Fun With Numbers
EOPM - Estimating APM with rates

(This is an old, crappier version of EOPM kept for nostalgia.  Click this link for the updated EOPM.)

Last year, Steve Ilardi and Aaron Barzilai created the most accurate Adjusted Plus-Minus (APM) numbers to date. For those unfamiliar with Adjusted Plus-Minus, it takes as input every 5-man lineup in the NBA, the opponent lineups they faced, and the point differential when those 10 players are on the court - for every rotation in the NBA for a given season. By employing multivariable regressions that are well above my abilites, Iladri and Barzilai used that data to separate each player from the effects of his teammates and opponents. What we’re left with is every player’s net effect, offensively and defensively, on the game’s score per 100 possessions.

I really like APM. It’s the only statistic I know of that can account for intangibles - elements of a player’s game that affect the score but don’t show up in a box score - such as tipping rebounds to teammates, forcing your opponent to take bad shots, boxing out your opponent’s rebounder, setting a good pick, and hustle plays.

That’s not to say APM is perfect. Read through Ilardi’s summary of his method, and you’ll see that APM isn’t exact. Because players play in a limited number of different lineups each year, it’s impossible to completely separate a player from every other factor. So each player’s APM comes with a level of possible error. Its other flaw is it doesn’t account for style of play. We saw last year that Shaquille O’Neal might be best suited for a slow offense (he struggled with the Suns, and the Suns are struggling again this year), so his APM might underestimate his value on a slower paced team.

As a result of those flaws, the APM for each player can vary uncomfortably from year to year. Back in 2005, Steve Rosenbaum (the creator of the first version of APM) attempted to minimize that variation by trying to find a relationship between Box Score Statistics and APM. Rosenbaum used a linear regression to fit per-40 minute statistics to his 2005 APM data, and he wrote his findings here.

EOPM (Expected Offensive Adjusted Plus Minus) - My Attempt at a Statistical Plus/Minus

I think I can improve on Rosenbaum’s Statistical APM in two ways. First, he fit his data to 2005 APM numbers, which had higher error levels across the board than the 2007 Ilardi APM numbers I’m using. Second, he used Per 40 Minute boxscore totals - rebounds per 40, assists per 40, steals per 40, blocks per 40 - which 1) don’t account for pace, or number of available rebounds, or number of available assists, and 2) leave for some illogical conclusions, such as “Players who go to the line more, holding the other variables constant, tend to be more effective on offense and defense. In fact, the effect is larger on defense.” While I don’t doubt Rosenbaum found a correlation between FTA and defensive APM, I don’t see how to apply it in projections. (Devin Harris dramatically increased his FTA per 40 this season, does that mean he became a better defender?)

I’ve fixed those issues in two ways. First, I’ve decided not to estimate defensive ability - there’s not enough data in a boxscore (steals, blocks, fouls, and that’s it) to logically rate defense. To me, it doesn’t make sense that a higher free throw rate would cause a higher defensive APM. And (another Rosenbaum conclusion) it doesn’t make sense to me that a higher steal rate would cause a higher offensive APM. So I’ve only used categories that logically correlate to offense. Second, I’ve used rates - TS%, AST%, OREB%, TOV%, USG% - instead of per 40 numbers to account for available opportunities and the size of the impact a player makes on his team’s offense (AST% and USG% especially).  With those changes, EOPM has a big benefit to me that Statistical Plus-Minus doesn’t have: 1) It focuses more on what a player means to his team, based on his role on the team, rather than a player’s effect in a vacuum, and 2) because it uses just five logical inputs, EOPM is easy to understand, calculate, and use in deeper analysis.

But does it mean anything?  Using every player who played at least half his team’s minus last year, I ran a regression using Ilardi’s 2007-2008 Offensive APM data as the dependent variable and a player’s TS%, OREB%, AST%, TOV%, and USG% (copied from basketball-reference, so TS% is between 0 and 1, the others are between 0 and 100) as the independent variables. The formula my Excel spreadsheet spit back is as follows.

Expected OPM = -18.2057+30.2201*TS%+0.128564*OREB%+0.183697*AST%-0.31078*TOV%+0.136568*USG%

Here are the EOPMs (click to view html spreadsheet) for every qualified player in the 2007-2008 season along with his actual Offensive APM and the difference between the two numbers. You can download the Excel spreadsheet below where you can tinker with the data.
EOPM Spreadsheet (click the link and save as)

A few notes:

1) The average per-player difference between Expected OPM and actual Offensive APM was 1.693, meaning if a player’s Expected OPM is +3.0, you can be fairly certain that his actual Offensive APM is between +1.3 and +4.7 (APM error rate aside). For a 5-variable system, I think that’s surprisingly accurate.

2) Some players were off by as much as 4 to 5 points, which on the surface isn’t good. But a closer look suggests that Expected OPM may have identified which players had an inflated or deflated Offensive APM because of APM error. Below are the players with the highest differential.

Jamario Moon, Luol Deng, Andrei Kirilenko, and Peja Stojakovic all ranked among the Top 25 Offensive players in basketball last year according to Offensive APM, yet none were their team’s 1st or 2nd offensive option. And maybe their rankings are accurate, but they sure seem off to me. And while Steve Nash and Devin Harris played well in ‘07-’08, I don’t think many would’ve ranked them as the #1 and #6 offensive players in basketball.  EOPM adjusted accoringly.

On the flip side, Offensive APM rated Jarrett Jack, Chris Kaman, Rajon Rondo, and David West as four of the worst offensive players in basketball. None of them were offensive stars last year, but they certainly didn’t seem to be in Bruce Bowen / Ben Wallace territory. David West’s Offensive APM rated 8.19 points below immoble teammate Peja Stojakovic - that especially didn’t seem right. Again, EOPM adjusted accordingly.

Maybe I’m (very) biased, but EOPM seems to pass the validity test moreso than Offensive APM, or PER, or ORtg, or any other standalone stat I’ve seen. 27 of the top 31 players on the list have been all-stars, and the other four (Jose Calderon, Deron Williams, Jason Terry, Andre Miller) had years deserving of consideration on playoff teams. Similarly, the names at the bottom - Ben Wallace, Jeff Green, Bruce Bowen, Sam Dalembert, Marcus Camby, Drew Gooden, Brendan Haywood - have gained a reputation as non-scorers, inefficient shooters and/or turnover machines.

3) To get a sense of which factors have the strongest effect on EOPM, I took the 1st and 3rd quartile rates of the players I analyzed (there are 130 players in my database, so I found the 32nd and 98th ranked players for each category), found the difference between them, then multiplied that difference by the rate’s coefficient in the EOPM formula.

That’s not a great way of showing how much a rate can affect EOPM - AST% can get to over 50% (Chris Paul, John Stockton), and four players had a USG% over 30% last season (LeBron, Kobe, McGrady, Carmelo). But either way, AST% stands above the others, and I find that interesting. For decades, the adage was “You need a big man to win a championship”, and I figured part of the reason was that a quality post scorer means more to an offense than a quality wing player or point guard. But Ilardi’s Offensive APM data shows otherwise. Several all-star caliber big men had surprisingly low Offensive APM numbers last season - Dwight Howard (+2.78), Shaquille O’Neal (+1.12), Andruw Bynum (+1.00), Yao Ming (+0.21), Al Jefferson (-0.07).

On the other hand, the best passing big men in the league had strong Offensive APMs - Kevin Garnett (+6.88), Tim Duncan (+4.39), Brad Miller (+2.08). That makes me rethink what a interior-scorer actually does to impact an offense, and “ability to pass” and “what does an assist actually mean” are two things I’ll be exploring in the coming weeks.

I also want to take a look at using EOPM to project team success, both in the regular season and playoffs, in much the same way Neil Paine did at the basketball-reference.com blog using Rosenbaum’s method. And I’m going to look at breaking EOPM down by position. The difference between Offensive APM and EOPM for point guards is higher than I want it to be (the error is partially due to the huge range in PG Offensive APM totals, from Kidd’s +10.01 to Jack’s -5.48), and perhaps there’s a way to lower that difference by accounting more for how a player contributes to his team.

For now, feel free to download the spreadsheet I used EOPM Spreadsheet (click the link and save as) if you’d like to play around with the numbers and knock the crap out of my method :) And if you’re just curious about a specific player’s EOPM, you can find the EOPM of any player in history by typing their rate data (straight from their basketball-reference page in the “Advanced” section) into the EOPM calculator below.  Make sure TS% is between 0 and 1 (example: .552), and the other rates are between 0 and 100 (example 20.6).

Player of the Day: Chris Gatling, -0.15 career EOPM

Fun With Numbers
What’s a Good Shot? Part I of II

NBA.com’s HotSpots breaks down field goal shooting by location of shot attempt. Here’s the HotSpots for my Bulls this season:

Chicago Bulls HotSpots Shot Chart, 2008-2009 Season

For those seeing HotSpots for the first time, the inner semi-circle statistic represents shots that were scored around the basket. The stats outside the biggest arc represent 3 pointers. And the Bulls are suprisingly really good at those, especially the straight-on 3.

The numbers aren’t perfect. As of today basketball-reference has the Bulls shooting 367/974 from 3-point range, while HotSpots has them at 365/956. But the numbers are close enough across the board to get a sense of what’s what. So let’s take a look at league-wide data, which I’ve converted into eFG% (FG% that awards an extra point for 3’s) by zone. Below is the data for every shot taken in the NBA this season.

Combined Shot Chart of Every NBA Team, 2008-2009 Season

Organizing that data by distance from the hoop:

(Note: basketball-reference has league-wide 3PT% at 36.6%, which correlates to a 54.9% eFG. I’ll be using that 54.9% in the next post.)

37.5% of NBA field goal attempts are from “mid-range” (long-twos + short jumpers), yet eFG% on those shots is under 40%. And you’d be hard-pressed to find anybody scoring efficiently from that range - Ray Allen is the best I could find, and he’s hitting at a 52.2% clip.

What about drawing fouls from that range? According to 82games.com, shooting fouls occur on only 2% of mid-range field goal attempts compared to about 17% on close-range attempts.

Makes you wonder why offenses would ever settle for a mid-range shot, much less settle 37.5% of the time. It would make sense that the best offensive teams take a high ratio of shots from efficient zones compared to inefficient zones. We’ll test it out next Monday.

Player of the day: Uwe Blab, career high 46.8 eFG% in 1986