## The Statistic the Timberwolves Forgot (Or Never Knew)

There was a lot of news about our favorite team over the weekend. The Wolves appear to be set to make a large (\$50m/4yrs?) offer to Batum. They're talking to Greg Stiemsma, Brandon Roy, and Jordan Hill. They've been interested in Jamal Crawford for a while now. These guys have at least one thing in common, and it isn't good news. More below the jump.

Plus/minus is a simple way of measuring what happens when a player is on the court. All you do is calculate the difference between the points scored by his team and his opponent, usually normalized to a per-100 possession basis. There's a big problem with this simple measure: lineups aren't random. For example, players in the starting lineup typically have better teammates and better opponents on the court with them. Adjusted plus/minus is a method for getting around this lineup bias. Here is a good article describing the basics of it. I'll try to do my best to explain it here as well.

Regression analysis is a statistical method that attempts to explain the effect of a set of variables ("independent", or "explanatory" variables) on an outcome (the "dependent" variable). The important thing to know about regression analysis is that it estimates the effect of each explanatory variable controlling for the effect of all other variables.

For example, suppose I estimated a model that tried to explain the price of a house as a function of the square footage, number of bedrooms, presence of central AC, etc. If I had data on the price and characteristics of enough houses, I could estimate how much each attribute affects the price. For example, I could estimate how much people will pay for an extra bathroom, controlling for the size and location of the house.

In our case, we want to know the effect of a specific player on his team's performance, controlling for the other players on the court (for both teams). Each observation represents a unique lineup combination (a "shift"), or five specific players on one team and five specific players on the other team. The dependent variable is that shift's margin, or the plus/minus per 100 possessions. The explanatory variables are indicator variables for all of the players who appear in any lineup. A particular player variable is "turned on" for observations in which the player appeared in that lineup. (There's some home vs. away stuff you can see in the link, but I won't get into it here.) Observations across one or two seasons are used to estimate each player's effect on his team's margin, controlling for the other players on the court with him.

Here's some crude intuition about how this works. Suppose a player only ever came on the court when LeBron James was on the court for the opponent. (They played the Heat a lot. Bear with me.) He's probably going to have a negative raw plus/minus, because LeBron and the Heat are pretty good. But suppose that he does OK, such that his lineups are -8 against LeBron while the rest of the league is -10 against him. The regression model will account for this, and assign the player a +2 adjusted plus/minus. (This is oversimplified, but I think it illustrates what the model is trying to do.)

In theory, adjusted plus/minus is the best statistic there is, because it accounts for everything the player does on the court, whether it makes its way into the box score or not. In practice, it's a very "noisy" statistic. That is, the standard errors of the player effects (which reflect the precision of the estimated effects) are large relative to the effects themselves.

This quote (from here) sums up the key problem well:

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.

One solution is to use more than one year of data, which provides more observations and more lineup combinations, improving the precision of the estimates. BasketballValue.com provides one- and two-year estimates. Where available, I use the two-year estimates because the standard errors are quite a bit lower than they are for the one-year models. Note that a player can be included in a two-year model even if they only have one year of data (i.e., rookies). That is, the second year of data helps estimate the contribution of the players who share the court with the rookie.

Adjusted Plus/Minus for the Wolves FA Targets

Why do we care about all of this? The table below shows the adjusted plus/minus values for the players the Wolves have targeted in free agency, all taken from BasketballValue.com. (Stiemsma only has a 1-year value available. I use Roy's one-year value from 2010-11 to account for the effects of injury.) The second column contains the adjusted plus/minus value and the third column contains the standard error of that estimate. As I wrote above, adjusted plus/minus tends to be a very noisy statistic. When the adjusted plus/minus is larger than the standard error, I start paying attention. A more strict statistical interpretation would pay attention to plus/minus values that are at least twice the standard error (in absolute value).

 Player Adj. +/- SE Batum -4.79 2.83 Stiemsma* -10.56 6.11 Crawford -2.96 2.54 Roy** -1.11 4.47 J. Hill -7.84 3.78

As you can see, the Wolves have picked out players who do not look good according to this statistic. (Is there a sort ascending versus descending confusion at Timberwolves headquarters?) Many of the values are large enough relative to the standard errors to be troubling. Batum's bad result isn't a one-year fluke. His estimate from last year (which used data from the 2009-10 and 2010-11 seasons) was just as bad.

Hollinger looked at these results (or results like them) and concluded:

Additionally, Batum's on-court, off-court data suggests he may be wildly overrated. While lauded for his defensive potential, there is little to no evidence that he's been an impactful defender in the actual games; in fact, there's mounting evidence that he may be lousy at defense. It's possible he becomes an \$11 million player, but he certainly hasn't been one up until now.

This gives me some concern about the financial commitment the Wolves are prepared to make to Batum. We want a top-notch complementary wing who gives us good perimeter defense. Batum has that reputation in some circles, but there's pretty good evidence that his play hasn't caught up.

The Good News

The Wolves have already acquired one wing in Chase Budinger. They are rumored to be attempting to trade for Pau Gasol. The good news is that both of these players look quite good according to adjusted plus/minus. The table below shows values for them and other potential targets (real or imagined in the world of Canis Hoopus).

 Player Adj. +/- SE Budinger 4.97 3.30 Gasol 9.54 3.77 Iguodala 2.74 3.39 D. Green 6.80 3.97 L. Williams -0.71 3.30 Gee -3.17 3.43 D. Brown -2.92 3.66 K. Martin -3.12 3.92 C. Lee 1.44 3.37

Many of these values fall into the "ignore" category because the standard error is larger than the plus/minus estimate. Budinger, Gasol, and Danny Green look very good. I don't love the result for Gee, Derrick Brown, or Kevin Martin, but again, the standard errors are big so I take them with a grain of salt.

The Wolves have already acquired one pretty good looking wing in Chase Budinger. They're trying to get Batum, and they're willing to pay big money to do it. There are some reasons for optimism there: the Wolves are apparently willing to spend to get better, Batum scores fairly efficiently, and he's young. However, his performance has tailed off in the last two seasons compared to his excellent 2009-10 campaign. In addition, his adjusted plus/minus (and the reputation he has among Blazers' fans for passive play) indicates that he might not be anywhere near the defender the Wolves probably think they'd be getting. The Wolves may very well be better off signing two lower profile free agents than spending the big bucks on Batum.

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