*"If they knocked two of our guys down, I'd get four. You have to protect your hitters."*

—

*"I hated to bat against Drysdale. After he hit you he'd come around, look at the bruise on your arm and say, 'Do you want me to sign it?'"*

—

In our last installment of Schrödinger’s Bat we began an investigation of hit batsmen by looking at the big-picture trends in the rate of hit batsmen since 1901. That exploration led to summarizing various theories that have been proposed over the years to explain the fluctuation of rates, including the physical hazard theory, the offensive context theory, the intimidation theory, the expansion theory, the new strike zone theory, and finally the aluminum theory. From among that group, we can say that the last one seemed to make sense for the recent upward trend that began circa 1985.

Although I promised that this week we’d scrutinize the differences in hit batsmen rates since the introduction of the designated hitter in 1973, and discuss the theories proposed to explain it, last week’s column generated such a large volume of email that I thought it would be worth spending one more column on the big picture before moving on to the DH era.

**Big Picture Trends Redux**

Let’s start off by addressing a few of the more prevalent reader questions regarding the bevy of big picture trends discussed last week. Indicative of the questions received was this one from reader Marc Stone, where Marc touches on two aspects of HBP trends that the article overlooked.

Nice job, Dan, but you left out one very useful comparison: how do changes in HBP compare to changes in BB rates and, to a lesser extent, K rates and pitches per PA.

Reader Ryan Tippetts echoed the second part of that question by noting:

My immediate thought, specifically regarding recent upward trends, was the modern trend of increased pitches per AB. Might it be as simple as because a batter sees more pitches he has more opportunities to be hit by a pitch?

Thanks to Ryan and Marc, and to all the other readers who had similar comments. I have to admit that neither looking at walk and strikeout rates nor at pitches per plate appearance in comparison with the rate of hit batsmen had occurred to me. But of course all three suggestions make a lot of sense:

- If pitchers are walking more batters at the same time they’re hitting more of them, that may be indicative of worse control (the “wildness theory”).
- If strikeouts are strongly correlated with hit batsmen, then perhaps a more aggressive hitting style (the “free swinger theory”), or the intimidation of the HBP, or even changes in the strike zone are playing a role.
- If pitchers are throwing more pitches overall, it does indeed provide more opportunity for hitters to get plunked (the “opportunity theory”) which in the end may be all that is required.

To see whether the wildness or free swinger theories shed any light on the question of changes in HBP rates over time, we can add unintentional walks and strikeouts per 1,000 plate appearances for each league to the graph we showed last week:

What you’ll notice is that up until around 1970, there appears to be some correlation between walk rate and HBP rate. Unfortunately, the correlation is the inverse of that which the wildness theory would predict. As walk rates increased from around 1920 through the late 1940s the rate of hit batsmen fell. As walk rates declined, the frequency with which batters were hit increased.

In other words, one might be inclined to conclude that there is a more or less constant rate at which pitchers put batters on for free via the HBP or unintentional walk, at least based on the graph from 1901 through 1970. While that’s an attractive idea, and akin to the offensive context theory discussed last week, you can’t simply add the two rates, since hit batsmen are so much less frequent than walks–as evidenced by the fact that in order to get both on the graph, the scale of HBP is per 1,000 PA while that for walks is per 100 PA. As a result, the number of runners that pitchers put on for free is driven almost entirely by the number of walks.

In any case, there appears to be no correlation over the past 35 years, as walk rates have been fairly steady, while the number of hit batsmen has increased dramatically.

On the other hand, the free-swinger theory appears more promising. Strikeout rate does correlate pretty strongly with the HBP rate since around 1950, and in the 1910-1925 period as well. In fact, from 1950 through 2005 the correlation coefficients are a very healthy .72 and .69 for the American and National Leagues respectively, which can be interpreted to mean that strikeout rates explain around 50% (.70

But as every statistics professor drums into the heads of his students, *correlation is not necessarily causation*, and before 1950 the correlation is much weaker–in fact, for the preceding 25 years the two rates were moving in opposite directions. As a result, one might argue that the free-swinger theory holds since 1950 because the normative hitting style became more aggressive, resulting in hitters diving over the plate more frequently, which in turns results in more hit batsmen. Under this interpretation, during the 1970-1984 period, free swinging was less in vogue, and pitchers reacted with fewer brushback pitches, resulting in fewer HBP.

An alternative theory noted by reader JMHawkins that would fit the same set of facts holds that an expanding strike zone, especially on the outside corner, forces hitters to stand closer to the plate and dive over it more frequently, resulting in more batters being hit. The expanded zone also happens to induce more strikeouts, so strikeout rate and HBP rate aren’t causally related, but both are related to this third factor. There is undisputed evidence that the strike zone expanded in 1963, and anecdotal evidence that the low outside corner became an increasingly rewarding target for pitchers in the last 20 years or so. As umpires reigned in the zone after the redefinition in 1969 and the increased scrutiny around 2001, both strikeouts and hit batsmen fell. This “fluctuating strike zone theory” then explains why strikeout and HBP rate seem to mirror each other.

In either case, we’d still need a theory to account for the preceding 25 years, when strikeouts rose and hit batsmen fell, although under the above theory it appears that those 25 years from 1925 to around 1950 are the exception and not the rule.

To be honest, I was initially most hopeful about the opportunity theory. It's pretty well known that the number of pitches per plate appearance has been on the rise, so it makes intuitive sense, but when we try to look at this theory, we run into the problem that we don’t have complete play-by-play data–and hence pitch counts–for most of baseball's history. Despite the recent and very welcome additions to the work being done at Retrosheet we are still missing the vast majority of the data required to complete the picture from 1901 through 2005; the 49 seasons that Retrosheet provides are often missing pitch sequence data.

Some alert readers (aka, the real stat geeks) may also be thinking that perhaps we could use pitch count estimators in order to estimate the number of pitches, and hence the rate at which batters are hit per pitch. Unfortunately, the basic estimators that are in use rely on constant multipliers for strikeouts and walks to estimate the number of pitches, and we’ve already taken those into account in the graph above. More complex estimators rely on estimates of balls-in-play rate (the percentage of pitches on which balls are put into play, which varies by league and year), which we don’t have historically. There are other factors that could also influence the result which models have difficulty capturing.

However, we can look at data we do have, and that's as far back as 1988. You’ll recall that during the 1988-2005 period HBP rates have more than doubled. What we find, however, is that during that time the number of pitches per plate appearance has risen only around 5%. So it doesn’t look like the opportunity theory explains at least the most recent upward trend.

Year P/PA 1988 3.60 1989 3.63 1990 3.64 1991 3.71 1992 3.68 1993 3.68 1994 3.75 1995 3.75 1996 3.75 1997 3.76 1998 3.70 2000 3.75 2001 3.72 2002 3.73 2003 3.74 2004 3.76 2005 3.73

**What do Trilobites and Jason Kendall Have in Common?**

Although the free-swinger and fluctuating strike zone theories (or some combination thereof) provides some insight, and the opportunity and wildness theories perhaps less so, the most often cited theory by readers not discussed in last week’s column is the “body armor theory.” A succinct explanation was provided by reader Jeff Bullington:

This would only affect the recent rise, but what about the increased use of body armor? Would this be the 'contra-intimidation theory'?

As Jeff noted, this is the polar opposite of the intimidation theory and holds that as hitters began to wear more and more protective gear, they’ve been less afraid of getting hit, allowing them to stand closer to the plate and be more aggressive about hanging in. It follows logically that pitchers would respond by upping the ante in an effort to move batters off the plate, and reclaim their rightful territory.

This idea is akin to the evolutionary arms race between predator and prey, whereby one species evolves stronger protection in response to selection pressure from predators as has been speculated for trilobites, which in turn leads to selection pressure on predators to evolve accordingly.

As arguments go, this is a particularly difficult one to measure quantitatively. What we can certainly see that the use of protective gear–such as hard elbow and shin pads–has increased in the past 20 years. One only has to look at the protection worn by

That said, in 2002 Major League Baseball began enforcing rules that limited the use of protective gear to players with medical exemptions, such as the one employed by

Whether coincidentally or not, the recent Kendall incident notwithstanding, the rate of hit batsmen has stabilized since that time. This was also immediately after the rate had reached its apogee in 2001, when the AL set its all-time record in hit batsmen per 1,000 plate appearances and the NL its highest total since 1901.

AL NL 2001 10.67 9.92 2002 9.90 9.17 2003 10.21 9.86 2004 10.40 9.60 2005 9.52 10.05

We can also note that although helmets have been mandatory for MLB players since 1956, ear flaps have only been enforced for players who reached the majors after 1983. Ear flaps do coincide with the recent upward trend, and although one can imagine there would be an attendant psychological boost for the hitter, it’s more difficult to believe that this relatively minor change would have had that large of an immediate impact. After all, players already in the league were allowed to use the old-style helmets, so the change was gradually phased in, and the head is the part of the body hit with the least frequency.

But this does provide the opportunity to sneak in a quick trivia question: Who was the last player to wear a helmet without an earflap in a game and in what year? (Wait for it, we'll get to the answer at the bottom of the column.)

So, whether or not body armor and the introduction of the ear flap is responsible for the twenty-year upward trend in HBP rates or not, an argument can be made that the crackdown on body armor has played a role in retarding the arms race.

**A Vicious Circle?**

Finally, reader Jake Slemp wrote to say that whatever the cause of an increasing or decreasing trend in hit batsmen, it would likely be self-sustaining and reinforcing. His reasoning:

After all, hit batsmen beget more hit batsmen within the same game, which often beget still more in subsequent games between the two teams…which beget more in those games, etc.

In other words, even a small increase in hit batsmen might form a feedback loop based on retaliation. This situation is often described in economic terms as a *virtuous* (if the results are favorable) or a *vicious* (if they are negative) circle, where each cycle continues the trend in the current direction until stopped by some outside force.

To look at this “vicious circle theory,” we can use play-by-play data for 2001 through 2005 to examine the distribution of games by the number of hit batsmen. We can then compare the actual distribution with what would be expected if the hit batsmen were distributed randomly (in a binomial distribution) given the overall rate of HBP and the average number of plate appearances per game. What we find when we do so is as follows:

```
HBP Games Expected
7 1 0
6 1 1
5 10 10
4 118 71
3 455 394
2 1626 1610
1 3980 4325
0 5953 5732
6191 6412
```

As you can see, the number of games where zero through two batters are hit are all pretty much in line with what would be expected. However, we do see that the frequency of three and especially four batters hit in a game surpass the numbers you'd expect, and there are fewer games with a single batter hit than expected. And of course this list provides the opportunity for a second trivia question: What teams were involved in the lone seven hit batsmen game of the past five years? (Again, answer appears at the bottom.)

What this confirms is that retaliation is a likely factor in hit batsmen. Games where we would otherwise expect two batters to be hit can quickly turn into games where three or four are hit. We already knew that intuitively, but what we need to know is whether or not increased retaliation is responsible for the increasing number of hit batsmen.

To look at this, we can calculate the expected number of games with various numbers of hit batsmen over four successive periods, starting in 1985.

Actual vs 1985-1989 1990-1994 1995-2000* 2001-2005 Expected 5+ 850% 246% 322% 104% 3 - 4 162% 125% 119% 123% 0 - 2 100% 100% 99% 99%

* Does not include 1997-1999.

As we saw with the 2001-2005 period, in all periods there are just about the expected number of games with zero, one, or two HBP. However, there are always more games than expected with three or four batters hit, and lots more with five or more hit.

While this confirms that retaliation within games is probably a persistent feature of hit batsmen, it doesn’t appear as if blatant retaliation has increased over the past twenty years. Keep in mind, the HBP rate has doubled during that time frame. If anything, it would appear there are slightly *fewer* beanball wars now than in the past, perhaps as a result of the double-warning rule put into effect in 1994. Note that this conclusion holds even if you assume that the increase in games with three or more hit batsmen is completely due to wildness (after all, it’s certainly true that when a pitcher hits one batter he’s more likely to hit another simply due to control problems).

What this doesn’t rule out is the idea that teams now employ a more subtle form of retaliation, whereby they will wait to take revenge in a subsequent series, and where the retaliation doesn’t escalate out of control. As a result, it would be possible that retaliation and escalation are to blame for the recent increase in hit batsmen, but it seems unlikely.

However, even if retaliation is not the cause of the increasing rate of hit batsmen, the body armor theory may provide the starting point for the vicious circle that was interrupted by the new rules, starting in 2002.

**Error on the Side of Caution**

If nothing else, I hope that we’ve highlighted that in an activity as complex as baseball, there are usually many factors that contribute to the big-picture trends that we see. That’s true for hit batsmen as well as the more visible trends, like the offensive upsurge of the last dozen years or so. If there is a lesson to be learned here, it’s probably that we should all be more cautious of simple explanations and easy answers.

Let’s wrap up with a couple of corrections from last week.

First, when discussing the expansion theory I noted that expansion would have a tendency to dilute talent in both leagues. While that’s true to some extent, I was reminded by our own Christina Kahrl that actually the 1992 expansion draft was the first time players from both leagues were available in an expansion draft. Prior to that, for example in 1977, the expansion teams could only choose unprotected players from their own league. And in that 1992 draft, AL teams were able to protect more players than NL teams; it was not until the 1997 draft that all teams were able to protect the same number of players.

Second, I noted last week that *major-leaguer* to be fatally injured by a beanball. Several minor leaguers were killed in the 1950s and 1960s including Otis Johnson in 1951.

—

Okay, so you waited, here are a couple of answers. For the first trivia question,

The answer to the second question: June 7, 2001 the A’s visited Anaheim to take on the Angels. In that game

#### Thank you for reading

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