A few weeks ago, I had the honor of presenting some research at SaberSeminar in Boston to a room full of sabermetric luminaries (also, Mike Ferrin was there!) and one guy who was really confused about why no one was talking about swords. The seminar itself was a joy but, of course, given that I was in Boston and within walking distance of Fenway Park, it would have been a shame not to go to a game. It was my first time at Fenway, and if you haven’t yet been, it is worth the trip from anywhere. TV doesn’t do it justice.

I sat in the center-field bleachers between Ben Lindbergh’s girlfriend and former Kansas City Royals pitcher Brian Bannister. Bannister was one of the first active major leaguers to talk about how he used sabermetric principles and also happens to be a really nice guy. We ended up discussing our proposals to our wives in buildings that no longer exist.

In the fifth inning of the game itself, David Ortiz did one of those things that David Ortiz does. He smacked a ball down the right-field line and, because this was Fenway Park, he wrapped it around the Pesky Pole for a home run. Brian remarked that he remembered a time when Ortiz hit a similar home run off him. He said that he felt angry after it happened, but that he realized he needed to calm himself down. Lindbergh, hearing our conversation, leaned over and asked Brian, “Well, what do you do if you’re a pitcher in that situation?” Brian just shrugged. His message was clear. What can you do, really? “I guess you just throw the next pitch,” I said.

From a perfectly rational point of view, “throw the next pitch” is the correct answer. You can’t do anything about what just happened. The home run will not be coming back (unless it’s at Wrigley) and whether you like it or not, there’s another batter walking toward the plate who gets to take his swings. Your job is to get him out. There’s no use crying over spilled milk. Of course, no one is perfectly rational. The sting of disappointment will be there. Baseball players are human. To suggest they would be completely unaffected is silly.

But does it affect the outcome for the next hitter? Brian might have been angry about giving up the home run to Ortiz, but when Manny Ramirez strode to the plate, did it affect his fastball? This is another case where announcers and fans alike love to play amateur psychologist. The camera zooms in on the pitcher’s eyes and everyone tries to tell how the pitcher was affected by the misfortune that just befell him. (For the record, Ramirez proceeded to fly to left and Kevin Youkilis struck out)

Why settle for amateur (read: usually awful) psychoanalysis when you can actually look at the data with a real-life psychologist? You know what that means …

Warning! Gory Mathematical Details Ahead!

I looked at all starting pitchers from 2009 to 2013 who had just given up a home run and compared it to times when the batter before had done something other than hit a home run, but there was no one on base (just like after a home run). Per usual, I restricted the sample to pitchers who had faced at least 250 batters that year when they faced off against a batter who had been to bat 250 times that year. Using the log-odds ratio method, I created a control for the chances that this particular batter-pitcher matchup would end in a strikeout (or a walk or another home run). I controlled for whether the pitcher had a handedness advantage and what his pitch count was at the start of the at-bat. Plug everything into a bunch of logistic regressions and see what happens.

What happens when a pitcher gives up a home run? He actually gets a little better. Sorta. Most outcomes move in the direction that the pitcher was hoping for. Strikeouts (remember, relative to expectations, once we’ve controlled for batter and pitcher talent, handedness, and pitch count) go up, walks go down, outs in play go up, and singles go down. The problem is that the p-values in those regressions are all teetering on the edge of significance, so we need to be careful in how much faith we put in them. However, the chances that a pitcher will give up an on-base event of any kind does significantly decrease, to the tune of about seven points of OBP. In general, pitchers may be angry with themselves after giving up a home run, but they respond to it by saying, “Well, I won’t make that mistake again.” Giving up a home run doesn’t make our pitcher invincible, but it does actually make him a little better.

It’s not just learning from the mistakes that led to a home run that make a pitcher better. I looked at what happened to pitchers after they gave up any sort of on-base event (other than a home run) in the prior at bat vs. recording an out. This time, I compared plate appearances in which the pitcher was pitching with runners on base. In that case, the previous at-bat ending in an on-base event again actually made the pitcher better than expectation (or the batter worse?). In this case, walk and strikeout rates fell while single and out-in-play rates bumped up a bit. The net effect was a slight (a couple of points) decrease in expected OBP, again controlling for batter-pitcher matchup, handedness, and pitch count.

But wait a minute. That’s what the average pitcher does. Shouldn’t we be a little more nuanced here? After all, there seem to be pitchers who fall apart after a home run. You can see it in their faces. Amateur psychology, confirmation bias, and cherry picking make for a dangerous cocktail. Let’s take this down to an N = 1 approach and look at what the data actually say.

Again, for 2009 to 2013, I isolated each pitcher-season and ran an individualized regression for each player-season. (That’s to say Bronson Arroyo’s performance from 2009 got its own regression, as did his 2010, 2011, 2012, and 2013 seasons.) Perhaps there are some pitchers who show a particularly big effect after giving up a home run?

After generating the regressions, I calculated what the model saw as the effect size for each player-season. In the same way that the average pitcher was seven OBP points better than expectation after a home run, maybe one player was four better and another was 12 worse. But before I really looked deep into the heart of those results, I first looked at whether this appeared to be a repeatable skill. After all, the average starting pitcher gives up a couple of dozen home runs per year. Given how long it tends to take pitching stats to stabilize, it seems silly to examine one season of data and take the pitcher's reaction in the batter after 20 or so home runs as evidence of much of anything. In fact, I ran an AR(1) intra-class correlation on the five years of data in my data set. (For the uninitiated, think of it as a year-to-year correlation, just that you can put five—or more—data points into it.) The intra-class came out just a hair away from zero. Same basic results for modeling what happens after an on-base event.

That means that even if you show that a pitcher in 2009 seemed to fall apart after giving up a home run, and that the next guy's at-bat in those 22 plate appearances hit .600 against him, it tells you nothing about what he will do next year. You haven’t identified a character flaw. You’ve identified an interesting case that is likely to be freak accident of small-sample weirdness.

Maybe All That Coaching Actually Worked …

The problem with amateur psychology is that it creates lovely two-dimensional models of human behavior:

Pitcher gives up home run  
Pitcher is sad  
But brave pitcher continues to throw ball good  
Hooray America!

Maybe that is the case for some players, but the truth is that we won’t be able to diagnose him from his performance.

What is interesting to note is that while we might not be able to tell much about individual pitchers, the aggregate numbers show something a little different than we might expect. Pitchers who give up home runs are actually a little better than we would expect in their next plate appearance. Or maybe the batters get worse: "Oh good, Ortiz already hit the homer, I can take this one off." While we might expect a pitcher who has just given up a home run to be in a short-term funk, the results suggest the opposite. Perhaps he corrects for some mistake he had been making. Maybe over the years, he’s heard, “Don’t let it get to you!” so many times from his coaches that he’s learned not to let it get to him.

The reality of human psychology is that it’s much more complex than simple models make it out to be. Yes, a pitcher is probably sad about giving up a home run, but while the hitter trots around the basepaths, he has a few moments to pull another skill out of his back pocket: the ability to calm himself down. It’s an easy thing for a coach to say, “Don’t let it get to you!” and yes, it does take some skill to do it, but the reality is that most adults manage to develop that particular skill. If they don’t, they generally don’t function well as adults, much less major-league pitchers. Instead, our pitcher got some very real feedback about something that obviously doesn’t work, and he is smart enough to incorporate that feedback.

There’s much more going on in a baseball game than fits in a bumper sticker–sized model.

Thank you for reading

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Would the fact that better hitters tend to hit more home runs cloud the numbers here? By that I mean it's more likely over the sample period that Miguel Cabrera hits a home run than the player that follows him and Miggy hits more home runs than most players so the odds of the 'following batter' hitting a home run, or even getting on base, would be decreased. Additionally, was any adjustment made for pitchers batting following a home run? A substantial percentage of home runs by an NL number 8 hitter are going to end up followed by an out. Last quirk, does pitching from the stretch rather than the wind-up impact the data at all?

Thanks and keep up the great work
That's a really important observation and probably the biggest issue in this sort of research.

The log-odds ratio method specifically corrects for that though. Going into each at-bat, the regression "knows" that the batter has a certain seasonal stat line and adjust expectations based on that knowledge.
Obviously, there are far more outs than hits to go around.

So if the first outcome is a hit, a big hit, a home run even, it's logical that the chances the next batter will make an out increases, not decreases.

Isn't that akin to regression to the mean.?
Well, through log-odds, we have a reasonable estimate of the probabilities of what is about to happen next. We know that Smith makes outs 70 percent of the time and that Jones induces outs 68 percent of the time. The rest is just a little #GoryMath.

In the strict sense, because we're looking retrospectively, we're working in a closed system, so the subset of "plate appearances that follow on-base events" is by definition a set where we've at least eliminated one plate appearance (immmediately before) that featured an on-base event. However, in a sample that large (5 years worth of play-by-play) the effect will be negligible.
Each AB is an independent event.
Here is the definition of an independent event: "An event whose occurrence or non-occurrence is not in any way influenced by the occurrence or non-occurrence of another event."

Okay, that works for flipping coins, but pitching a baseball game does not result in a series of random outcomes.

Prior events in baseball can easily influence the outcome of subsequent events, for good or for ill. Even the fact that a pitcher eventually tires as he faces more batters can easily have an affect upon upcoming at bats.
A great read Russell, thanks!
Enjoyed this.

Does the fact that some pitchers will get removed from the game after a home run have a significant survivorship effect?
Thanks. I only looked at starters, so at that point, if he is removed, the regression doesn't care any more. It's possible that if you have a game where the starter's last act is giving up a three-run shot (and obviously, his manager thinks that he is completely gassed), you could make the case that's a survivorship bias, but that's a problem of missing data bias at that point, because we don't know what would have happened if his manager had left him in.
In the insurance world, we call this behavior "getting religion." After a driver has a DWI and the driver's insurance rates skyrocket, the driver minds his p's and q's and pays more and has significantly better driving habits (so no drinking and driving). Driver finds religion, insurer pockets profits. Similar with doctors who have bad habits. The doc has an incident, pays beaucoup bucks for insurance for a couple of years, does real well, goes back to the standard market. My company (who loves to insure these troubled physicians) has made money and wishes the doc well in the return to a standard insurer.

Another example of life imitates baseball.