Adjusted Plus-Minus Ratings:
New and Improved for 2007-2008

Like most devoted NBA fans, we enjoy looking at boxscores. They’re often incredibly informative. And yet boxscores fail to capture many important elements of play – lockdown defense, screens, rotations, hustle plays, and so on – that can affect the game’s bottom line

That’s why we believe it’s necessary to “think outside the boxscore” in order to adequately measure each player’s true value to his team. The plus-minus statistic, which tracks all changes in scoring while each player is on the court, is one promising approach that’s gained traction in recent years. But this metric suffers from a key drawback: each player’s rating is heavily dependent upon the quality of his on-court teammates. Even a mere role player on a great team (e.g., Glen Davis) typically has a higher plus-minus rating than a superstar on a bad team (Dwyane Wade). Fortunately, this limitation can be overcome through the use of advanced mathematical techniques, which make it possible to isolate the unique effects of each player on the court. In other words, we can statistically adjust each player’s plus-minus rating to account for the simultaneous impact of all his teammates and opponents [1]. Hence the name: adjusted plus-minus [2].

At first blush, the metric might even seem like the “holy grail” of basketball statistics – a single measure that captures the precise effect of each player on his team’s bottom-line scoring margin. But it, too, has a major drawback: as a mathematical estimate, each adjusted plus-minus rating contains measurement noise, i.e., a margin of error.

It’s important, therefore, to get this noise (error) level as low as possible, and we’ve taken an important step in that direction with the present set of ratings. Specifically, we’ve used five seasons’ worth of data (provided by 82games.com) – weighted very heavily in favor of the 2007-2008 season – to disentangle the individual effects of teammates who frequently appear on the court at the same time. As a result, we are able to present below the most accurate (low-noise) adjusted plus-minus ratings ever to appear in the public domain. In addition, we’ve modeled separately each player’s impact on offense and defense, treating these as completely independent variables. (Both innovations are explained in detail below the fold.)

Accordingly, the tables that follow include offensive, defensive, and total (net) adjusted plus-minus ratings for each player who logged at least 300 minutes in the 2007-2008 season, with high-minutes (> 2000 total regular-season minutes) and low-minutes players listed separately. These ratings reflect each player’s points contributed per 100 possessions, as compared with the league average. As an added bonus, both tables are sortable by adjusted plus-minus ratings, team, and position.

Noise Reduction
As noted above, adjusted plus-minus ratings – as mathematical estimates – tend to be noisy. For example, Ilardi’s 2006-2007 ratings had a typical standard error of about 3.0, which implies a margin of error for each player estimate (at a 95% confidence level) of roughly +/- 5.9 points [3]. Obviously, the presence of so much noise greatly limits the usefulness of the metric. To take a telling example: Chris Paul had a 2006-2007 rating of + 4.1 pts (per 40 minutes), with a margin of error of roughly +/- 5.9; so, we can only say with 95% confidence that his performance was somewhere between that of a sub-mediocre player (-1.8) and one of the top players in the league (+10.0)!

But where does the noise come from, and how can it be eliminated? Mostly, it results from the fact that teams tend to put the same players on the court together at the same time. That is, many players’ minutes are strongly inter-correlated, so the underlying adjusted plus-minus model has a hard time disentangling individual player effects at a high level of accuracy. In addition, the number of unique observations (i.e., lineups) of a given player in a single season is surprisingly small [4], typically under 1,000.

However, there exists a straightforward solution to both of these problems: use multiple seasons’ worth of data. How many seasons of data are needed? As many as possible. With only one season, the standard errors are very high – typically around 5.0 points per 100 possessions (with corresponding margins of error of roughly +/- 10.0 points). Such ratings, of course, are highly suspect. With two seasons of data, the noise level drops by about 40%, but it’s still uncomfortably high. The picture, however, gets appreciably clearer with each additional season that’s added to the model.

In this article, we present adjusted plus-minus ratings that utilize five seasons’ worth of data – ratings with an unprecedented lack of noise. As you’ll see, the average standard errors have now dropped down to about 1.1 points per 100 possessions [5], 80% lower than the errors based on only one season. This makes them dramatically more useful.

Of course, many readers will want to know how a given player performed last year (2007-2008), not how he performed on average over the past five seasons. Fortunately, the inputs to the adjusted plus-minus model can be weighted to emphasize data from the most recent season. This approach increases the noise level slightly in comparison with an equal weighting of all seasons, but it still reduces it substantially. Accordingly, the ratings herein can be regarded as estimates of 2007-2008 player performance, with data from the previous four seasons included in the model at fractional weightings [6] for the purpose of reducing estimation errors.

Separating Offense from Defense
We’ve also taken another important step toward enhancing the usefulness of the adjusted plus-minus metric by including a separate analysis of each player’s performance on the offensive and defensive sides of the ball. To do so, we’ve constructed a separate variable to model each player’s possessions on offense (Xi) and defense (Di), respectively. Thus, for each unique 10-player lineup that appears in each game, two separate lines of data are generated for use in the ensuing regression model: one line reflects scoring efficiency (points per 100 possessions) during those possessions when the home team is on offense and the away team on defense, and one represents the converse scenario (i.e., possessions in which the away team is on offense and the home team on defense). The model can be rendered as follows: