More on pitcher fatigue

A few weeks ago, I started to take a look at the 100 pitch limit on starting pitchers or more to the point, pitcher fatigue more generally.  I was able to model a set of regression equations that predicted what an average hitter would look like against an average pitcher at different pitch counts.  Sure enough, as a pitcher throws more pitches, the expected AVG/OBP/SLG numbers begin to inch their way up.  While that’s a good start, there might be other factors around fatigue that would be useful to look at.
Certainly, pitchers are somewhat less effective when they are tired, but a high pitch count within the game is not the only way that a pitcher can become tired.  A few other factors could conceivably be involved.  The pitcher could not have gotten proper rest (or alternately, he could have gotten a lot of rest).  He might have thrown a lot of pitches last game and that’s having a carry-over effect.  He might have a lot of “mileage” on that arm this year.  And then there’s a certain strategic fatigue.  Not only is a pitcher throwing a lot of pitches, but eventually, he comes around to re-face some of the same batters again.  Certainly, those batters have learned something from the first (or second) time around.  Then again, maybe he’s learned something.
Gory detail alert.  Like in my previous study, I’m using a similar methodology, primarily based on binary logit regression.  I’ve taken my dataset from 2000-2006 and selected only those pitchers who were starters.  Using the odds ratio method, I’ve calculated what the odds ratio of different outcome events were of the particular pitcher/batter matchup (and taken the natural log of that odds ratio).  That expected outcome entered into the regression is a nice way of controlling for batter and pitcher quality.  I still have the pitch count from the beginning of the plate appearance entered into the regression.  So, I’ve set up a pretty stringent statistical test.  In order for a variable to make a signficant impact, it will have to show that it predicts to the odds in the regression while vying for variance with the batter/pitcher quality control and raw pitch count.  Let’s see if any of the other “fatigue” variables actually do have any predictive power over any of the outcomes I’ve looked at.
First, let’s look at days of rest.  I coded for how many days it had been since the pitcher’s last start using some basic date arithmetic.  However, I only looked at cases where the “rest number” was 10 or less.  The reason?  A pitcher who makes starts ten or more days apart is likely to have been a spot-starter (and may have pitched in long relief in the interim) or a guy who got sent down to AAA and then came back to the majors (and probably pitched at AAA), or was on the DL (and let’s be careful with guys coming off the DL).  10 days allows for the occasional skipped start or rain out or off day or All-Star break that stretches things beyond the usual 5th day.  While better pitchers are more likely to be in that bucket (they pitch more frequently, because they’re better), we’re already controlling pretty well for pitcher quality.
I coded all plate appearances as being (or not being) a K, BB, HBP, ROE, HR, 1B, XBH, or out-in-play.  I then used that as the basis for my binary logit with the batter/pitcher control and pitch count entered (which was significant for all the regressions I ran throughout this article), plus the days of rest that the pitcher had, and the interaction term between pitch count and days of rest.  Maybe there’s an additional penalty in throwing extra pitches if the pitcher hasn’t had proper rest.  Did the amount of rest predict to a greater or lesser likelihood of any of the outcomes under study?  One of them (and it technically wasn’t significant, probability on that one was actually .09)  Home runs.  Better rested pitchers have a lowered chance of giving up a HR.  Interestingly enough, adding in rest accounted for all of the variance that had been previously accounted for by pitch count.  Pitch count was reduced to absolutely zero effect.  (And I don’t just mean “not significantly different from zero.”  I mean zero.)  Rest did not have any impact on any of the other outcomes after pitcher/batter matchup quality and pitch count were controlled.
Next, let’s look at how much mileage is on the pitcher’s arm.  I looked at how many pitches the pitcher in question had thrown in that season up to that point.  (The beauty of my program is that I was able to get a real time number at each batter!)  Plus, I added in pitch count and the interaction between pitch count and mileage.  (Gory detail: I restricted my sample to pitchers who had a maximum rest number of 10 for the whole season.  This allows me to weed out guys who spent time in AAA and collected more mileage that would not be reflected in the MLB data file.  I also made sure that the pitcher in question had started one of the team’s first ten games of the season.  This kicks out all the guys who started the year at AAA.)  Does mileage have an effect?  Yeah.  In the last article, I found that as pitch count goes up in a game, walks actually become less likely.  The same type of effect holds true for mileage on the arm.  Walks become less likely as the season wears on, after controlling for expected outcomes and pitch count within the game.  But that’s it for mileage.
What about how many pitches the pitcher threw in the last game.  Maybe there’s a hangover effect for pitchers who go deep into a game on some of these variables.  Same basic set up as above statistically.  Again, there’s an effect on walks, and as the pitcher throws more pitches in his previous start (had to have been less than seven days previous), which would likely make him a tinge more tired today, the less likely it was that he would give up a walk.  Tired pitchers seem to either not fool around with pitches or make more hittable mistakes.  Instead, there was an effect where more pitches the game before led to more singles in the next game.  There’s also a small interaction effect for being hit by a pitch (becomes more common with more fatigue, but it’s not much of an effect).
Then there’s the number of times that the pitcher has gone through the batting order.  I coded for how many times in this game the pitcher had faced this particular batter and put that into the regression.  Again, here I’m coding for batter and pitcher quality, and in-game pitch count (which is going to be somewhat correlated to “times through lineup”.)  So, now we’re looking at the effects of multiple times matching up in the same day in near isolation.  What happens as the game wears on and the lineup turns over a few times?  There were some pretty strong effects for both walks and strikeouts.  The probability of both went down (for what it’s worth, so did the probability of a hit batter.)  What went up?  The chances of some sort of out on a ball hit in play.  It’s hard to tell whether this one is a cause or an effect.  What type of pitcher is going to be allowed to pitch that deep into a game?  One who keeps the pitch count low.  If that’s the goal, high K and BB totals aren’t going to do much for you.  And if you’re getting a bunch of outs on balls in play, the manager will probably let you stay in for a while because the perception is that the pitcher deserves the credit for that.
A few notes on what wasn’t significant.  I found no significant effects for extra base hits (doubles and triples) or for reached on errors.  (Sometimes errors just happen.)  The strongest effects were found in the DIPS relevant stats: K, BB, HR, and HBP.  It makes sense that if those are the things that a pitcher controls, then those are the things that would be most prone to be affected by fatigue.  However, there were some effects for generating outs in play, although that more means that instead of striking batters out, pitchers are seeing the ball put into play more, and about 70% of all balls in play are turned into outs.  I smell a follow up study looking only at balls in play.  But, it looks like the more a pitcher tires, the more likely it is that he’s going to need some help from the seven friends behind him (and the catcher) to get him his outs.  So, the hallmark of a tired pitcher is one who is seeing the ball put into play more so than he would normally expect, given everything else we know about him.

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