To illustrate the calculation and utility of the new metrics, and to quantify the influence of space on shooting efficiency in the NBA, we analyzed a database recording every shot taken during the 2011–2012 NBA regular season. Data were obtained from ESPN.com; shot location data were available on a game by game basis in publicly available text xml files, though no information was provided concerning collection methodology or accuracy. This database includes Cartesian coordinates for over 141,000 field goal attempts, and detailed attribute information including who took the shot and whether or not the attempt resulted in a made basket. All analysis was conducted in R v. 3.0.2 (R Core Team 2013) and used the classInt (Bivand 2013), RColorBrewer (Neuwirth 2011), sp (Bivand et al. 2013), and weights (Pasek 2012) libraries. To construct a robust estimate of league-wide local field goal probability, we used EB rate estimation, as documented in Section 2. The 2011–2012 raw NBA field goal rate is mapped in Figure 1A, while the EB rate inferred surface is shown in Figure 1B. It is visually apparent that the EB rate surface is much smoother than the raw rate, as much of the noise present in the raw rates has been removed. Raw rates with small numbers of shots, typically in locations far from the basket, were affected substantially: EB estimated rates were as much as 0.94 below the raw rate and as much as 0.42 above the raw rate. However the mean rate difference (EB-raw) was 0.003, with half the rate differences falling between –0.035 and 0.057. Distinct spatial patterns are apparent in the EB estimated rate surface. Rates are high very close to the basket, but fall as distance increases to about 15 feet. There is an arc at about this distance with higher rates; this arc is wider at the free throw line and on the left side. At distances >20 feet the smoothed rates resume their descent, with generally extremely low rates more than 30 feet from the basket. Along the three point line several higher spots emerge: in the corners and on the right elbow. Shots taken from behind the backboard are low-percentage, particularly on the figure’s left: shots from these locations evidently must be attempted with one’s left hand, which may explain the discrepancy. The EB field goal rate surface is not left-right symmetric, though the broad pattern is persistent.

Figure 1 Mapped shooting patterns from the 2011 to 2012 NBA regular season: (A) raw field goal rate surface; (B) empirical Bayesian smoothed FG rate estimate, based on all shots taken in that season. Color/shade scales represent the same rates for both surfaces.

We generate local shooting percentage for individual players and calculate LPPS, LPD, and LSSE, as described above. Each local metric is calculated for each 1-square foot in which shots were taken, and summaries of their values across the court were collected to provide single, global shooting metrics for each player (SSE and POLA). Finally, graphics showing local shooting ability are constructed for selected players. The utility of the introduced metrics, particularly SSE and POLA, is evaluated in three case studies. The first contrasts the shooting performance of two NBA players during the 2011–2012 season in detail; the second is a league-wide analysis to determine the most and least effective shooters; the third offers an assessment of the spatial shooting ability of two players. Throughout these studies the spatial metrics are compared with non-spatial measures of shooting ability.

We first illustrate the new shooting evaluations by comparing two NBA players: Dwight Howard and Steve Nash. During the 2011–2012 NBA season Steve Nash took 554 shots and Dwight Howard took 721 shots. They were very different players, with very distinctive spatial shooting tendencies. As a center, almost all of Dwight Howard’s shots were taken close to the basket; as a point guard, Steve Nash’s shots were more scattered, generally occurring much further from the basket. The traditional metric to compare these player’s shooting effectiveness is field goal percentage (FG%); Howard was second in the league in FG%, shooting 57.3%, while Nash was 11th, at 53.2%. Using effective FG%, which accounts for three point shooting, Nash ranked third at 58.1%, while Howard, who attempted no three point shots, ranked fourth at 57.3%. Points per shot, an efficiency metric, could not distinguish between these shooters, as both scored at a rate of 1.16 points per shot.

All of these measures neglect to account for the striking spatial differences between Howard and Nash’s shooting tendencies. Figure 2A illustrates the expected points per shot surface, based on the 2011–2012 EB estimated field goal rate for the league as a whole. This rate is multiplied by the points per shot earned from each location on the court to produce the surface in the figure. Cells intersected by the three point line are adjusted by the proportion of three point shots vs. two point shots attempted. This map reflects the same spatial patterns observed in Figure 1B, but with the impact of the shot value factored in. The shot constellations of Howard and Nash are overlain on this map in Figure 2B. It is visually apparent that almost all of Howard’s shots were in areas of where the league has relatively high EPPS values; this was not the case with Nash. To quantify this striking visual contrast, spatial shooting metrics were calculated for both players.

Figure 2 (A) Expected points per shot (EPPS) map for the 2011–2012 NBA regular season, obtained by multiplying the rate estimates in Figure 1B by the local points per field goal; (B) Shot maps for Steve Nash (orange diamonds) and Dwight Howard (blue x’s) overlain on a map of expected points per shot.

Steve Nash’s 554 attempts result in 520.9 expected points, an average of 0.94 expected points per shot (EPPS); Dwight Howard’s 721 attempts result in 727.2 expected points, an average of 1.01 EPPS. By itself, this analysis simply reveals that one should expect fewer points per shot from Nash than Howard, given their respective shot constellations. Next we compare these estimated point totals with actual points accrued by the shooter as a means to assess their shooting efficiency relative to their cohort of NBA peers. Steve Nash’s shots actually resulted in 645 points, an average of 1.16 points per shot (PPS). His spatial shooting effectiveness (SSE) was 0.23 points per shot, meaning that he scored over a fifth of a point per shot more than would be expected from his shot constellation. Nash accrued 124.1 points more than expected (POLA), indicating that he shot much better than an average NBA shooter from these locations. Dwight Howard’s 721 shots actually resulted in 832 points, an average of 1.15 PPS; scoring 104.9 POLA. His SSE was 0.15 points per shot, a full third less than Nash’s. Interestingly, both shooters accumulated points at nearly the same rate: 1.16 points per shot; however, Nash’s PPS might be considered more impressive in the sense that it is much larger than his EPPS value of 0.94. However, a weighted difference of means t-test of each player’s LSSE distributions resulted in a t-statistic of 1, with an associated *p*-value of 0.32, indicating no support for the hypothesis that Nash’s SSE is larger than Howard’s.

For the league-wide analysis, we calculated PPS, EPPS, and SSE values for the 250 individual players that attempted at least 250 field goals during the 2011–2012 NBA regular season. includes summary statistics for these three key variables. Players’ points per shot distribution (free throws excluded) ranges more widely than EPPS, but has a similar mean. Spatial shooting effectiveness (SSE) ranges over more than half a point per shot, indicating considerable individual variability in scoring efficiency, even while accounting for each player’s spatial constellation of shots. That said, substantial variability in points scored is explained by spatial position: the standard deviation of total points from field goals for this group of players is 232, while the standard deviation of the difference in total points from expected points is just 47.

Table 1 2011–2012 NBA regular season shooting statistics for 250 players.

reports the players with the ten highest and ten lowest EPPS values in the NBA during the 2011–2012 regular season (team and position data from http://www.nbastuffer.com). Players with the highest EPPS values have constellations with the highest field goal percentage, centered in the highest PPS areas in Figure 2A. Players with the lowest EPPS values are taking more shots in the low-percentage areas of that figure; for example, Kobe Bryant is popularly known for taking difficult shots; his position in the table appears to support this. indicates players with the highest and lowest points per shot averages, excluding free throws. This is a non-spatial measure of shooting efficiency.

Table 2 2011–2012 NBA players with highest and lowest EPPS values.

Table 3 2011–2012 NBA players with highest and lowest PPS values (free throws excluded).

Taken individually, and indicate the expected and actual scoring effectiveness of individual players’ shooting constellations. These metrics are combined to identify spatial shooting effectiveness (SSE) as a global measure of relative shooting prowess. shows the ten best and worst SSE values in the NBA during the 2011–2012 season. Steve Novak scored 0.32 points per shot better than the league average from his shooting constellation, while Tony Douglas scored 0.21 points less per shot from his. The narrow range of SSE is noteworthy: just one percent of high-frequency NBA shooters are able to score more than two-tenths of a point more per shot than the NBA average for their constellations, while no player is able to remain high-frequency (>250 FGA) while shooting worse than 0.21 points per shot below the average for their constellations. also prints 95% confidence intervals for weighted paired t-statistics evaluating whether the observed SSE is >0 (for players with positive SSE) or <0 (for players with negative SSE). Perhaps unsurprisingly, the ten highest and lowest SSEs had confidence intervals that did not include zero. Out of the 250 players evaluated, the SSE confidence intervals of 187 spanned zero, 38 players had a positive SSE with 95% confidence intervals above zero, while just 25 players had a negative SSE with confidence intervals entirely below zero. Intervals are relatively wide: those of the top 10 players all overlap, as do those of the bottom 10 players.

Table 4 2011–2012 NBA players with highest and lowest SSE values and accompanying 95% confidence intervals.

Figure 3 illustrates the relationship between EPPS and PPS for all players in the 2011–2012 season with at least 250 field goal attempts. These two components comprise the spatial shooting effectiveness (SSE) metric, and this scatterplot provides some insight into how players end up where they do. A player with an average points per shot and an average shooting constellation will end up in the middle (Jameer Nelson claims that spot on the graph). Players who take shots from higher percentage locations on the floor will be at the top of the graph, while those who hit shots at a higher percentage will be on the right. The SSE gradient runs from right to left on a diagonal. By inspection, the three players with the lowest SSE scores have very different shooting profiles. Lamar Odom’s shot constellation of his EPPS was of average difficulty, but he scored at a low rate on those shots. In contrast, Toney Douglas, the lowest-ranking SSE shooter, had a relatively difficult shot constellation, while Tristan Thompson scored at a higher rate per shot, but his EPPS was relatively high. Similar patterns are evident at the high end of the SSE gradient. Steve Novak had the best SSE value on the strength of a very high points per shot metric over a relatively high EPPS score. Curry and Nash trailed him, as both scored fewer points per shot on much lower EPPS shooting constellations. Tyson Chandler had the highest point per shot score of the season, but because his expected points per shot were quite high, his SSE ranks only ninth.

Figure 3 Points per shot vs expected points per shot, 2011–2012 NBA season. Selected outlying players are labeled. Diagonal lines are SSE contours.

Steve Novak’s SSE was tested against those of each other player using the weighted t-test described in Section 3. Novak’s league-leading SSE was significantly above the SSE of all but 10 players: the 9 below him on the SSE top 10 and the 11th-ranked SSE player, Jason Smith.

An important question for any new metric is whether its information content is largely captured by an existing metric. For SSE, a natural comparison is with effective field goal percentage. The correlation between these two metrics in this study is 0.80: substantial but not perfectly linear; their relationship is depicted in Figure 4. EFG by itself is not able to distinguish between players like Nikola Pekovic and Steve Nash, who have similar EFG percentages (58.1% vs. 56.4%). However, their spatial shooting effectiveness is quite different, with Nash being one of the best in the NBA and Pekovic being a below-average shooter for his constellation (0.23 vs. –0.01). A comparison of their SSE distributions with the weighted t-test revealed that Nash’s SSE was significantly greater (*p*-Value 0.002).

Figure 4 Spatial shooting effectiveness vs. effective FG percentage, 2011–2012 NBA season. Selected outlying players are labeled.

Figure 5 shows maps of POLA for two players, Kevin Garnett and Greg Monroe. The players are superficially similar shooters using nonspatial metrics: Garnett scored 785 points from the field with a 50.5 EFG%, and he averaged 1.01 points per shot. Monroe scored 814 points with a 52.1 EFG% and 1.04 PPS. From the maps though, it is clear that they have different constellations and markedly different success relative to the NBA average from those constellations. Garnett’s points above league average was 101.2 points, while Monroe’s was –48.7. A weighted t-test of the difference of these player’s POLA scores resulted in a t-value of 10.5 and a near-zero p-Value: Garnett scored significantly more effectively from his constellation than Monroe did from his. This map reveals more about the source of Greg Monroe’s shooting difficulties: his local POLA in the cell directly under the basket was –48. Monroe was effectively an average shooter from the rest of his constellation, but at the basket he hit 85 of 148 shots (57%) at 1.15 PPS. The EB field goal percentage for that cell is 74%, with an EPPS of 1.47. Monroe apparently had trouble scoring effectively right at the basket in comparison to the rest of the league.

Figure 5 Maps of points above league average (POLA): (A) Kevin Garnett; (B) Greg Monroe, 2011–2012 NBA season. Square size indicates number of shots taken from that location, while hue indicates POLA value for that spot. Garnett’s summed POLA value was 101.2 points, while Monroe’s was –48.7.

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