With interest in statistical applications to sports creeping from the blogosphere to the mainstream, more writers than ever are interested in metrics that can more accurately summarize or predict player and team skill.
This is, by and large, a good thing. Smarter writing is better writing. A downside, however, is that writers without a formal training in statistics are forced to discuss concepts that can take more than a semester’s work of undergraduate or graduate training to flesh out. That’s difficult, if not impossible and unfair.
One such topic that comes up across sports is the concept of regression toward the mean. Here are a few examples of headlines:
Regression to the mean can be a bitch! (soccer)
Clutch NFL teams regress to the mean (football)
Beware the regression to the mean (basketball)
30 MLB players due for regression to the mean (baseball)
Avalanche trying to stave off regression and history (hockey)
In each case, the regression (i) sounds scary, (ii) applies to over-performance, not under-performance, and (iii) is striving really hard to reach an exact target, in these examples a vaguely specified ‘mean.’
From a statistical perspective, however, regression toward the mean requires strict assumptions and precision, the context of which are almost never discussed in practice. As a result, examples that refer to a regression to the mean may be ill-informed, and are often best described by a similar sounding but more relaxed alternative.
Using the notation and descriptions in Myra Samuels’ 1991 paper in the American Statistician, “Statistical Reversion toward the mean: More universal than regression toward the mean,” here’s a quick primer through the context of sports.
What is regression towards the mean?
Let X and Y be a pair of random variables with the same marginal distribution and common mean µ. Most often in sports, X and Y are simply team/individual outcomes that we are interested in measuring and describing. For example, X could be the batting average of a baseball player through July 1, and Y his average from July 2 through the end of the season. In this example, µ represents that player’s probability of getting a hit.
The definition of regression toward the mean is based on the regression function, E[Y|X = x]. That is, conditional on knowing one variable (X = x), what can we say about the other? Formally, regression toward the mean exists if, for all x > µ,
µ < E[Y|X = x] < x,
with the reverse holding when x < µ.
This is a fairly strict requirement. For an outcome above a player or team’s true talent, we can expect that the ensuing outcome, on average, will lie in between µ and the original outcome. Linking to linear regression, for any initial observation x, the point prediction of y is regressed towards an overall mean representative of that subject. However, y will still exhibit some natural variation above and below the regression line; some points will fall closer to the mean, and others further away.
There are easy pitfalls when it comes to applying regression toward the mean in practice. The most common one is assuming that what goes up must come down. For example, assuming that players or teams become more and more average over time is not regression toward the mean. A second misinterpretation is linking regression toward the mean with the gambler’s fallacy, which entails assuming that a team or player that was initially lucky is then going to get less lucky. This is also not true. The probability of a fair coin landing heads, given that it landed heads five, ten, or even fifteen times in a row, remains at 0.5. Such misinterpretations are frequent in sports, particularly when describing team performance with respect to point spreads or performance in close games.
While its easy to confuse regression toward the mean with such scenarios, there’s good news, in the form of some easy to understand alternatives.
What’s the better alternative?
To start, replacing ‘regression’ with ‘reversion’ relaxes the assumptions presented above while still implying that extreme observations are more likely to be followed by less extreme ones. More often than not, when writers speak of regression to the mean, using reversion is sufficient and accurate. Furthermore, Samuels proves mathematically that ‘regression toward the mean implies reversion toward the mean, but not vice versa.’ Namely, reversion is a more relaxed alternative; the conditional mean of the upper or lower portion of a distribution shifts, or reverts, toward an unconditional mean µ.
For example, in the headlines listed above, good soccer teams, MLB players hitting for high numbers, and the Colorado Avalanche were all more likely to revert to a more standard form that was indicative of their true talent. No regression equation is necessary.
In addition to generally being a more appropriate term, use of the word reversion has a side benefit, in that it is more interpretable when applied to outcomes that initially fell short of expectations. It is recognizable, for example, to expect an MLB batter hitting 0.150 to revert to form; meanwhile, it doesn’t make sense to claim that the same MLB batter will regress, given the negative connotations of the latter.
And is it regression/reversion ‘to’ or ‘toward’ the mean?
Well, it depends. While increased use of the word reversion is part of the solution, more precise writing should also consider both the outcome of interest and that outcome’s expected value. For example, here are two examples of over-performance:
Mike Trout hits 0.500 in his first ten games of the season.
Mike Trout tosses a coin 10 times, landing nine heads.
And here’s the same sentence to describe our future expectations – can you tell which one is accurate?
Mike Trout’s batting average will revert to the mean.
Mike Trout’s ability to land heads will revert to the mean.
In the first example, the outcome of interest is Mike Trout’s probability of getting a hit. Because we can comfortably say that Mike Trout is better than the league average hitter, while his batting average is going to come down, it is reverting towards an overall average, but not to the overall average.
Meanwhile, unless Trout can outduel the Law of Large Numbers, I can comfortably say that in the long term, his observed ability to land heads will revert to a probability of 0.5. In this silly example, the second statement is the more precise one.
Anything else worth discussing?
Well, maybe. In searching for some of the examples used above, I found it strange how little was written of the one word that tends to encompass much of a players’ performance above or below his or her true talent.
The obvious aspect linking, say, the Colorado Avalanche winning games while being outshot and Mike Trout tossing coins and landing heads, is that each was on the receiving end of some lucky breaks. So while we expect some type of reversion to or towards a more traditional performance, that’s to no fault of the Avalanche or Trout. With outcomes that are mostly (or entirely) random, variability above or below the league average is simply luck. As a result, there’s nothing for the Avalanche, Trout, or even us to be scared/beware of. We wouldn’t tell Trout to fear a balanced coin, nor should we tell Avalanche fans to beware of reversion towards a more reasonable performance given their teams shot distribution.
The issue here lies not in a distinction between regression and reversion, but a deeper and more serious problem; humans have a poor grasp of probability. In sports (and likely in other areas of life), lucky outcomes are all too often touted as clutch, while unlucky players or teams are given the label of chokers. It’s standard practice to use terms like savvy to describe the Patriots win over Seattle, for example. A more skilled writer, however, would perhaps recognize that the Patriots were on the better end of a 50-50 coin toss, from more or less the start of the game all the way until the end (in more ways than one; the game closed as a near pick-em at sports books. Even bettors couldn’t nail down a winner).
Writes Leonard Mlodinow in The Drunkard’s Walk,
the human mind is built to identify for each event a definite cause and can therefore have a hard time accepting the influence of unrelated or random factors.
It’s difficult and counterintuitive to describe an outcome in sports as lucky. However, that’s what many of them are are.
So while it may sound trendy to toss around terms like ‘regress to the mean,’ it is often more accurate, and certainly more simple, to propose that some luck was involved in the initial outcome. As a result, a decline from overperformance is nothing more than a player or a team, much like a coin tosser no longer landing heads five times in a row, not getting as lucky as they initially had been.