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.):

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

Coefficients:
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:

Coefficients:
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: http://bit.ly/wboxscores

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.

NBA playing style spectrum

Many conversations about sports revolve around comparisons of quality — team A is better than team B, player X is the best of all time, this draftee will help his team more than that one, etc. For this type of discussion, many metrics exist, both qualitative and quantitative, one of which is BoxScores, developed here at the Arbitrarian. Other conversations center around similarity–team C plays like team D did in the 1980s, player Y is a taller, faster player Z, etc. The Arbitrarian has spent substantial time investigating this type of comparison as well, using statistical proximity and network diagrams. Yet another characterization, somewhat more general than direct similarity comparisons, is that of type, or style. While playing style has been discussed here, and style markers can be seen everywhere in my work in the form of various colorations, I would like to develop the idea a little more fully, and present a novel graphical visualization of the concept applied to NBA players.

Very rudimentary factor and cluster analysis I performed a long time ago indicated that there are distinctions in the data between players who tend to try to score a lot, those who play a “smaller” game, and those who play like “big men.” In terms of the NBA’s tracked counting statistics, this translates to a differentiation between those who specialize in points and field goal attempts, rebounds and blocks, and steals and assists. I have chosen to call each of these three tendencies Scorer, Perimeter, and Interior, and collectively they form the SPI Style Trichotomy.

Calculation

To identify each player’s style is conceptually simple, but computationally somewhat more complex. Essentially, one sums each player’s fga + tr + bk + as + st, and determines what percentage of the total each SPI factor constitutes:

• Scorer percentage = fga / (fga + tr + bk + as + st)
• Perimeter percentage = (as + st) / (fga + tr + bk + as + st)
• Interior percentage = (tr + bk) / (fga + tr + bk + as + st)

These numbers are interesting on their own, but for the calculation of an index of style, they require further manipulation. In the league as a whole, the Scorer percentage is around 50%, the Perimeter percentage around 20%, and Interior 30%. Thus, if using these percentages, the vast majority of players would appear to be very scoring-centered. My concern here, in constructing a useful index, is to identify player propensities relative to other players, and for that, I calculate the percentile of each player’s percentages.

• Scorer index = percentile(Scorer percentage)
• Perimeter index = percentile(Perimeter percentage)
• Interior index = percentile(Interior percentage)

Thus, even though the maximum Scorer percentage in a season might be close to 75% while the maximum Perimeter percentage is closer to 25%, the players with the highest percentages in the sample under consideration will be assigned an index value of 1. Players with median values on a percentage will have an index value of 0.5, and so on. The percentilization normalizes accross style tendencies and player subpopulations, and has the added virtue of scaling from 0 to 1.

Interpretation

Thus we have a set of three numbers for each player which can be used to characterize his playing style. The numbers easily translate to more qualitative descriptions. A player with a SPI triplet of (0.8, 0.2, 0.7) is an interior scorer, without much perimeter production. A player with this triplet (0.1, 0.7, 0.75) is anything but a scorer, sometimes called a “glue” guy. Someone at (0.5, 0.5, 0.5) produces the league median of each type, which is different from a player whose percentages are 33%, 33% and 33%. Such a player would have a relatively lower Scoring index, for example.

Since each individual is characterized by three variables, their SPI type can be plotted in three dimensions. Unfortunately, three dimensions are difficult to convey on a computer screen, so here is a plot which depicts Perimeter indices along the X-axis, Interior indices on the vertical axis, and Scoring indices as the size of the point.

(Click to enlarge)

Historical application note: Since steals and blocks have not been kept for the entirety of the history of professional basketball, players from earlier eras may have slightly skewed SPI values. While percentages and indices can still be calculated based only on fga, tr, and as, it is not difficult to see that leaving out blocks and steals, in comparison to eras in which those defensive statistics are included, will tend to skew players from an earlier era more toward the Scoring type. Unfortunately, without substantial era-specific correction, this effect is unavoidable. However, the sorting still manages to work well, especially if this detail is kept in mind when making certain cross-temporal comparisons.

Presentation

One of the advantages of using three sub-indices to construct the overall SPI Trichotomy is the convenient translation of index values to color. The three primary colors of light are Red, Green and Blue, and when combined in certain proportions, it is possible to generate infinite gradations of color (see Wikipedia). This means that each SPI triplet for each player can be represented as a single color. This aids understanding and comparison, as it is much easier to keep in mind that a certain player is a deep red than that his SPI triplet is (0.9, 0.1, 0.2), or that a player is a medium grey than that his triplet is (0.45, 0.53, 0.55). Further, a greenish-blue player is easily paired with another greenish-blue player, without having to specifically compare each of the players’ three index values. The human eye is capable of extremely high-resolution discernment, and using a single color to represent three numerical values takes advantage of this.

Here is the above plot, with color added according to RGB values derived from each player’s SPI indices, as you can see, “blueness” increases from bottom to top, “greenness” from left to right, and “redness” varies with the size of the point. The top-right corner is aqua or cyan, while the bottom left is mostly reddish, due to an absence of green and blue.

(Click to enlarge)

Unfortunately, this presentational format leaves a lot to be desired. Since each player can be represented by just one color, can we do better than a pseudo-3-dimensional plot? The answer is yes and no: No, because to ensure that the hue, saturation, and value of each color are captured, we still require three variables (see Wikipedia); yes, because most of what we are interested in here is hue–the underlying color for each player, red, yellow, green, aquamarine, vivid tangerine, indigo, etc. The other two components of HSV color space, saturation and value, allow us to see how “pure” the hue is, which in our basketball application, translates to how “pure” an individual’s playing style is.

The advantage of a conversion from RGB to HSV is that, by combining the S and V, we can represent the entire playing style spectrum in a format resembling a color wheel. This is the most straightforward and useful format for presentation, and this graphic is the big payoff:

Click to view in a Google Maps format. Also available at the easy-to-remember http://bit.ly/spi

As you can see, each of these NBA greats is aligned at a certain angle and distance from the center (I used polar coordinates as a basis for this plot), and this allows us to identify relatively similar players, player’s “opposites,” clusters, and other interesting observations. In this graphic, players with greater MEV (Model Estimated Value) are larger, and this allows comparisons along radii, as in X plays in a similar manner to Y, but is more valuable/productive.

Vocabulary

There are several ways our statistical vocabulary can be expanded via the SPI Style Trichotomy. The first is that we can characterize the degree to which a given player is Scoring/Interior/Perimeter-focused by reporting their index value. The second is that we can describe the player’s color–“He’s a deep blue defensive center,” or “Shane Battier is not chucker! He’s a cyan-colored scorer’s opposite type.” Finally, we can approximate a vector–I suggest the convention of overlaying the hours of a clock over the spectrum diagram (to indicate vector direction), with 12 o’clock at the very top, 1:00 at the interior’s opposite position, 3:00 at the scorer’s position, 4 o’clock bisecting the scorer and perimeter’s opposite position, etc. Distance from the center of the diagram (the length of the vector) indicates the degree to which a player fits exactly into their playing style–those players whose games are more balanced are closer to the center, players whose games are more specialized or narrow are further from the center. So, one could say, for example, “Dikembe Mutombo is a pure 7 o’clock,” or, “Michael Jordan was between about a two and three for most of his career, but in 88-89, he shifted closer to a pure 11.” And so forth.

Feedback

I would be very interested to hear any and all comments–does the trichotomy make sense? Is it useful? Are the S/P/I typologies a reasonable first division? Is it helpful to have a more continuous, yet still quantifiable, tool to describe player type than just position labels? Do you like the idea of equating, for example, a shoot-first point guard with a yellow-green 12 o’clock player? Does the graphic offer any new insight, confirm your subjective observations, or conflict with your opinions? I will be following this post up with many more using this methodology and this type of display, I hope you will come back often, or possibly subscribe.

Predicting the future, by analogy

Many times before, I’ve posted network diagrams which I suggest highlight objective similarities between athletes, according only to their statistical production. I’ve also noted that one of the most common discussions, especially around the draft and its aftermath, is that which attempts to identify which current or past professional player is most similar to which draftee. This is done, I believe, to convey some idea of playing style, but also, I think, to convey some idea of an individual’s potential. If a collegiate or recent draft pick gets compared to Michael Jordan instead of Zan Tabak, it means that the comparer thinks the rookie is more of a scoring wing player than a non-scoring center type, and that he has the potential to be a very good player in the NBA, rather than a very good player in Europe.

Thus, I thought it would be useful to do this same sort of comparison, but statistically, rather than subjectively. The main problem I encountered is that one cannot just add a college player’s statistics to a database of pros, match them, and expect the results to be valid. A player who scores 28 ppg in college could turn out to be a prolific scorer in the NBA, but he may also turn out to be Adam Morrison. Even comparisons of two players’ statistics across NCAA teams, I would submit, is shaky, given that college teams are so variable in terms of playing styles and abilities. Nevertheless, that it what I have chosen to do: Compare the collegiate statistical profile’s of some of this year’s draftees to those of other recent draftees, and suggest the inference, by analogy, that their professional careers will be similar to those whom their college careers match. I understand that this is fraught with tenuous connections and weak connections, but given my personal data limitations and relative lack of patience and time, this is what I’ve come up with:

Statistical Proximity of Selected NCAA Basketball Players [pdf]

Incidentally, player vertices are scaled according to their per-game MEV (Model-Estimated Value-similar to the calculation for BoxScores), and colors are according to the Playing Style Trichotomy outlined here. I find it interesting that the algorithm matches Michael Beasley with Kevin Durant, who just had a ROY season. Derrick Rose isn’t directly connected to anyone spectacular, though he is only two degrees of separation from Chris Paul, which is good company. OJ Mayo is tied to Ben Gordon, who is off to a promising start in the NBA, and Rodney Stuckey is most closely matched to Dwyane Wade (perhaps the Pistons used similar methodology in making their pick). Anyway, I’m sure many of you will gain greater insight from the graphic than my own descriptions, so please fill me in with a comment.

Mr. Consistency

Who are the most consistent scorers in the NBA? This is a question of some interest for those who participate in fantasy leagues, as consistency might be a virtue in determining the value of a player on your roster. For various reasons, a player might be worth more to you if they score 20 points every game, rather than alternate between 10 and 30 every other game. Further, some measure of consistency may highlight a player’s ability to impose their will on a game: a player able to get his scoring in, regardless of the opposition, could be said to be more of a game-defining player.

I’ve managed to estimate, for players since the 86-87 season, each individual’s mean points per 48 minutes, as well as the standard deviation of said statistic, and thus the coefficient of variation (sd/mean) and 95% confidence interval. Here’s a spreadsheet of the top (634) players in the league, by mean pts/48, sorted by coefficient of variation. Thus, the players at top could be said, in some way, to be more consistent scorers than those at the bottom.

Most consistent scorers, 1986-2008

Below is another way to view the same question. Using each player’s mean and standard deviation pts/48, along with the sample size, we can construct a 95% confidence interval for our estimate of their true mean. In the graphic linked below, each player is ranked by their mean pts/48, and the x-axis indicates how they fare under this measure of scoring. Each mean is surrounded by a line indicating the 95% confidence interval. This means, essentially, that we can be 95% sure that the player is within the span of their colored line. For players with smaller samples or greater variance, the error bars will be wider.

NBA Pts/48 min means with error bars

As you can see, some players have no error bars at all–this means that they only have one observation. Others’ error bars go down past zero. This means that we can be 95% sure that their mean pts/48 is in a range that includes zero, which doesn’t tell us very much. Anyway, here is the same graphic, for the 2007-08 season only:

Note that Carl Landry (#73) has a greater variance than most players around him, but he ranks as a better per-48 scorer than Shaquille O’Neal.

Finally, here’s a regular-season 2007-08 graphic for players’ MEV (or model-estimated value, using regression-derived regression weights like those seen here). Landry does even better here (18th), in terms of his mean, but his confidence interval is very large. This estimate suggests, though, that at worst, he’s about as good as Odom, Andre Miller, and Kirilenko; while at best, he is in rarified air. Keep in mind that this is still just a 95% confidence interval, so statistically, there’s still a 1 in 20 chance the true mean isn’t even in this interval. All should be taken with a grain of salt. One of the things I like most about this presentation is that it’s a per-minute stat, which controls for playing time (although not pace), but still reminds us that estimates for those players with little playing time should be taken with large grains of salt, and might not really mean much of anything. Josh McRoberts, for example, is probably not the 406th, much less the 6th, most valuable player in the NBA, even though his simple arithmetic mean indicates as much–his confidence interval reminds us of this, while maintaining the simple ordering.

I suppose this is also the public debut of any sort of official MEV ordering for 2007-08. I’d be interested to hear what people thought about this… this is something similar to Berri’s estimates, but I think the weightings are a little more appropriate. Let me know in the comments if they seem, at least, per-minute, to be reasonable estimates and orderings of player value.