# You shall know our velocity

This article borrows its title from a book by Dave Eggers, but it could more aptly be named after an earlier work by Eggers entitled, “A Heartbreaking Work of Staggering Genius.” The work of genius, however, was not my own, by derived a brilliant hypothesis put forth by John R. Mayne in 2010. Mayne emailed to alert me of this piece earlier this year, after my release of SIERA (Skill-Interactive ERA) at FanGraphs, and I recently tested it. Despite initial pessimism, I was shocked by what I found.

Everyone now knows how important velocity is for a pitcher. For years, pitching coaches scolded amateurs about over-reliance on velocity. Prepubescent pitchers are lectured about Greg Maddux, told that movement and location are more important than a couple digits on a radar gun.

I’m no PITCHf/x expert, but everything I’ve read by those capable of studying that data says that velocity is actually very important, perhaps more important than movement and location after all. It’s hard to throw a 100 mph fastball that is easy to hit, and you have to be Jamie Moyer to get away with an 80 mph, lukewarm heater. Even a few ticks in the ones column of a radar gun can make world of difference.

However, until very recently, I believed that a proper study of a pitcher’s peripherals could tell you which of two guys with a 92 mph fastball has the superior arm, and I also believed that two pitchers with the same SIERAs with different fastball speeds were no different in future skill level.

When discussing SIERA’s ability to adjust for pitcher control of BABIP, Dave Cameron once noted that velocity may explain some of the missing pieces of the puzzle that correlated with both strikeout and BABIP skills. However, I found that if you control for peripherals, age, year, and role, then knowing a pitcher’s velocity is not useful.

In fact, running a regression on all of these, you will actually get an insignificant and positive coefficient of .00035 on velocity; in other words, a 3.0-mph increase in velocity with the same characteristics will correspond with a BABIP that is a full point higher!

When Mayne emailed me with this suggestion, I expressed my skepticism, but I was thinking about the idea the wrong way. Mayne was talking about projections in that article—predicting the future. What I now found was that knowing a pitcher’s velocity tells you about his potential to improve the statistics that express skill level better.

If you just run a regression of a pitcher’s ERA next season on his ERA from the current season, you get the following equation:

ERA_next = 2.76 + .368*ERA

Include velocity, and you get:

ERA_next = 9.49 + .327*ERA – .073*velocity

This formula says that, of two pitchers with the same ERA last season, the one who threw faster is more likely to improve. That’s not surprising. We know that a pitcher was probably more capable if he threw faster, so he probably had better peripherals and worse luck if he had the same ERA and more velocity. Right?

Actually, let’s take a closer look at the pitcher’s true skill level and replace his ERA with his SIERA to see what happens. If you run a regression of a pitcher’s ERA next season on SIERA from the current season, you will get the following equation:

ERA_next = 1.21+ .733*SIERA

However, if you run a regression of ERA next season on SIERA and velocity, you get the following result:

ERA_next = 4.52 + .677*SIERA – .034*velocity

Both coefficients are statistically significant at the 99.9 percent level. In words, this means that a 2.9-mph increase in velocity will correspond with a 0.10 lower ERA, even if you know the pitcher’s SIERA from the previous season.

What’s going on here? Well, the pitchers who throw faster are doing something better than others with the same peripherals. What is that? I looked at various components of pitcher performance to find the answer and found why Mayne’s hypothesis was accurate.

Suppose you know a pitcher’s strikeout rate. In this case, you can predict his future strikeout rate next year very well:

K%_next = 3.87 + .764*K%

However, once you know that pitcher’s velocity, you have a lot more information.

K%_next = -16.1 + .701*K% + .233*velocity

Verbally, this mean that if you have two pitchers with the same strikeout rate the previous year, the pitcher who throws 4.3 mph faster will strike out one percent more batters the following year than the pitcher who throws slower.

What about walks?

BB%_next = 2.864 + .644*BB%

BB%_next = 0.237 + .638*BB% + .296*velocity

In the case of walks, more velocity actually portends an increase in free passes.

However, if you start to include more terms, its significance disappears. Higher velocity is just correlated with other variables that are related to increases in walk rates, such as relief role, age, and strikeout rate itself!

Including strikeout rate in the regression on next year’s walks renders the velocity coefficient insignificant (p = .224), while it remains very significant (p = .000) in the regression on next year’s strikeouts:

BB%_next = 1.04 + .0151*K% + .6361*BB% + .0179*velocity

K%_next = -15.9 + .6984*K% + .0752*BB% + .2254*velocity

Controlling for both rates, more speed foreshadows an improvement in strikeout rate. Including a slew of other variables (results omitted for brevity) did not alter this conclusion.

If you look at BABIP, you start to see more of an effect of a good fastball. If you try to predict BABIP next season using only this season’s BABIP, and then try to do so with BABIP and velocity, you can create a clearer picture:

BABIP_next = .238 + .191*BABIP

BABIP_next = .283 + .191*BABIP – .00050*velocity

Velocity helps predict next season’s BABIP pretty well, though this effect is somewhat minimized when considering the effect on other variables.

The rate of home runs per fly ball is another metric that is mostly determined by luck but incorporates some skill as well. Velocity actually corresponds well with a decreased rate of home runs per fly ball, even in the same season.

Running a regression of home runs per fly ball while incorporating peripherals with interactions, season, year, age, and role, we will still get a coefficient of -.00066 on velocity. This means that a pitcher who gives up 3.0-mph in velocity will yield one fewer home run every 500 fly balls. It’s not a big deal, but it’s statistically significant.

It also matters because the coefficient only goes down to -.00063 when changing the dependent variable to next year’s home run-per-fly ball rate. The skill is something that shines through over time, revealing an ability to get hitters out that gets behind the luck mashed in with other statistics.

However, if we simply check how much velocity adds to HR/FB itself in predicting next year’s HR/FB rate, we can see that:

(HR/FB%)_next = 8.39 + .186*(HR/FB)

(HR/FB%)_next = 20.17 + .173*(HR/FB) – .129*velocity

Knowing velocity is important for this as well.

Velocity is an even bigger deal than we thought, and Mayne hit the nail on the head. Not only do pitchers who throw faster succeed more often, but they improve more as well. It foretells a higher strikeout rate, lower BABIP, fewer home runs per fly ball, and a subsequently lower ERA than other pitchers with similar yearly statistics.

Incorporating velocity into projection systems would appear to be not only a useful tool, but perhaps a pivotal one in better understanding the importance getting the ball to the batter sooner has on getting him out.

This is certainly not unique to this particular article, but correct me if I’m wrong – most of the excel-based linear regression calculations of p-values and the like are a simple H0 v. Ha hypothesis test, right? So a p-value only tells you an effect is non-zero, not how big or small an effect is, right? I.e., you would need to re-run these tests against a different null hypothesis (something other than zero) before making those size-of-effect comparisons, and at the very least you need confidence intervals to get a real look at how much these effects overlap. I also… Read more »

A brief thought on ideas and execution. I think I read on a game development blog (probably Sarah Northway’s) that there’s an axiom that “Ideas are worthless.” I don’t have the tools to execute the ideas, which means that I often can’t prove the ideas, which means they are worthless. I’m sure scores of people inclusive of me developed something like support-neutral win-loss – but Michael Wolverton did the legwork. He won the internet, and rightly so. (Or, on Big Bang Theory last night, dude has an idea for 3-D glasses for all television. How does it work? “I don’t… Read more »

@Nate: Thanks for your comment, Nate. I think you might be mixing up a couple of concepts. These aren’t H0 vs. Ha p-tests, really, though the regression output does provide p-stats for the coefficient that do test H0=0. The coefficients are the center of a confidence interval of approximate effect, but you could run tests if I had reported the standard errors. When the p-stat is =.001 exactly (and many of these are <.001), then the bottom of the confience interval is 63% of the way to 0. (So if the coefficient is 1.00 and the p=.001, then the 95%… Read more »

Thanks for the clarifications – a few comments/questions, if you don’t mind. I enjoyed that you even included the p-values at all, as I’ve read a lot of articles that just drop the regression equations and a R-value as if that’s all there was to it. Hopefully these aren’t too stat-nerdy: 1. I guess I find that info about the confidence interval more helpful than an isolated p-value. Taking your example from above about a 2.9 mile per hour increase* leading to a .10 decrease in future ERA (with a p of .001), it seems as though with the confidence… Read more »

Confidence intervals & R^2 (Stata didn’t reported Adjusted R^2 for these, but the increases in R^2 are sufficient to see the difference) For ERA on ERA & velo, the velo confidence interval was (-.088,-.057) and the increase in R^2 by including velo was .1072 to .1389. For ERA on SIERA & velo, the velo confidence interval was (-.049,-.019) with R^2 increasing from .1850 to .1911. For K% on K% & velo, the velo CI was (.0018,.0029) with R^2 going from .5629 to .5770. For BB% on BB% & velo, velo CI: (.0000349,.0005571) with R^2 going from .3859 to .3871. (the… Read more »

Many of my fantasy leaguemates are now reading this. Can we screen them out?

(Seriously, Matt, thanks for the followup. Having the degree of the effect quantified is huge, and I don’t think this has been done before. A big step forward – and while my ideas were surely worth something, they weren’t worth as much as the execution.)

—JRM

Colleague: This will just inflate your ego.

JRM: How is that possible?

Matt interesting article. You guys are way above in the math—too many years since college.

I do not understand the statement “This means that a pitcher who gives up 3.0-mph in velocity will yield one fewer home run every 500 fly balls” Does this mean a pitcher with lower velocity gives up less HRs per flyball than high velocity pitcher? It seems to be opposite of the rest of the article—Velocity is good.

@Jack I meant to say that a pitcher with higher velocity gives up fewer home runs. Sorry about that. @Nyet The R^2=.5 benchmark is random, and it doesn’t really matter here. There is a lot of variation in pitching outcomes that are random but have larger influences, but that only serves to make it more important to have a fundamental understanding of what matters over time. One season of data may induce an R^2 of less than .5, but a few seasons will beat .5. Half a season will do even less. The .5 level is not unique—income regressions generally… Read more »

I disagree that .5 is “random;” you probably mean arbitrary, but it seems sensible that you would want to have a goodness of fit such that more variance is explained by the model than left unexplained. That doesn’t seem arbitrary to me, but perhaps you mean something else. And I didn’t say it wasn’t important, I said it wasn’t useful. If I am trying to project forward for two players who are equal in all respects but then learn that the model with velocity explains 19% of the variation instead of 18.5, there is an exceedingly small chance (rough calc,… Read more »

Nyet, the goal is not always to increase the R^2. In graduate school, I was taught to stop focusing on R^2 much at all. If I wanted to increase my R^2, I could do it by throwing in a host of other variables. In unreported regressions to check for the robustness of my conclusions, I included variables like age that increased the R^2 somewhat, and I could have included park effects, quality of a team’s defense, etc. And if I wanted to study other aspects of pitching (something I have done in other articles and others have done in many… Read more »

Thanks for the explanation. I was under the impression that to counter the increased R^2 by just adding adding variables one should use the adjusted R^2 to evaluate the fit. But your explanation about those other variables not being correlated to these variables on velocity does make it sound like you could just leave them embedded in the “error” and not worry about them messing up your coefficients for velocity and the like. Still, I didn’t say (or at least didn’t mean to say) that the conclusion was useless; I said it was not (practically) useful. Looks like it’s very… Read more »

Alright, forgive me if I’m misunderstanding … but for every regression you’ve listed here except For K% on K% & velo, the R^2 is under (and often well under) .5. So in all of those cases, more of the variation is explained by error and/or hidden variables than the explained by variables accounted for in the model? Am I missing something? I just don’t understand how those models are useful if they don’t even get at half of the variance. And I don’t get how knowing the velocity helps if a difference is more explained by error than velocity. I… Read more »

Let me try this another way. Teams on the free agent market behave as though one run in expectation is worth about $500K. If we know that a given starting pitcher is worth 20 runs above replacement +/- 20 runs in a given season, they will pay about $10MM for his services, knowing that it may be worth anywhere from $0-20MM to them. If this is a SP expecting 180 IP, then the difference in between .10 of ERA is about two runs. So take a pitcher who appears to be worth about 20 runs above replacement looking at his… Read more »

Thanks again for your explanation; I hope your “let me try this another way” qualifier doesn’t mean you think I’m daft. I completely get that is the standard interpretation, sure. I understand that, and I understand why; it’s the “what are you gonna do, might as well take the bet that would pay off in the long run” response. Which is a perfectly sensible long-range multiple iteration game theoretical approach to the situation. I even understand that descriptively, that’s how actors on the free agent market behave. What am I asking is whether that makes sense given that these decisions… Read more »

I like the analysis here, but I’d have to agree with Nyet that the impact of velocity data on ERA prediction accuracy is smaller, not larger, than expected. A 1 mph difference in fastball velocity is only worth 0.03 rpg of skill that isn’t already being captured by peripherals? If I’m a GM, that tells me that I need to focus more my statistical models and less on velocity. That said, intuitively I do value velocity in pitching prospects. I would be interested to see if velocity has a larger effect on long-term player projections (rather than just the very… Read more »