Post
by **steveshea** » Thu Nov 07, 2013 9:31 pm

Below are several excerpts from Chapter 7 of our book. That chapter is on Lineup Entropy. It doesn't present the "sexy" metrics such as those approximating an individual player's overall value. (We do that earlier in the book with our metrics: Efficient Offensive Production, Defensive Stops Gained, and Approximate Value) Instead, it presents a tool for analysis (that may only appeal to the true stat nerds). I hope you enjoy.

7

Lineup Entropy

Miles plays basketball for the Predictables. In the first half of each game this season, the Predictables played Miles, Jimmy, Jason, Jacob, and Jordan. In the second half, the team played Michael, Marcus, Marvin, Mario and Mo. In the same season, Gavin played for the Unpredictables. At every stoppage of play, the coach of the Unpredictables randomly selected five players from the team of ten to play. If you were told at some time in the season, Miles was on the court, how difficult would it be to predict his teammates? Now, consider the same question for Gavin.

Teams in the NBA live between the predictable and unpredictable extremes described above. Exactly where each team stands on the spectrum of possibilities is important for our statistical analysis. Recall our classification of top down and bottom up metrics. A top down metric is one that measures a player’s contributions by looking at the contributions of entire units on the court. For example, a player’s plus-minus is the points scored by the player’s team while he is on the court minus the points scored by the opponent in the same time frame. Top down metrics, and in particular plus-minus, have a major weakness. They are biased towards players that play with good teammates.

...(omitting discussions, charts and examples regarding plus-minus, plus-minus per 48, and net plus-minus per 48)...

Statistically, the Unpredictables’ mixing of lineups is ideal for trying to distinguish the contributions of the individual from that of his teammates. The Predictables’ strategy is the worst possible statistically. While we do not see either extreme reached in the NBA, there are situations where players play a significant amount of time together. For example, Mario Chalmers played 1922 of his 2068 minutes with LeBron. In other words, Chalmers played 93% of his time with LeBron. Our metrics (and everyone else’s) say LeBron is pretty good. There is an obvious opportunity for Chalmers’ plus-minus numbers and net plus-minus numbers to be inflated. This is why Chalmers is sixth in net plus-minus when other Miami players such as Bosh and Battier drop out of the top 20.

If we want to consider top down metrics, we need to pay attention to the amount of time that players play with each of their teammates. For a specific player, we can look at their proportion of time with each teammate. However, to do this for all players would be tedious at best. It also does not give us a method to quickly identify players that might be particularly prone to team pull biases. This chapter is about taking all of the percentages of a player’s playing time with each teammate and combining them into one number that well represents the degree to which that player mixes with different teammates. This will be called Lineup Entropy. Here, the more teammates the player plays with and the more evenly distributed the time with those teammates, the better.

To be clear, our new Lineup Entropy is not the sexy new metric everyone likes to read about. In this Chapter, we will not be producing top 20 lists with Jordan, Magic and LeBron. Instead, Lineup Entropy is about laying a proper foundation for and assigning confidence to top down metrics. It is about better understanding where stats such as adjusted plus-minus or net plus-minus might be misleading. Let us give an analogy that Boston sports fans will appreciate. Like stapling Curt Schilling's ankle before the famed bloody sock game in the 2004 ALCS, Lineup Entropy is ugly and essential.

What is Shannon’s Entropy?

To quantify the diversity in lineups of an individual, we will use Shannon’s entropy from information theory. Our use of entropy here is motivated by its application as the measure of diversity in other settings. It has been used by entomologists (bug scientists) to study the variety of insects in a swamp and economists to study concentration in industries . We now explain Shannon’s entropy formula and how it can be used for these types of applications.

...(omitting explanation of, motivation for, and formal definitions of Shannon's Entropy and Lineup Entropy (LE))...

Our intuition told us that plus-minus per 48 was overrating players like Chalmers, Perkins, Sefolosha, and Haslem because these players played for excellent teams. These players also had very low LE. As a point of reference, Chalmers’ LE of 2.97 is in the 5th percentile of our data. In other words, 95% of players mix at a higher rate than Chalmers did in 2012-13. We knew these players played for good teams. Low LE tells us that these players are playing almost all of their minutes with a select few players on that good team.

...

Team Lineup Entropy

We can also define Lineup Entropy for the entire team. The team number allows an efficient and objective means of monitoring a team’s situation. Allow us to give a somewhat depressing analogy. If an individual is not feeling well, he or a doctor might take his temperature. The temperature alone cannot determine the cause of the illness. However, it can be a means of classification, an indication that something is amiss. In a more serious scenario, a doctor might monitor a patient’s respiratory or heart rate. The rates themselves do not give an answer, but abnormal numbers again can indicate malfunction of some sort. Team Lineup Entropy (TLE) is a measure of the degree of mixing of linemates for the entire team. Teams will have different baseline TLEs as TLE can be a reflection of coaching style and the make-up of the team. For example, the Spurs rely on second string units often as coach Gregg Popovich is excellent at resting his aging core of Duncan, Ginobili and Parker. Like monitoring a patient’s heart rate, when TLE hits the extremes or varies a great deal from the team’s baseline, it reflects a serious shift in the team and suggests that something has gone wrong. One possible cause is an injury to a star player.

...