# Is There a Hit-by-Pitch Hangover?

One of the things I’ve been curious about recently and have on my list of research questions is what the ramifications of a hit-by-pitch are in terms of injury risk—basically, how much of the value of an HBP does the batter give back through the increased injury risk? Today, though, I’m going to look at something vaguely similar but much simpler: Is an HBP associated with an immediate decrease in player productivity?

To assess this, I looked at how players performed in the plate appearance immediately following their HBP in the same game. (This obviously ignores players who are injured by their HBP and leave the game, but I’m looking for something subtler here.) To evaluate performance, I used wOBA, a rate stat that encapsulates a batter’s overall offensive contributions. There are, however, two obvious effects (and probably other more subtle ones) that mean we can’t only look at the post-HBP wOBA and compare it to league average.

The first is that, ceteris paribus, we expect that a pitcher will do worse the more times he sees a given batter (the so-called “trips through the order penalty”). Since in this context we will never include a batter’s first PA of a game because it couldn’t be preceded by an HBP, we need to adjust for this. The second adjustment is simple selection bias—not every batter has the same likelihood of being hit by a pitch, and if the average batter getting hit by a pitch is better or worse than the overall average batter, we will get a biased estimate of the effect of the HBP. If you don’t care about how I adjusted for this, skip to the next bold text.

I attempted to take those factors into account by computing the expected wOBA as follows. Using Retrosheet play-by-play data for 2004–2012 (the last year I had on hand), for each player with at least 350 PA in a season, I computed their wOBA over all PA that were not that player’s first PA in a given game. (I put the 350 PA condition in to make sure my average wasn’t swayed by low PA players with extreme wOBA values.) I then computed the average wOBA of those players weighted by the number of HBP they had and compared it to the actual post-HBP wOBA put up by this sample of players.

To get a sense of how likely or unlikely any discrepancy would be, I also ran a simulation where I chose random HBPs and then pulled a random plate appearance from the hit batter until I had the same number of post-HBP PA as actually occurred in my nine year sample, then computed the post-HBP wOBA in that simulated world. I ran 1000 simulations and so have some sense of how unlikely the observed post-HBP performance is under the null hypothesis that there’s no difference between post-HBP performance and other performance.

To be honest, though, those adjustments don’t make me super confident that I’ve covered all the necessary bases to find a clean effect—the numbers are still a bit wonky, and this is not such a simple thing to examine that I’m confident I’ve gotten all the noise out. For instance, it doesn’t filter out park or pitcher effects (i.e. selection bias due to facing a worse pitcher, or a pitcher having an off day), both of which play a meaningful role in these performance and probably lead to additional selection biases I don’t control for.

With all those caveats out of the way, what do we see? In the data, we have an expected post-HBP wOBA of .3464 and an actual post-HBP wOBA of .3423, for an observed difference of about 4 points of wOBA, which is a small but non-negligible difference. However, it’s in the 24th percentile of outcomes according to the simulation, which indicates there’s a hefty chance that it’s randomness. (Though league average wOBA changed noticeably over the time period I examined, I did some sensitivities and am fairly confident those changes aren’t covering up a real result.)

The main thing (beyond the aforementioned haziness in this analysis) that makes me believe there might be an effect is that the post-walk effect is actually a 2.7 point (i.e. 0.0027) increase in wOBA. If we think that boost is due to pitcher wildness then we would expect the same thing to pop up for the post-HBP plate appearances, and the absence of such an increase suggests that there is a hangover effect. However, to conclude from that that there is a post-HBP swoon seems to be an unreasonably baroque chain of logic given the rest of the evidence, so I’m content to let it go for now.

The main takeaway from all of this is that there’s an observed decrease in expected performance after an HBP, but it’s not particularly large and doesn’t seem likely to have any predictive value. I’m open to the idea that a more sophisticated simulator that includes pitcher and park effects could help detect this effect, but I expect that even if the post-HBP hangover is a real thing, it doesn’t have a major impact.