Advice on making the most of basketball three-point shot data

Authors: George Terhanian1

¹George Terhanian founded Electric Insights after holding executive positions at The NPD Group, Toluna, and Harris Interactive. He has also served on boards or advisory groups for several organizations, including the US National Academy of Sciences, the Advertising Research Foundation, and the British Polling Society. He is known for conceiving how to make survey data, including pre-election forecasts, more accurate through statistical matching methods.

Making the most of basketball three-point shot data

ABSTRACT

This study’s primary goal is to help National Basketball Association (NBA) and other basketball teams worldwide increase their three-point shooting accuracy and decrease their opponents’, a key to winning more games.  A related goal is to explain how a combination of good data, logistic regression analysis, likely effects reporting in probabilities or percentage points, and self-serve simulation can improve communication among data analysts, basketball coaches, and players, and enhance each group’s effectiveness.  Logistic regression analysis of 32,511 NBA three-point shots shows six factors affect the three-point shooting percentage: closest defender’s distance to the shooter, time left on the 24-second shot clock, whether the player shot after dribbling or catching the ball, game period, shot distance, and venue.  In the past, data analysts conveyed the results of such analyses to coaches and players using terms such as regression, logits, and odds.  Some NBA executives say doing so again would be disastrous.  An alternative is to emphasize probabilities and percentages in communication and create self-serve simulators coaches and players can use to predict how changes in critical factors affect three-point shooting percentages.  NBA and other teams worldwide can apply this approach to new and existing datasets they maintain, enhance, and build.

Key Words: self-serve simulation, predicted probabilities, logistic regression, likely effects reporting, psychotherapy

INTRODUCTION

The National Basketball Association (NBA) releases specific three-point shot characteristics, such as shooter name and shot distance.  Aside from the 2014-15 season’s first 903 of 1,230 games (and 2015-16’s first 631, though the latter data are no longer publicly available), the released data exclude a variety of individual shot characteristics such as the closest defender’s distance to the shooter, a crucial defensive effectiveness measure (14).  Teams are said to consider the excluded characteristics proprietary.  As Mike Zarren, assistant general manager and chief legal counsel for the NBA’s Boston Celtics, explained, “You can’t share stuff with other teams…We are not at an equilibrium point where all the teams know what everyone else is doing.  There are some advantages that some teams have over others” (15) (51:47). 

The analyses here use the 2014-15 shot dataset, the last and largest single-season one containing full shot data that is publicly available.  The main goal is to help NBA and other basketball teams worldwide increase their three-point shooting accuracy and decrease their opponents’.  Teams that do so should win more games.  A related goal is to explain how a combination of good data, logistic regression analysis, likely effects reporting in probabilities or percentage points (e.g., “Shooting off the catch rather than the dribble is associated with a two-percentage-point increase in our three-point shot make percentage.”), and self-serve simulation can improve communication among data analysts, basketball coaches, and players, and enhance each group’s effectiveness.  NBA and other teams worldwide can apply this approach to new and existing datasets they maintain, enhance, and build.  Aspects of the approach are also portable to many other issues and areas where the key outcome variable is binary (26).

This paper has seven additional sections (excluding references and other ancillary information).  The first summarizes basic rules and strategies for NBA basketball, highlighting the importance of the three-point shot.  It also explains why data analysts seeking to communicate effectively with coaches and players should consider using non-technical language.  The second section describes the three-point shot data used in this paper’s analyses.  It then provides the rationale for relying on logistic regression analysis for model building and prediction.  The third section reports the results of the analyses and suggests how data analysts might share them with coaches and players.  It also explores why academic researchers tend not to report likely effects in probabilities or percentage points.  The fourth details how data analysts can build self-serve simulators that report likely effects in probabilities or percentage points.  The limitations of this paper’s analyses are discussed in the fifth section.  The next-to-last section describes how teams might apply the approach described here, while the final section provides concluding remarks.

NBA Basketball: Basic Rules and Strategies

NBA games have two teams with five players competing for four 12-minute periods (excluding possible five-minute overtime periods).  To score, a team needs to shoot the ball through the basket.  With the clock running, a successful shot is worth three or two points, depending on the shooter’s distance from the basket.  The clock stops for free throws, which are uncontested 15-foot shots worth a single point awarded for specific infringements.  One can calculate each shot’s expected value (EV) by multiplying its potential value by its average make percentage.  For the 2022-23 regular season, the expected value of a three- and two-point shot was almost identical: 1.08 points (3*.36) for a three-pointer and 1.10 (2*.55) for a two-pointer.  Each free throw’s expected value was .78 points (1*.78) or 1.56 for a more typical pair (3).  A recent example shows why the expected value measure can be strategically important.

In the second round of the 2020-21 playoffs, the Atlanta Hawks shocked the heavily favored Philadelphia 76ers, coming from behind to win the seven-game series four to three.  The Hawks’ decision to foul Ben Simmons repeatedly to force him to shoot free throws contributed to the victory.  As Hall-of-Fame player Earvin “Magic” Johnson observed, “…it fueled the Hawks’ comeback” (13).

Simmons shot just 33% (15 for 45) from the free-throw line for the series, far below his 61% (and the league’s 77%) regular season average.  Simmons’s 33% figure suggests the Hawks expected him to score only .66 points for two free throws in a series in which his team made 40% of its three-pointers (for an expected value of 1.22 points) and 52% of its two-pointers (for a 1.05 expected value).  That means the Hawks expected to gain .56 points (1.22 – .66) for a replaced three-point shot and .39 points (1.05 – .66) for a replaced two-pointer with the foul Simmons strategy.  Perhaps more notably, it may have affected Simmons’s decision-making.  To his team’s detriment, Simmons chose not to attempt an open lay-up or dunk with 3:30 remaining in game seven (4), arguably for fear of getting fouled and having to shoot free throws (21, 27).

Overstating three-point shooting’s significance is difficult.  In 2022-23, the Toronto Raptors, Charlotte Hornets, and Houston Rockets won 41, 27, and 21 (of 82) regular season games, too few to qualify for the post-season playoffs; their three-point shooting percentages of 34%, 33%, and 33% were the league’s worst.  The Philadelphia 76ers, Golden State Warriors, and Los Angeles Clippers won 54, 44, and 44 games, enough to compete in the playoffs; they were top performers in three-point shooting at 39%, 39%, and 38%.  These data and separate multi-season analyses (18, 20) suggest that winning in the NBA hinges heavily on making (and defending) three-point shots. 

Clear Communication 

An excellent statistical model is “a simplified version of reality, like a street map that shows you how to travel from one part of a city to another” (28) (p. ix).  But that map will not help you find your way if it includes esoteric terms or unfamiliar signs or symbols.  Likewise, if data analysts use uncommon language when giving advice, coaches and players may feel lost.  Mike Zarren would agree.  If Celtics’ data analysts were to apply logistic regression to three-point shot data, he would tell them to communicate what they learn “without using the word regression because that’s a disaster” (15) (11:18).  Terms like logits, standard deviations, odds, odds ratios, and z scores also would be off-limits.  Zarren does not believe coaches and players are unintelligent.  Even good data analysts can find aspects of logistic regression challenging.  That is why DeMaris (7) (p. 1,057) observed, “…there is still considerable confusion about the interpretation of logistic regression results.”  And why Gelman and Hill (11) (p. 83) commented, “…the concept of odds can be difficult to understand, and odds ratios are even more obscure.”

Washington Wizards’ assistant coach Dean Oliver’s views on clear communication resemble Zarren’s.  “When I directed quantitative analysis for the Denver Nuggets and would prepare stuff for coaches,” he said, “there were actually very few numbers in there.  It was usually words because it was easier for them to absorb…” (15) (48:54). 

An alternative to avoiding numbers is to report key predictor variables’ likely effects with familiar ones like probabilities and percentages—the NBA reports various descriptive statistics and cross-tabulations on its website, emphasizing percentages, hence coaches’ and players’ familiarity. 

Methods

Data

The NBA has used technology to gather detailed player performance data since the 2013-14 season via SportVU, then Second Spectrum.  The analyses here use SportVU data, described as “real-time and innovative statistics based on speed, distance, player separation, and ball possession for comprehensive analysis of players and teams” (25).  How did the SportVU system work?  In each arena’s rafters, six cameras recorded information throughout each game in .04-second intervals, producing 25 images per second.  A computer algorithm then plotted the locations of the ball, basket, and 10 players.  SportVU delivered data and reports to each team and the league as a last step.

As noted earlier, the NBA made available SportVU raw, shot-level data—including the defender distance variable—for three-quarters of the 2014-15 regular season.  (The NBA also made available raw, shot-level data early in the 2015-16 season before discontinuing the practice entirely in January 2016.  The latter dataset is no longer publicly available.)  The 2014-15 dataset (17)—the last and largest single-season one publicly available—contains 21 variables and 128,069 three- and two-point shots, as described in the Appendix.  After making minor changes (e.g., removing two-point shots), the remaining three-point shots totaled 32,511—11,426 makes and 21,085 misses—taken from October 28, 2014, through March 4, 2015.

Analysis Method 

Logistic regression models the relationship between a binary outcome (e.g., made or missed three-point shots, or nearly anything with a yes or no interpretation) and, typically, several predictor or explanatory variables.  It is ideal for identifying and estimating the effects of actions to increase or decrease the size or proportion of the group of interest, specifically, made three-point shots.  It can also predict each three-point shot’s probability of belonging to the “made” rather than the “missed” group.  Many academic researchers consider it “the standard way to model binary outcomes” (11) (p. 79), “dominating all other methods in both the social and biomedical sciences” (2) (para. 1).

RESULTS
The final logistic regression model comprises one dependent and six predictor variables.  The predictor variables were selected based on their relationship with the dependent variable, one another, theory, availability, and their effect on the model’s predictive accuracy.  Below are descriptions of the seven variables and brief explanations for how they may differ from the original ones described in the Appendix.

1. ShotResult: The dependent variable: whether the shooter made the shot. (Values: 0=Missed, 1=Made; Original variable: Fgm)
2. DefDist: The closest defender’s distance to the shooter in feet (ft.). Basketball players and coaches recommended a four-category variable after discussions and preliminary analyses. (Values: 1=0-3 ft., 2=3-6 ft., 3=6-9 ft., 4=9+ ft.; Original variable: Close_Def_Dist)
3. ShotClock: The number of seconds (secs.) on the 24-second shot clock. Analyses showed steep drops in the make probability at the 4- and 2-second marks, thus the decision to create a variable with three categories. (Values: 1=0-2 secs., 2=2-4 secs., 3=4+ secs; Original variable: Shot_Clock)
4. Catch: Whether the shooter took the shot off the catch or dribble. The original variable reported the number of dribbles the shooter took before shooting. Basketball players and coaches recommended a two-category variable after discussions and preliminary analyses. (Values: 1=Off Catch, 2=Off Dribble; Original variable: Dribbles)
5. Period: The game period when the shot was taken, with fourth period and overtime shots pooled because of their similar make percentages. (Values: 1=1, 2=2, 3=3, 4=4+; Original variable: Period)
6. ShotDist: The distance in feet from the center of the basket to the shooter. Basketball players and coaches recommended a four-category variable after discussions and preliminary analyses. (Values: 1=22-24 ft., 2=24-25 ft., 3=25-26 ft., 4=26+ ft.; Original variable: Shot_Dist)
7. Venue: Whether it was a home or away game for the shooter’s team. (Values: 0=Away, 1=Home; Original variable: Location)

Table 1 reports the logistic regression analysis results, notably, standard information such as logit coefficients, odds, z scores, and a measure of statistical significance (i.e., p>z).  It also reports useful non-standard information such as frequencies, (predicted) probabilities, and expected values.  The rationale for reporting standard and non-standard information, to borrow from the statistician Frederick Mosteller, is to “let weaknesses from one…be buttressed by strength from another” (16) (Ch. 4, p. 116), a concept he referred to as “balancing biases.”  As envisioned, data analysts can rely on standard information when building and evaluating logistic regression models, and non-standard when communicating the results and their implications to coaches and players.

Note. n=32,511.  Log pseudolikelihood, starting value: -21,078.18; final value: -20,827.69.  Likelihood ratio (degrees of freedom=13): 498.44, p > chi² = 0.00. Tjur R²: 0.014; McFadden R²: 0.012.  Stukel chi²(1) = 4.10, p > chi² = 0.043

Standard versus Non-Standard Interpretations

Table 1 shows that the defender distance variable (DefDist) affects the outcome variable.  A standard interpretation would emphasize odds ratios and statistical significance:

Controlling for other variables’ effects, three-point shots taken with the closest defender 9+ feet away have a:

• 60% higher odds (i.e., 1.6/1) of going in than those taken with the closest defender 0-3 feet away,
• 24% higher odds (i.e., 1.6/1.29) than those with the defender 3-6 feet away, and
• 10% higher odds (i.e., 1.6/1.46) than those with the defender 6-9 feet away.

Each effect is statistically significant, as their z scores show.

Although the standard interpretation is correct from a technical standpoint, coaches and players may not understand or act on it, given Zarren’s and Oliver’s comments (as well as those of DeMaris, Gelman, and Hill).  Now consider a non-standard interpretation (that relies on Table 1’s non-standard information).  Note that each percentage’s associated expected value is in parentheses.

All else unchanged, the percentage of three-point makes would decrease from 35% (1.05 pts.) to:

• 29% (0.86 pts.) with the defender always 0-3 feet away from the shooter, and
• 34% (1.02 pts.) with the defender always 3-6 feet away.

It would increase from 35% to:

• 37% (1.11 pts.) with the defender always 6-9 feet away, and
• 39% (1.17 pts.) with the defender always 9+ feet away.

NBA coaches and players would probably prefer the non-standard interpretation.  Arguably, reporting the likely effect in percentage points instead of odds is more intuitive and actionable (26, 30). 

Calculating Each Shot’s Make Probability

Another number to note in Table 1 is the constant of -1.46 logits which translates to a predicted make probability of 19% (0.56 pts.).  The -1.46 number represents a three-point shot with the lowest value on each predictor variable:

• Defender 0-3 feet away
• 0-2 seconds on the shot clock
• Off the catch
• First period
• Shot distance of 22-24 feet
• Away game

An implication is that it is possible to calculate the predicted make probability of each of the 32,511 shots.  Such information can spark curiosity and foster improved performance for a player scrutinizing his own (or opponents’) shot data.  For example, Row 1 of Table 2 reports the logit coefficients associated with the first three-point shot Klay Thompson of the Golden State Warriors attempted in 2014-15.  In the third period of an away game versus the Sacramento Kings with 4.6 seconds on the shot clock, Thompson missed from 22 feet off the catch with the defender 3.9 feet away.  As the column titled Prob shows, that shot’s predicted make probability was 38% (.38*100), calculated by applying the following formula to select Table 2 numbers: exp (sum of logit coefficients + constant)/ (exp (sum of logit coefficients + constant) +1).

Upon closer examination, Thompson could have asked the team’s data analysts how that shot’s make probability would have changed had the defender been 9+ rather than 3.9 feet away.  To respond, an analyst could have replaced the DefDist logit coefficient of 0.25 with 0.47, the one corresponding to a 9+ feet value.  As shown in Row 2, the make probability would have risen to 42%, a four-percentage-point increase or likely effect. 

Thompson next might have asked how shooting off the dribble rather than the catch would have affected the 42% probability.  After replacing the Catch logit coefficient of 0 with-0.09, an analyst could have reported that the probability would have dropped to 39%, as Row 3 of the Prob column shows. 

Thompson, an excellent shooter, would probably work to improve specific aspects of his shooting if he had such data for all his three-point shots (31).

Row	DefDist	ShotClock	Catch	Period	ShotDist	Venue	Cost	Total	Prob
1	0.25	0.77	0	-0.05	0	0	-1.46	-0.49	0.38
2	0.47	0.77	0	-0.05	0	0	-1.46	-0.27	0.42
3	0.47	0.77	-0.09	-0.05	0	0	-1.46	-0.36	0.39

Predicting the Likely Effect of Multiple Changes to Multiple Predictor Variables

Coaches thinking more broadly might focus on all 32,511 shots and ask analysts to predict the likely effect of multiple changes to the values of multiple predictor variables. Building on the Thompson example, analysts could approach the task by conceptualizing changes as scenarios.  Below, and graphically in Figure 1, are three illustrative ones.

Scenario 1. Players take all 32,511 three-point shots with the defender 9+ ft. away.  

Prediction: 39% of all three-pointers will go in, an increase of four percentage points compared to the 35% baseline, translating to 1,297 more makes and 12,723 total ones.

Scenario 2. Players take all 32,511 three-point shots:

• with the defender 9+ feet away 
• from 22-24 ft. away from the basket

Prediction: 42% of all shots will go in, a three-percentage-point gain vs. Scenario 1.  This translates to 808 more makes and 13,531 total makes.

Scenario 3. Players take all 32,511 three-point shots:

• with the defender 9+ ft. away 
• from 22-24 ft. away from the basket
• with 4+ seconds on the 24-second shot clock

Prediction: 43% of all shots will go in, an increase of another percentage point compared to Scenario 2, translating to 370 more makes and 13,901 total ones.

Each scenario’s likely effect results from all-or-nothing simulation.  How does it work?  For any predictor variable, such as Catch, data analysts select one target value—either “Off Catch” (occurring 75% of the time) or “Off Dribble” (25%).  Assume they choose “Off Catch,” with a logit coefficient of 0, as Table 1 shows.  For the 8,127 “Off Dribble” shots, they would replace the coefficient of -0.09, also shown in Table 1, with 0 and calculate the new likely effect: 158 more made three-pointers for the season, translating to 11,584 total makes. 

Adopting a fine-tuning approach is another possibility.  After examining the frequency distribution of the Catch values, analysts could specify a new distribution, such as 92% “Off Catch” and 8% “Off Dribble,” ensuring the total sums to 100%.  They would keep the original 24,384 “Off Catch” values (i.e., 75%) and change the -0.09 coefficient to 0 for another 2,600 selected randomly from the original 8,127 “Off Dribble” values to achieve the 92:8 ratio.  The change would result in 11,530 made three-pointers, 54 less (i.e., 11,584-11,530) than if players had taken all shots off the catch.

If coaches and players embrace simulation, there could be too many scenarios for data analysts to handle.  To stay ahead of demand, they could build self-serve simulators tailored explicitly for coaches’ and players’ use.  Finding prototypes in academic research will be a struggle, however, arguably because of the non-linear relationship between logits and probabilities (26, 30) and its dampening effect on reporting likely effects in probabilities or percentage points.  Figure 2 plots illustrative logit and probability values to cast light on that relationship.

Note how a one-logit increase from zero to one on the x-axis corresponds to a .23 probability increase (from .5 to .73) on the y-axis.  Yet a one-logit increase from four to five (or minus 5 to minus 4) translates only to a tiny probability increase.  As shown in Table 1 (and later in Table 3), it is still possible to report the effect of a predictor variable, x, on a binary outcome, y, in probabilities or percentage points (e.g., a one-unit change in x is associated with a three-percentage-point increase in y, all else being equal).  Arguably, it is also sensible to do so, not least because NBA players make roughly 35% of their three-point shots and the relationship between logits and probabilities is reasonably linear between .2 and .8 on the probability scale, as Figure 2 shows.  But in more extreme cases, as Figure 2 suggests, the effect size will depend heavily on the value of y and the values of the model’s other predictor variables.  More precisely, the size of the effect will decrease near 0 and 1.  As a result, x’s effect on y in probabilities percentage points “…cannot be fully represented by a single number” (19) (p. 23).  That may be why some logistic regression experts (6-8) have advised against using probabilities or percentage points to report and interpret logistic regression coefficients’ overall effects.  It also may be why most major statistical software packages do not produce effects in probabilities or percentage points through pre-packaged procedures or built-in modules.  As an unintended consequence, some data analysts seeking guidance likely have had to fend for themselves.           

A GUIDE TO BUILDING SELF-SERVE SIMULATORS
Data analysts can use this guide to build simulators that report likely effects in probabilities or percentage points.  (For convenience, references are made to the three-point shot data used in this paper’s analyses, although the guide is general and should work across areas of interest.)  Several steps are involved in the process:

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