Daily Box Score 8/28: Burying OPS

I admit, I was once put under the spell of OPS. It was so easy to calculate, and seemingly so much better than its peers at the time, that it seemed absurd that more people didn't use it. 

OPS was first introduced to the sabermetric community at large by Pete Palmer and John Thorn, in their classic tome The Hidden Game of Baseball. But you know what? OPS is bad. It destroys information that is essential to player evaluation. 

Please, allow me to demonstrate.

Correction Appended

Table of Contents

What Got Me Thinking
Calculus Class
The List
Discussion Question of the Day

What Got Me Thinking

Andy over at the Baseball-Reference blog wrote a post about Ryan Howard. He was following up on a piece he wrote at the end of last season. In that 2008 post, he wrote:

First, check out this list of lowest OPS+ values for a guy with 47+ HR. It's been done 92 times, and only twice has a guy been under 130. Think about his OPS+ of 121...does that seem like a number that a guy leading all of baseball in HR and RBI should have?

A large part of it is his low batting average. Two years ago, he hit .313 and slugged .659 with an OPS+ of 167. This year, he's batting only .248 and slugging .537. That's 65 points off his batting average and 122 points off his slugging average. His extra bases are roughly the same--the difference is actually about 30 fewer singles, and the fact that he's got more ABs this year.

And then just yesterday, he wrote this:

Howard's walk totals have dwindled over the last 4 seasons, from 108 and 107 in 2006 and 2007, down to 81 in 2008 (in more games) and projecting to fewer than 80 this year. You can see this trend in his BB% right here. Of note is that his walk dropoff seems to be due mainly to fewer intentional walks, going 37, 35, 17, and just 3 this year.

Anyway, this is the second straight year that Howard will finish with monster HR and RBI numbers but an OPS+ under 140 (although he's historically had massive Septembers, especially last year with 11 HR and 32 RBI.)

So, according to Andy, Howard's low OPS+ are because of his batting average and his walk rate falling. Sounds pretty bad, right?

Only thing is, Ryan Howard is still a good hitter. He's been worth 22.9 batting runs above average this year, and 19.9 last year. There can be no doubt that he is not as valuable as he was in 2006, when he was worth 60.6 (!) batting runs above average. So why the low OPS+ numbers?

Calculus Class

It's been a while since I've taken a calculus class, but since people seem to enjoy watching me flail around with numbers, I thought I would indulge those folks. So let's try some basic calculus.

OPS, as we all know is on base percentage plus slugging percentage. In equation form:

OPS = OBP + SLG

Now, since I'm lazy and didn't use a calculator for this (consider this your warning), I made some simplifying assumptions. I'm ignoring HBPs (with apologies to Plunk Everyone) and SFs. So, 

PA = AB + BB

OBP = (H+BB)/PA

SLG = TB/PA

Rewriting, we get

OPS = (H+BB)/PA + ((TB/(PA-BB))

All I did was substitute the definitions into the original equation. Everyone with me so far? Cause here's where it gets a little complicated. I took the derivative of the above equation with respect to walk rate (BB/PA). That required some slight rewriting:

OPS = H/PA + BB/PA + ((TB/(PA-PA*(BB/PA)))

Okay, we're starting to accumulate an impressive number of parentheses, which in my view are like bacon. Better to err on the side of more. Now, to take the derivative, we'll need the Quotient Rule. I'll leave the bloody details as an exercise for the reader, but feel free to check my work. Here's the resulting derivative function:

OPS'(BB) = 1 + (TB*PA)/(PA^2*(1-BB/PA)^2)

This formula tells us how sensitive a player's OPS is to changes in his walk rate. The higher the value, the more a shift in walk rate will affect his OPS.

The List

Ok, so now that I've done some mathematics, let me present you with a list, selected more or less at my whim, of how various players rate. I've used career statistics. In decreasing order of OPS'(BB):

1. Albert Pujols 0.73
2. Ryan Howard 0.67
3. Hanley Ramirez 0.59
4. Mark Reynolds 0.58
5. Pablo Sandoval 0.57
6. Joe Mauer 0.55
7. Chris Davis 0.52
8. Jimmy Rollins 0.47
9. Ichiro 0.46
10. Mark Bellhorn 0.46
11. Pete Rose 0.45
12. Erick Aybar 0.40

[Correction: Pablo Sandoval's numbers were incorrect due to a clerical error. I regret the mistake.] Keep in mind these are career statistics, and they seem to be pretty closely related to slugging percentage. That is to say, the higher the slugging percentage, the more sensitive a player's OPS is to changes in walk rate. But I am not sure I have figured out yet why Pablo Sandoval tops the list of players I have selected.

In any event, the fact that this doesn't work out to be close to (if not) the same number for each player suggests to me that OPS may be unfair on ol' Ryan Howard. After all, he's got a wOBA of .382. 

Discussion Question of the Day

I know I didn't give you many links today, so I appreciate your patience. Now is your chance to get me back. Where have I gone completely wrong (as I'm sure I have)?

Also, I'd like to apologize in advance to my high school calculus teacher.