## The Law of Home Runs

Frank Stephenson of Division of Labour sent me a link to this the other day, but I haven’t had time to read it because I’ve been working on the The Hardball Times Annual (have you reserved your copy yet?). And now that Tyler at Marginal Revolution has posted a link to it this morning, I need to get on the ball and get this out there.

“Has Home Run Hitting Changed in Major League Baseball” is a paper by UC-Irvine economics professor Art De Vany who runs a fantastic blog that I was previously unaware of. Here is a summary of his conclusion:

There is a lot of speculation about steroid use in MLB, but the evidence is mostly anecdotal, misleading and incomplete. It is surely not an adequate basis for public policy to 1. assume that there is an increase in home runs, and 2. to assume that steroids are the explanation. The first statement is incorrect, there has been no increase. That makes point two vacuous. There is no need to invoke an external explanation like steroid use when there is no change to be explained.

The same law of home runs holds now that held 40 years ago. Year to year differences in home runs require no explanation; they are all within the variation of the outcomes under the stable probability distribution of home runs. The burst of new records does not require an external explanation like steroids; they are part of the pattern that comes from the nature of the law of home runs.

The pace of new records in recent years is due to the extraordinary accomplishments of three prodigious hitters. We have lucky enough to see three Babe Ruth’s in this generation. Hitters such as these may never appear again. You cannot take an ordinary player and turn them into home run hitters of the accomplishment of Bonds, McGwire, and Sosa by dosing them with steroids. It may even be harmful. Home run hitting of that magnitude is human accomplishment at its highest, as incomparably rare as the work of Einstein or Wagner.

Even greater performances are possible because the long upper tail of the law of home runs gives them positive and non-vanishing probability. The law of home runs says that the probability that Babe Ruth’s record of 60 home runs would be broken is 0.0109, about one in a hundred. Given enough time and hitters, it was almost sure to fall. Barry Bonds’ record of 73 will be harder to break. The probability that his record will be broken is 0.007206, about seven in a thousand.

This might be the most important sabermetric paper written this year. Certainly, it has the biggest policy implications. Please, read it. I’ll be doing so shortly. The media needs to be aware of it.

Addendum: I had a few minutes to read through the paper, and I think it’s quite interesting. However, I’m going to have to think about it some more. I think he’s right, and it’s a very good paper. His idea is so simple, that I can’t believe no ones thought of it before. It seems that the best ideas are often like this. Great performances are rare and very unpredictable. We just happened to witness some of them very recently, that’s all. And what up-tick in home runs among all players that we do observe is explained by other factors.

### 3 Responses “The Law of Home Runs”

1. J. Cross says:

The “Law of HR Hitting” is new (as far as I know) and interesting. In the past I’ve always seen HR’s treated as a normal distribution. This is also the most thorough treatment of the issue I’ve ever scene. There’s a lot of excellent stuff here but what happens to the crux of his argumnent when we know that McGwire, Sosa and Bonds DID take steroids. (Admittedly, we’re not 100% sure of this yet)

Steroids do not come into the picture, nor is there any need to invoke explanations that go beyond the natural variation of home run hitting, at bats, chance, and the laws of extreme human accomplishment.

Sure, we don’t NEED to invoke steroids to explain the raw HR data or Bonds, McGwire and Sosa. Of course, you could just as easily say that we don’t NEED to invoke extreme human accomplishment when the simple explanation that they took steroids will do.

It follows that steroid use either is not wide-spread among MLB players or it is ineffective in increasing home run power.

… or that the HR rate would have fallen if it were not for steroids. I acknowledge his point however, that steroids haven’t had a large effect on league HR rates. Still, what if 5% of the hitters are taking steroids and it’s increasing their HR production by 20%? That would only amount to a 1% increase for the league, below the level we’d notice. Or, IMO more likely, what if steroids are hurting some players while helping others including helping some players drastically? Would we then notice if a fraction of the players were taking steroids? Also, it’s not only the hitters taking steroids.

I think the purest measure of power hitting is home runs per hit. This takes base on balls, at bats, number of games played, hitting average and other factors out of the measure. With that said, I also develop the numbers for home runs per at bat and home runs times per batting opportunity (at bats plus base on balls)…I chose home runs per hit to get a sense of whether or not current hitters hit with more power than earlier hitters. Home runs per hit strikes me as a good measure of this; if hitters are more powerful, then of the hits they do get, more of them should be home runs.

I think HR per ball in play would make more sense. Perhaps even HR/fly balls would make more sense. Not sure it matters though.

I graphed home run hitting in percentiles over the time period of 1959 to 2004 in Figure 1. The figure shows home runs per hit in the 20th, the 50th, the 70th, the 90th percentiles of all hitters and the most home runs hit in a year by a single player. Only the maximum shows a peak during the McGwire, Sosa, and Bonds great years.

Isn’t that what you’d expect from the “juiced” model?

Body building, and the slow twitch fiber composition that it produces, could not produce the power and speed that Mark McGwire, Barry Bonds, and Sammy Sosa exhibited in their prime years.

I don’t think he really establishes this in his paper nor do I think it’s true. Especially since we know that McGwire and Bonds were intense body builders. He’s also stepping more than a little out of his area of expertise here.

2. Guy says:

The first sentence of the paper says “Home runs per game in MLB have not changed in over 40 years.” But this is clearly not true. From 1965 to 2005 HR/G increased about 25% in both leagues (NL: .81 to 1.01; AL: .85 to 1.06). That’s a BIG change, especially in terms of the resulting probability for extreme performances.

So the whole paper appears to be built on a falacy. Given that, is reading this paper worth the time investment? (not a rhetorical question)

3. JC says:

You need to read the whole paper first. You wouldn’t stop after reading the first contradictory line of A Tale of Two Cities would you? There’s more to the argument than that, much more. And a lot of it rests upon the nonnormal distribution of home runs.