Understanding FG% and Rebounding in Player Efficiency Ratings

In our previous post we tackled, at length, how John Hollinger’s Player Efficiency Rating (PER) is calculated and identified how different aspects of a player’s game is weighted. This leads us to question certain components of PER. Should I prefer a scorer or a rebounder? How do these terms interact? What about team affiliation? In this post, we tackle a few of these components to help understand what PER truly says about a player when combined with their traditional statistics; of which PER is obtained.

For example, every rebound is reduced by the league rate of rebounding. For the 2016-17 NBA season, this amounted to reducing all defensive rebounds by 76.7% and reducing all offensive rebounds by 23.3%.

Offensive Rebounders are Better.

Consider two players who are identical in every way. Same number of minutes, same number of points scored with identical field goal (two’s and three’s as well) and free throw percentages, same number of steals, same number of steals, and same number of rebounds. These two players should have the same PER right? Not a chance.

Klay Thompson is Better Off on the Warriors than the Suns

First off, if these two players are on different teams, then they are affected by their team’s performances. There are two parts of  PER, which I coined the “What the Heck” factor given by

For these two identical players, the league assists, field goals made, and free throws made totals are going to the be the same. However, suppose one player is on the Golden State Warriors and the other player is on the Phoenix Suns. Let’s denote the percent of team field goals that came off assists as

Applying a little algebra and we see that the difference between these two players with identical stats across the board has a difference of unadjusted PER (uPER)

First we see the difference between Golden State and Phoenix’ assist to field goals made ratio washing out as a difference between these players. To give insight, Golden State had 70.5266% of field goals come off assists. Alternatively, Phoenix managed only 49.0520%. This difference sits at .214746. This means the Warriors player will have a boost of the first term by twenty-one percent. We call this a boost, only if the first term is indeed positive.

For reference, the 2017 NBA season witnessed 55,600 assists, 96,061 made field goals, and 43,883 made free throws. This results in

This is only positive if the player makes 2.518625 more field goals than free throws. For the 2017 NBA season, this is only true for 238 of the 486 players. The most common type of players on this list are three point specialists such as Kyle Korver (6.2895), Danny Green (6.5185), and Wayne Ellington (6.2432); mid-range shooters who rarely draw fouls such as Meyers Leonard (4.1714), Leandro Barbosa (4.3000), and Boris Diaw (5.6154); and terrible free throw shooters such as Thaddeus Young (8.0444), Sam Dekker (5.3412), and Andre Drummond (3.5255).

This means if the players we are comparing score more field goals than free throws than the 2.51 required rate; then the Golden State Warrior will obtain a higher PER just by being on the Warriors. The amount difference? Let’s suppose this player is Klay Thompson. Then the Golden State Warrior bump in uPER is 2.4945. This amounts to roughly an extra field goal made; which will result in roughly an increase in 0.0769 in PER.

What this means is… if a Phoenix Suns player replicates Klay Thompson’s 2016-17 NBA season; they are docked nearly 0.08 PER point due to playing on the Phoenix Suns.

Now How About Those Rebounds

Now that we have seen just by playing on a certain team, a player’s PER can be affected by roughly 0.08; let’s see the impact of rebounds. In this exercise, we return to the identical players who have the identical stat lines, except for breakdown of rebounds. In the 2016-17 NBA season, there were 256 of the 486 NBA players with 164+ rebounds (2 per game). Our example of Klay Thompson above yielded 285 rebounds on the season. Thompson’s rebounds break down as 49 offensive rebounds and 236 defensive rebounds. The amount of contribution to uPER for rebounds is given by

The estimated points per terminated possession, v, is a league average calculated points scored per estimated number of possession. While the estimated number of possession is grossly off, despite corrections given by Basketball-Reference (still off by a few hundred possessions) and Squared 2020, we calculate the league points per possession factor as 1.062323 (259,753 points scored over 244,514.2 estimated possessions).

The defensive rebound rate, d, is the percentage of league rebounds that resulted in defensive rebounds. This factor is the simplest of the three to compute in the PER calculations and is given by 82,109 divided by 107,046 for a value of 0.767044.

This means that Klay Thompson’s rebounds contributes to an extra 98.3403 uPER. In our article breaking down PER, we noted that all rebounds are down-weighted by the league average. Using a little algebra and ignoring v, we saw that this is equal to defensive rebounds minus league-average-weighted-defensive rebounds plus offensive rebounds minus league-average-weighted-offensive rebounds:

What this means is that Offensive Rebounds are more important than defensive rebounds! Let’s check this calculation. Let’s assume that Phoenix Suns player who has identical stats to Thompson exists. Furthermore suppose that this Suns player has instead of 236 defensive rebounds, they have 250 defensive rebounds. This yields only 35 offensive rebounds. This will result in a uPER contribution of 90.3984. This is another 8 solid points less than Klay Thompson’s uPER attributed to rebounding.

This means that Klay Thompson gets an extra ~0.33 PER points from being a stronger offensive rebounder, weaker defensive rebounder, and playing on the Warriors instead of the Suns.

This indicates approximately a 10 player jump in PER despite having same summary statistics except a 15 rebound change in OREB/DREB.

Volume Shooters are Rewarded

And you don’t have to be that good…

Next we look at how field goal percentage is taken into account. The terms involving field goals made (FGM) and field goals attempted (FGA) are taken into account. To understand how FGM and field goals missed (FGA – FGM) are rewarded/punished, we look at “breaking even” in field goal shooting percentage. This percentage would then indicate how well a shooter must shoot before seeing their PER increase.

To start, we take a look at the terms associated with FGM and FGA. These are given by

Note that the factor, f, is the what the heck factor which is merely 0.600493. While league field goals made appears twice in the denominator of f, the contribution of a player is fairly negligible. For instance, consider the worst field goals made players: Aaron Harrison, Ben Bentil, Danuel House, Gary Neal, Patricio Garino, and RJ Hunter. All these players scored zero field goals and admit zero contribution to these denominators. On the flip side, Russell Westbrook lead the league with 824 made field goals. His contribution will affect this factor by .001 and affects field goal percentages by less than .0001. This means that we can effectively treat this factor as a constant.

Next, we see that FGM appears for a player embedded in Team FGM. This is a place where a player can change the field goal percentage. The largest percentage, of which, is again Russell Westbrook with 824 of this Oklahoma City Thunder’s 3237 field goal makes. This is over a quarter of field goal attempts taken. This will actually affect field goal percentage necessary for finding a zero contribution to uPER. In this case, the maximum, we find the change in FG% is less than 1.5%. This means that if we treat this percentage as a constant for each team, the field goal percentage will effectively be within one percent.

Time for Heavy Duty Math…

Now if we write the above model to being zero contribution (Break Even) to uPER and eliminate constant terms that attribute to effectively plus-minus one percent of resulting FG%, we obtain the model

The function F(t) is the factor, f, times the assists per field goals made for each team. This will be a function of a respective team. Now, let’s break down FGM by its definition FGM = 3PM + 2PM. Similarly, we break down FGA by its definition FGA = 3PA + 2PA. We then obtain an equation in terms of 3PA, 2PA, 3PM, and 2PM. We can collect like terms to obtain

Now we perform the trick of multiplying by one. This will eliminate two variables (more so will hide them cleverly) and give us two equations with two unknowns; hidden as one equation. I will build off the previous two steps to show where the work comes in:

This last line is actually two equations two unknowns. The rationale for this is due to the fact that no player can attempt less than zero three point or two point attempts.

Hence, the only way this quantity can be zero is if both 3PA and 2PA are zero or the coefficients are zero. Let’s assume the latter, as we are interested in finding field goal percentage that results in a net zero gain. Let’s isolate this equation and look at each coefficient.

This equation is the important one. First, if the field goal percentage for each type of field goal attempt is above the “break even percentage” then the uPER increases by the number of attempts of that player. This directly shows that PER favors volume shooters provided they shoot above a “break even” percentage. Note from the equation that this is team-dependent.

Second, this equation yields the break even value for each team. Anything above this percentage results in positive PER. The better the field goal percentage, the higher the PER per attempt.

Solving for both three point and two point percentages, we obtain

Calculating this for every team for the 2016-17 NBA season, we obtain the break even field goal percentages for every team:

“Break Even” field goal percentages for both 2FG% and 3FG%. Anything above these percentages will be positive multipliers to the number of relative attempts.

What this shows is that a team effectively needs to shoot better than 33% from two-point field goal range and better than 23% from three-point field goal range. These are terrible numbers! Anything above these percentages will result in positive PER per attempt. Let’s drive this point home.

Suppose a player on the New Orleans Pelicans shoots 38% for two’s and 25% for three’s. This player will obtain a multiplier of 0.149639 per two-point field goal attempt and 0.063147 per three-point field goal attempt. Let’s compare this player to Shaun Livingston of the Golden State Warriors. Livingston, obtains 0.614088 per two-point field goal attempt and 0.385922 per each three-point attempt.

This means that for every two-point attempt Livingston takes, the Pelicans player can take 4.10 two-point attempts to look as good in PER as Livingston. Similarly, the Pelicans player can take 6.11 three-point attempts per single attempt from Livingston.

How does this translate into the game?

As Livingston plays roughly 18 minutes per game and takes a mere 4.2 field goals per game, the Pelicans player ends up taking 17.42 field goal attempts per game. In terms of uPER, these two players are identical: Pelicans Player 2.5654 vs. Livingston 2.5564. Playing over the course of 18 minutes per game, this means the Pelicans player will be at roughly 40-50% usage.

Taking this one step further, Livingston can account for approximately 4.6 points over the course of approximately 4 possessions. Similarly, the Pelicans player will have accounted for approximately 6.6 points over the course if approximately 17 possessions. This means that the Pelicans defense must stop 13 extra possessions in order to compensate for a player who looks better than Livingston in PER.

The real question is… what coach would be letting this Pelicans player do this to his team? Certainly not Alvin Gentry, as he still has his job in tact!

More importantly, this accounting for PER becomes more important in the ranking of players as volume shooters are favored more than others. The above example is cartoonish in nature, but exploits the very real fact that a player can improve their PER by sinking their team through volume shooting.

Conclusions

In three careful examinations of PER, we find that PER benefits players on teams that pass well, make more than 2.5 times as many field goals than free throws, and tend to gather more offensive rebounds than defensive rebounds. We also identified that PER heavily benefits volume shooters and if they manage to be at minimal terrible shooters, that can use that to inflate their numbers to look better than their All-Star and All-NBA quality peers.

This doesn’t show that players with high PER are terrible, however. What we do say is take this measure with a grain of salt and surround it with qualifying metrics. We quickly see that this measure favors perimeter shooters, as well as poor free throw shooters who score a sizable amount of points from the field. The above explanations capture these rumblings.

How else do you see PER?

Note: (Edit 5 Dec) Thanks to Richard Yannow (@RichardYannow) for correction on the previously reported Hypothetical Pelicans example.

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