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Last week at FanGraphs, Eno Sarris posted an article that made me think. Sarris talked to a few players about the idea of hitting the ball out in front of the plate as a way to generate power. Aggression leads to home runs. Several of them cited the idea that instead of waiting back on a ball, sometimes it’s best to just go out and get it. Sarris then provided some data that showed that exit velocities, ideal launch angles, and home run probabilities are at their highest when a batter makes contact a couple of inches in front of home plate.

It’s an interesting data point, although it's just one point of light that needs more data around it before we start prescribing behavioral changes. After all, there are some other explanations that could be true. It’s possible that players with more power just tend to also be more aggressive hitters. It’s possible that if a hitter swings early with the confidence that must be behind a swing that connects out in front of the plate, it’s because he felt that his pre-pitch guess was correct based on whatever information he was able to gather out of the pitcher’s hand, and what we’re really seeing is the results of a correct guess.

Maybe we also need to consider that the only balls in the data set are those that were contacted. Our batter went out and got the ball one time, but what about the seven other times when swinging out front meant that he didn’t wait back long enough to see that the ball was about to break and he was left with just a silly-looking swing and miss?

But on the other side, this is something, and maybe the idea of being aggressive really is a good way to generate more power. It’s the sort of thing that requires more evidence. Well …

Warning! Gory Mathematical Details Ahead!

I pulled some stuff out of the archives for this one. The first sabermetric writing that I ever did was to create what is now a rather quaint and outdated metric for plate discipline. In my defense, I wrote it in 2007, which pre-dates even PITCHf/x data. It was powered by pitch-by-pitch outcome data (swinging strike, called strike, ball, etc.) available on Retrosheet and used signal detection metrics to model plate discipline. Plate discipline was essentially defined as “strikes are bad” but the nice thing about signal detection theory is that it produces two different numbers that each tell us something interesting about a hitter.

In a classic signal detection theory framework, there might or might not be a “signal” present in the environment. The classic case is trying to design a test for a disease. We want a test that says “yes” if the disease is present and “no” if it's absent. Since there are two ways for the test to be correct (correctly identifying the disease, correctly saying no when it's absent), it means that there are two different ways for it to be wrong (the dreaded Type I and Type II errors that you’re not totally sure if you remember which is which from your intro stats class).

In the case of the disease detector, the test can say that a person is infected when in reality, they are healthy (Type I) or the test can say that they are healthy, though they are actually sick (Type II). Ideally, we want a test that makes no errors, but perfection is hard to come by. Just ask Rich Hill. There is a number that can be calculated within signal detection theory that tells us how well the test performs in minimizing errors (the measure’s sensitivity), but another one that tells us which type of error (I vs. II) the test is more likely to make (response bias).

In my strike zone metric, I used whether or not the batter swung as the response I was looking for. A Type I error is essentially a swinging strike (the batter responded, even though that was a bad idea) and a Type II error is a called strike (the batter did not respond, even though he should have). That leaves us with a response bias metric that tells us something interesting. A response bias of 1.0 is perfectly balanced. A player will always have some mistakes on his resume, and some players will be better at strike zone judgment than others, but a player who has a response bias over 1.0 is more likely to swing than the population, and in doing so, he’s going to hit a few extra balls, but he will probably purchase those extras with a few extra swings-and-misses.

A player who has a response bias under 1.0 is a “patient” hitter, who will let a few go by. The marginal called ball that he gets because of his passive ways is bought at the price of a few extra called strikes. But that response bias itself will tell us how aggressive a hitter is, taking out the noise of whether he’s actually good at avoiding strikes. A hitter who swings 95 percent of the time and misses on all of them is just a hacker. A guy who swings 95 percent of the time, but who can put the ball into play on many of those swings, is a very different hitter.

The response bias metric is not the same thing as the measure that Sarris used to define “aggression,” and I’m OK with that. In fact, I consider it a feature, rather than a bug. The two do speak to a similar mindset. If a player’s first instinct is “swing” he will have a lot of swings and misses, but he’s going to be there to meet a few extra pitches ahead of the plate in the putative power zone that Sarris identifies.

Now that we’ve defined aggression, we need a test to see whether it's actually related to power. I used simple HR/FB rate on a seasonal level, using all players from 2012-2016 (minimum 250 plate appearances). The longitudinal aspect is important. We know that certain players are just better at hitting the ball over the fence than others. When we do cross-sectional analyses between aggression and power hitting, you can’t rule out a spurious correlation. What we really want to see is that within the same hitter, as aggressiveness goes up or down, power hitting goes along in the same direction, relative to what came the year before.

I used a technique known as mixed linear modeling, which I think is particularly helpful for these questions. In this case, I have up to five data points for each hitter, but because I know that the ability to hit for power is mostly in the grasp of the hitter himself (rather than a function of the pitchers whom he faced) we know that each of those five measures will be correlated with each other. (For the initiated, I used an AR(1) covariance matrix to model that. The rest looks like a simple regression with our response bias variable for a player’s season predicting his HR/FB rate.)

The answer: Yeah, the two are related, and we can be a little more confident that it’s not just some sort of cross-sectional weirdness. Players who show more “aggression” in that they swing and miss a little more than they should, do hit more balls out of the yard when they do hit fly balls.

But … using the same technique, we see that they also strike out more, walk less, and make less contact per swing. I believe the expression is “selling out for power” or “swing hard in case you hit it.” So, while being a bit more aggressive probably does lead to more power, it has some unfortunate side effects.

B-E Tentative?

There’s other evidence that at least points toward the idea of aggression leading to more power. When looking at the home run surge a few months ago, I found that it was accompanied by players being more willing to swing and miss, although when they did make contact, it was more likely to be on a fastball. It’s possible that what we’re seeing is hitters sitting on a specific pitch (in this case, the fastball), and when they see it, going to get it.

If you’re not worried about waiting that extra half beat to let the ball go deeper in the zone so that it might reveal a bit more of its flight path, you might as well swing with all your might. And in a world where the ball itself seems to, uh, carry a little more than it used to, the reward when you make contact is a little greater. This is one of those articles where we don’t solve the problem, but we can at least see a few strains of evidence all pointing in the same direction.

Whether it makes sense to actually prescribe players to try to be more aggressive is another matter. Sure, being more aggressive is arguably just a matter of a different mindset, but there’s another methodological issue that we have to consider. Maybe all of this points to the fact that the players who tried being more aggressive are the ones who had reason to believe that they might actually get something out of it. The problem with baseball is that it stubbornly refuses to behave like a randomized controlled trial. But from what evidence we have, we can’t rule out the idea that aggression is the secret to power. We just can’t really say that it is yet, either.