games. If you scoff at these two leading the league, refer to NBA.com's raw +/- for 1998 where both players lead the league. Jordan is only 6th here and near teammate Kukoc. Strangely, Kukoc's defensive RAPM is one of the highest for the season; there's perhaps a weird interaction between those two and Pippen's missed games. This is where a prior-informed version would excel: assume Jordan has a superstar impact, Harper and Kukoc with more modest impacts, and crunch the numbers from there.
As for the leaders for each category, Barkley was nearly +5 just on offense, while interestingly Olajuwon was a paltry -0.8. This is likely a one-year hiccup due to limited data, but it's an intriguing note. Shaq was second, as that was one of his many seasons destroying the league: a 58.7 TS% with a 33 usage. Reggie Miller's shooting brings him in at number three, and MVP Malone is a close fourth. Detlef Schrempf was fifth, as the Sonics survived the loss of Kemp.
On defense, there were a few bizarre players in the top spots, but there were also the usual suspects. Mutombo won the DPOTY award, and RAPM agrees: he has the top spot. Yet Ben Wallace, and David Robinson were third, and fifth, respectively, while Kukoc and Jaren Jackson were second and fourth. As for an unheralded guy, Aaron McKie was a close sixth. He's known for being a tough defender, but not Defensive Player of the Year worthy. He was traded midseason (Pistons to the 76ers), but both teams were not enormously better with him; both actually only slightly improved on defense when he was on the team. Also, Duncan won Rookie of the Year in one of the best rookie seasons ever; RAPM agrees.
With two consecutive season files completed for RAPM, a prior-informed version can now be created for this season. Hopefully, some of the strange results like Kukoc a +4 on defense will be eliminated or reduced. Look for a prior-informed version to be posted soon along with a version of RAPM informed by a statistical plus/minus. The great MVP debates for Jordan and Malone now have more information, and it's exciting to have these advanced metrics for legends including those two players and young, explosive Shaq.
Click here for the link to the spreadsheet.
Edit: I fixed a problem in the matchup file producing better results for version 2.0.
Wednesday, October 30, 2013
2013 Retrodiction: How Player Metrics Predicted Wins
As the 2014 season starts, let's not forget the past and what we can learn. NBA player metrics are more popular than ever, but which ones do we trust? How useful are they? One method for evaluating metrics is to use the past values for each player to predict team wins in the current season. For example, for the Lakers we use Dwight Howard's PER in 2012 along with his minutes in 2013, we use Kobe Bryant's PER in 2012 along with his minutes in 2013, and so on until we have every player.
The metrics
PER: John Hollinger's invention. Explained here. Uses box score stats to create a player per-minute productivity value. Values usage highly. Adjusted for pace.
Win Shares: Uses box score stats to calculate individual wins. A large team defense factor is evenly divided among players. Values efficiency.
To rate the various metrics, we can use what's known as a root-mean square error. Sum all the squared differences between wins and predicted wins, and then take the square root. The effect of this is to penalize the biggest errors, and then to calculate the root mean (the final units are in wins, not squared wins.)
Which metric won the 2013 retrodiction? xRAPM blew away every other metric. A root-mean squared error under 6 is extremely good for a prediction for the regular season. Although knowing the minutes distribution is a huge advantage, it's worth noting that the best analysts and Vegas topped out at 5.94 for the 2013 season. (Interestingly, while PER did not perform well, Hollinger's own predictions did.) For the worst performing metric, that is Wins Produced. Only PER approaches how poorly Wins Produced did, but that's not with the "real" wins. Most metrics had minor differences in how they predicted real wins and Pythagorean wins, but PER "lucked out" that real wins trended much closer to its own prediction. Win Shares does extremely well and beats out the other adjusted plus/minus methods, but we won't stop here.
Since you're really not testing anything on teams with low roster turnover, I used the percentage of minutes from new players excluding rookies to calculate squared errors at three points: no new players, half the minutes from new players, and a completely new team. Why is this important? Box score metrics assume defense is explained by rebounds/blocks/steals, and since a good defender will force more missed shots (i.e. rebounds), it's hard to test this without the player switching teams.
The results are shown in the table below. The most relevant line is 50% because the team with the most roster turnover topped out at 59.3%. The "Win" metrics do very well when the roster doesn't change, which isn't surprising because their metrics are built from offensive/defensive team measures, but they start to perform horribly with heavy roster change. Weirdly, PER doesn't change much when the roster does for real wins; there was something kooky going on with PER trying to explain teams with roster change (case in point: the Rockets gained Asik, who PER underrates because he's a defensive player who doesn't shoot well, but the Rockets underperformed their point differential.)
The RAPM metrics are rough in estimating wins on teams with no roster turnover, but this makes sense: plus/minus models are known for their noise. However, they do much better with roster turnover, especially the metrics that use priors: xRAPM and RAPM. This is despite completely whiffing on the Lakers, which the box score metrics did not do, but that team struggled with injuries and chemistry.
PER does the best at 100% minutes from new players, but, again, the 50% line is more relevant and it's more the case that PER was horrible at teams with no roster change, making them look better when projected out to 100. As a last note, non-prior forms of RAPM are generally disregarded, but they still appear to be as predictive as widely used box score metrics like Win Shares.
How to translate the metrics to wins
For Win Shares and Wins Produced, the work has already been done. Just multiply by minutes.
PER has a companion stat for wins: EWA (estimated wins added.) It's explained here:
VA: Value Added - the estimated number of points a player adds to a team’s season total above what a 'replacement player' (for instance, the 12th man on the roster) would produce. Value Added = ([Minutes * (PER - PRL)] / 67). PRL (Position Replacement Level) = 11.5 for power forwards, 11.0 for point guards, 10.6 for centers, 10.5 for shooting guards and small forwards
EWA: Estimated Wins Added - Value Added divided by 30, giving the estimated number of wins a player adds to a team’s season total above what a 'replacement player' would produce.
Plus/minus stats are translated to wins using the Pythagorean method. The formula is points scored^14/(points scored^14+points allowed^14). The expected plus/minus of the team is translated into points scored versus points allowed relative to the league average.
Rookies are assumed to be heavy negatives. The average value for rookies was estimated from past values.
WS/48 mins: 0.05
PER: 13
WP/48 mins: 0.05
Plus/minus (RAPM's): -1.96
The metrics
PER: John Hollinger's invention. Explained here. Uses box score stats to create a player per-minute productivity value. Values usage highly. Adjusted for pace.
Win Shares: Uses box score stats to calculate individual wins. A large team defense factor is evenly divided among players. Values efficiency.
Wins Produced: Like Win Shares, uses box score stats to calculate individual wins. Instead of a large team defense factor, heavily values rebounding. Also values efficiency.
RAPM: Ridge-regression adjusted plus minus. Fundamentally based on whether or not a player's team scores more often or allows less points when he's on the court. Adjusts for pace, strength of schedule, among other factors. The "ridge" part regresses heavily to a prior value based on possessions, i.e. it's hard for a player to move away from his prior if he rarely plays.
-npi RAPM: Non-prior informed RAPM. The priors are all set to 0.
-Vanilla: Uses the previous three seasons of npi RAPM. Most recent seasons are weighed more heavily.
-RAPM: The prior is the previous season's RAPM. The starting point is the first dataset used (2001.)
-xRAPM: A mix of RAPM and a statistical plus/model using box score stats and height.
To rate the various metrics, we can use what's known as a root-mean square error. Sum all the squared differences between wins and predicted wins, and then take the square root. The effect of this is to penalize the biggest errors, and then to calculate the root mean (the final units are in wins, not squared wins.)
Which metric won the 2013 retrodiction? xRAPM blew away every other metric. A root-mean squared error under 6 is extremely good for a prediction for the regular season. Although knowing the minutes distribution is a huge advantage, it's worth noting that the best analysts and Vegas topped out at 5.94 for the 2013 season. (Interestingly, while PER did not perform well, Hollinger's own predictions did.) For the worst performing metric, that is Wins Produced. Only PER approaches how poorly Wins Produced did, but that's not with the "real" wins. Most metrics had minor differences in how they predicted real wins and Pythagorean wins, but PER "lucked out" that real wins trended much closer to its own prediction. Win Shares does extremely well and beats out the other adjusted plus/minus methods, but we won't stop here.
PER
|
Win Shares
|
Wins Produced
|
npi RAPM
|
Vanilla RAPM
|
RAPM
|
xRAPM
|
|
Real wins
|
7.00
|
6.62
|
7.95
|
7.51
|
7.27
|
6.74
|
5.67
|
Pyth. wins
|
7.98
|
6.93
|
7.75
|
7.41
|
7.23
|
7.11
|
5.85
|
The results are shown in the table below. The most relevant line is 50% because the team with the most roster turnover topped out at 59.3%. The "Win" metrics do very well when the roster doesn't change, which isn't surprising because their metrics are built from offensive/defensive team measures, but they start to perform horribly with heavy roster change. Weirdly, PER doesn't change much when the roster does for real wins; there was something kooky going on with PER trying to explain teams with roster change (case in point: the Rockets gained Asik, who PER underrates because he's a defensive player who doesn't shoot well, but the Rockets underperformed their point differential.)
The RAPM metrics are rough in estimating wins on teams with no roster turnover, but this makes sense: plus/minus models are known for their noise. However, they do much better with roster turnover, especially the metrics that use priors: xRAPM and RAPM. This is despite completely whiffing on the Lakers, which the box score metrics did not do, but that team struggled with injuries and chemistry.
PER does the best at 100% minutes from new players, but, again, the 50% line is more relevant and it's more the case that PER was horrible at teams with no roster change, making them look better when projected out to 100. As a last note, non-prior forms of RAPM are generally disregarded, but they still appear to be as predictive as widely used box score metrics like Win Shares.
% of mins. from new
players
|
PER
|
Win Shares
|
Wins Produced
|
npi RAPM
|
Vanilla RAPM
|
RAPM
|
xRAPM
|
|
Real wins
|
0
|
41.3
|
-3.6
|
3.1
|
19.7
|
16.3
|
24.8
|
16.4
|
50
|
53.5
|
71.4
|
98.3
|
77.7
|
74.1
|
57.4
|
41.4
|
|
100
|
65.6
|
146.3
|
193.4
|
135.7
|
131.9
|
90.0
|
66.4
|
|
Pyth. wins
|
0
|
43.7
|
0.2
|
15.7
|
21.7
|
26.2
|
22.1
|
20.6
|
50
|
75.3
|
75.9
|
85.9
|
74.1
|
67.4
|
67.0
|
42.2
|
|
100
|
106.8
|
151.7
|
156.2
|
126.5
|
108.7
|
111.9
|
63.8
|
How to translate the metrics to wins
For Win Shares and Wins Produced, the work has already been done. Just multiply by minutes.
PER has a companion stat for wins: EWA (estimated wins added.) It's explained here:
VA: Value Added - the estimated number of points a player adds to a team’s season total above what a 'replacement player' (for instance, the 12th man on the roster) would produce. Value Added = ([Minutes * (PER - PRL)] / 67). PRL (Position Replacement Level) = 11.5 for power forwards, 11.0 for point guards, 10.6 for centers, 10.5 for shooting guards and small forwards
EWA: Estimated Wins Added - Value Added divided by 30, giving the estimated number of wins a player adds to a team’s season total above what a 'replacement player' would produce.
Plus/minus stats are translated to wins using the Pythagorean method. The formula is points scored^14/(points scored^14+points allowed^14). The expected plus/minus of the team is translated into points scored versus points allowed relative to the league average.
Rookies are assumed to be heavy negatives. The average value for rookies was estimated from past values.
WS/48 mins: 0.05
PER: 13
WP/48 mins: 0.05
Plus/minus (RAPM's): -1.96
Thursday, October 10, 2013
Introducing: 1990's RAPM
Background for what RAPM is: +/- was a revolution for the NBA because it allowed a completely new method at evaluating players. You look at how a team scores and defends with you on the court and without you. When you set players as variables, you can use regression to calculate player impact. It's a full scope view of what matters in a game: outscoring your opponent. However, it's noisy for a number of reasons. One is that some player combinations are rare (this is known as collinearity.) Another is that the models don't deal well with players with low minutes, as they don't have enough of a sample for an accurate estimate and will often produce a ludicrous result just to "fit" the data better.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
After some major work, I'm finally ready to display the '97 RAPM numbers. Given the nature of the data source, where only last names are used, making it tricky to figure out who's on the court for teams with players who share last names, it's been a labor intensive process. There are also a number of random errors that crop up. Regardless, the work has been fruitful, and I'll work on the seasons following 1997 when I can. The results are shown below.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
Back in the earlier days of the stat movement, there was something called the laugh test: if Shaq was not the top ranked player, something was wrong with your metric. Since this is the 90's, this is the Jordan test. I was relieved to find Michael Jordan near the top of the leaderboard, as well as other stars like MVP Malone third. But Christian Laettner and Terry Mills? Well, first of all non-prior informed RAPM often produces wacky results. For example, the fourth through sixth ranked players in the 2002 list (non-prior informed) from the popular RAPM site stats-for-the-nba.com were Eduardo Najera, Devean George, and Ryan Bowen. Laettner played every game for the Hawks, who were one of the best defensive teams in the league: so perhaps without a missed game there wasn't enough data to show his "true" defensive value.
Nevertheless, credit should be given to him and the forgotten Terry Mills, a beefy three-point shooting power forward. He was a bench player on a good team yet had the 12th best net +/- rating, according to stats.NBA.com. Putting up a great raw +/- can come by accident by hitching yourself to better players, only playing when they do; but he came off the bench for a playoff team and has a raw rating nearly two times as great as anyone else on the team. There's something to be said about Terry Mills' value in '97.
Another surprising result was Bo Outlaw's outstanding defensive +/-. It was the best rating a hair ahead of DPOTY Mutombo (retroactively justified here) with defensive legends Mourning and Ewing not too far behind. Even at 21 years-old, Garnett shines once again in +/-, just as he did 16 years later in the 2013 season. As for offense, Jordan, Pippen and Mookie Blaylock, and of course Terry Mills, are near the top, but MVP Malone takes the mantle. As for an underrated player RAPM likes, Hornacek is a close sixth. The sweet-shooting guard had one of the best jump shots in the game and could pass well for an outside shooter. He's also one of the most accepting of advanced stats for former players and coaches.
In the future, I'll work on the 1998 season, as well as creating a statistical +/- prior to test out for this season and other fun items. (Since play-by-play data is unavailable before '97, the results using pure +/- are limiting. Thus, a model that creates priors for every player should provide more reliable results, instead of blindly regressing every player toward 0. Players who only play around 50 minutes will be rated as 0's in RAPM, even though they're probably a lot worse, and the model shouldn't assume the prior for a guy like Michael Jordan is 0.)
Click here for the link to the spreadsheet.
Edit: On updating the possession/lineup parser and adding missing games, some of the values here changed.
In simple terms, RAPM deals with this by introducing a heavy dose of regression to the mean. While traditional adjusted +/- creates a model by minimizing error (the difference between the actual points per possession scored/allowed and the expected), RAPM also minimizes the coefficients in the model using a lambda term. The coefficients are reduced toward the "prior," which can be set as zero or as a set of prior values (like the previous season's result.) Players with few possessions/minutes will have results close to their priors because their sample size isn't big enough to prove to the model they're more or less valuable.
After some major work, I'm finally ready to display the '97 RAPM numbers. Given the nature of the data source, where only last names are used, making it tricky to figure out who's on the court for teams with players who share last names, it's been a labor intensive process. There are also a number of random errors that crop up. Regardless, the work has been fruitful, and I'll work on the seasons following 1997 when I can. The results are shown below.
*When you reference the spreadsheet, try to include the version number. This will reduce future discrepancies.
Back in the earlier days of the stat movement, there was something called the laugh test: if Shaq was not the top ranked player, something was wrong with your metric. Since this is the 90's, this is the Jordan test. I was relieved to find Michael Jordan near the top of the leaderboard, as well as other stars like MVP Malone third. But Christian Laettner and Terry Mills? Well, first of all non-prior informed RAPM often produces wacky results. For example, the fourth through sixth ranked players in the 2002 list (non-prior informed) from the popular RAPM site stats-for-the-nba.com were Eduardo Najera, Devean George, and Ryan Bowen. Laettner played every game for the Hawks, who were one of the best defensive teams in the league: so perhaps without a missed game there wasn't enough data to show his "true" defensive value.
Nevertheless, credit should be given to him and the forgotten Terry Mills, a beefy three-point shooting power forward. He was a bench player on a good team yet had the 12th best net +/- rating, according to stats.NBA.com. Putting up a great raw +/- can come by accident by hitching yourself to better players, only playing when they do; but he came off the bench for a playoff team and has a raw rating nearly two times as great as anyone else on the team. There's something to be said about Terry Mills' value in '97.
Another surprising result was Bo Outlaw's outstanding defensive +/-. It was the best rating a hair ahead of DPOTY Mutombo (retroactively justified here) with defensive legends Mourning and Ewing not too far behind. Even at 21 years-old, Garnett shines once again in +/-, just as he did 16 years later in the 2013 season. As for offense, Jordan, Pippen and Mookie Blaylock, and of course Terry Mills, are near the top, but MVP Malone takes the mantle. As for an underrated player RAPM likes, Hornacek is a close sixth. The sweet-shooting guard had one of the best jump shots in the game and could pass well for an outside shooter. He's also one of the most accepting of advanced stats for former players and coaches.
In the future, I'll work on the 1998 season, as well as creating a statistical +/- prior to test out for this season and other fun items. (Since play-by-play data is unavailable before '97, the results using pure +/- are limiting. Thus, a model that creates priors for every player should provide more reliable results, instead of blindly regressing every player toward 0. Players who only play around 50 minutes will be rated as 0's in RAPM, even though they're probably a lot worse, and the model shouldn't assume the prior for a guy like Michael Jordan is 0.)
Click here for the link to the spreadsheet.
Edit: On updating the possession/lineup parser and adding missing games, some of the values here changed.
Wednesday, September 18, 2013
Initial Plus/Minus Numbers for the '97 Season
I've been silent for a few weeks, working on a bigger project, and the fruits of that labor are appearing. A few months ago, the NBA released play-by-play dating going back to the 1996-97 season, allowing for the possibility of more advanced basketball stats beyond the typical box score counting stats. NBA.com already lists detailed player shooting data (percentages and totals by distance), but the Holy Grail was plus/minus -- it had become the most respected advanced stat out there, beating out other metrics in future, out of sample predictions, especially even further into the future when more guys move from team to team.
When no one stepped up to the plate to break down the play-by-play data, I decided to do this myself. The problem was the data only included last names, making it difficult to discern exactly who was on the court. However, home and away teams have separate columns, so the problem is reduced to only teams where two players have the same name. There are 6 such teams: Portland with Rumeal and Cliff Robinson; New York with Buck and Herb Williams; New Jersey with another set of Williams's, in Jayson and Reggie; Phoenix with Mike and Chucky Brown (no, not that Mike Brown); Denver with LaPhonso and Dale Ellis, as well as Brooks and LaSalle Thompson; and Indiana with Dale and Antonio Davis. Fortunately, the Brown's never played in the same game, so they were easily separated. The rest of the player pairs were successfully separated with the exception of the Davis brothers, who are similar anyway, and the Ellis's, who remain the largest problem. For the matchup file, I combined the pairs as one player; and as such, the initial results below should be taken with a grain of salt, though mostly for players on those two teams (Denver and Indiana.)
Without further ado, here's a table of the top players for 1996-96 by adjusted plus/minus including the playoffs with a minutes cut-off of 250:
When no one stepped up to the plate to break down the play-by-play data, I decided to do this myself. The problem was the data only included last names, making it difficult to discern exactly who was on the court. However, home and away teams have separate columns, so the problem is reduced to only teams where two players have the same name. There are 6 such teams: Portland with Rumeal and Cliff Robinson; New York with Buck and Herb Williams; New Jersey with another set of Williams's, in Jayson and Reggie; Phoenix with Mike and Chucky Brown (no, not that Mike Brown); Denver with LaPhonso and Dale Ellis, as well as Brooks and LaSalle Thompson; and Indiana with Dale and Antonio Davis. Fortunately, the Brown's never played in the same game, so they were easily separated. The rest of the player pairs were successfully separated with the exception of the Davis brothers, who are similar anyway, and the Ellis's, who remain the largest problem. For the matchup file, I combined the pairs as one player; and as such, the initial results below should be taken with a grain of salt, though mostly for players on those two teams (Denver and Indiana.)
Without further ado, here's a table of the top players for 1996-96 by adjusted plus/minus including the playoffs with a minutes cut-off of 250:
Rank
|
Player....................
|
Adj. +/-
|
St. Err.
|
Minutes
|
PER
|
WP/48
|
1
|
Mookie Blaylock
|
15.79
|
7.01
|
3056
|
20.4
|
0.197
|
2
|
Tim Hardaway
|
15.64
|
6.79
|
3136
|
20.8
|
0.198
|
3
|
Michael Jordan
|
13.92
|
5.48
|
3106
|
27.8
|
0.283
|
4
|
Latrell Sprewell
|
13.13
|
6.79
|
3353
|
19.7
|
0.115
|
5
|
Patrick Ewing
|
13.08
|
5.78
|
2887
|
21.3
|
0.163
|
6
|
Terry Mills
|
12.92
|
5.82
|
1997
|
16.4
|
0.148
|
7
|
Hakeem Olajuwon
|
12.81
|
6.46
|
2852
|
22.7
|
0.154
|
8
|
Greg Anthony
|
12.81
|
7.75
|
1863
|
16.6
|
0.090
|
9
|
Kevin Garnett
|
12.49
|
5.87
|
2995
|
18.2
|
0.116
|
10
|
John Stockton
|
12.07
|
16.05
|
2896
|
22.1
|
0.226
|
11
|
Tyrone Hill
|
11.79
|
6.82
|
2582
|
17.8
|
0.184
|
12
|
Mitch Richmond
|
10.94
|
6.38
|
3125
|
21.6
|
0.166
|
13
|
Stanley Roberts
|
10.80
|
8.94
|
378
|
14.1
|
0.060
|
14
|
Melvin Booker
|
10.65
|
9.31
|
430
|
8.8
|
0.014
|
15
|
Shaquille O'Neal
|
10.62
|
5.88
|
1941
|
27.1
|
0.197
|
16
|
Gary Payton
|
10.52
|
6.74
|
3213
|
21.8
|
0.193
|
17
|
Nate McMillan
|
10.28
|
6.41
|
798
|
14.1
|
0.158
|
18
|
Alonzo Mourning
|
9.87
|
5.38
|
2320
|
21.6
|
0.174
|
19
|
Litterial Green
|
9.77
|
10.10
|
311
|
13.5
|
0.119
|
20
|
Jerome Kersey
|
9.70
|
4.95
|
1766
|
12.3
|
0.102
|
*Per 200 possessions (roughly a full game)
**This is the entire '97 season plus the playoffs with the exception of 11 missing games.
***Players with under 250 minutes were combined into one variable. That coefficient, by the way, was -5.94 with a st. error of 7.14.
****Playoff possessions are weighted twice as much (i.e. they're twice as important as regular season ones.)
****Playoff possessions are weighted twice as much (i.e. they're twice as important as regular season ones.)
The important thing to note about adjusted plus/minus data is that the estimates are not precise: there are usually huge ranges for the predictions. The standard errors for the guys in the top 20 table are around 5 to 8, meaning Ewing, for example, isn't significantly "better" than Mourning (by adj. +/-.) With such high variation, what use are these results? For one, it's more evidence to use in evaluation of historical (and even some current) players. One year plus/minus is a little wacky, but once you're armed with a few years of data and better techniques like ridge regression you can find patterns and judge which players have consistently high, or mediocre, impact. For wacky results, you may have noticed three non-entities in the top 20: Stanley Roberts (Shaq's former teammate from LSU), Melvin Booker, and Litterial Green. That's pretty normal in one year adjusted plus/minus, as it's the biggest weakness (low minute guys.) As a sanity check, homecourt advantage was calculated as +3.29.
If you're wondering how a certain star ranked, I put the most notable guys in the table below:
Rank
|
Player....................
|
Adj. +/-
|
St. Err.
|
Minutes
|
PER
|
WP/48
|
22
|
Christian Laettner
|
9.36
|
6.38
|
3140
|
19.1
|
0.177
|
23
|
Scottie Pippen
|
9.28
|
5.42
|
3095
|
21.3
|
0.203
|
25
|
Horace Grant
|
8.82
|
5.41
|
2496
|
17.3
|
0.148
|
27
|
Kendall Gill
|
7.69
|
5.75
|
3199
|
19.6
|
0.132
|
31
|
Vlade Divac
|
7.44
|
6.68
|
2840
|
17.9
|
0.123
|
33
|
Chris Webber
|
7.12
|
5.77
|
2806
|
21.8
|
0.159
|
37
|
Hersey Hawkins
|
6.68
|
5.95
|
2755
|
17.6
|
0.190
|
47
|
Detlef Schrempf
|
5.56
|
5.43
|
2192
|
18.3
|
0.174
|
48
|
Clyde Drexler
|
5.54
|
5.94
|
2271
|
19.9
|
0.172
|
51
|
Rasheed Wallace
|
5.38
|
5.63
|
1892
|
18.4
|
0.163
|
52
|
Anfernee Hardaway
|
5.15
|
5.54
|
2221
|
21.4
|
0.175
|
54
|
Jeff Hornacek
|
5.06
|
7.58
|
2592
|
18.8
|
0.190
|
55
|
Sam Cassell
|
4.98
|
5.40
|
1714
|
18.4
|
0.108
|
56
|
Jason Kidd
|
4.91
|
4.89
|
1964
|
16.9
|
0.107
|
58
|
Karl Malone
|
4.76
|
8.60
|
2998
|
28.9
|
0.268
|
70
|
Derrick Coleman
|
4.33
|
5.63
|
2102
|
17.3
|
0.076
|
77
|
Kenny Anderson
|
3.94
|
6.73
|
3081
|
19.5
|
0.193
|
79
|
Toni Kukoc
|
3.90
|
4.91
|
1610
|
20.2
|
0.204
|
80
|
Reggie Miller
|
3.76
|
6.89
|
2966
|
20.2
|
0.200
|
82
|
Dikembe Mutombo
|
3.54
|
6.76
|
2973
|
19.0
|
0.183
|
94
|
Kevin Johnson
|
3.22
|
3.22
|
2658
|
22.9
|
0.211
|
95
|
Eddie Jones
|
3.07
|
3.07
|
2998
|
17.3
|
0.154
|
100
|
Allen Iverson
|
2.94
|
2.94
|
3045
|
18.0
|
0.065
|
108
|
Grant Hill
|
2.32
|
2.32
|
3147
|
25.5
|
0.223
|
113
|
Vin Baker
|
2.23
|
2.23
|
3159
|
20.1
|
0.127
|
122
|
Arvydas Sabonis
|
1.97
|
1.97
|
1762
|
21.8
|
0.205
|
141
|
Kobe Bryant
|
0.27
|
5.24
|
1103
|
14.4
|
0.079
|
160
|
Dennis Rodman
|
-0.76
|
5.20
|
1947
|
13.9
|
0.148
|
173
|
Shawn Kemp
|
-1.48
|
5.64
|
2750
|
20.7
|
0.174
|
197
|
Anthony Mason
|
-2.30
|
6.52
|
3143
|
18.9
|
0.173
|
214
|
Rik Smits
|
-2.93
|
5.24
|
1518
|
18.3
|
0.105
|
222
|
Damon Stoudamire
|
-3.22
|
6.59
|
3311
|
18.1
|
0.110
|
252
|
Terrell Brandon
|
-4.84
|
6.58
|
2868
|
21.5
|
0.181
|
256
|
Chris Mullin
|
-5.24
|
5.82
|
2733
|
17.6
|
0.124
|
273
|
Rony Seikaly
|
-6.49
|
5.66
|
2615
|
18.3
|
0.125
|
275
|
Tom Gugliotta
|
-6.52
|
6.37
|
3131
|
19.0
|
0.103
|
284
|
Dominique Wilkins
|
-6.96
|
5.43
|
1945
|
19.6
|
0.083
|
288
|
Rod Strickland
|
-7.21
|
9.72
|
2997
|
19.7
|
0.141
|
324
|
Shareef Abdur-Rahim
|
-10.45
|
5.88
|
2802
|
17.4
|
0.049
|
Non-traditional stars like Hornacek and Pippen rate well by this metric. It's disappointing to see Grant Hill, whose prime was cut short, and Sabonis with just decent numbers. I was hoping they'd have monster impact. It's interesting to see who doesn't fare well, however -- the old star Wilkins is a net negative, Kemp is as well, and a young Shareef Abdur-Rahim, often called a guy who put up good stats on a bad team, brings up the rear. I am surprised, however, by Dennis Rodman's low ranking. This is Rodman the Rebounder on his second title run, although his numbers weren't great in the playoffs. It's shocking to see Malone so low given his MVP, but both he and Stockton has high standard errors, suggesting there were problems untangling the two guys since they both never missed a game.
Remember that one year adjusted plus/minus stats are volatile. It's normal to see a guy in his prime with a terrible +/- one season and then rebound with an excellent one next season. If the roster rotations were rigid, it's impossible for this method to pick up on which guy actually deserves the credit on the court. With another year of data behind it, the plus/minus stats (prior-informed with RAPM or two year adj. +/-) will improve by a large margin. RAPM deals better with low minute guys, which can change some of the numbers completely like a really complicated maze of dominoes.
And why did I set a minutes requirement of 250 for the tables? The weakness of plain adjusted plus/minus is that it dumbly guesses absurd estimates with guys who have few minutes. For example, without a minutes cutoff you get results like Evric Gray with an earth-shattering +27.2 ... in 42 minutes all season. Gary Grant and Jack Haley were another two guys ranked above legends like Olajuwon, though David Robinson was +14.6 in 147 minutes. Since they have so few possessions, these low minutes guys are basically play-doh to fill in any cracks to minimize the squared error. This is where ridge-regression (RAPM) excels: a heavy dose of regression to the mean (or prior.) If you're going to have a high rating, you'd better prove it with many possessions or a high rating the previous year.
I'll clean up the data more and tackle RAPM further, but for now some initial results were worth posting.
Edit: I used Rosembaum's preferred minutes cutoff of 250. Before I was using something really low.
Second edit: found some problems with how the name for the player pairs were being assigned, plus I eliminated all but two player pairs (hence the label "initial" results.) As a result the numbers are completely revamped.
Edit: I used Rosembaum's preferred minutes cutoff of 250. Before I was using something really low.
Second edit: found some problems with how the name for the player pairs were being assigned, plus I eliminated all but two player pairs (hence the label "initial" results.) As a result the numbers are completely revamped.
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