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January 16, 2012
Adding to Value Add
Four Pettinella Score

by Drew Cannon

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One of my favorite basketball writers is SI's Luke Winn, so when he was referencing a recently created statistic called Value Add this summer, I was intrigued. It comes from a Marquette-centric blog called Cracked Sidewalks, and the then-Top 100 rankings in the stat were posted a few weeks ago.

In theory, Value Add does something that I clumsily do mentally all the time, rank players based on their individual offensive rating, percentage of team possessions used, and the quality of defenses faced. Something about it always bothered me, though, and recently I finally figured out what it was.

Value Add is calculated like this:

O = offensive rating
R = replacement level, here set at a constant of 0.9435
P = percentage of possessions used
S = team defensive strength of schedule

Value Add = (O/(RS) - 1)*P

Or, to put the above into words:

1. Take the offensive rating of the player and divide it by the estimated points per possession a replacement major-conference player would earn (set at 0.9435 times the team's opposing defenses' adjusted defensive efficiencies, hypothetically the equivalent to a typical high-major ninth or tenth man). This gives you the percentage improvement the player brings as compared to his replacement when the player is responsible for a possession.

2. Then subtract one (to make, for example, 1.06 mean "six percent improvement") and multiply the remainder by the percentage of team possessions the player used. This leaves you with, hypothetically, the amount an offense is improved with the inclusion of the target player, relative to a replacement player.

3. There is a multiplication of this number by percentage of minutes played, which will be ignored for the time being.

I found all of this easier to comprehend when I imagined a team comprised of four replacement players and a "target" player. I'm calling this the Four Pettinella Score (PET for short), in honor of my personal favorite replacement player of all-time, former Virginia forward Ryan Pettinella, and, most importantly, in honor of his comedic masterpiece of a free throw stroke. (The angle on that video does not do him justice. At his best, Pettinella looked more uncomfortable shooting free throws than any human has looked doing any activity. This was just the best I could find.)

If you add one to Value Add and multiply it all by replacement level, you get this:

Four Pettinella Score = P*(O/S) + (1 - P)*R

In other words, take a player's offensive rating, adjust for the defenses he faced, and weight it by his possession usage. Then take all the other possessions he leaves for his teammates and multiply that by replacement level offensive efficiency.

Which is all well and good, but, at the moment, PET still ranks players identically to Value Add.

It was when I put everything into these terms, though, that I recognized what it was about Value Add that had been rubbing me the wrong way: It treats replacement level as a constant.

Everyone agrees that maintaining a certain level of efficiency becomes more difficult as usage increases. If you go from shooting 50 percent at ten shots per game, you're likely to shoot less than 50 percent at 14 shots per game because the four additional shots are all more difficult than the ten you were shooting previously. Offensive rating must be considered in conjunction with possession percentage -- otherwise, the stat would be claiming Steve Kerr as the greatest NBA offensive player ever. And John Pudner at Cracked Sidewalks built this right into the Value Add equation.

But when you look at this conventional wisdom in the PET form, it's clear that this truth isn't being applied to the situations when other players have the ball. What we need instead is a sliding scale, where using more possessions improves the efficiency of the replacement level teammates. (Technically, in the Value Add formula, it applies to league-average teammates, but the logic is identical.)

So at first I did essentially what Pudner did, with a few minor tweaks, to find replacement level. I took all the major-conference players from the last two years who played at least 15 games and got between 4.7 and 11.2 minutes per game (typically the ninth and tenth men). Then I tried to fit a regression on defense-adjusted offensive rating in terms of possession percentage.

When coaches do their jobs, though, they don't let kids outshoot their ability over the course of a season, and as a result my regression line came out flat. So rather than pick through hundreds of game-by-game records, I borrowed a rule of thumb from someone who already did, Eli Witus. (See Witus's seminal article on usage and efficiency.) According to this postulate, for each one percent increase in usage rate a player's offensive rating can be expected to drop by 1.25 points per 100 possessions.

I took the mean of the data that I had (an offensive rating of 92.4 and a usage rate of 15.9) and applied Witus's rule of thumb to find that:

Replacement OE = 1.123 - 1.25*%Poss

That gives you the following table, which looks awfully good to me.

  %Poss   ORtg
    5     106.1
    10    99.8
    15.9  92.4
    20    87.3
    25    81.1

Finally, I chose to ignore the percentage of minutes played factor. I do this because including it smashes together all the reasons for missing time: Injury, eligibility, coach's discretion, foul trouble, fatigue, etc. We shouldn't be trusting this instrument completely, anyway -- one of the most important uses of the eye test is to determine the importance of blindingly obvious differences like these.

So we use the equation for replacement OE above to estimate the replacement level of the other four players on the court, to whom we give equal chunks of the possessions left on the table by the target player. That leaves us with the official formula for Four Pettinella Score (which, again, is only building off the work done by Pudner at Cracked Sidewalks):

PET = P*(O/S) + (1 - P) * (1.123 - 1.25*P)

So much for the nuts and bolts. Let's meet back here tomorrow for the fun part: applying our new metric to the real world. Which offensive player adds the most value nationally? I'll have that answer for you tomorrow.

Drew Cannon is a college student and a regular contributor to Basketball Prospectus. Follow him on Twitter at @DrewCannon1.

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Drew Cannon is an author of Basketball Prospectus. You can contact Drew by clicking here or click here to see Drew's other articles.

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