The Basketball Impact and Efficiency Rating (BIER) is an advanced stats model formula developed by Bob Boozer Jinx perpetrator-in-chief J.D. Mo; it’s based on many of the box score analysis principles found in Dean Oliver’s brilliant book, *Basketball on Paper* and Basketball-reference.com’s Win Share model. In function, however, BIER works like John Hollinger’s “Game Score” calculation, only it doesn’t use Hollinger’s values for made shots, missed shots and rebounds. In other words, BIER doesn’t reward volume scorers or arbitrarily devalue rebounds; the BIER model is Frankenstein, constructed from the best parts of Oliver’s “points produced” and Hollinger’s PER models, with extra detail added to hone in on a definitive player rating.

**Wait, what is this BIER thing? ***(Article moved from “BIER ratings leaders at the break” post.)*

A player’s BIER rating is most often expressed in Bob Boozer Jinx blogs as a per36 minute, pace adjusted stat. The values plugged into the formula are from Basketball-reference.com’s per36min tables. BIER can also be expressed as a per game stat without the minutes equalized, and then as a full season stat (Real Adjusted BIER) which takes into account what percentage of the entire season a player has played. Lebron James hasn’t missed a game this season (2017-18), so his per game BIER and RABIER (no, I’m not trying to lampoon RAPM*¹ here) will be the same number. But usually, the BIER expressed is the per36 minute result, always pace adjusted.

**Why points per 36?** The purpose of BIER is to compare players within a season, and also to compare players’ career numbers. Who’s better, Lebron or Bird? Who’s having the better season, Lebron or Giannis? Is Anthony Davis putting up better all around numbers than James Harden? And so on. The super-duper stars play about 36 mins per game, so using per36 stats makes the most sense. Also, by pro-rating every player’s stats to 36 mins, BIER relates which part-time players are having the highest *impact, “Impact” being an important, nominal part of BIER*.

**Do we really need another advanced box score metric?** Of course we don’t *need* it, not really, and it can be a real headache if you want to get down to the nitty-gritty of assist rates and other details (like the value of missed free throw rebounds). But *you* don’t have to worry about the headaches because the editorial board chains J.D. Mo to his laptop a few times a month and has him sort it all out until *he’s* got a headache. *(Unfortunately, the board can’t always get him to write anything, or a blog like this one would have appeared last year at the All-Star break, the first time he ran the whole league through the BIER model).*

All hubris noted, the justification for BIER — beyond the smooth acronymical homonymousness it shares with the German word for beer — can be found in the flaws of other models such as Hollinger’s infamous PER (brought to you by ESPN), NBA 2K, and many fantasy scoring systems, where volume scoring is valued over efficient scoring. PER at its core commits a major statistical no-no in that it bases missed shot value on offensive rating (points per possession) instead of on basketball’s basic 2-point scoring system. The main factor defining the offensive rating is whether or not shots go in the bucket, so there’s a causal relationship there. If most of the league shoots poorly — which it did in the dead ball era — it follows that possession value drops — which it did, all the way down to 1.04 pts/poss.) — and a little less is deducted with each miss. Bad shooting is thus rewarded. Conversely, good shooting is punished in PER because it’s undervalued to begin with, then leads to higher offensive ratings — which PER uses to set a greater deduction for missed shots.

Oh, the adjustments made to correct for this problem! And once those are done, PER remains relevant only within each season. When you hear NBA analysts criticizing advanced metrics for “tailoring” to the changing game, PER is what they’re talking about. The arbitrary downgrading of rebounding within the PER formula is another problem, which Oliver, unfortunately, actually supports in *Basketball on Paper*. It’s as though Hollinger chose Kobe’s side in the Kobe-Shaq conflict of the early 2000s and form-fit his formula for volume shooting guards and small forwards, at a time when Shaq was so dominant it was almost boring. One hears less talk of PER now as advanced stats geeks and saber-metrics pros have moved on to better measures —

### the search for the Holy Grail model of Basketball analytics goes ever onward.

Why not simply keep shooting independent of offensive rating (its result) and create a scoring system that conforms to basketball’s 2-point basic scoring system? BIER does this by letting the rebounding rate decide what to deduct for a missed shot and multiplying REB% by two to create 2-point values out of the percentages, similar to the Win Score/Game Score model used at Basketball-reference.com.

**For example**: The current league-wide rate that defenses rebound missed shots is 77.6%, the all-time high in the NBA. Because percentages are on a whole-of-one value, multiply .776 x 2 to conform to basketball’s 2-point scoring base and get 1.552 — this is the hit a shooter takes when missing a shot in the 2017-18 season. The defensive player who grabs the rebound is awarded (2 – 1.552 = 0.448), and the possession ends. Conversely, if the offense grabs the missed shot, the rebounder who saved the possession has effectively canceled out the missed shot, so that’s + 1.552 for the rebounding player. The -1.55 stays with the shooter as the possession continues.

A -1.55 may seem like a big deduction for a missed shot, but with offensive rebounding at an all-time low, the numbers are merely a reflection of the game. Also, with a record number of 3-pointers attempted, there is often greater reward (3 pts) to go with the greater deduction for missing (-1.55), another appropriate reflection of the changing game. When the 3-pointer was little-used in the early 1980s and one-third of missed shots were rebounded by the offense, the scoring reward was almost always lesser (2 points) and so was the risk (-1.33 for a missed shot). A shooter who misses 2 out of 3 three-pointers in today’s game will end up in the negative, but that’s OK. The league 3-point average is at 36.1%, so the 33% three-baller is more likely than not digging his (or her) team a hole against the opponent’s better shooters.

In theory and in practice then, BIER mirrors what probably happened on the scoreboard during the game. It . . . makes . . . sense *(and I can’t help but imagine Dean Oliver reading this someday, pausing now to sip his coffee, and coming to this same awesomely inescapable conclusion).*

**In practice** (using current league-wide numbers): if a shooter goes 3 out of 10 from 3-point-land, the shooter scores +7.782 in both PER and BIER (after a basic assist adjust, which will be covered in sections ahead) — but PER deducts only 5.9 for the misses to keep our volume shooter ahead + 1.88 even though the shooter is (probably) losing the game for his team. BIER deducts 7 x 1.55 for the misses = -10.85, leaving our volume shooter minus 3.07. By then, the shooter is possibly on the bench or, if still in the game, being frozen out by his point guard, who’s looking for more efficient scoring. Kobe Bryant shot like this often, but Kobe got to the line 8-10 times a game in his prime and did a lot of other things on the court to help his Lakers win. Other so-called stars were not so active on the court, and PER helped the volume shooters of the deadball era (Kobe, Tracy McGrady, Paul Pierce, Michael Redd et. al) look a lot better on paper than they were.

*(Up until recently, the NBA used PER (aka EFF) as a featured stat on its own website, and journalists around the league began to follow suit. Thankfully, the NBA has adopted its own Player Impact Estimate (PIE) to replace PER/EFF as the featured advanced stat on NBA.com player pages. PIE is entirely different from BIER or PER, and is like the “Total Points of Awesomeness” system a friend of mine came up with a few years ago, where you add the good stuff in the box score, subtract the bad stuff and the result is the player rating. The NBA adds up a team’s Total Points of Awesomeness and expresses each player’s share as a percentage of team TPA. Not very scientific, undervalues missed shots, but still pretty cool.)*

The rebounding rates used in BIER adjust year-to-year with the different rates for each season (for career stats, an average of league rates during the player’s career is used) and there are rare instances where it is a good idea to adjust a team’s in-season rebounding % above the league average. Russell Westbrook and the OKC Thunder are a great example of this — the Thunder lead the league in offensive rebounding at 28.1%, so when Westbrook misses, J.D. Mo deducts the OKC rebounding number (2 – (2 x .281) = 1.44) instead of the leaguewide number.

*At this point it should be noted that BIER in-season numbers are adjusted for both the pace and the assist rates of each team, and are expressed per 36 mins. The Golden State Warriors play a fast pace and have one of the highest assists rates in the history of the NBA, so the pace is adjusted in the formula and GSW shooters take a bigger assist-adjust hit (this doesn’t make Klay Thompson look good in BIER but it can’t be helped). *

It’s not a good idea to adjust for rebounding rates except in rare circumstances such as OKC, where rebounding is part of their offensive strategy. Most teams don’t crash the boards like OKC does (and it would be another statistical no-no to reward good offensive rebounders with a lower OREB value), so the league-wide rates for rebounds are always used, and missed shot value adjusted only in rare cases Again, missed shot value = DREB% x 2, the result being the same value used for OREBs. For OKC J.D. Mo is using (1 – ThunderOREB%) x 2 for Westbrook’s missed shot value and the leaguewideDREB% x 2 for Thunder offensive rebounders.

Let’s introduce some terms and the first part of the formula.

- Made shot value = 2 or (3 for 3-point shots), one for free throws
- Points (or scoring) is expressed at the beginning of the equation as ptsper36
- Missed shot value is 1.55 (using leaguewide rebounding rates)
- First part of the equation is: ptsper36 – [(field goal attempts – field goals made) x 1.55]

**The assist adjustment** enables any stat geek to run the numbers using the box score stats, avoiding the problem of needing to know which made shots were assisted and which were unassisted. The goal here is that the player who sets up a made shot with a pass should be rewarded with a portion of the made basket value — sticking to basketball’s basic 2-point system. No need to award greater value to assists that lead to 3-point baskets, a simple 0.7 value will do.

**Why 0.7?** Many assists aren’t worth 0.7 points, being the result of the ball moving around the perimeter to open shooters or simply an exercise of typical team offense — but some assists (like lob passes to dunking 7-footers or behind-the-back passes on the break) are worth the price of admission, so the hope is that it balances out. The 0.7 value seems sensible and fair, is greater than the value of a defensive rebound and less than the value for steals, and is the standard value used in other systems, so I adopted 0.7 for BIER.

So if Westbrook dishes out 10 assists in a game, a +7.0 is added within the BIER formula. But those 7 points can’t come out of thin air — they have to come from somewhere, and the obvious place is the scorers. But we can’t take 0.7 away from every scorer because not every basket is assisted. Not knowing which baskets in a box score are assisted or unassisted means the formula has to assign an assist rate to calculate an average assist adjustment for every made shot. This can be the leaguewide 2017-18 assist rate of 0.58, a basic 0.6 or a more detailed rate depending on how much work one feels like doing.

*(This is why the editorial board chains J.D. Mo to his laptop and has him calculate the assist rates for each team as well a special assist rate for point guards. Each point guard or high assist player (like point forwards Lebron James, Giannis Antetokounmpo and Draymond Green) in the BIER rankings charts has a separately calculated assist rate. About 64% of Westbrook’s shots are unassisted, so when the assist adjustment in BIER is applied to Westbrook (as part of the 2-point system), the assist rate of (1 – .6325 = 0.3675) is used, the lowest assisted basket rate I’ve seen this season.)*

The calculation of the assist adjust = made shots x 0.7 x assist rate. Westbrook makes 9.5 shots per 36, so multiply 9.5 x 0.7 x 03675 = 2.444. This is subtracted from Westbrook’s pts per 36 at the start of the equation:

25.3 – (assist adjust of 2.444) – [(FGA 21.4 – Made FGs 9.5) x 1.44 (the detailed OKC missed shot value)] = 5.724

Because Westbrook misses a lot of shots (11.9 per 36) and shoots just 44.4%, he’s doing a lot of work to end up +5.724, while a 54% shooter like Lebron James, Anthony Davis or Giannis Antetokounmpo will generate a higher + value given fewer shot attempts. This is the efficiency side of BIER. All three forwards rank higher than Westbrook in the BIER Top 20.

**How are three point shots calculated in BIER?** A missed or made threes calculation is not part of the formula because they are accounted for in the per36 points scored at the beginning of the formula. In the BIER spreadsheets, J.D. Mo includes columns for 3-pointers made and attempted, and the resulting % because he wants to see what a player’s shooting from downtown, but the formula treats missed 3-pointers no differently than missed layups. Both the points and the made FGs are already accounted for in the formula, and missed threes are calculated along with the 2-point misses in FGA-FGs.

**Free throws attempted and made are divided by two, right?** Yep, and the next step after figuring out the assist adjust and the missed shot deduction is to calculate the missed free throws deduction. A missed first FT in a two-shot trip to the line is always rebounded by the team offense and a 2nd (or 3rd) missed FT is almost always rebounded by the defense — so how do we account for this with rebound rates? The only available science out there seems to be a study by 82games.com that said 86.1% of missed free throws are rebounded by the defense. This a rate for a contested missed free throw rebound, obviously, and it’s an older study from the early 2000s when offensive rebounding was more prevalent — but it’s the only research I’ve found. I couldn’t think of any justification for reducing the deduction for missed free throws by taking into account the unknown % of missed first FTs (box scores don’t give us this) so I err on the side of the available science and use the 86.1% rebound rate, though I suspect it’s probably a few % points low. One of these days, J.D. Mo will be chained to his laptop and made to do a credible Missed Free Throw Rebounding study, but not today.

Here’s what the free throw calculation looks like:

(Free throws attempted – Free throws made)/2 x (2 x 0.861) or (FTA – FTM)/2 x 1.722

Westbrook attempted 6.9 FTs per 36 and made 5 through games of Feb. 15, so we calculate his misses, divide by two to convert one point FTs to the basic 2-point framework of the game, then multiply the misses by the 2-point scale defensive rebounding rate (FTA 6.9 – FTM 5) = 1.9 / 2 to result in 0.95 x 1.722 = 1.636

The missed FT value is also subtracted from the points per 36 at the beginning of the formula, Westbrook’s scoring per36 numer is cut down to a + 4.088. Put it all together and the scoring efficiency part of the formula looks like this (using leaguewide rebounding rates, not the special OKC rate):

PTSper36 – (FGM x 0.7 x team AssistRate) – [(FGA – FGM) x 1.55] – [(FTA-FTM)/2 x 1.722]

The rest of the formula is fairly straightforward. Values of offensive and defensive rebounds, assists, steals and blocked shots are added — turnovers and fouls are subtracted.

- Offensive rebounds (OREBs) = 1.55
- Defensive rebounds (DREBs) = 0.45
- Assists = 0.7
- Steals = leaguewide points per possession = 1.086
- Blocked shots = 0.57 x leaguewide pts per possession = 0.61
- Turnovers = – (leaguewide pts per possession) = – (1.086)
- Fouls: For ever foul committed, the opponent ends up shooting 0.852 free throws. Remarkably, this rate has changed only by one-thousandths over the last 40 NBA seasons.

Here’s the entire “unadjusted” formula:

PTSper36 – (FGM x 0.7 x team AssistRate) – [(FGA – FGM) x 1.55] – [(FTA-FTM)/2 x 1.722] + OREBs x 1.55 + DREBs x 0.45 + Assists x 0.7 + STLS x 1.086 + Blocks x 0.61 – TOVs x 1.086 – PFs x 0.852

**All BIER ratings are pace-adjusted** to 100 possessions per game, so the BIER number of a player who plays at a pace of 95 will adjust up 100/95 — the pace played is the denominator. This number (1.05 in this example) is multiplied by the “unadjusted BIER” arrived at in the above formula. Westbrook’s unadjusted BIER through Feb. 15 was 11.38. The Thunder play at a 96.1 possessions per game rate, so to get to BIER100 multiply 11.38 x 100/96.1 = 11.84.

**The highest career BIER player rating in NBA history** is an estimated 18.98 recorded by Wilt Chamberlain (1960-1973), estimated because box score stats were not complete until 1977 when turnovers were added. Offensive/Defensive rebounds, blocked shots and steals were added to box scores in 1974. The highest career BIER since 1977 belongs to Magic Johnson at 17.49 and then Charles Barkley with a 17.18, the best career rating for a forward in NBA history. Sir Charles was a very efficient scorer who passed the ball more than most remember, and really was the Round Mound of Rebound (his 3.9 OREBS per 36 were absurdly great).

Michael Jordan posted a career BIER of 14.58. No other shooting guard comes close. Larry Bird (14.51) and Lebron James (13.95 at the start of this season), the never-ending comparison, lead the small forwards of NBA history (though both did/have done time at power forward) with Lebron due to close the gap some with a 15+ BIER this season. Kevin Durant (12.68 entering 2017-18) is so far a Top 5 BIER small forward in NBA history, just ahead of the great mid-range scorer, Adrian Dantley, but still a full point behind Julius “Dr. J” Erving (13.59).

Chris Paul (13.8 – 2nd PG behind Magic), James Harden (11.16 – 5th SG) and Anthony Davis (14.65 – 2nd PF behind Barkley) are the other current players who began the season as Top 5 All-Time career BIER-rated players at their respective positions, though Davis (in just his 5th year) hasn’t played long enough to be ranked All-Time. Steph Curry (career 11. 49), in his 9th season, has the tenure, and could possibly move ahead of Oscar “the Big O” Robinson into 5th on the All-Time point guards list, for the time being anyway.

BIER may be tough on players who don’t make shots or do much other than shoot, due to the current high 1.55 deduction for missed shots, but isn’t that what an advanced box score rating system should be? The NBA’s very best players are having no trouble posting historic numbers in BIER or any other metric. If the average and below average players, and the one dimensional shooting guards and small forwards, are revealed in BIER for what they are, the system is doing what it was intended to do.

*— J.D. Mo*

**¹* *RAPM is “Real Adjusted Plus-Minus”, a metric frowned upon by many and one the Bob Boozer Jinx strongly suggests should not be used for player analysis.*