{"id":590,"date":"2011-05-20T10:40:07","date_gmt":"2011-05-20T15:40:07","guid":{"rendered":"http:\/\/godismyjudgeok.com\/DStats\/?p=590"},"modified":"2011-05-20T10:47:51","modified_gmt":"2011-05-20T15:47:51","slug":"a-review-of-adjusted-plusminus-and-stabilization","status":"publish","type":"post","link":"http:\/\/godismyjudgeok.com\/DStats\/2011\/nba-stats\/a-review-of-adjusted-plusminus-and-stabilization\/","title":{"rendered":"A Review of Adjusted Plus\/Minus and Stabilization"},"content":{"rendered":"

As I prepare to release my first work based on the Adjusted Plus\/Minus and derivative methods, I felt it would be wise to write a plain-English review of the state-of-the-art of Adjusted Plus\/Minus and its derivatives, or at least what is known in the public domain.<\/p>\n

What is Plus\/Minus?<\/h3>\n

Plus\/Minus, at its core, simply compares how the team fares with the player on the court with how the team fares with the same player off the court.\u00a0 If the team does better with the team on the court than off, the player is probably pretty good.\u00a0 If the team does better with the player on the bench, the player is probably not as good.<\/p>\n

The concept of Plus\/Minus originated, I believe, with hockey.\u00a0 It was first used there in the 1950s, according to Wikipedia<\/a>.\u00a0 Since the 1960s it has been in common use.<\/p>\n

Plus\/Minus was adapted to basketball some time thereafter; I have not been able to track down when it was first used.<\/p>\n

For the 2007-2008 season, the NBA began tracking Plus\/Minus in box scores.\u00a0 Results may be found on NBA.com<\/a> (along with player pairs, trios, etc.) for every year since 05-06.\u00a0 BasketballValue.com<\/a> has complete results (termed “1 year unadjusted overall rating”) for every year since 07-08.<\/p>\n

Issues with Raw Plus\/Minus<\/h3>\n

It has long been recognized that raw Plus\/Minus is not particularly indicative of each player’s actual performance or value.\u00a0 The largest issue is that the stat is totally context-dependent: if I was on the court with Lebron James, Dwyane Wade, and Chris Bosh, I’d probably have a pretty good Plus\/Minus despite the fact I shouldn’t be playing at all.\u00a0 And even if I was a really good player, I still wouldn’t have good raw Plus\/Minus numbers if I were playing with Jamario Moon, Anthony Parker, Ramon Sessions, and J.J. Hickson.\u00a0 In addition, the opponents matter as well.\u00a0 Some players just play in garbage time against the other team’s scrubs.\u00a0 Their Plus\/Minus numbers may be pretty good, but only because they were playing against Brian Scalabrine!<\/p>\n

This big issue, the dependence of Plus\/Minus on the other 9 players on the court, gave rise to a new statistic aimed at adjusting for this problem: Adjusted Plus\/Minus.<\/p>\n

Adjusted Plus\/Minus: the Basics<\/h3>\n

Each period of time when a group of 10 players are on the floor may be expressed as:<\/p>\n

ObservedEfficiencyDifferential = P1 + P2 + P3 + P4 + P5\r\n                                 - P6 - P7 - P8 - P9 - P10 + HCA<\/pre>\n
where P1 to P5 are on one team, and P6 to P10 are on the other.<\/p>\n
Next, compile that equation for EVERY matchup of lineups through the whole season.\u00a0 You now have somewhere around 60,000 independent equations and somewhere around 400 unknowns (the number of players that played that year).\u00a0 If you solve this system of equations for P1, P2 … Pn , weighting each equation by the number of possessions played in that stint, you have normal, raw APM.<\/p>\n
However, some players have a very small sample size, so they may muddy the final results by basically taking up all of the residual in the few minutes they played.\u00a0 Many APM approaches, such as that used by BasketballValue.com<\/a> (the site to go to for APM results), lump the rarely-used players into an “other players” bucket and just average them out.\u00a0 This should improve the estimates on the players that are actually rated.<\/p>\n
Let me digress here to point the reader to the definitive series on Adjusted Plus\/Minus, mostly over at 82games.com:<\/p>\n