Regression toward the mean

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In simple English, regression toward the mean refers to the fact that those with extreme scores on any measure at one point in time will probably have less extreme scores the next time they are tested for purely statistical reasons. Scores always involve a little bit of luck. Many extreme scores include a bit of luck that happened to fall with or against you depending on whether your extreme score is extremely high or extremely low.

For example, imagine we want to examine students who score a perfect 100% correct and a perfect 100% wrong on the SAT. Both of these scores are hard to get. To get 100% wrong, you would have to be wrong 100% of the time and even when you guess the answer, your guesses are wrong 100% of the time.

1. Some of those people who scored 100% right guessed on some of the questions and guessed right. Conversely, some of those who scored 100% wrong guessed on some of the questions and guessed wrong.

2. The odds of all your guesses being right or wrong is pretty slim. By chance alone, the next time you flip a coin to decide which answer to pick, you will probably not guess 100% right or wrong again. If you guessed right 100% of the time last time, you probably won't be so lucky the next time (on average) and your score will drop a bit (towards the average, or mean). In fact, luck can't take you any higher than 100% right. If luck moves you at all, it moves you down (towards the mean). Similarly, if you scored 100% wrong the first time, you probably won't be so unlucky the next time. Again, you can't do worse than 100% wrong. So, by chance alone your score will go up the next time you take the SAT.

Contents 1 History 2 Ubiquity 3 Mathematical derivation 4 Regression fallacies 5 In sports 6 References 7 External links

[edit] History

The clearest early example of regression towards the mean was perhaps Galton's diagram in his article in Journal of the Anthropological Institute, vol. 15 (1885), p.248). It concerned the "Rate of Regression in Heriditary Stature", and compared children's height with their mid-parent height. (All heights are expressed as deviations from the median.) He summarises this results by putting in the diagram:

• When mid-parents are taller than mediocrity (by which he means the median), their children tend to be shorter than they

and

• When mid-parents are shorter than mediocrity, their children tend to be taller than they

which is as good a summary of the "regression effect" as we are likely to get.

Galton also drew the first regression line. This was a plot of seed weights and was presented at a Royal Institution lecture in 1877. Galton had seven sets of sweet pea seeds labeled K to Q and in each packet the seeds were of the same weight. He chose sweet peas on the advice of his cousin Charles Darwin and the botanist Joseph Dalton Hooker as sweet peas tend not to self fertilise and the seed weight varies little with humidity. He distributed these packets to a group of friends throughout Great Britain who planted them. At the end of the growing season the plants were uprooted and returned to Galton. The seeds were distributed because when Galton had tried this experiment himself in the Kew Gardens in 1874, the crop had failed.

He found that the weights of the offspring seeds were normally distributed, like their parents, and that if he plotted the mean diameter of the offspring seeds against the mean diameter of their parents he could draw a straight line through the points — the first regression line. He also found on this plot that the mean size of the offspring seeds tended to the overall mean size. He initially referred to the slope of this line as the "coefficient of reversion". Once he discovered that this effect was not a heritable property but the result of his manipulations of the data, he changed the name to the "coefficient of regression". This result was important because it appeared to conflict with the current thinking on evolution and natural selection. He went to do extensive work in quantitative genetics and in 1888 coined the term "co-relation" and used the now familiar symbol "r" for this value.

In additional work he investigated geniuses in various fields and noted that their children, while typically gifted, were almost invariably closer to the average than their exceptional parents. He later described the same effect more numerically by comparing fathers' heights to their sons' heights. Again, the heights of sons both of unusually tall fathers and of unusually short fathers was typically closer to the mean height than their fathers' heights.

[edit] Ubiquity

It is important to realize that regression toward the mean is unrelated to the progression of time: if we work with standardized measures , the fathers of exceptionally tall sons also tend to be closer to the mean than their sons.

The original version of regression toward the mean suggests an identical trait with two correlated measurements with the same reliability. However, this character is not necessary, unless any pair of predicting and predicted variables had to be viewed with an identical potential trait. The necessary implicate presumption is that the standard deviations of the predicting and the predicted are the same to be comparable, or have been transformed or interpreted to be comparable.

One later version of regression toward the mean defines a predicting variable with measurement error which impairs the predicting coefficient. This interpretation is not necessary. For example, in the original case the measurement error of length could be ignored.

[edit] Mathematical derivation

If random variables X and Y have standard deviations S_X and S_Y and correlation coefficient ρ the slope of the regression line is given by

A consequence of this is that a change of 1 standard deviation in X is associated with a change of ρ SDs in Y. Unless X and Y are perfectly correlated ρ will be less than unity. Thus, for a given value of X the value of Y that would be predicted by the regression line is always fewer SDs from its mean than X is from its mean. Regression to the mean will occur if | ρ | < 1, so in practice it always occurs.

Note that regression toward the mean is more pronounced the less the two variables are correlated, i.e. the smaller |r| is.

The phenomenon of regression toward the mean is related to Stein's example.

[edit] Regression fallacies

Misunderstandings of the principle (known as "regression fallacies") have repeatedly led to mistaken claims in the scientific literature.

An extreme example is Horace Secrist's 1933 book The Triumph of Mediocrity in Business, in which the statistics professor collected mountains of data to prove that the profit rates of competitive businesses tend toward the average over time. In fact, there is no such effect; the variability of profit rates is almost constant over time. Secrist had only described the common regression toward the mean. One exasperated reviewer likened the book to "proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals".

A different regression fallacy occurs in the following example. We want to test whether a certain stress-reducing drug increases reading skills of poor readers. Pupils are given a reading test. The lowest 10% scorers are then given the drug, and tested again, with a different test that also measures reading skill. We find that the average reading score of our group has improved significantly. This however does not show anything about the effectiveness of the drug: even without the drug, the principle of regression toward the mean would have predicted the same outcome. (The solution is to introduce a control group, compare results between the group to which drugs were administered and the control group, and make no comparisons with the original population. This removes the bias between the groups compared.)

The calculation and interpretation of "improvement scores" on standardized educational tests in Massachusetts probably provides another example of the regression fallacy. In 1999, schools were given improvement goals. For each school, the Department of Education tabulated the difference in the average score achieved by students in 1999 and in 2000. It was quickly noted that most of the worst-performing schools had met their goals, which the Department of Education took as confirmation of the soundness of their policies. However, it was also noted that many of the supposedly best schools in the Commonwealth, such as Brookline High School (with 18 National Merit Scholarship finalists) were declared to have failed. As in many cases involving statistics and public policy, the issue is debated, but "improvement scores" were not announced in subsequent years and the findings appear to be a case of regression to the mean.

The psychologist Daniel Kahneman referred to regression to the mean in his speech when he won the 2002 Bank of Sweden prize for economics.

I had the most satisfying Eureka experience of my career while attempting to teach flight instructors that praise is more effective than punishment for promoting skill-learning. When I had finished my enthusiastic speech, one of the most seasoned instructors in the audience raised his hand and made his own short speech, which began by conceding that positive reinforcement might be good for the birds, but went on to deny that it was optimal for flight cadets. He said, "On many occasions I have praised flight cadets for clean execution of some aerobatic maneuver, and in general when they try it again, they do worse. On the other hand, I have often screamed at cadets for bad execution, and in general they do better the next time. So please don't tell us that reinforcement works and punishment does not, because the opposite is the case." This was a joyous moment, in which I understood an important truth about the world: because we tend to reward others when they do well and punish them when they do badly, and because there is regression to the mean, it is part of the human condition that we are statistically punished for rewarding others and rewarded for punishing them. I immediately arranged a demonstration in which each participant tossed two coins at a target behind his back, without any feedback. We measured the distances from the target and could see that those who had done best the first time had mostly deteriorated on their second try, and vice versa. But I knew that this demonstration would not undo the effects of lifelong exposure to a perverse contingency.

[edit] In sports

Statistical analysts have long recognized the effect of regression to the mean in sports; they even have a special name for it: the "Sophomore Slump." For example, Carmelo Anthony of the NBA's Denver Nuggets had an outstanding rookie season in 2004. It was so outstanding, in fact, that he couldn't possibly be expected to repeat it: in 2005, Anthony's numbers had slightly dropped from his torrid rookie season. The reasons for the "sophomore slump" abound, as sports are all about adjustment and counter-adjustment, but luck-based excellence as a rookie is as good a reason as any. Of course, not just "sophomores" experience regression to the mean. Any athlete who posts a significant outlier, whether as a rookie (young players are universally not as good as those in their prime seasons), or particularly after their prime years (for most sports, the mid to late twenties), can be expected to perform more in line with their established standards of performance. The trick for sports executives, then, is to determine whether or not a player's play in the previous season was indeed an outlier, or if the player has established a new level of play. However, this is not easy. Melvin Mora of the Baltimore Orioles put up a season in 2003, at age 31, that was so far away from his performance in prior seasons that analysts assumed it had to be an outlier... but in 2004, Mora was even better. Mora, then, had truly established a new level of production, though he will likely regress to his more reasonable 2003 numbers in 2005. Conversely, Kurt Thomas of the New York Knicks significantly ramped up his production in 2001, at an age (29) when players typically start to play more poorly. Sure enough, in the following season Thomas was his old self again, having regressed to the mean of his established level of play. John Hollinger has an alternate name for the law of regression to the mean: the "fluke rule," while Bill James calls it the "Plexiglass Principle." Whatever you call it, though, regression to the mean is a fact of life, and also of sports.

Regression to the mean in sports performance produced the "Sports Illustrated Jinx" superstition, in all probability. Athletes believe that being on the cover of Sports Illustrated jinxes their future performance, where this apparent jinx was an artifact of regression.

[edit] References

• J.M. Bland and D.G. Altman. "Statistic Notes: Regression towards the mean", British Medical Journal 308:1499, 1994. (Article, including a diagram of Galton's original data, online at: [1])

• Francis Galton. "Regression Towards Mediocrity in Hereditary Stature," Journal of the Anthropological Institute, 15:246-263 (1886). (Facsimile at: [2])

• Stephen M. Stigler. Statistics on the Table, Harvard University Press, 1999. (See Chapter 9.)

[edit] External links

• A non-mathematical explanation of regression toward the mean.
• A simulation of regression toward the mean.
• Amanda Wachsmuth, Leland Wilkinson, Gerard E. Dallal. Galton's Bend: An Undiscovered Nonlinearity in Galton's Family Stature Regression Data and a Likely Explanation Based on Pearson and Lee's Stature Data (A modern look at Galton's analysis.)

• Massachusetts standardized test scores, interpreted by a statistician as an example of regression: see discussion in sci.stat.edu and its continuation.

• Kahneman's Nobel speech [3]