## GLOSSARY

PTC

Player Total Contribution (PTC) is an index to evaluate the performance (production) of basketball players which is based only on box-score data, and it has been validated using several procedures (Martínez, 2019).

PTC = 1 PTS + 0.91 BLK + 0.58 DRB + 0.92 ORB + 0.86 STL + 0.48 AST + 0.23 FD  – 0.91 MFG – 0.57 MFT – 0.86 TOV – 0.23 PF

Where: PTS: points made; BLK: blocks made; DRB: defensive rebounds; ORB: offensive rebounds; STL: steals; AST: assists; FD: fouls drawn. MFG: missed field goals; MFT: missed free throws; TOV: turnovers; PF: personal fouls made.

PTC can be easily computed by game (PTC/G) or by minutes played (PTC/MP), just dividing PTC by games or minutes, respectively.

ePTC/MP

Estimated Player Total Contribution (per minute) when fouls drawn are unknown (Martínez, 2019).

Fouls drawn are an important variable for determining indexes measuring global performance of basketball players. Draw fouls means in some cases the opportunity to go to the free throw line, and in every case the threat of expulsion (by accumulating fouls) for the player who make the foul. Therefore, fouls drawn contribute to the evaluation of performance of basketball players, although the weight of such variable is different depending on the system of evaluation considered. For example, the Spanish ACB League system of evaluation weights fouls drawn with 1 point (equivalent to 1 point scored), and the Player Total Contribution (PTC) index, weights fouls drawn as 0.23 (with respect to 1 point scored).

However, if we try to achieve comparison of historical performance of basketball players, we face the problem of lack of data of fouls drawn in previous seasons. For example, in the official NBA stats site there is no reliable data of fouls drawn previous to the 2005/2006 season. For the ACB Spanish League stats site, there is no reliable data of fouls drawn previous to the 1990/91 season.

Using this simple equation we may get ePTC/MP:

ePTC/MP = 1.0512 PTC/MP

PTCpred

Predicted PTC is an index to consider disparate minutes played. The rationale of this index is explained in a paper that is currently under review in a journal. The paper will be available online if it is accepted. I show here the abstract:

This paper introduces a way to relativize per-game Player Total Contribution PTC/G by minutes played. It is a form of obtaining a prediction of PTC/G (PTCpred) in order to compare the performance of basketball players who have played different minutes along the season. A sample of 5060 NBA players was collected and analyzed using linear and non-linear variables. In addition, a Maxima code for computing standard error is provided. Results shows that PTCpred could be valuable for analysts, media and fans. Finally, several limitations are discussed in order to avoid mistakes in its practical use.

LPI (League Player Impact)

This is a very simple effect size index which is computed as the PTC/MP of each player minus the median of the PTC/MP of the league (once filters are applied). This index can be useful to compare the impact of players of different decades and from disparate leagues, beacuse its reflects the relative impact of each specific player considering the performance conditions of each league.

Non-linear difficulty curve

One of the awards that the NBA grants every season is the most improved player (MIP) prize. This prize is awarded to the player who has grown more in performance from one year to other. A panel of sportswriters give their votes; the election, consequently, is subjective. However, sportswriters use to ground the decision on some key statistics about the production of the player and the evolution from one year to another. The comparison of performance
between two years use to be done through the raw comparison and the percentage comparison.
In this paper (Martínez, 2019), I have proposed a method to evaluate the improvement of basketball players from one season to the following, considering the difficulty of the achieved performance. After adjusting a non-linear function and employing the Lagrange interpolation, a difficulty curve is obtained. By integrating between the performance of the previous season and the performance of the current season, a final coefficient (difficulty area) is provided. This coefficient is empirically based on the probabilities of performance of players, and provides an alternative criterion to decide which players has been the most improved.

+/- Error

When evaluating the percentage of field goals made or free throw made, it is important to take into account the number of attempts. We consider that when a player attempts 100 shots during a season, that figure is enough to reflects its true skill (true percentage). Therefore, above 100 shots there is no sampling error, because N es large enough to show the true percentage. However, below 100 shots there is an error that should be considered. We approximate the error using the binomial distribution and the standard error of the sample percentage employing a gausssian approach to compute a 95% confidence interval. Therefore, for a maximum variance case p=50%. with 50 shots attempted the error would +/- 10%, i.e. the 95% CI would be (40% – 60%). We consider this error as the maximum admisible to evaluate shooting. Therefore, players with a number of attempts below 50 are excluded in the ranking of best shooters.

Some

Filters

Some filters are necessary to achieve a correct ranking of players. Players have to play at least 33% of the games (at the moment of the ranking) and at least have played 12.7% of the available minutes. Low sample sizes sometimes provide non-reliable data, so those filters are needed.

Median performance and median salary

When the distribution of data is assymetric, median is often much more interesting than mean to describe data. In the following picture the distribution of salaries is showed.

Mean is $6,998,587 but median is$2,964,840. Therefore, highest salaries boost the mean, and mean provides a distorded draw of the real distribution. Median is, consequently, preferred.

Average player

As the units of PTC are arbitraries, the best way to compare the performance of players is referring that performance to an average player. Sometimes we will employ de mean (instead of median) of PTC/G and PTC/MP to characterize the average player, because their distributions are close to a Normal, and their means and medians are not very disparate.

PTC is grounded on the factors determining production in each game (Martínez, 2012), and is built thanks to a relationship of equivalence. There is a way to transform PTC in wins produced or wins added. Although the procedure is not truly reliable, it provides a better interpretation of the numbers.

To achieve this aim we have considered the values of PTC of all the NBA teams from 1996/97 to 2018/19, and the number of wins. Through a linear regression we may propose:

$Wins=\alpha&space;+\beta&space;PTC+u$

Results are the following:

$\widehat{\alpha&space;}=-34.85$

$\widehat{\beta&space;}=0.0159$

$R^{2}=0.30$

It is true that the explained variance is no very good, but here the interest is mainly focused in the prediction of the number of wins (again is not a very reliable prediction).

In order to overcome some problems related with its interpretation, the best way to procceed is to relativize those values using the mean of the distribution. Therefore, we will avoid problems with the intercept $\widehat{\alpha&space;}$ during each moment of the season (the wins produced would be negative during several months).

Consequently, the PTCwins index is computed as follows:

$(PTCwins)_{i}=\widehat{\alpha}&space;+\widehat{\beta&space;}(PTC)_{i}-\overline{PTCwins}$

PTCwins for each player is the number of estimated wins added compared to the mean of the league. A positive PTCwins means that a player produces more than the average, and a negative PTC wins means just the opposite.

VAL_ACB

The “Valoración ACB”, i.e. the index based on box-score employed by the Spanish ACB league. It is not based on theory and it has a lot of methodological and conceptual flaws. I provide this index to compare the missclassification that could be happen using this index.

This index counts every box-score action with the same weight and, for example, also considers block against as a negative point. Consequently, if a player miss a shot he scores -1, but if has been blocked he scores -2!!. Another nonsense is the fact that assists have the same weight that points. It is a complete nonsense but (sadly) it is widely employed.