I mentioned in my last post on calculating adjusted plus/minus that the next thing I wanted to do was split it into offensive and defensive adjusted plus/minus. Lior and Cherokee_ACB had some good suggestions about how to do that in the comments, but first I wanted to see if I could replicate Dan Rosenbaum’s original methodology. This was a little tricky because Dan didn’t spell out his process in detail, but after some trial and error I think I’ve been able to duplicate what he did. As a result, I’m able to calculate 2007-08 player rankings for offensive and defensive adjusted plus/minus, metrics that have not been available publicly since Rosenbaum last presented them in 2005.

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Adjusted plus/minus is a way of rating players first developed by Wayne Winston and Jeff Sagarin in the form of their WINVAL system (more here). The basic idea is simple. For each player, it starts with the team’s average point differential for each possession when they are on the court (sometimes referred to as the player’s on-court plus/minus). This gives a number showing how effective the player’s team was when they were in the game. The problem with using this to evaluate individual players is that it is biased in favor of players who play alongside great teammates (and players who play against weak opponents). This can be seen by looking at the 2007-08 leaders in on-court plus/minus, which can be seen here (the Overall Rtg, On column) or here (the On Court/Off Court, On column). Kendrick Perkins rode his teammates’ coattails to the second highest mark in the league. Adjusted plus/minus uses regression analysis to control for these biases by controlling for the quality of the teammates a player played with and the opponents he played against.

That sounds easy enough, but it’s actually kind of complicated, and the specifics of WINVAL were never made public (Mark Cuban reportedly was paying a handsome sum to use the system for the Mavs). Thankfully, in 2004 Dan Rosenbaum spelled out the details of the methodology in an article. He called his version adjusted plus/minus, and released a series of analyses using the metric (here and here). Eventually Dan was hired to consult for the Cleveland Cavaliers, but because he had spelled out the methodology others were able to duplicate his work for future seasons. David Lewin published rankings for the 2004-05 and 2005-06 seasons, and Steve Ilardi and Aaron Barzilai have done the same for the 2006-07 and 2007-08 seasons (up-to-date ratings can be found here).

I always wanted to try to calculate adjusted plus/minus on my own, but I was intimidated. I figured that I didn’t know enough about running regressions and that I didn’t have the data, software, or computing power to run such a large analysis. But I finally sat down and tried to do it a few days ago, and I discovered that it’s not that difficult. Using Dan Rosenbaum’s description of his method, publicly available data from BasketballValue, Excel 2007, and the free statistics program R, I was able to set up and run the whole thing in less than an hour. Here’s how I did it.

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I’m still working on that follow-up post on regression to the mean, but in the meantime I wanted to put up a post comparing various player rating systems. For the most part this will be a subjective rather than objective evaluation of the metrics, along the lines of Dean Oliver’s “laugh test” (as in, “a rating system that thinks Dennis Rodman was better than Michael Jordan doesn’t pass the laugh test”). I think looking at how players are rated differently in various systems can tell us a lot about both those players and those rating systems.

#### The Player Ratings

I took a look at seven popular player ratings. Two basic linear weights metrics based on boxscore stats - John Hollinger’s Player Efficiency Rating (PER), and Dave Berri’s Wins Produced (WP). Two metrics built on Dean Oliver’s individual offensive and defensive ratings - Justin Kubatko’s Win Shares (WS), and Davis21wylie’s Wins Above Replacement Player (WARP). And three plus/minus metrics based on team point differential while the player is on the court - Roland Beech’s Net Plus/Minus (Net +/-), Dan Rosenbaum’s Adjusted Plus/Minus (Adj +/-), and Dan Rosenbaum’s Statistical Plus/Minus (Stat +/-). For the purposes of comparison I looked at the per-minute (or per-possession) versions of all these metrics (e.g. WP48 instead of WP, WSAA/48 instead of WSAA, WARPr instead of WARP).

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Recently, 82games put up some pages listing the top five-man lineups from this past regular season in terms of plus/minus and points scored and allowed per possession. You can find similar rankings on BasketballValue. I wanted to go a step further and adjust each lineup’s ranking based on the quality of the opposing lineups that it faced during the season.

To do this I started with lineup data from BasketballValue. To adjust each lineup’s offensive rating, I calculated a weighted average of the season defensive ratings of all the opposing lineups that that lineup faced. These defensive ratings were weighted by the number of possessions the original lineup played against that defensive lineup. This meant that for each lineup I had its offensive rating and its average opponents’ defensive rating. I subtracted the second from the first to get an adjusted measure of the lineup’s offensive production. So if a lineup had a good offensive rating but played against poor defensive lineups, its rating was decreased, while if a lineup had a poor offensive rating but played against good defensive lineups, its rating was increased.

The adjustments I made were only one level deep. In college football ranking systems you sometimes see similar multi-level adjustments for strength of schedule that take into account a team’s record, its opponents’ records, and its opponents’ opponents’ records. The same thing could be done here - I’m adjusting each team’s offensive ratings for their opponents’ defensive ratings, but I could first adjust the opponents’ defensive ratings for *their* opponents’ offensive ratings. Theoretically, one could do this infinitely, and I think the results would ultimately be similar to what you’d get from a regression-based method like Dan Rosenbaum uses for his adjusted plus/minus. But I’m just going to do one level of adjusting, partly because it can be calculated pretty quickly with some pivot tables in Excel, and partly because you just don’t gain that much the deeper you go. This is because over the course of a season, things tend to even out, and most lineups end up facing a similar mix of good and bad opposing lineups. The variance in opponents’ defensive ratings is a lot less than the variance in lineup offensive ratings, and the variance in opponents’ opponents’ offensive ratings would be even smaller.

Below are the adjusted rankings for offensive rating, defensive rating, and point differential. I excluded lineups that played together for less than 200 offensive possessions (or 200 defensive possessions). “ORtg” is the lineup’s offensive rating (points per 100 possessions), “oppDRtg” is the weighted average of the defensive ratings of the opposing lineups faced. “offDiff” is the additional points scored per 100 possessions over what would be expected based on the quality of the defenses faced. “DRtg”, “oppORtg”, and “defDiff” are the defensive counterparts to those stats. “totDiff” is the sum of “offDiff” and “defDiff”, which represents the additional point differential per 100 possessions over what would be expected based on the quality of the offenses and defenses faced.

#### Best and Worst Offensive Lineups:

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