Monday, December 10, 2012

The Regression Fallacy

Consider a teacher, keen to apply rational techniques to maximize the effectiveness of his didactic program. For some time, he has been gathering data on the outcomes of certain stimuli aimed at improving his pupil's performance. He has been punishing those students that under-perform in discrete tasks, and rewarding those that excel. The results show, unexpectedly, that performances were improved, on average, only for pupils that received punishments, while those that were rewarded did worse subsequently. The teacher is seriously considering desisting from future rewards, and continuing only with punishments. What would be your advice to him?

First note that a pupil's performance in any given task will have some significant random component. Luck in knowing a particular topic very well, mood at the time of execution of the task, degree of tiredness, pre-occupation with something else, or other haphazard effects could conspire to affect the student's performance. The second thing to note is this: if a random variable is sampled twice, and the first case is far from average, then the second is most likely to be closer to the average. Neglect of this simple fact is common, and is a special case of the regression fallacy.

If a pupil achieves an outstanding result in some test, then probably this as partly due to the quality of the student, and partly due to random factors. It is also most likely that the random factors contributed positively to the result. So a particular sample of this random variable has produced a result far into the right-hand tail of its probability distribution. The odds of a subsequent sample from the same probability distribution being lower than the first are clearly the ratio of the areas of the distribution, either side of the initial value. These odds are relatively high.

Imagine a random number generator that produces an integer from 1 to 100 inclusive, all equally probable. Suppose that in the first of two draws, the number 90 comes up. There are now 89 ways to get a smaller number on the second draw, and only 10 ways to get a larger number. In a very similar way, a student who performs very well in a test (an therefore receives a reward) has the odds stacked against them, if they hope to score better in the next test. The regression fallacy, in this case, is to assume that the administered reward is the cause of the eventual decline in performance.

The argument works exactly the same way for a poorly performing pupil - a really bad outcome is most likely, by chance alone, to be followed by an improvement. This tendency for extreme results to be followed by more ordinary results is called regression to the mean. It is not impossible that an intervention such as a punishment could cause improved future performance, but the automatic assumption that an observed improvement is caused by the administered punishment is fallacious.

Another common example comes from medical science. Its when my sinusitis is worst that I sleep with a freshly severed rabbit's foot under my pillow. I almost always feel better the next morning.

These before-after scenarios are a special case, as I mentioned. In general, all we need in order to see regression to the mean is to sample two correlated random variables. They may be from the same distribution (before-after), or they may be from different distributions.

If I tell you that Hans is 6 foot, 4 inches tall (193 cm), and ask you what you expect to be the most likely height of his fully-grown son, Ezekiel, you might correctly reason that father's heights and son's heights are correlated. You might think, therefore, that the best guess for Ezekiel's height is also 6', 4", but you would be forgetting about regression to the mean - Ezekiel's height is actually most likely to be closer to average. This is because the correlation between father's and son's heights is not perfect. On a scale where 0 represents no correlation whatsoever, and 1 indicates perfect correlation (knowledge of one necessarily fixes the other precisely1), the correlation coefficient for father-son stature is about 0.5. Reasoning informally, therefore, we might adjust our estimate for Ezekiel's still unknown height to half way between 6' 4'' and the population average. It turns out we'd be bang on with this estimate. (I don't mean, of course, that this is guaranteed to be the fellow's height, but that this would be our best possible guess, most likely to be near his actual height.)

The success of this simple revision follows from the normal (Gaussian) probability distribution for people's heights. The normal distribution can be applied in a great many circumstances, both for physical reasons (central limit theorem), and for the reason that we often lack any information required to assign a more complicated distribution (maximum entropy). If two variables, x and y are each assigned a normal distribution (with respective means and standard deviations μi and σi), and provided that certain not-too-exclusive conditions are met (all linear combinations of x and y are also normal), then their joint distribution, P(xy | I), follows the bivariate normal distribution, which I won't type out, but follow the link if you'd like to see it. (As usual, I is our background information.) To get the conditional probability for y, given a known value of x, we can make use of the product rule , to give

P(x | I) is the marginal distribution for x, just the familiar normal distribution for a single variable. If one goes through the slightly awkward algebra, it is found that for xy bivariate normal, y|x is also normally distributed2, with mean

and standard deviation

where ρ is the correlation coefficient, given by

Knowing this mean and standard deviation, we can now make a good estimate of how much regression to the mean to expect in any given situation. We can state our best guess for y and its error bar.

We can rearrange equation (2) to give

which says that the expected number of standard deviations between y - given our information about x - and μy (the mean of y when nothing is known about about x) is the same as the number of standard deviations between the observed value of x and μx, only multiplied by the correlation coefficient. A bit of a mouthful, perhaps, but actually a fairly easy estimate to perform, even in an informal context. In fact, this is just what we did when estimating Ezekiel's height.

When reasoning informally, we can ask the simplified question, 'what value of y is roughly equally as improbable as the known value of x?' The human mind is actually not bad at performing such estimates. Next, we need to figure out how far it is from the expected value of y (μy) and reduce that distance from μy by the fraction ρ, which again (with a bit of practice, perhaps), we can also estimate not too badly. In any case, an internet search will often be as complicated as any data-gathering exercise needed to calculate ρ more accurately.

Here are a few correlation coefficients for some familiar phenomena:

Life expectancy by nation (data from Wikipedia here and here):

life expectancy vs GDP per capita: 0.53
male life expectancy vs female life expectancy: 0.98

IQ scores (data from Wikipedia again):

same person tested twice: 0.95
identical twins raised together: 0.86
identical twins raised separately: 0.76
unrelated children raised together: 0.3

Amount of rainfall (in US) and frequency of 'flea' searches on Google: 0.87
(from Google Correlate)

With the above procedure for estimating y|x, we can get better, more rational estimates for a whole host of important things: how will our company perform this year, given our profits last year? How will our company perform if we hire this new manager, given how his previous company performed? What are my shares going to do next? What will the weather do tomorrow? How significant is this person's psychological assessment? Or criminal record?

In summary, when two partially correlated, random variables are sampled, there is a tendency for an extreme value of one to be accompanied by a less extreme value for the other. This is simple to the point of tautology, and is termed regression to the mean. The regression fallacy is a failure to account for this effect when making predictions, or investigating causation. One common form is the erroneous assumption of cause and effect in 'before-after' type experiments. Rabbits' feet do not cure sinusitis (in case you were still wondering). Another kind of fallacious reasoning is the failure to regress to the mean an estimate or prediction of one variable based on another known fact. For two normally distributed, correlated variables, the ratio of the expected distance (in standard deviations) of one variable from its marginal mean to the actual distance of the other from its mean is the correlation coefficient.

[1] Note: there are also cases where this condition holds for zero correlation, i.e. situations where y is completely determined by x, even though their correlation coefficient is zero. Lack of correlation can not be taken to imply independence, though if x and y are jointly (bivariate) normal, lack of correlation does strictly imply independence.

[2] I've been a little cavalier with the notation, but you can just read y|x as 'the value of y, given x.' Here, y is to be understood as a number, not a proposition.


  1. Somewhat related: Keep practicing at a sport, and you will see that, with time, your personal best gets better and better. At least part of this can be explained purely by statistics, i.e. with no improvement from practicing: there is a random element to each performance, and the more one samples this distribution, the likelier it is to get a high value.

    1. Yes. More generally, I think its fair to say that most sports commentators build their careers on an ability to exploit the regression fallacy to the max.