the numbers

I tried to pick an easier productivity metric to start, and at least at first glance, the NBA Efficiency metric looks to be that.

NBA.com evaluates all players based on the efficiency formula: ((Points + Rebounds + Assists + Steals + Blocks) - ((Field Goals Att. - Field Goals Made) + (Free Throws Att. - Free Throws Made) + Turnovers)).

However, when I started giving it more thought and reading what others had to say about it, I started to realize that it was much more complicated than I first imagined.  Two points in particular are interesting to discuss here: first is the value applied to scoring opportunities and the second is the value of gaining or keeping possession of the ball.  For example, points, assists, field goals, field goal attempts, free throws, and free throw attempts would all be associated with scoring.  On the other hand, rebounds, steals, blocks and turnovers all relate more to possession of the ball.

This raises the question of how to value scoring and possession relative to each other when trying to maximize a team’s wins.  How much is a rebound worth compared to a field goal?

David Berri speaks of this issue on his blog.  Some people criticize Berri for giving too much value to rebounds, and that the value a rebound receives in a calculation such as the NBA Efficiency Rating is much more accurate.  One of Berri’s arguments against this is that it is important to appropriately account for shooting percentage.  For example, if points are weighted too much relative to possessions (rebounding, steals, blocks, turnovers), then this will give an advantage to prolific shooters rather than “efficient” shooters.

For example, if one rebound is equivalent to one point, as it basically is in the NBA Efficiency metric, then a player will break even shooting 25% from 3-pt range, and 33% from 2-pt range.  If he hits one out of four 3 pointers, he gets 3 points and the opponent can acquire 3 rebounds, which cancel out the points.  Therefore, by taking more shots from any range, the frequent shooter will increase his NBA Efficiency Rating despite not being a great percentage shooter.

I am not saying that I feel that Berri’s analysis is bullet-proof either, but I just wanted to point out some of the interesting issues that I learned while diving into the statistic known as the NBA Efficiency Rating.

I haven’t had a post in a while on the topic of basketball analytics, and I feel that a good way to get back into it would be to go through several player productivity metrics over the next couple posts and attempt to identify their strengths and weaknesses.  Some metrics that I hope to cover include John Hollinger’s Player Efficiency Rating, the NBA Efficiency metric, David Berri’s Wins Produced per Player, the Alternate Win Score, and possibly more.

I don’t necessarily hope to come up with one metric that is better than the others or a new and improved metric.  In such a dynamic sport as basketball, to thoroughly assign a metric to a particular player is a difficult task, and I am hoping to shed some light on what is currently out there.

Here is an interesting chart I found on basketballprospectus.com that depicts both distance of shots per game and the field goal percentage of those shots. It is important to note that shots within 5 feet from the basket have not been included in this chart. This makes sense because those shots are easily the most variable ranging from a difficult off-hand hook shot to an easy dunk or layup.

A link to Ken Pomeroy’s article can be found here or by clicking the image above.

This then leads itself to the argument of how much a team should focus on the 2 or 3 point shot. After all, a 3-pointer is worth 50% more than a 2-pointer, and it seems that the percentage of those shots are higher than shots taken from the three point line all the way in to 5 feet from the basket. However, it is important to realize that many of those 5 foot shots are probably much more contested.

I would argue that the 3 point shot definitely has its benefits and its place; however, although I haven’t worked up any of the data, I feel that the number of positive results that can come from driving to the basket might be higher. For one example, there is a chance of a foul, both with and without hitting the shot. Either way, the offensive player is sent to the free throw line, and the defensive player is one foul closer to leaving the game and subsequently has to play more cautiously.

Game theory would argue that some combination of both 2 and 3 point shots would lead to the most optimal outcome; however, finding this balance is the real challenge. From longer rebounds on a 3 pointer leading to more offensive boards to getting the and-1 from under the basket, both have their time and place. The trick is finding when that is.

After last weekend, when Stanford split against the Arizona schools—loosing to ASU in overtime and winning a close game against Arizona two days later—I heard many remarks around campus and read articles online that discounted the now #9 ranked Cardinal’s ability. Losing a game on the road in the strong Pac-10 conference is far from an immediate sign of weakness.

First, it is important to create some perspective on the situation. It is really easy to forget that college basketball is composed of athletes who are about twenty years old. Although they may receive an intense amount of scrutiny and national attention, these players are still young and figuring out how to balance the game, the attention, and the pressure that comes along with expectations. This is one possible explanation for the outcome against Arizona State. Stanford had just acquired their #7 ranking, after rising to that level from around #20 in about 3 weeks time. I would think that the team felt a whole new level of expectations over their shoulders after those rankings came out.

As of February 19th, John Gasaway from Basketball Prospectus wrote an article that talks about the major conferences in NCAA basketball and quantifies their possessions per 40 minutes, their points per possession, their opponents points per possession, and their efficiency margin. Stanford is far from lacking in the points per possession category at 1.04 points; however, the much more impressive statistic is that they only allow 0.93 points per possession from their opponents. This is by far the best defensive performance in the Pac-10 (UCLA is next highest at 0.97). Also, this ranks in the top couple teams of all the major conferences. It is this statistic that helps explain Stanford’s 21 wins so far this year.

Therefore, when people can’t imagine how Stanford would play against the run and gun offenses of Memphis, UNC or Duke, they wouldn’t—they would slow down those offenses. A showdown between Stanford and Duke would look very different from the game a couple weeks ago between UNC and Duke. Unlike in that game, Stanford wouldn’t be taking too many shots with 25-30 seconds remaining on the 35-second shot clock, and they would try to lock down Duke, not take too many risks on defense, and better control the pace of the game.

Stanford does look different from many of the other teams in the top 10; however, they still belong to be there. Defense can win games just as much as offense can.

On 82games.com, Ed Peterson writes a very interesting article trying to quantify how the 2004-2005 Sacramento Kings do as a team and as individuals with specific play types. Plays range from fouls, offensive rebounds, and driving into the paint, to cutting and pick and rolls. The usefulness of such analysis assuming it is all accurate and appropriately calculated is very high because it would allow coaches to run certain plays for certain players and even run specific offenses or defenses depending on which 5 players were on the court at any given time. Knowing that a player has had more success off a pick and roll jumper over a catch and shoot play would be very applicable, especially considering that it might be easier to get a shot after a pick and roll rather of a set shot.

Unfortunately, quantifying such plays is not easy. Realizing that Peterson’s study is a good start to play type analysis, I hope that it eventually lists more than about 10 plays, out of which several are a combination of several plays. It may be more helpful if a shot off the dribble was categorized differently from a pick and roll jumper. Also, I feel that it would be very interesting to include offensive styles as a category. For example, how does the triangle offense, or the princeton offense work for certain individuals and specific teams? In addition, it may be interesting to study different defenses with respect to plays run by the offense. For example, for different zone defenses, such as a 2-3, 3-2, 1-3-1, etc, how well do individual plays work against them?

One other comment I had regarding Peterson’s study is about the RTG category (a rough “points per 100 possessions” number). For a category such as “Away from ball / intentional fouls”, how is possible to have a RTG number greater than 200. It seems that if a player was fouled away from the ball or not shooting the ball, that player should maximally be able to earn 2 free throws per possession and therefore 200 points in 100 possessions.

Measuring production by play type is very challenging because there are so many variations of plays within a basketball game, excluding how many different plays may occur within one possession. That being said, such analysis could be very helpful to players and coaches at any level.

In the below table (Gilovich et al, 1985) from the 76er’s 1980-1981 season, we can see that weighted means of hitting a shot seems to increase when the player has missed his last shot (or several shots). This is contrary to any belief that the hot hand may exist. However, again because of the binomial model, this will not take into account the idea that after hitting a couple shots, a player’s confidence could increase and he may start taking more challenging shots.

[click table to enlarge]

In the model proposed by Gilovich, Vallone, and Tversky in their 1985 paper, “The Hot Hand in Basketball: On the Misperception of Random Sequences”, they use a binomial shooting model to evaluate the existence of a hot hand. Their conclusion that the hot hand does not exist was based on the non-significant p-values of the null hypothesis (the only player with significant p-values in the chart above is Daryl Dawkins). However, as is documented in other literature, non-significant p values do not confirm that the binomial model itself is accurate. I am not arguing here that the hot hand exists or does not, but I do want to point out that the data presented by Gilovich is not entirely conclusive because the act of shooting is much more complicated than modeled.

reference: Sun, Y., Detecting the Hot Hand: An Alternative Model. cogsci.northwestern.edu.

The debate of whether there exists a hot hand in basketball formally began in academic literature in 1985 with a paper by Gilovich, Vallone, and Tversky. They based their argument on the bionomial model that a given shot resulted in a hit or a miss—similar to a coin flip ending in heads or tails. Using this model, they found no evidence for a positive correlation between the outcomes of successive shots. Good shooters are bound to go on streaks where they hit several in a row and where they miss several in a row. Statistics and probabilities will be much more stable in the long run. However, this doesn’t prevent people from thinking that if you flip a coin only 4 times, 2 will end in heads and 2 in tails. Realistically, such a small localized sample is incredibly difficult to predict, and if their conclusion is correct, many players, coaches, fans and bettors are not making the best decision in scenarios where a player apparently has a “hot hand.” Coaches will keep players in who are hitting more shots and other teammates are more likely to pass to the individual who has just hit 4 in a row.

To contribute more numbers from Gilovich’s paper, after a group was told of a hypothetical 50% shooter, those sampled believed he would hit 61% after just making one, while he would hit only 42% after just missing one. According to Gilovich, the next shot should be another random independent event and the likelihood of hitting that shot should be 50% irregardless of the last one. They go on to discuss the probability of a hit shot based on a player’s past of hits and misses, as well as the stability of players’ percentages across games. In both of these studies, they found that there is no statistically significant impact of a hot hand in basketball.

Supporting their use of a binomial model in this situation, Gilovich, Vallone and Tversky claim that such a model is equivalent to a more complicated process that takes into account shot difficulty. They claim that this is acceptable because each shot is randomly chosen from a group of shots ranging in difficulty from a dunk to a turn-around, fade-away 3-pointer. This is one part of their argument where I am still not convinced, and I feel that much of their conclusions rest on this fact—that a basketball shot can be modeled binomially.

I have begun reading several papers on the idea of a “hot hand” in basketball. Most of what I have read suggests that such a notion does not exist and it is a bad decision for coaches or players to make any game time choices based on this idea. However, being an amateur player, and having my shot being one of my strongest assets in any pick-up game, I feel that if I have made several jump shots in a row, the probability that I will hit the next one is higher than my average shooting percentage. In other words, I feel I am more likely to hit a shot after hitting several right before. The cause may be that I am more confident resulting in better shooting form, I am more locked-in on the rim, or I am just more focused.

Unfortunately, the above reasons I listed for feeling that I am more likely to hit my next shot are very difficult to quantify. In the next couple posts, I am going to try to look more into this issue and see if I can come to some resolution.

Berri D, Brook SL, Frick B, Fenn AJ, and Vicente-Mayoral R. 2005. The Short Supply of Tall People: Competitive Imbalance and the National Basketball Association. Journal of Economic Issues 39(4):1029-1041.

Berri’s argument is summarized first:

[Used several times through this article is the Noll-Scully competitive balance measure. This value compares the actual performance of a league to the performance that one would expect if the league were maximally competitive. Therefore, the smaller the value, the less the deviation of the actual vs. ideal performance and the league is more competitive. The ideal performance of a league is calculated by dividing the mean winning percentage by the square of the total regular season games played.]

Using this measure, Berri et. al. continue to evaluate the competitive balance of many sports leagues. The most competitive sport is soccer, followed by football, and then hockey. Usually, both baseball and basketball have provided a standard deviation of winning percentage that is more than twice the ideal. Also important to note is that there is relative consistency of the Noll-Scully competitive balance measure within leagues from a single sport. Such a result leads one to believe that competitive balance is dependent on the sport being performed.

The article goes on to provide an evolutionary biology cause for this variation in competitive balance between sports. When there is more variation of talent within a league, more imbalance will occur; however, as more and more players reach their biomechanical limits of performance, the competitiveness of the league should also increase. This last fact depends on the elite athletes having similar biomechanical limits. As a result, it becomes more understandable why soccer is much more competitive than basketball. In basketball, so much of one’s biomechanical limit is a function of their height, while people of most heights can play soccer.

In the 2003-2004 season, 30 percent of players in the NBA were 6 feet 10 inches or taller. On the other hand, 97.9 percent of young adult males are six feet three inches or smaller. The height requirements in the NBA reduce the number of available players because no amount of work will make an individual taller. To strengthen the claim that height is a major factor leading to the competitive imbalance in the NBA, one can find that performance of frontcourt players is more varied than that of backcourt players. Frontcourt players rely more on their height and thus there is a shorter supply of them.

Differing thoughts:

There are some things that need to be considered before arriving at a firm conclusion whether Berri is correct in his analysis or not.

Phil Birnbaum makes a couple very interesting points with respect to this issue. He says that if height is just considered another skill like passing well or shooting straight, then similar to most skills possessed by professional athletes, those athletes will exhibit skill levels far to the right of that distribution. He also points out that not only do many teams have players who are close to only 6-feet tall, but because tall people are noticeable in a crowd, most if not all teenagers at or about 6’5” will be encouraged to try their hand at basketball. This is not the case for the rest of the population.

Beyond these points, it is important to consider the amount of luck that is incorporated into any sport. For example, if a team or individual can win by “getting lucky”, the competitive balance in that league will probably be higher. When a soccer game is tied 0-0 or 1-1 going into the last ten minutes of the match, either side can get lucky. However, when the forth quarter of a basketball game rolls around and one team is already up by 20 pts, it might take a miracle for the other team to come back. This goes back to why several sports involve playoffs of teams playing a best of 5 or 7 series to determine who is best. This provides a smaller chance for the “lucky” team to win. March Madness is in fact madness because one 40 minute competition could favor a confident team with a shooting night that is a standard deviation above their mean.

As it stands, I am not convinced that basketball or the NBA has a competitive imbalance because of a short supply of tall people. Height may play a role in the league’s competitiveness, but it seems difficult to believe that it is the dominant reason.

Reviewing: Beech, Roland. “NBA ‘Points in the Paint’”. 82games.com. 18 June 2007. <http://82games.com/pointsinpaint.htm>.

In this article, Roland Beech attempts to evaluate the importance and validity of the “points in the paint” statistic in basketball. He begins by pointing out the following three errors in this stat: (1) free throws are not accounted for, (2) turnovers and shooting percentage are ignored, and (3) there is no efficiency measure for points in the paint activity. I would also like to add a forth piece that the statistic does not include, and this is the difficulty of the shot and/or execution of the offense. A lay-up over a 6-foot guard is a much easier shot than a hook over Yao Ming.

However, this being said and noting that the “points in the paint” statistic is flawed in many ways, I feel that it still provides a rough calculation of a team’s effectiveness. I understand Beech’s explanations that defensive points in the paint have a much larger impact than offensive points in the paint; in other words, preventing your opponent from scoring in the paint is more valued that you scoring there on offense. Also, if you compare the performance of the top fifteen teams in net points in the paint versus the bottom fifteen, the top fifteen only have 3-4% more wins from the 05-07 seasons than the bottom fifteen. Therefore, only nominally scoring more points in the paint is not extremely valuable over the course of a season.

Throughout the paper, Beech is trying to determine if winning the “points in the paint” statistic wins games. However, there arises a question of whether this is an instance of causation or correlation. For example, if you look at teams’ records during the 06-07 season when they outscore their opponent by 11+ points in the paint, only 6 out of 30 teams have losing records in that category. And more importantly, the combined win-loss record of all teams during the same season when outscoring their opponent by 11+ points in the paint was 304-141. From this analysis it is clear that scoring more points in the paint and winning games is correlated; however, it is unclear that scoring more points in the paint causes victories. Having a good game or playing a weaker opponent could indirectly lead to more points in the paint.

Therefore, although it is something to be aware of, if I was coaching a team, I wouldn’t make a focus of a game to score in the paint. Instead, I would focus on running a sound offense and getting the best shot opportunities possible. As a result, the team would probably win the points in the paint differential anyway.