On the 100 pitch limit

Who decided that starting pitchers were only allowed to throw 100 pitches?  OK, so we can use the eyeball test to see that pitchers usually aren’t usually as sharp in the seventh inning as they were in the first.  And there surely must be a point where it would behoove the manager to make that walk out to the mound to politely ask the starter to leave, but what’s the deal with 100 pitches?  It’s a nice round number, but does it have any real validity?
For the rest of this article, let’s assume that we are talking about Game Seven of the World Series and that we don’t need to think about burning out this particular pitcher’s arm for five years down the road.  (Or alternately, let’s pretend that Dusty Baker is the manager.  Sorry to all my Cub fan readers out there… that’s probably a sore subject.)  In other words, I’m only interested in what happens during this game and no other.  When does a pitcher start to lose his mojo?  Is it at 100 pitches?  More?  Fewer?  Again, the eyeball test tells us that it’s something of a gradual process, but if my goal as the manager is to win the game, when would I be best served to take my starter out in favor of a reliever?  (Depends on the bullpen, I know.)  Let’s assume that my pitcher is league average and that the opponent’s line up is made up of 9 league average hitters.  (How did they make it to the World Series?)
There are seven basic outcomes to an at-bat.  Of course, it’s a bit more complex than this, but in general, you either strike out, walk, single, double, triple, hit a home run, or make an out on a ball in play.  You can only choose one of the above.  I took my giant 2000-2006 data base and isolated all the plate appearances (750K+) involving the starting pitcher.  Since pitching changes are rarely made within an at-bat, I calculated the starter’s pitch count at the beginning of each appearance (so he starts out at zero on the first batter and goes from there.)
If you don’t want the gory details, skip the next paragraph.
How to tell if pitch count actually makes a difference (and how much of one at that) in determining which outcome the batter will choose.  First, I calculated the rate at which the batter did the first six outcomes over the course of the year (BB, K, HR, 1B, 2B, and 3B, although I combined 2B and 3B into one simple XBH category) when facing the starter.  I then calculated the rate at which the pitcher allowed each event for the year overall.  I turned those probabilties into odds ratios (OR = p / 1-p), and used the odds ratio method to figure out what was the odds ratio of the probability of that outcome for this batter pitcher matchup (xOR / lgOR = bOR / lgOR * pOR / lgOR).  I took the natural log of that expected odds ratio.  I then plugged that into a binary logit regression, which already works with the natural log of the odds ratio anyway.  Ideally, the coefficient of the ln(OR) predicting in a binary to the outcome should be about 1.0 (since in theory, it’s the same thing.)  But, I also added pitch count into the regresstion to see what happened.  My efforts gave me five regression equations.  (If anyone wants them, e-mail me.)  Again, the actual probabilities will vary with the actual hitters and pitchers, but here I’m assuming league average all around. 
Some of the numbers that I calculated were a little off due to the fact that I ignored HBP, sac bunts, and reached-on-errors (plus a few other things).  Individually, they are hard to model since they are low-frequency events (something that binary logit has a hard time with), but totalled up, they were enough to throw things off a bit.  For example, in 2006, the league average OBP was .337, although most of the values that my model calculated were in the .305 range.  UPDATE: See below, I re-did the numbers and now they make sense.
(One really interesting finding before we go on.  Did you know that walks actually become less likely as the pitcher throws more pitches?  ‘Tis true.  Fewer walks, but more hits.)
Let’s take a look at strikeouts as an example.  The league average starter struck out 15.88% of the batters he faced in 2006 (and conversely, the league average batter struck out 15.88% of the time when facing the starter).  I used the regression equation that the data generated and figured out what the expected probability for a strikeout was at each pitch count.  Again, I’m assuming that we’re dealing with an average hitter facing an average pitcher.  Below I’ve listed the pitch count (at the start of the at-bat), and the probabilities that the confrontation between a league average pitcher vs. a league average batter would result in a strike out.

Pitch count  K%
0                   .1731
10                 .1688
20                 .1646
30                 .1605
40                 .1564
50                 .1525
60                 .1486
70                 .1448
80                 .1411
90                 .1375
100               .1339
110                .1304
120                .1271
By about 100 pitches, the average pitcher has lost about 4 percentage points from his strikeout rate, which is about a quarter of what he had to begin with.  I did the same for walks, singles, home runs, and extra base hits.  Since there’s about a 9:1 ratio of doubles to triples in baseball, I sliced the extra base column up that way.  I calculated AVG/OBP/SLG expected for an average hitter facing an average batter.  Again, these numbers are lower than might be expected due to some of the methodological problems I ran into.  If I have a moment I might try to correct for it.
UPDATE: I had some time and re-ran the numbers (I also found another boo-boo that I made.)  These numbers below are updated and they make much more sense.  Again average hitter vs. average pitcher.
0     .2545 .3248 .4023
10   .2568 .3263 .4068
20   .2591 .3277 .4113
30   .2614 .3291 .4158
40   .2637 .3306 .4203
50   .2659 .3320 .4249
60   .2682 .3334 .4294
70   .2704 .3348 .4340
80   .2726 .3362 .4385
90   .2749 .3376 .4431
100 .2771 .3390 .4477
110  .2793 .3404 .4523
120  .2815 .3418 .4569
As the game wears on, every ten pitches seems to tip the OBP balance toward the batter by about a point and a half or about .15%.  At some point, there comes a time where there is a more effective option in the bullpen.
Now, what’s the point of using a statistical model rather than a simple look at the actual outcome data chopped up by pitch count?  There are a few advantages.  One is that actual outcome data has a selective sampling issue.  Pitchers who are more likely to be allowed to throw 100 pitches in a game are likely to be better pitchers, or at least pitchers having better games.  We can simultaneously control for the differences in pitcher (and batter) quality while looking at the effects of pitch count.  Using a model also allows us to project out to see what happens at pitch counts that pitchers don’t often get to.  Perhaps we might build a case that a pitcher would actually be OK to throw 150 pitches (again, concerns about his future aside.)
I consider this to be a first stab at an area that hasn’t really been studied much (and I plan on taking a few more stabs).  Pitchers do get tired and it does seem to affect their performance.  How much?  Does it matter what his body build is?  How much rest does he really need?  Is it a different ballgame for relievers?  And I’m just doing it with Retrosheet.  Once Pitch F/X gets going full speed, we can have some real fun looking at this.


15 Responses to On the 100 pitch limit

  1. tangotiger says:

    Great job! I posted a comment on my blog (click name).

  2. jwjays says:

    Hm …
    What’s causing what, though? Some pitchers are more effective at higher pitch counts than other, obviously, but what if they are more effective at those higher pitch counts because they are allowed to get there more often? Their arms are used to pitching with that count so can stay more effective there?
    Take, say, Roy Halladay. Halladay is a great pitcher at high pitch counts, and the manager comes for him when he reaches about 120 pitches or is labouring. Taking the 76-100 pitch count effectiveness, would it not be reasonable to think that Halladay would be less effective in those counts if he had a strict 100 pitch limit as opposed to that 120 limit that he usually has? It would be interesting to look at effectiveness vs. percentage of average pitch count and see if it varies less than the effectiveness vs. pitch count that you have calculated.
    I’m not very convinced that this 100 pitch limit guideline has much value, personally.

  3. Pizza Cutter says:

    There are a few other issues that I hope to investigate. Body-type (body-mass index?), rest, mileage (how many pitches has he thrown all year), and practice (how many pitches does he usually throw? Perhaps his arm has built up strength?)

  4. What interests me is where this idea came from. Why 100? Why not 90? Or 110? Or 105? Why is it that we automatically assume 100 pitches should catalyze a series of red flags thrown up. Maybe there are some psychological reasons there? Hint, Hint.
    In any event, awesome job.

  5. hilarie says:

    I think you ought to run that same PC AVG OBP SLG chart against times facing starting pitcher in the game, without respect to pitch count. Comparing the two, you might have something. Or an indication that there isn’t anything. Maybe isolating pinch hitters in the model would help.

  6. Josh says:

    I know this is completely irrelevant to the point of the article, and I apologize for that, but couldn’t reaching on an error be included as a basic outcome of an at-bat?
    On topic, It also would be interesting to know if there’s any correlation between the manager and the length of his starter’s starts. Are guys like Dusty Baker and Grady Little really more prone to leaving a guy out there, or is that just perception?

  7. Pizza Cutter says:

    Josh, the updated numbers now include ROE. As to the questions about managers, that’s something that can be tested. In fact, I might be able to do that in fairly short order…

  8. […] When does a starting pitcher lose his stuff? – (Source) […]

  9. dan says:

    A note on the Shandler USA Today article…. he didn’t provide a link. I remember reading it (both of them, or was it 3?), but I can’t find them now to go back and read. Not that big a deal, but I’m just lookin out for ya
    Anyways, congrats on the shoutout.

  10. Pizza Cutter says:

    From 2000-2006 the following managers had the highest average pitch counts (min 50 games managed):
    Ozzie Guillen: 100.20
    Johnny Oates: 98.98
    Dusty Baker: 98.17
    Terry Francona: 98.11
    Bob Melvin: 97.58
    Easiest on the arms: Bob Boone, Jimy Williams, and Lloyd McClendon.
    Grady Little came in middle of the pack at 92 and some change.

  11. dan says:

    Well managers with better pitchers will have higher average pitch counts (in general). Bad pitchers get knocked around and are taken out after 60 pitches while good ones go deep into the game throwing over 100.

  12. Pizza Cutter says:

    A sensible hypothesis.

  13. kevin says:

    The effect might be a little weaker than your raw data suggest, as the first 10 pitches are against the opponents (presumably) best hitters. Maybe even the first 20. So the dropoff isn’t quite as steep. Very interesting work.

  14. Pizza Cutter says:

    Kevin, I controlled for the fact that those first ten pitches are likely to be against better hitters. The numbers I posted assume an average hitter and pitcher, but the point is that the pitcher will actually hold any played under his expected numbers early in the game, while a batter will exceed his usual numbers later in the game.

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