Game 4 between Utah and L.A. last night provides an interesting data point in considering the usefulness of +/- as a basketball stat. We’ve all seen instances where a player who contributes very little statistically ends up with a phenomenally high +/- rating, while the team’s stars are rated as having played poorly. Occasionally, this even happens for a player on the losing team. The reason, of course, is that a player’s +/- rating is, by definition, the point differential between two teams accrued while that player is on the court. So, if you’ve got a scrub on the court for precisely the five minutes that his superstar teammate reels off 15 consecutive points while the other team is held scoreless, he ends up with a +15 rating; the superstar gets a +15 for that five-minute stretch too, but since he probably plays the rest of the game too, his score is likely diluted by game-end.
With that in mind, it’s interesting to read J.A. Adande’s recap of Game 4 with a scorecard on hand. What we see is that Kobe Bryant, who scored 38 points on 66% shooting in 39 minutes, ended the game with a +4 rating, the same as Luke Walton, who chipped in 9 points on 50% shooting in 19 minutes. Sasha Vujacic had a +9 for his contributions of 9 points on 33% shooting in 17 minutes. The guys who fare the best are Lamar Odom (10-15-6 line on 40% shooting in 41 minutes, for +20) and Pau Gasol (13-10-1 in 41 minutes, for +15). Clearly, those numbers are all over the place, but what’s weird is how Kobe ended up with just a +4, given how much everyone insists he dominated the game. The following excerpt from Adande’s article helps to explain this:
Bryant came out firing. … He scored the Lakers’ first 11 points and 13 of their first 15, hitting 6-of-8 shots in the first quarter. … But the rest of the Lakers went 2 for 10 and Utah led, 25-20. The turning point actually came when he was on the bench at the start of the second quarter and the Lakers fielded a lineup of Shannon Brown, Sasha Vujacic, Luke Walton and starters Lamar Odom and Pau Gasol. It was the reserve players who hit three consecutive 3-pointers to take the Lakers from a seven-point deficit to a two-point lead.
A-ha! Kobe, may have scored a full one-third of the team’s points over the course of the game, but he was sitting out during a key stretch. Worse, while he was lighting it up in the first quarter, everybody else’s ineptitude was costing him in the +/- stakes.
This situation captures almost perfectly the beauty and the shortcomings of +/-: on the one hand, it zooms in on what matters most — winning — without ascribing arbitrary weightings to individual statistics like points, rebounds and assists while still incorporating the value of intangible contributions. On the other hand, the value of the individual is so tightly tied to who is on the court with him that +/- as a comparative tool becomes worthless, since it reflects more the overall quality of the team and the ability of the coach to construct effective rotations. It’s sort of the classic curse of single-number metrics.
One is tempted to abandon +/- as the ‘holy grail’ statistic at this point, but it’s really a very good idea in as far as cutting out the pseudo-science of sports statistics and focussing on what actually leads to wins. One wonders if the problem is ’snapshotting’ +/- ratings at the end of the game, which necessarily throws out all the information that a real-time +/- score would have. Behold:
What you’re seeing is a graph of realtime +/- ratings for the teams (ie, overall) as well as for various players over the course of the entire game. The overall team rating is just the point differential between L.A. and Utah (I’m a Lakers fan); for the individual players, you can see how their +/- rating was related to that of the team through time. The flat regions (eg, Luke Walton from the beginning of the game until about 7:30 in) are periods when the player’s +/- didn’t change — ie, he was either off the court, or he was on, but neither team scored. You’ll also see some periods when everyone’s rating moves together — eg, the plotted players’ +/- ratings all fell between 10:30 and 14:30 — which implies that they were all on the court (in the 10:30 - 14:30 period, the falling ratings mean Utah made a run).
The plot makes it immediately apparent how, for example, Luke Walton ended up with the same +/- rating as Kobe: he simply sat out the last 10 minutes of the game, when Utah was cutting L.A.’s lead. This, of course, raises some interesting questions, such as how starting and ending games ends players’ +/- ratings; from this example, it looks like garbage time can hurt the winning team’s ratings (though conversely, Utah’s players’ ratings benefited from garbage time), while starting can be good or bad for one’s score depending on who starts the game stronger (ie, there’s no bias there, whereas there is for garbage time).
Of course, since one now has a nice time-series describing the players’ contributions (as encapsulated by +/-), one can try computing a single-number metric of performance (though, obviously, YMMV*) by simply computing the correlation between the team’s overall +/- and the players’ individual ratings. Note that this number really only makes sense in the context of the team’s performance — ie, a player with a perfect score on the Kings is clearly worse than a player with a perfect score on the Lakers, since correlation doesn’t differentiate between covariance that leads to team success or failure. That said, let’s take a look at what this ‘+/- correlation’ metric looks like for Game 4:
I’ve shown both team’s players on this one chart, so what you’re seeing is who contributed to what ultimately proved to be a Lakers’ win as a normalised score based on players’ full real-time contributions. So if you were to say a dude is absolutely indispensable to a team’s success (ie, the MVP), you would expect him to have, over the course of the season, a +/- correlation of almost +1. We see that, for example, for Pau Gasol, implying he was on the court for almost every significant period of the game (even if he didn’t contribute as much as Kobe in raw numbers). The implications for Andrew Bynum are also interesting, since a correlation that low means he basically didn’t matter much as far as the team’s ultimate success is concerned. I’m really itching to see what it says about Shane Battier ;)
Unfortunately, it’s not the end-all, be-all of metrics because by construction the amount of playing time a player receives will bias it, and because it’s basis in the overall winning ability of the five players on the court means it doesn’t break down how much of that success is attributable to any individual player. However, it does roughly put players where we expect them in the pecking order, and gives a rudimentary sense of how much responsibility a team’s results individual players should be ascribed. And there are ways to improve this metric, a topic which really deserves another post (although here’s an obvious extension: define the overall team +/- to always be in favour of the team that wins).
In summary, what complaints we have against +/- ratings might be resolvable by analysing players’ +/- ratings as they evolve through time. In abstract terms, +/- seems a much more objective metric than the weighted-average metrics in more common use (eg, PER, Wages-of-Wins), which make assumptions that borne out only by a couple of decades’ history.
I’m attaching the spreadsheet containing my data as well in case anyone’s interested in taking a look at it. If I get time — and if there’s interest — I might compile this sort of data for all the games next year and put it on the web. Let me know what you think.
Utah-L.A. Game 4, Playoffs 2009 - Data
Utah-L.A. Game 4, Playoffs 2009 - Scorecard
- I have trouble using this phrase in polite conversation now, but hopefully we’re all on the same page as to why I’m using it!





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