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Can we detect the use of steroids statistically? Nate attempted one approach in the piece reprinted below, which was originally published on March 30, 2005.

It has become increasingly difficult to take a neutral position on the steroid issue.

My past four weeks have featured three bookstore signings, four fantasy drafts, two chats here at BP, a neverending series of radio gigs, and innumerable conversations with friends and readers about everything from Mark McGwire's testicles to the aesthetics of the newly-installed green seats at U.S. Cellular Field. At each juncture, I have been asked a lot of questions about steroids, and I have done everything short of pulling a Sammy Sosa and hiring a mum-mouthed interpreter to evade them.

It's not just that I want to avoid pouring more fuel on the fire, though as far as I'm concerned, the Crusading Sports Journalist belongs in about the same pit of hell as the Philandering Televangelist, the Embezzling Senator, and the Reality TV Show Host. Nor is it that, somewhere deep down, it really does disappoint me that at least some of my favorite baseball players have been gaining a competitive advantage ingesting or injecting substances that can't be purchased at your local Walgreens.

Rather, it is the fact that I am a performance analyst, and steroids are an extremely difficult issue to discuss from the standpoint of performance analysis. We don't have more than a handful of confirmed or admitted positive results. We don't have any real idea of the differential effects that steroid use would have on pitchers and hitters. We don't have any sense for the scope and the duration of steroid use. We sure as hell don't have a reliable control group. In short, any sort of analysis based on examining the career paths of individual players is likely to be somewhere between fruitless and utterly misleading.

But if the media frenzy on the issue is to be believed, then greater sanctions against steroid use–whether in the form of suspensions, fines, or public scorn–are going to have an awful lot of impact on the way the 2005 season plays out. A substantial number of star players are going to fall flat on their face. The standings are going to shift in rapid and unexpected ways as the cookies-and-cream teams differentiate themselves from The Clear-and-The-Cream ones. The entire San Francisco Giants lineup is going to develop pituitary tumors. Biceps are going to tear; elbows are going to shatter. The fans are going to stay away, the marketing deals are going to evaporate, and NASCAR is going to become the new national pastime. (Never mind that baseball has always been unpredictable, that star players have always seen their performances crater without any apparent reason, that fans are gulping up tickets with unprecedented gusto, or that baseball just this week signed a huge, new marketing deal with one of the world's largest corporations.)

It will go without saying that I don't expect any of that to happen. I don't expect a substantial change in offensive levels, nor a disproportionate number of individual collapses. I do expect lots and lots of surprises, things that PECOTA could never have seen coming with a processor the size of the Hubble Space Telescope, but not any more than there usually are.

I don't expect any of those things to happen because we do have some evidence on this issue, and the evidence does not support the prevailing opinion.

Let's take a step back for a moment. Suppose that the predominant media opinion on the subject of steroids is correct: a substantial number of players are using steroids, and steroid use results in substantial and bifurcating improvements to player performance. We will call this the Steroid Gap Theory. What would we expect the corresponding impact on the game's competitive ecology to look like?

It might be the case that offensive levels would rise, if more hitters than pitchers were using steroids, or if the benefits of steroid use were more profound for hitters than they were for pitchers. But this would not be the distinguishing mark of steroid use; offensive levels cycle upward and downward all the time, and they have since the very origin of the game. Rather, the distinguishing mark would be that variance in player performance would increase. If some players, be they hitters or pitchers, were gaining a new and substantial competitive advantage, while others were remaining in place, then we'd expect a greater amount of differentiation between the best-performing players and the worst-performing players, in the same way that, say, placing every child west of the Mississippi into a top-tier private school and every child east of the Mississippi into a forced labor camp would increase the differentiation in nationwide SAT scores.

Take the simplest possible case: we have two baseball players of essentially identical ability, and we have an increase in their collective performance. Let's say that these players are named Jose and Ozzie, and that they each hit 20 home runs last season.

We could achieve the increase in one of two ways. Either the increase could be global, affecting each player equally, or it could be differential, affecting one player but not the other:

            Type of Improvement
      Previous   Global   Differential
Jose    20 HR     25 HR     30 HR
Ozzie   20 HR     25 HR     20 HR

In both cases, the average number of home runs hit by the players would be the same at 25. However, in the differential case, the standard deviation in home run rates would be larger.

This is not merely an artifact of there being only two players in the league, or the players having started from the same initial performance level. We can conduct the same thought experiment over a larger and more diverse group of players, and come up with the same result.

I took a list of the season-ending home run totals for all major league players who accumulated at least 400 at bats in a season between 1988 and 1992, the last years before the recent upswing in offensive levels that began with the expansion year of 1993. There were around 580 players on this list; the average number of home runs hit was just under 14, and Cecil Fielder had the best individual season at 51 home runs.

I then inflated the home run statistics the list in one of several different ways:

  • Global Impact. All players' home run output was increased by 10%.
  • Weakly Differential Impact. Fifty percent of players had their home run output increased by 20%, while the other fifty percent experienced no change. The players receiving the performance boost were assigned at random.
  • Differential Impact. Twenty percent of players had their home run output increased by 50%, while the other eighty percent experienced no change. The players receiving the performance boost were assigned at random.
  • Strongly Differential Impact. Ten percent of players had their home run output doubled (increased by 100%), while the other ninety percent experienced no change. The players receiving the performance boost were assigned at random.

Note that it's the latter two examples that correspond most strongly with the Steroid Gap Theory. The Global Impact case, meanwhile, would correspond with an improvement that all hitters might benefit from, like an increase in the liveness of the ball, while the Weak Differential case might correspond with a more widespread phenomenon like improvements in health and training methods.

Here is what the data look like after we perform those alterations:

                             Average    St. Dev
Pre-Inflation Era            13.8 HR     9.7 HR
Post-Inflation Scenarios:
Global Impact                15.2 HR    10.7 HR
Weakly Differential Impact   15.1 HR    10.8 HR
Differential Impact          15.1 HR    11.0 HR
Strongly Differential Impact 15.1 HR    11.4 HR

In each case, both the average and the standard deviation in the number of home runs increase. However, while the average increases by about the same amount in each scenario, the degree of increase in the standard deviation is greater when the inflation is applied more differentially. Put simply: players gaining a differential advantage should lead to differential results.

While we cannot, at least at this point, reliably decipher the impact that steroids might have on an individual's statistics, we can compare the results of our thought experiment to actual leaguewide statistics. To start with, let's look at seasonal means and standard deviations in home run rates before the "Juiced Era" offensive explosion began. In the chart below, I've plotted the mean and the standard deviation number of home runs hit per 650 plate appearances in each National League season between 1961 and 1992 (I've excluded the strike season of 1981; the short season has the effect of artificially increasing the standard deviation). Both the mean and the standard deviation are weighted based on plate appearances.

This time period incorporates a number of important changes to the game, like the influx of Latin American players, the development of the modern bullpen, various iterations in the size of the strike zone, and a correspondingly diverse array of leaguewide power levels, ranging from the modern deadball era of the mid-60s, to the "Juiced: The Prequel" year of 1987. In spite of all of that, the relationship between average home run rates and the standard deviation in home run rates remains highly linear, and highly predictable: as the average increases, the standard deviation increases proportionately.

We know, of course, that average home run rates increased markedly as of about 1993. What we don't know is how the impact was distributed: was it confined to just a subpopulation of hitters, or were the benefits conferred to more or less everyone? If the former is true, as the Steroid Gap Theory posits, then we'd expect not only the league average to increase, but also the standard deviation to increase disproportionately on top of it.

The evidence renders a clear verdict to the contrary.

The dashed line represents a linear projection outward of the trend that we established using the older data set. The red points represent actual National League power levels in the "Steroid Era"–the years from 1996 through 2004 (I've excluded the transitional year of 1993, as well as the strike-shortened seasons of 1994 and 1995). Standard deviation increases as we go up the chart, so we'd expect the new data points to be above the dashed line if the Steroid Gap Theory is correct, indicating that the juicing players have caused variance to increase disproportionately with the gain in power output itself.

As it happens, not only has the increase in the standard deviation failed to keep a proportionate pace with the increase in home run rates, but it has actually decelerated. That is, while offensive output has increased substantially, the playing field has become comparatively more level. Last season, for example, about 19.3 home runs were hit per 650 plate appearances in the National League, with a standard deviation of 11.9. Compare that to 1970, when just 15.6 home runs were hit per 650 PA–about a 20 percent decrease from contemporary levels–but the standard deviation was actually a bit higher, at 12.3.

This is far from a perfect experiment. But at the very least, it is highly problematic for the Steroid Gap Theory. If just a substantial minority were benefiting from steroid use, and the benefit were predictably and markedly positive, then we'd expect the differentiation between the haves and the have-nots to have increased. That differentiation has in fact increased on an absolute level, but it has decreased relative to what we would expect given the overall environmental improvements that all hitters are benefiting from, be those in the form of expansion, a lively ball, a smaller park, the birth of Jimmy Haynes, or what have you.

It is imperative to note that, while these results are inconsistent with the Steroid Gap Theory, they are not inconsistent with the notion that some players have used steroids, or the sentiment that steroid use is a problematic thing for the game. It is irrefutably the case that some players have used steroids, and as far as I am concerned, it is just as irrefutably the case that it will be to everyone's benefit to increase the rigorousness of testing programs, and the penalties for steroid use. What is required, however, is a different framework for understanding the impact that steroids might have had on the game's statistics–an alternative to the Steroid Gap Theory. I will propose three such hypotheses that are both more consistent with the data, and more consistent with my underlying sense of reality:

  • The Isolated Incident Theory. There are some players using steroids, possibly including some of the game's best-known sluggers. However, the number of players who both use steroids and receive substantial and prolonged benefits from them is very small, and the increase in offensive levels has been brought about by other mechanisms. While steroids might have a profound impact on the statistics of a few players, they have not benefited enough players to trigger a measurable change in the competitive ecology of the game.
  • The Superscrub Theory. The results described above would also be consistent with the notion that, while steroid use might be fairly widespread, its benefits have mostly been conferred upon the have-nots, rather than the haves. That is, it is the mostly the "enhanced" performance of players who once had little power that is boosting the league averages, rather than incremental improvements by the game's naturally elite power hitters.

    You will sometimes hear versions of the Superscrub Theory in the media, generally taking the form of exasperated complaints about "all these scrawny second basemen hitting all these damned home runs," or something similar. However, most of the hysteria, particularly in light of the recent Congressional hearing, has been focused on implications of steroid use by the game's star players, and the corresponding consequences for longstanding offensive records. Why I find the Superscrub Thoery compelling is that it is consistent with a couple of economic explanations for how steroids might differentially impact players:

    1. Incentives. The players who might have the most to gain from steroid use are not established stars with large, guaranteed contracts and endorsement deals, but rather those players on the fringe, for whom a good season and an attendant longer-term contract could make the difference between a lifetime of relative poverty and a secure income stream deep into their lives. There are at least a few anecdotal examples, like Phil Garner's recent admission that he considered using steroids for a brief period at the end of his career, when his skills were in remission, that work to support this notion. An exception might be in contract years, when good, established players would also have a substantial incentive to use steroids.
    2. Marginal Returns. While players of all ability levels might be equally disposed to use steroids, the substances might confer more benefit on those that have more to gain. That is, it is easier for steroids to turn a relatively weaker, smaller player into a bigger, stronger player than it is for steroids to turn a player who is already very big and very strong into some sort of ubermensch. The players who are already at the extreme end of the bell curve in terms of their physical development would receive diminishing returns from steroids.

    Finally, it is worth remembering that there are political and rhetorical points to be scored by accusing star players of using steroids, rather than forgettable scrubs. Consequently, if a list of known positives develops through some means other than a random and universal testing program, it is likely to be a biased list. Jose Canseco can push a lot of books by accusing Rafael Palmeiro of using steroids; the same wouldn't be true if he'd singled out Lance Blankenship.

  • The Big Hairy Mess Theory. While performance-altering substances do exist, there is not a fine line between improved nutrition, legal supplements, their quasi-legal variants, and explicitly illegal steroids. Moreover, the benefits of these substances is not universally positive, but will vary substantially based on the particular substances that a player takes, his training habits, and his underlying physiology. In some cases, the impact might trigger a tipping point and be substantially positive, but in many others it will be marginal, and in other cases still, like that of Jeremy Giambi, it might be deleterious. While "steroids" might be responsible for some of the global gain in offensive levels, their impact on the competitive ecology of the game is ambiguous, and not readily distinguishable from the more routine sorts of discrepancies that have been present from the first days of the game, like differences in equipment or coaching.

It is my belief that the latter theory is closest to the mark. There is clearly something going on–but it is not producing the sort of predictable impacts that everyone expects. Nor, because of the complexity of the underlying chemistry, are we likely to see substantial changes in the game's statistics resulting from efforts to curtail use of these substances.

In other words, apart from an increase in the Scoville Scale of your local baseball column, 2005 is likely to be business as usual. And that is a wonderful thing.

Thank you for reading

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So we're looking at cross sectional variance here... What about within-player longitudinal variance? It could be that the users are hitting more home runs and the nonusers are retiring and getting demoted to the minors. In other words, selection into the major leagues is highly endogenous to steroid use. In that scenario, maybe everybody in the majors has more home runs... And the non-steroid users lose roster spots and don't really impact your cross sectional variance.

Trying to think of the best approach to showing this... I guess one way to show evidence for this story would be that player performances vary more relative to their own ex ante expected performances. So based on history up to 1995, you expect so many home runs from a player in 1996, and the variance of his actual home runs in 1996 relative to what you expected is larger than that variance would've been using a similar forecasting method for 1983 using historical data through 1982.

The sophisticated version might be a panel data model of player level home run totals with player and year fixed effects. Use, say, 1970-2012. Then you test the null hypothesis that the variance of the white noise errors is larger for the 1993-2006 period than for the preceding and following years. If you reject the null that the variance increases for the steroid years, that supports the "steroids mattered" theory.