New Blog

Though this blog is no longer being updated, I am now writing on political science and information visualization at

For the record

Point projection: Obama’s percent of the nationwide two-party vote:


Even more quiet

See here. It’s going to be even more quiet around here than usual. Thanks to everyone for reading and contributing your ideas.

Assigning credit to Team USA

During the NBA Finals, I made an effort to estimate player contributions to the final score, using Model-Estimated value, and a metric which I unimaginatively called “Credit.” MEV is a linear-weighting player productivity measure (read about it here), and Credit (which I’ve modified somewhat since the Finals coverage), attempts to divide credit (or blame) for a team’s success among individual players:

Player Credit = Player MEV / Team Total MEV * (Team Points / (Team Points + Opponent Points))

This way, total Credit for all players on both teams sums to one in every game, and players on teams that win by a lot are allocated more Credit to divide amongst themselves, whereas in tight games, each team has closer to 0.50 Credits to attribute. In the spirit of the upcoming Olympic Games, I hope to return to semi-regular coverage, though not necessarily in the form of any actual posting. Rather, I will endeavor to update, after each game leading up to and played in Beijing, MEV and Credit statistics for each member of Team USA. Each game’s statistics, as well as cumulative stats can be found here:

I hope you find it useful and insightful over the next few weeks.

Change of venue

Folks, starting today, I’ll be writing a weekly post on Thursdays, over at Hardwood Paroxysm. I’m likely to continue posting here, although frequency might drop a little bit. If you would like to continue subscribing to The Arbitrarian’s RSS feed, but would also like to read my posts at HP, I’ve put together a joint feed in Yahoo! Pipes that will do just that. Thanks for all of your support, readership, and insight–having readers makes blogging fun.

Here’s the feed: Abitrarian Everywhere

Estimating team chemistry

I posted recently to introduce a new method of characterizing basketball playing styles, which I call the SPI Style Trichotomy. The advantage of this methodology is, among other things, that it is an objective, performance-based means of characterizing player type that offers substantially more nuance and accuracy than the traditional position adjectives.

Well, today I’m going to take a step back from this seamless, continuous spectrum perspective, and impose some order in order to investigate the value of each playing style. Since the SPI characterizations themselves are productivity- and value-independent, it may be of interest to see the degree to which employing a player who plays a given style can add to team success. My first step was to identify, for each player-season, which of seven arbitrary playing style categorizations they most closely match. A quick look at the SPI Spectrum Graphic indicates that I’ve already “named” six spokes–each of the pure SPI styles, plus their opposites. For this post (and possibly into the future), I will refer to these six spoke-categories as (counter-clockwise from the 3 o’clock position) Pure Scorer, Perimeter Scorer, Pure Perimeter, Scorer’s Opposite (though catchier, “Defender” is too bold, and inaccurate), Pure Interior, and Interior Scorer. Note that I could have made any number of categories here, and that one of the positives of the SPI System is the lack of such arbitrary distinctions–nevertheless, for the purposes of running a regression, I’ve categorized them. Each player’s SPI numbers were used to identify the spoke to which they are closest, and for a given season, this is the category into which that player is lumped. To the six already mentioned above, I added a seventh identifier, “Mixed,” for those players who were closer to the center of the diagram than any of the six style archetypes. To give an idea of the results of the sorting, here is a table presenting the top 50 players for each archetype:

Exemplars of each SPI7 Style

The ranking was derived by summing each player’s BoxScores over seasons during which he was classified under a given archetype–thus, this isn’t a “Best-ever” list, necessarily–just a list of familiar players and their categorization.

Following this categorization process, I calculated, for each team-season, the sum of minutes played by players fitting in to each of the seven categories. Thus, the 07-08 Blazers featured 8,307 minutes of playing time from Interior Scorers–coming mostly from Aldridge, Outlaw and Webster, but with contributions from James Jones and Von Wafer. In fact, the team sums are pretty interesting in and of themselves, so I added that table as a second sheet to the Google Doc linked above: SPI7 Team Sums.

From here, I ran very basic regression analysis. I was hoping to identify the (relative) value of a minute played by each archetype. Thus, I regressed the team minute sums on team win totals (from the 52-53 season, onward, except 1999). This is a very simplistic analysis, but it yielded interesting results (in the variable names, SS is the Pure Scoring, SP is Perimeter Scoring, etc.):

Min       1Q   Median       3Q      Max
-31.4014  -8.4507   0.7324   8.6521  33.1204

Estimate Std. Error t value Pr(>|t|)
SSmin 0.0013482  0.0002170   6.213  7.2e-10 ***
SPmin 0.0016794  0.0001653  10.161  < 2e-16 ***
PPmin 0.0024116  0.0001986  12.142  < 2e-16 ***
PImin 0.0027014  0.0002147  12.580  < 2e-16 ***
IImin 0.0026275  0.0002129  12.344  < 2e-16 ***
ISmin 0.0019478  0.0001317  14.785  < 2e-16 ***
MMmin 0.0019719  0.0001493  13.204  < 2e-16 ***
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 11.92 on 1170 degrees of freedom
(3 observations deleted due to missingness)
Multiple R-Squared: 0.9217,     Adjusted R-squared: 0.9212
F-statistic:  1967 on 7 and 1170 DF,  p-value: < 2.2e-16

Each coefficient is significant, and we may gain some insight by comparing the magnitude of these coefficients. The least valuable archetype (in this extremely superficial analysis which should be taken with several hundred salt grains), is the Pure Scorer, who adds 0.0013 wins per additional minute played. The most valuable (surprisingly?) are the Scorer’s Opposite types–Kevin Garnett, Shane Battier, Kirilenko, etc. who add roughly double that number of wins per minute played. The rest you can figure out easily from the regression output. As a biased observer, with my own subjective preferences, I like these results a lot: one-dimensional scorers, adored by causal fans, but disdained by me, are identified as less valuable than the glue guys and lockdown defenders, etc. who focus on things other than scoring (although as Garnett and Barkley show, they can score, too). Keep in mind that this output is somewhat hastily done and only somewhat less hastily thought-through, but the results are certainly interesting.

Another question regarding these playing types might concern the combinations of types which are most effective. From a team-building standpoint, when considering the draft, trades, or free agent acquisitions, such an investigation might prove useful. Using the same set of data as above, I ran another regression, this time using only the interactions of each team’s minutes-by-type sums. In other words, instead of seven independent variables, there are now 21: one for each combination of archetypes, Pure Perimeter/Scoring Interior, Scorer’s Opposite/Pure Scorer, etc. The interaction means that the minutes for each of the two categories are multiplied together, and this is the value included in the regression. The output is as follows:

Estimate Std. Error t value Pr(>|t|)
SSmin:SPmin -1.956e-07  9.312e-08  -2.100 0.035909 *
SSmin:PPmin -5.331e-08  1.227e-07  -0.434 0.664067
SSmin:PImin  5.307e-07  1.203e-07   4.413 1.11e-05 ***
SSmin:IImin  5.360e-07  7.759e-08   6.908 8.12e-12 ***
SSmin:ISmin  3.934e-07  5.915e-08   6.652 4.46e-11 ***
SSmin:MMmin  1.263e-07  1.066e-07   1.184 0.236560
SPmin:PPmin -1.104e-08  9.102e-08  -0.121 0.903440
SPmin:PImin  5.573e-07  5.492e-08  10.146  < 2e-16 ***
SPmin:IImin  4.618e-07  4.015e-08  11.499  < 2e-16 ***
SPmin:ISmin  3.647e-07  3.967e-08   9.195  < 2e-16 ***
SPmin:MMmin  2.264e-07  6.264e-08   3.614 0.000314 ***
PPmin:PImin  6.224e-07  1.213e-07   5.132 3.37e-07 ***
PPmin:IImin  4.720e-07  9.427e-08   5.007 6.39e-07 ***
PPmin:ISmin  2.489e-07  8.016e-08   3.105 0.001948 **
PPmin:MMmin  4.288e-07  6.213e-08   6.902 8.43e-12 ***
PImin:IImin -3.740e-07  1.114e-07  -3.358 0.000811 ***
PImin:ISmin  1.617e-08  9.902e-08   0.163 0.870291
PImin:MMmin  1.937e-07  9.729e-08   1.991 0.046721 *
IImin:ISmin  1.150e-07  7.674e-08   1.498 0.134291
IImin:MMmin  2.654e-07  8.736e-08   3.038 0.002432 **
ISmin:MMmin  2.622e-07  6.672e-08   3.930 9.00e-05 ***
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 12.15 on 1156 degrees of freedom
(3 observations deleted due to missingness)
Multiple R-Squared: 0.9196,     Adjusted R-squared: 0.9182
F-statistic:   630 on 21 and 1156 DF,  p-value: < 2.2e-16

Note: If you thought the above regression was methodologically shaky, this one is even worse! But, nevertheless, it’s interesting to look at. Here, the coefficients are much more difficult to interpret, so I would recommend focusing mainly on whether or not they are significant (indicated by *s), and the sign attributed to the variable. It appears as though PP/PI combinations are especially fruitful, while PI/II is a deadly combination… Anyway, that’s more than enough for one post, but please feel free to add your own insights, and especially your criticisms!

The best of the WNBA, updated daily

Borrowing from the extremely useful DougStats, and making use of Google Docs, I present a (more-or-less) daily-updated list of the 100 most valuable WNBA players, using the BoxScores methodology.

WNBA Top 100 BoxScores

easy URL:

As I said, this ought to be updated more-or-less daily, and should serve as an easy, quick reference to see the state of the WNBA. For the uninitiated, BoxScores attempt to estimate the value of each individual player in terms of contributions to team success, and the unit being estimated is wins. Thus, as of this posting, we can see that rookie Candace Parker leads the league with a 3.76 BoxScore/wins created, followed by Lindsay Whalen with 3.34… odd that Whalen failed to make the Olympic team, being that she is the second most valuable WNBA player in the league right now.

Note: Since the regression coeffeicients employed in Model-Estimated Value (MEV) were fitted for the NBA, it is unclear whether or not their values translate identically to WNBA play–that is, a steal in the WNBA may be worth more than in the NBA, or less, etc. However, based on the work of others, I’m assuming there is relatively little difference between the leagues on this front, and since the formula is applied evenly across all WNBA players, I will assume that the differnces even out on average. Let me know if you find this list useful, and whether or not the ranking seems to mesh with your own subjective perceptions.