If you’ve ever looked at Mike Trout’s Baseball Reference page, you’ve seen a lot of black ink, indicative of leading the league in some offensive category. Trout is currently leading the league in a lot of such categories — and he has done so in the past. He’s likely to lead the AL in homers for the first time this year, and as noted recently by Ben Lindbergh, that would be the 10th category out of 17 basic ones at Baseball Reference that he’s led the league in at least once in his career. Only about two dozen players have led the league in more of these 17 categories at some point in their careers.
Trout is also leading the league in another category that would also be a first for him: hit by pitch (HBP). He’s been plunked 15 times this season (editor’s note: now 16), one more than Shin-Soo Choo as of this writing. HBP is not what you would call a sexy stat, but it does have value. It’s as valuable as a walk — in fact, per FG’s linear weights, a little more so, apparently because it occurs slightly more in base-out states that have higher run expectancy (RE) than average.
HBP, like walks, are valuable because they put a runner on base, where he has a chance to score, while simultaneously avoiding an out, thus giving another batter a chance. They’re a good thing. But they also come with a risk. If a batter is hit by a pitch, he could suffer a significant injury and miss time. The resulting lost time is lost value. This raises an obvious question: is the increased value gained by getting hit by pitches greater than the potential risk of losing value to injury? If it is — and one would assume that in this stat-savvy age, it would have to be — by how much?
I’m going to begin analysis of this benefit-risk relationship with Trout for a few reasons. First, he’s probably the most popular player on this website, as any article about him always has our attention. Second, because he’s currently the league leader in the stat, we would expect that his risk of getting injured would be greater than that of other players. Just how great is that risk?
Most important, though, is that Trout turns out to be a perfect test case. Since he is by far the most valuable player in baseball, there are two extreme factors that actually mitigate against him when getting hit. On the one hand, he stands less to gain from HBP relative to not getting hit than other batters, because his average value, his baseline, is much greater than anyone else’s. Though Trout produces just as much run value — in the context-independent framework of FG’s linear weights — when he gets HBP as any other player, because his value when he doesn’t get HBP is much higher than anyone else’s, HBP has less added benefit for him than it does for others.
At the same time, Trout’s great value also hurts him more on the risk side. Any games he does lose to injury cost more in value than they do for other players. Trout averages more WAR per game or per PA than any of his peers — indeed, as I have pointed out in a previous post — more than anyone other than the great Babe Ruth. So his time spent healing an injury accumulates more lost WAR.
Because of these unique conditions, we can say that if any player has potentially less benefit than risk from getting hit by a pitch, it would be Trout. Whatever the benefit/risk ratio is for HBP, it will be the lowest for him. Any other batter will definitely have a higher one.
The added benefit of HBP
The analysis is fairly simple. We need to determine both the value added by getting hit by a pitch, and the risk of injury, in terms of value lost. I begin with the first. FG’s Seasonal Constants page tells us that the current value of a HBP is .719 runs. This means that the wOBA of a player who gets HBP in a single PA is .719. We can then use the formula to convert wOBA to runs above average (RAA), runs above replacement (RAR), and wins above replacement (WAR; I will be using FG values, such as fWAR, throughout):
RAA = [(wOBA – lgwOBA)/SF] * PA
The league wOBA is currently .321, while the scaling factor (SF), an arbitrary number used to adjust wOBA values so that they appear on a scale similar to that of on base percentages (OB), is 1.15. The result is .346 RAA for a single PA, and the total RAA is 5.19 for 15 PA.
To get RAR, we must add replacement runs. From the value tab in Trout’s FG pages, we see that he currently has 17.8 replacement runs in 563 PA. It follows that he accrued .47 runs in his 15 HBP PA. So his total RAR is 5.19 + 0.47 = 5.66. We then divide this by runs/wins (10.34), again, from the Seasonal Constants page, to get .55 WAR.
So Trout has gained .55 WAR through getting hit by a pitch 15 times. However, we can’t leave it at that. We need to know how much value he added by being hit by the pitch, as compared to the value he would have produced if he hadn’t been hit by the pitch. For his career, Trout is averaging .0140 WAR/PA overall, of which .0135 WAR is from offense. To get the latter value, I multiplied .0140 by the fraction of total RAR that resulted from offense, that is:
(Offensive RAA + Positional RAA + League RAA + Replacement RAA)/total RAR
It’s somewhat arbitrary whether, or how, to include positional runs, league runs, and/or replacement runs, since they accumulate in every game and could be considered as much as a part of defense as offense. Obviously positional runs are directly associated with defense. However, all of these run values are calculated directly from a player’s PA, so I will consider them part of his offensive contribution.
Also note that Offensive RAA includes baserunning. I want to include these runs, though they are fairly minor relative to the total, because baserunning opportunities only arise because of on-base events like HBP. In this sense, any player’s base runs are generated by his positive batting events.
So in an average 15 plate appearances, Trout would produce .0135 * 15 = .20 WAR. If we subtract this from the .55 WAR produced from HBP, we get .35 WAR. This is the WAR Trout has produced above his own average by getting hit by pitches this season.
The approach I have used is somewhat similar to the logic underlying the tOPS+ values that Baseball Reference uses to compare a player’s OPS in some defined fraction or split of his PA to his overall OPS. The tOPS+ metric is expressed as a ratio, and we could, if we wanted, do the same thing with Trout’s HBP production. Since he produces .55 WAR in 15 PA when getting HBP, and .20 WAR in the same number of PA on average, he produces 2.75 times as much value as he does on average when getting hit by a pitch. The OPS+ metric used by by Baseball Reference actually correlates very highly with FG’s wRC+, which is based on linear weights and wOBA, so we could say that Trout’s tOPS+ when getting hit by a pitch is 275.
One normally uses tOPS+ only for splits that are determined before each PA, not from the results of that PA. But the principle is the same, and this value emphasizes how valuable a HBP really is. However, for this analysis, it’s the difference, rather than the ratio, that matters.
The average risk of HBP
Every HBP carries some finite risk of injury to the batter. The question is, how much is this risk? To estimate it, I will refer to data presented in a 2016 study of injuries resulting from HBP over a five-year period, from 2011-2015: Getting Hit by Pitch in Professional Baseball: Analysis of Injury Patterns, Risk Factors, Concussions, and Days Missed for Batters, by C.L. Camp et al. The study made use of data from both MLB and the minor leagues (MiLB), but I will refer only to the former.
The relevant findings of this study are:
- There were 361 injuries resulting from 9783 HBP events in the time period (Note: the number of HBP reported by this study is far greater than that reported by FG in this time period, 7834. I don’t know the source of the discrepancy, but will use the authors’s numbers). Thus, 3.7% of all HBP resulted in an injury.
- There were a total of 4216 days lost as a result of these injuries, or an average of 11.7 days lost per injury. Since 3.7% of HBP resulted in an injury, we can estimate the probability of suffering at least one injury following 15 HBP as 1 – .96315 = .432. This assumes that every HBP event has an equal risk of injury, independent of any other event. But how many days lost does this correspond to? The value of 43.2% is the probability that at least one injury will occur, but there is also a finite probability that two or even three injuries will occur. The math looks like this:
One injury: .327 * 11.7 days = 3.83 days
Two injuries: .088 * 23.4 days = 2.06 days
Three injuries: .0015 * 35.1days = .53 days
Total average risk: 6.4 days
After I went through this process, however, I realized there was a much faster and simpler way to determine the average risk. A single HBP has a .037 probability of resulting in an injury, and the average time missed from an injury is 11.7 days. So the average time missed from a single HBP is .037 * 11.7 days = .043 days. This relationship holds regardless of how many additional HBP occur; so for 15 HBP it’s .043 x 15 = 6.45 days. Though the probability of multiple injuries increases as HBP increase, they all result from the fixed relationship of time lost to a single HBP — assuming only that every HBP occurs independently of every other one.
Thus Trout’s 15 HBP have exposed him to an injury risk that would cost him about six games on average. He could of course suffer no injury — which appears to be the case so far this season — or he could suffer an injury (or injuries) that sideline(s) him longer. But on average, how much WAR does Trout risk losing?
For his career, he has averaged .0614 WAR/game; this is total WAR, including defense. We need to consider this, since of course if he misses time, both his offense and defense are lost to the team. The 6.4 games lost thus correspond to .39 WAR. We previously saw that he has produced .35 WAR more than his average by getting HBP 15 times. This value is a little less than the WAR he would risk losing by getting HBP this many times, by .04 WAR. Expressed as a ratio, benefit/risk = 0.90.
Thus Trout actually faces an average risk of losing more value than he gains from being hit by a pitch 15 times. What about other players? As I noted earlier, the benefit/risk ratio is less for Trout than for less valuable players, so let’s compare him with some others.
As a first example, let’s consider a player who is arguably the second best after Trout: Mookie Betts. Betts has the second highest career WAR/game of any active player, after Trout, of .046. His WAR/PA, after adjusting for offense, is .081 — about 60% of Trout’s value. Following the same calculations used for Trout, Betts would produce an added benefit of .55 – .12 = .43 WAR when HBP 15 times, while running a risk of 6.4 * .046 = .29 WAR. (Note: We can assume that the replacement runs for Betts or any other player will be the same as Trout’s for the same number of PA). So Betts would have a net benefit of .43 – .29, or about .14 WAR. Thus, unlike Trout, Betts — and remember, he’s one of the most valuable, if not the most valuable, players other than Trout — accumulates a net positive WAR. His benefit/risk ratio is 1.48.
The difference is even more dramatic for a less valuable player. Consider Trout’s teammate, Kole Calhoun. Calhoun has a career wRC+ of 106, indicative of a slightly above average hitter. He has produced .0031 offensiveWAR/PA, and .016 total WAR/game. His added benefit from 15 HBP is about .50 WAR, while his injury risk is about .10 WAR. So his net benefit is .40 WAR, and his benefit/risk ratio is 5.0.
The cumulative risk of HBP
I have performed this analysis on 15 HBP, because that’s Trout’s current total. What if he were to be hit by more pitches? The value added would increase, of course, but so would the risk for injury. How would these two factors change relative to each other?
It turns out their relationship for any given player wouldn’t change at all. I’ve already pointed out that the average risk of injury and time missed per HBP are fixed, as well as independent of the number of HBP. The same relationship holds for the added value of a HBP. The run value of a HBP is fixed for any given season, and so is the offensive WAR/PA of any player, since we’re using career values. The difference between these two must therefore also be fixed, meaning it rises in proportion to the number of HBP.
So everything is fixed! The added benefit of a single HBP is fixed, the average risk of a single HBP is fixed, and the difference between the two is fixed. This means that the benefit/risk ratio is fixed. For Trout, it’s less than 1, at .90. For Betts, let alone any less valuable players, it’s more than one.
The difference between less than 1 and greater than 1 is critical, because it determines whether a player is likely to add to his total net benefit or to his total net risk going forward. We have seen that Trout’s net risk is .04 WAR for 15 HBP. For 30 HBP, it would be .08 WAR, and so on. Conversely, Betts’s net benefit for 15 HBP is .14 WAR. For 30 HBP it would .28 WAR. Once a player’s benefit/risk ratio is determined, he is more or less locked into that, unless his baseline value changes dramatically.
Mike Trout better be careful! While he’s accumulated about half a WAR from being hit by pitches this season, about a third of a WAR over his expected production, analysis of injuries following HBP suggests that he runs the risk of giving it all back, and slightly more. The good news, such as it is, is that his benefit/risk ratio does not change with the number of times that he’s hit by a pitch. The bad news is that because it doesn’t change, the absolute value of his net risk, in WAR lost, increases linearly with the number of HBP.
Of course, the stats I’ve based injury risk on have been derived from a large pool of players. Every player is different, and some no doubt have less injury risk than others. Craig Biggio holds the modern career record for HBP, with 285, and to my knowledge, he never suffered a major injury from any of them that resulted in missing games. Assuming the probability of 3.7% reported by Camp, et al., the odds of this happening are about 1 in 50,000. There have been about 20,000 players in the history of MLB, and less than 100 of them were even hit by 100 pitches (using the same analysis applied to other players above, Biggio produced about 8.75 extra WAR from all his HBP, about 13% of his career total).
With respect to Trout, he holds his hands relatively high when batting. Perhaps this reduces the chances of getting hit there. Camp et al. report that after head and neck injuries, the most frequent ones following HBP are to the hands and forearms, and these have the highest average number of days lost.
I have also only considered the average and most probable risk. Given the run value of a HBP, and other seasonal constants, the WAR benefit of a HBP is more or less fixed. The potential loss of value, though, can vary greatly. Batters can break a wrist or hand when hit by a pitch, and as I just noted, lose weeks or months of time. The risk of such serious injuries will clearly increase with number of HBP. But while many of us can probably recall a player enduring this — Paul Goldschmidt, Giancarlo Stanton, and Freddie Freeman are recent cases that come to my mind — injuries requiring surgery, according to the Camp, et al. study, are quite rare. They reported that only about 3% of injuries fell into this category, or about one in a thousand HBP.
In any case, Trout appears to be the only active player for whom the injury risk suggests a net loss of value in continuing to be HBP. This is precisely because he produces so much more value than others. Even a very valuable player like Betts gains more value from HBP than he risks losing to injury, on average, and for less valuable players, the benefit/risk ratio as well as absolute net benefit is far greater.
As a rough guide, a player needs to average more than 9 WAR per season to be at this kind of disadvantage. A value of 9 WAR will reduce his average injury risk to .35 WAR, and unless he is a very poor defensive player, or DH, and his offensive WAR exceeds his total WAR, his added benefit would not be less than .35 WAR. One more way in which Mike Trout is one of a kind. In this case, he may be too good for his own good.