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.

Reviewing: Berri D. 1999. Who is ‘Most Valuable’? Measuring the Player’s Production of Wins in the National Basketball Association. Manage. Decis. Econ. 20:411-427.

Logically, there are many people who will need to understand what players are more productive than others in order to determine who should play, what players should be acquired, or what trades should be consummated. In this study, Berri attempts to draw a correlation between a player’s statistics and team wins. One award, the IBM award, essentially is the summation of an individual’s positive statistics less his negative statistics. Unaccounted for in this method s that various statistics carry different values (for example, a steal may be more valuable than a rebound).

Berri makes the argument that a team’s scoring is affected most by how the team acquires the ball, its ball handling efficiency, and the likelihood of converting possessions into points. Give this theory, the following team and player statistics were studied:
points-per-shot
opponents’ points-per-shot
free throw percentage
opponents’ free throw percentage
free throw attempts
offensive rebounds
defensive rebounds
assist-turnover ratio
opponents’ assist-turnover ratio
turnovers
opponents turnovers
field goal attempts
personal fouls

Interesting to note in the above list of statistical categories is that some are team statistics while others are individual players statistics. After mixing these two general groupings, it may be more difficult to determine an individual’s independent contribution to the team.

Another big factor that Berri included was the team’s tempo arguing that statistics will vary based on the tempo. A team playing at a faster clip will likely have more opportunities to increase its box score values. Taking team tempo into account while studying player performance through the above list of statistical categories, Berri decides that assists and personal fouls have limited impact on a player’s contribution to team wins. This was determined by calculating each statistic’s marginal value, which should indicate the impact each has on wins. The six ratios used in the equation to determine marginal values were points-per-shot, free throw percentage, assist-turnover ratio, and then the opponent’s equivalent to the previous three. In the paper, it was unclear how Berri went from these statistics to the marginal impact of three-point field goals made, two-point field goals made, assists, turnovers, free throws made, and free throws missed.

Although most of Berri’s reasoning is logical, some of his results seem counter-intuitive. Most significantly, it is hard to understand how assists and personal fouls can have little impact on an individual’s contribution to a team. First, out of the last three MVP’s in the NBA from the 2004-2005 to the 2006-2007 season, an assist expert, Steve Nash, won the MVP two times taking his team to higher levels in the playoffs than in the previous seasons. Also, an assist is only awarded after a successful basket; therefore assists should correlate highly with points. Berri does not seem to consider causation; for example, in the case of Nash, the fear of the pass may force defenders to play him differently, allowing him an increased opportunity for a jumper or a drive. Second, a player’s personal fouls can dictate how much playing time he gets. For example, receiving two personal fouls in the first quarter generally forces the coach to bench that individual until the second quarter. The case is similar when a player receives 3 personal fouls in the first half. Given these circumstances, a lower personal foul total would have a positive effect on quantity of playing time.

I will use this space (crossposted on www.babyhook.com and numbers.babyhook.com) to share my opinions on topics from sports analytics. Most of what I talk about will surround basketball; however, it will not be limited to purely that one sport.

There are many questions in sports–especially the hard questions—of which I feel we can come closer to answers by logically analyzing the numbers and previous statistics. I hope to address the following plus more: a player’s worth, issues of competitive balance, understanding specific stats, star power, coach and ref biases, and fan power.