## Tuesday, December 29, 2020

### Splitting defensive credit between pitchers and fielders (Part I)

(Update, 2020-12-29: This is take 2. I had posted this a few days ago, but, after further research, I tweaked the numbers and this is the result. Explanations are in the text.)

-----

Suppose a team has a good year in terms of opposition batted ball quality. Instead of giving up a batting average on balls in play (BAbip) of .300, their opponents hit only .280. In other words, they were .020 better than average in turning (inside-the-park) batted balls into outs.

How much of those "20 points" was because of the fielders, and how much was because of the pitcher?

Thanks to previous work by Tom Tango, Sky Andrecheck, and others, I think we have what we need to figure this out. If you don't want to see the math or logic, just head to the last section of this post for the two-sentence answer.

------

In 2003, a paper called "Solving DIPS," (by Erik Allen, Arvin Hsu, Tom Tango, et al) did a great job in trying to establish what factors affect BAbip, and in what proportion. I did my own estimation in 2015 (having forgotten about the previous paper). I'll use my breakdown here.

Looking at a large number of actual team-seasons, I found that the observed SD of BAbip was 11.2 points. I estimated the breakdown of SDs as:

7.7  fielding talent
2.5  pitching staff talent
7.1  luck
2.5  park
--------------------------
11.0  total

(If you haven't seen this kind of chart before, the "total" doesn't actually add up to the components unless you square them all. That's how SDs work -- when you have two independent variables, the SD of their sum is the square root of the sum of their squares.)

OK, this is where I update a bit from the numbers in the previous version of this post.

First, I'm bumping the SD of park from 2.5 points to 3.5 points, to match Tango's numbers for 1999-2002.  Second, I'm bumping luck to 7.3, since that's the theoretical value (as I'll calculate later).  Third, I'm bumping the pitching staff to 4.3, because after checking, it turns out I made an incorrect mathematical assumption in the previous post.  Finally, fielding talent drops to 6.1 to make it all add up.  So the new breakdown:

6.1  fielding talent
4.3  pitching staff talent
7.3  luck
3.5  park
--------------------------
11.0  total

----

We can use that chart to break the team's 20-point advantage into its components. But ... we can't yet calculate how much of that 20 points goes to the fielders, and how much to the pitchers. Because, we have an entry called "luck". We need to know how to break down the luck and assign it to either side.

Your first reaction might be -- it's luck, so why should we care? If we're looking to assign deserved credit, why would we want to assign randomness?

But ... if we want to know how the players actually performed, we *do* want to include the luck. We want to know that Roger Maris hit 61 home runs in 1961, even if it's undoubtedly the case that he played over his head in doing so. In this context, "luck" just means the team did somewhat better or worse than their actual talent. That's still part of their record.

Similarly here. If a team gets lucky in opponent BAbip, all that means is they did better than their talent suggests. But how much of that extra performance was the pitchers, giving up easier balls in play? And how much was the fielders, making more and better plays than expected?

That's easy to figure out if we have zone-type fielding stats, calculated by watching where the ball is hit (and sometimes how fast and at what angle), and figuring out the difficulty of every ball, and whether or not the fielders were able to turn it into an out. With those stats, we don't have to risk "blaming" a fielder for not making a play on a bloop single he really had no chance on.

So where we have those stats, and they work, we have the answer right there, and this post is unnecessary. If the team was +60 runs on balls in play, and the fielders' zone ratings add up to +30, that's half-and-half, so we can say that the 20-point BAbip advantage was 10 points pitching and 10 points hitting.

But for seasons where we don't have the zone rating, what do we do, if we don't know how to split up the luck factor?

Interestingly, it will the stats compiled by the Zone Rating people that allow us to calculate estimates for the years in which we don't have them.

------

Intuitively, the more common "easy outs" and "sure hits" are, the less fielders matter. In fact, if *all* balls in player were 0% or 100%, fielding performance wouldn't matter at all, and fielding luck wouldn't come into play. All the luck would be in what proportion the pitcher split between 0s and 100s.

On the other hand, if all balls in play were exactly the league average of 30%, it would be the other way around. There would be no difference in the types of hits pitchers gave up, which means there would be no BAbip pitching luck at all. All the luck would be in whether the fielders handled more or fewer than 30% of the chances.

So: the more BIP are "near-automatic" hits or "near-automatic" outs, the more pitchers matter. The more BIP that could go either way, the more fielders matter.

That means we need to know the distribution of ball-in-play difficulty. And that's the data that we wouldn't have without the development of Zone ratings now keeping track of it.

The data I'm using comes from Sky Andrecheck, who actually published it in 2009, but I didn't realize what it could do until now. (Actually, I'm repeating some of Sky's work here, because I got his data before I saw his analysis of it.  See also Tango's post at his old blog.)

Here's the distribution. Actually, I tweaked it just a tiny bit to make the average work out to .300 (.29987) instead of Sky's .310, for no other reason than I've been thinking .300 forever and didn't want to screw up and forget I need to use .310. Either way, the results that follow would be almost the same.

43.0% of BIP:  .000 to  .032 chance of a hit*
23.0% of BIP:  .032 to  .140 chance of a hit
10.3% of BIP:  .140 to  .700 chance of a hit
4.7% of BIP:  .700 to 1.000 chance of a hit
19.0% of BIP:          1.000 chance of a hit
---------------------------------------------
overall average: really close to .300

(*Within a group, the probability is uniform, so anything between .032 and .140 is equally likely once that group is selected.)

The SD of this distribution is around .397. Over 3900 BIP, which I used to represent a team-season, it's .00636. That's the SD of pitcher luck.

The random binomial SD of BAbip over 3900 PA is the square root of (.3)(1-.3)/3900, which comes out to .00733. That's the SD of overall luck.

Since var(overall luck) = var(pitcher luck) + var(fielder luck), we can solve for fielder luck, which turns out to be .00367.

6.36 points pitcher luck (.00636)
3.67 points fielder luck (.00367)
--------------------------------
7.33 points overall luck (.00733)

If you square all the numbers and convert to percentages, you get

75.3 percent pitcher luck
24.7 percent fielder luck
--------------------------
100.0 percent overall luck

So there it is. BAbip luck is, on average, 75 pitching and 25 percent fielding. Of course, it varies randomly around that, but those are the averages.

What does that mean in practice? Suppose you notice that a team from the past, which you know has average talent in both pitching and fielding, gave up 20 fewer hits than expected on balls in play. If you were to go back and watch re-broadcasts of all 162 games, you'd expect to find that the fielders made 5 more plays than expected, based on what types of balls in play they were. And, you'd expect to find that the other 15 plays were the result of balls being having been hit a bit easier to field than average.

Again, we are not estimating talent here: we are estimating *what happened in games*. This is a substitute for actually watching the games and measuring balls in play, or having zone ratings, which are based on someone else actually having done that.

------

So, now that we know the luck breaks down 75/25, we can take our original breakdown, which was this:

6.1  fielding talent
4.3  pitching staff talent
7.3  luck
3.5  park
--------------------------
11.0  total

And split up the 7.3 points of luck as we calculated:

6.36 pitching luck
3.67 fielding luck
--------------------------
7.3  total luck

And substitute that split back in to the original:

6.1  fielding talent
3.67 fielding luck
4.3  pitching staff talent
6.36 pitching staff luck
3.5  park
--------------------------
11.0  total

Since talent+luck = observed performance, and talent and luck are independent, we can consolidate each pair of "talent" and "luck" by summing their squares and taking the square root:

7.1 fielding observed
7.7 pitching observed
3.5 park
----------------------
11.0 total

Squaring, taking percentages, and rounding, we get

42 percent fielding
48 percent pitching
10 percent park
--------------------
100 percent total

If you're playing in an average park, or you're adjusting for park some other way, it doesn't apply here, and you can say

47 percent fielding
53 percent pitching
---------------------
100 percent total

So now we have our answer. If you see a team's stats one year that show them to have been particularly good or bad at turning batted balls into outs, on average, after adjusting for park, 47 percent of the credit goes to the fielders, and 53 percent to the pitchers.

But it varies. Some teams might have been 40/60, or 60/40, or even 120/-20! (The latter result might happen if, say, the fielders saved 24 hits, but the pitchers gave up harder BIPs that cost 4 extra hits.)

How can you know how far a particular team is from the 47/53 average? Watch the games and calculate zone ratings. Or, just rely on someone else's reliable zone rating. Or, start with 47/53, and adjust for what you know about how good the pitching and fielding were, relative to each other. Or, if you don't know, just use 47/53 as your estimate.

To verify empirically whether I got this right, find a bunch of published Zone Ratings that you trust, and see if they work out to about 42 percent of what you'd expect if the entire excess BAbip was allocated to fielding.  (I say 42 percent because I assume zone ratings correct for park.)

(Actually, I ran across about five years of data, and tried it, and it came out to 39 percent rather than 42 percent. Maybe I'm a bit off, or it's just random variation, or I'm way off and there's lots of variation.)

-------

So what we've found so far:

-- Luck in BAbip belongs 25% to fielders, 75% to pitchers;

-- For a team-season, excess performance in observed BAbip belongs 42% to fielders, 48% to pitchers, and 10% to park.

-------

That 42 percent figure is for a team-season only. For an individual pitcher, it's different.

Here's the breakdown for an individual pitcher who allows 700 BIP for the season.

6.1  fielding talent
7.6  pitching talent
17.3  luck
3.5  park
---------------------------
20.2  total

The SD of pitching talent is larger now, because you're dealing with one specific pitcher, rather than the average of all the team's pitchers (who will partially offset each other, reducing variability). Also, luck has jumped from 7.3 points to 17.2, because of the smaller sample size.

OK, now let's break up the luck portion again:

6.1  fielding talent
7.6  fielding luck
7.6  pitching talent
15.5  pitching luck
3.5  park
---------------------------
20.2  total

And consolidating:

9.75 observed fielding
17.3  observed pitching
3.5  park
---------------------------
20.2  total

Converting to percentages, and rounding from 31/69:

23%  observed fielding
73%  observed pitching
3%  park
---------------------------
100%  total

24%  observed fielding
76%  observed pitching
---------------------------
100%  total

So it's quite different for an individual pitcher than for a team season, because luck and talent break down differently between pitchers and fielders.

The conclusion: if you know nothing specific about the pitcher, his fielders, his park, or his team, your best guess is that 25 percent of his BAbip (compared to average) came from how well his fielders made plays, and 75 percent of his BAbip comes from what kind of balls in play he gave up.

------

Here's the two-sentence summary. On average,

-- For teams with 3900 BIP, 47 percent of BABIP is fielding and 53 percent is pitching.

-- For starters with 700 BIP, 24 percent of BABIP is fielding and 76 percent is pitching.

------

Next: Part II, where I try applying this to pitcher evaluation, such as WAR.