# Hitter Aggression, Visualized

Earlier this year I introduced a way of analyzing hitters’ plate coverage. Plate coverage is fine and interesting, but it measures *outcomes*. More often we want to understand a hitter’s *approach*; that is, their plate discipline*. *We have metrics like Z-Swing% and O-Swing% to measure a hitter’s discipline, and we can visualize this discipline with heat maps. But I was interested in a more fine-grained analysis: what’s the size of the box a hitter thinks they can do damage in? And how can we compare this box between players?

Aggressive hitters think they can do damage on pitches inside a large box; patient hitters focus on a smaller box and attempt to do damage there. Think of the cover of Ted Williams’ book, *The Science of Hitting*, which shows his ideal plate discipline:

This book came out in 1971, yet still guides players today. Kris Bryant’s father trained his son on Williams’ advice growing up. Kris made his dad (and Williams) look smart, as he has recorded a 143 wRC+ in the majors, and won the 2015 National League Rookie of the Year and 2016 NL Most Valuable Player awards.

I was curious how much plate discipline matters today. So I did some research and took a few interesting side trips and detours.

### Defining Aggression

To study plate discipline, I defined a metric called Aggression Factor in the following manner:

- Using Statcast data for each hitter in each year, I found the median absolute deviation along the horizontal and vertical axes of pitches they’d swung at.
- I multiplied these distributions together to get a sense of the box, in square inches, a hitter thinks they can do damage in.
- I divided this number by the total number of swings and multiplied by 100, to normalize among hitters with different amounts of playing time. This resulting number is the hitter’s Aggression Factor (AF). Lower AF’s indicate the hitter swings mostly at pitches in one area; higher AF’s indicate a player attempts to reach pitches in a variety of locations.
- I filtered out hitter-seasons with fewer than 400 swings, to prevent role players from polluting the sample too much.

As an example, consider Joey Votto and Justin Bour. Both are left-handed NL first basemen. In 2017, Votto has an AF of 4.2; Bour has an AF of 12.5. The plots below show pitches they’ve swung at:

Bour’s plot shows he’s the more aggressive hitter. Votto’s plot demonstrates the patience for which he’s known.

### Least- and Most-Aggressive Hitters

The scatterplots above show that AF does a good job of identifying selective and aggressive hitters. But what is a metric without leaderboards? The following graph shows the least aggressive hitter-seasons since 2015:

Did you need a chart to tell you that Votto is patient at the plate? Probably not. But you might’ve needed one to understand that Jed Lowrie is, at least this year. And you can see how Andrew McCutchen became more aggressive in 2016, perhaps leading to his disappointing season, before going back to his selective roots this year.

One takeaway: these hitter-seasons are all really good. The mean wRC+ of this group is a whopping 135. What does that say about the correlation between aggression and plate discipline? Let’s dig a bit further by looking at the *most* aggressive hitter-seasons in this time period:

These seasons aren’t as good as our patient ones. No one wants to be saddled with Chase d’Arnaud’s 76 wRC+ in 2016 or Orlando Arcia’s 64. Then again, having Gary Sanchez’s 174 wRC+ in 2016 or Devon Travis’ 136 in 2015 wouldn’t be so bad.

Overall though, the mean wRC+ of this group is a less-impressive 97. That’s significantly less than the 135 shown by the more patient hitters. So it seems there may be some kind of correlation here. To the scatterplot!

The equation *wRC+* = 125.9 – (2.8 * *AF*), with *p* < 2.2e-16, fits this data pretty well:

- The
*r*value of .35 indicates a non-trivial relationship between aggression and offensive output. - The R^2 of .127 indicates that AF explains 12.7 percent of the variance in a hitter’s wRC+. This means AF alone explains 77 points of wRC+, given the dataset’s variance of 608 wRC+. That’s a lot of wRC+ for just one variable to explain.

A small change in discipline can make a reasonable impact to a player’s offense. Narrowing your focus by one square inch at the plate can gain you almost three points of wRC+. That’s not a *lot*, but as the inches add up, the effect becomes more noticeable. Let’s put it another way. Narrow your focus by six square inches and you can go from a league-average hitter to one with a nearly a 117 wRC+.

Is this adjustment worth the effort? Each player has to decide for themselves. But know that GM’s and managers are paying attention. The R^2 of .127 (and its effect of 77 wRC+) and the lack of players with high AF’s tell me the front office and managers are doing a good job of turning discipline into offense.

I also broke this dataset into deciles, where Decile 1 contains the least aggressive hitters (AF’s from 4.1 to 5.6) and Decile 10 contains the most aggressive hitters (AF’s from 13.7 to 21.3), and computed the mean wRC+ in each decile.

The results align with common sense: offense decreases as hitters get more aggressive. Also note the steep cliff after the first decile. Offense drops off quickly at first, then gradually, as a hitter becomes more aggressive.

### Aggression Over Time

You, the intelligent baseball fan, surely know that 2017’s strikeout rate of 21.6 percent is the 12th consecutive season of either an increase or no change at all. You’re not the only one who’s noticed. “Hitters are swinging for the fences” declares a headline in the *Portland Press Herald*. Last December, Rian Watt wrote at FiveThirtyEight that today’s hitters are “channeling their inner Vladimir Guerrero.” And former pitcher Al Leiter, as quoted by Tyler Kepner, noted recently in the *New York Times* that “you don’t see a two-strike approach anymore.”

We’re all correct. At least since 2015, hitters have expanded the strike zone more and more. The following ridgeline plot shows how hitters’ AF’s have changed over time:

Aggressive hitters are taking over baseball. Note the following:

- Since 2017, the peak of the AF curve has moved to the right, in the direction of increased aggression. This means the typical player in 2017 is more aggressive than the typical player in 2015.
- After the peak, the shallower downward slope in 2017 indicates players with an AF in the 8–12 range are getting more playing time than they were in 2015. I’m uncertain whether this is because so many players are aggressive that managers have no choice but to play them or because managers are increasingly okay with giving plate appearances to aggressive hitters. Perhaps both.
- Hitters are becoming more similar to each other. 2017’s inter-quartile range (IQR) of 4.00 is less than 2015’s 4.34 IQR.

If the size of the called strike zone was increasing, I could forgive hitters for jumping at pitches everywhere. But the strike zone has actually decreased in size.

### Aggression by Base-Out State

I used base-out information to see the states in which hitters are most aggressive:

I’d long thought that hitters are most aggressive in a bases-loaded, no-outs situation. Turns out that’s true, but it’s not the whole truth: hitters expand the zone much more when there are no outs and there’s *only* a runner on third. In a first-and-third and bases-loaded situation, I suspect hitters hold back a bit because they’re cognizant of the double play possibility.

Hitters become less aggressive as the outs pile up. Interestingly, aggression drops at different rates depending on the baserunner situation:

- Aggression drops rapidly when there’s only a runner on third and no outs becomes one out.
- Aggression drops slowly if there are runners on second and third.

### Who’s the Calmest Under Pressure?

Some hitters control their aggression better than others. By taking the ratio of hitters’ AF’s in the four most-aggressive base states (NYY, YYY, NNY, and YNY) to their AF’s in the four least-aggressive base states (YYN, NYN, YNN, and NNN), regardless of the number of outs, we can see who remains calm under pressure and who gets jumpy when RBI opportunities are staring them in the face.

Curt Casali, Raul Mondesi, and Tony Wolters change the least when there are ducks on the pond. They become more aggressive than usual, but only by a factor of between 4 and 5. The average player becomes more aggressive by a factor of 10.

We can’t say that about the following hitters:

These guys mostly are low-power, high-contact hitters. Are you facing Manuel Margot with a runner on third base and no outs? Throw the ball anywhere near the strike zone and he’ll chase. He gets over 20 times more aggressive than he is otherwise. Travis Jankowski and Tyler Collins get similarly jumpy.

Know who else does? The best player in baseball. Mike Trout always looks so calm and collected, doesn’t he? He never seems to get rattled. But now we can see what’s going through his head when he has a chance to knock a man in from third. He’ll expand the strike zone considerably compared to less-tense situations. Maybe that’s just knowing when to pounce — maybe it’s something he needs to work on.

### The Modern-Day Ted Williams

So: who’s the modern-day paragon of plate discipline? You probably guessed it’s Votto, and you’d be right … if you limit yourself to left-handed hitters like Williams. Otherwise, the plate discipline crown (since 2015 anyway) goes to another star hitter:

This list doubles as a list of really good hitters in the past three years. That’s no surprise given what we’ve seen about plate discipline and hitter aggression. I challenge you to find a scrub on this list.

Kris Bryant’s father would be happy to see his son’s name here. Plate discipline matters: GM’s and managers look for it in today’s game, and players who possess it tend to stand out on offense. Ted Williams was right. Get a good pitch to hit, and good things will happen.

### References & Resources

- Ted Williams and John Underwood,
*The Science of Hitting* - Billy Witz,
*The New York Times*, “Kris Bryant Takes Lessons From Ted Williams’s Batting Bible” - Kevin Thomas,
*Portland Press Herald*, “On Baseball: With strikeout stigma gone, hitters are swinging for the fences” - Rian Watt, FiveThirtyEight, “Pitchers Won’t Throw Strikes, So Batters Are Getting Better At Hitting Bad Pitches”
- Tyler Kepner,
*The New York Times*, “With Home Run No. 5,694, Baseball Sets a Record With Time to Spare” - Jon Roegele, The Hardball Times, “Midseason 2017 Strike Zone Review”

Very interesting stuff. But a point about Cutch. You speculated that his poor 2016 season might have been related to being more aggressive. But his AF in that year was still below 5; when the scale apparently goes up to 20 or more, it’s hard to argue that a difference of less than 1 in AF makes much difference in offensive production.

But then, your final graph shows Cutch and some others with AF values of 4. Is there a difference between “aggression factor” and “aggressive factor”? I’m confused.

“The R^2 of .127 indicates that AF explains 12.7 percent of the variance in a hitter’s wRC+. This means AF alone explains 77 points of wRC+, given the dataset’s variance of 608 wRC+. That’s a lot of wRC+ for just one variable to explain”

The first sentence is accurate. The rest is misleading, as AF explains 77 points of the variation, not of the metric itself. You do appear to interpret it correctly, but the way you’ve explained it here is confusing.

It’s a minor issue in an otherwise solid article.

Thanks for the tip, I appreciate it 🙂

I’m posting again because there’s a problem with the edit function. In the second paragraph of my first post, I’m trying to say that the first graph says the lowest AF are > 4, whereas the last graph indicates they are < 2. I understand that the earlier graph lists the lowest seasons, while the latter is overall for the three year period, but the data still are inconsistent. If you have an overall AF of < 2 for that period, you must have had at least one season with a value that low.

I think the answer is in the fact that AF is sq. inches / # swings. Moving from 1 season of data to 3 seasons increases the # of swings (since the sample size increases) and thus leads to the smaller numbers you see in the last graph vs. the earlier ones.

Is there a way to pull this data for a specific player? Would love to see how Joey Gallo ranks given his high K and BB rates and unprecedented HR total vs hits total.

Yeah, Gallo’s 2017 AF of 7.12 is more aggressive than about 38% of the 330 batters in the 2017 sample. In this respect he’s similar to Odubel Herrera, Mike Napoli, Orlando Arcia, Paul DeJong, Martin Maldonado, and Mitch Moreland.

This was just superb.

Any way to do this to see AF as it correlates to LI?

Thank you 🙂 Yeah I would love to do LI and spent some time looking for a straightforward calculation. Will do a follow-up if I find it.

Hello Ryan.

I certainly agree that a r^2 of 0.127 is nothing to sniff at.

So I tried to replicate your calculations, but I couldn’t get close.

Any chance you could give a more precise description of what you did? (How did you calculate median absolute deviation? Something like _mad_ in R or did you roll your own? What do you mean when you say you ‘multiplied [the x & y] distributions together’?) A worked example for just one batter-season would be really helpful.

Sure. here’s a genericized version of the R code, using a data frame where batter_name is the batter’s name, game_year is the season, px_in is the horiz. location of the pitch, and pz_in is the vertical location of the pitch.

plate_discipline_overall <- savant_data %>%

group_by(batter_name, game_year) %>%

summarize(num_swings = n(),

horiz_disc = round(mad(px_in), 3),

vertical_disc = round(mad(pz_in), 3)) %>%

mutate(discipline_sq_in = horiz_disc * vertical_disc,

aggression_factor = discipline_sq_in / num_swings * 100) %>%

filter(num_swings >= 400)

However, I can see three reasons for the trend change, all linked: the juiced ball, money paid for HR, and pitcher usage. The juiced ball makes for more HR and more K’s; if I’m a hitter I see the same hit going out that used to be caught, I swing more to get that result. Owners pay for HR and care less about K’s (Though facts are that teams that K less win more, always). Juiced ball encourages aggressiveness, money does too. Starting pitchers are pulled faster now, bigger bullpens, so one would expect pitchers to be dominating more but they aren’t: juiced ball is harder to throw, more injuries, so no advantage.