#imperfect correlation

Last Uncertainty Wednesday I wrote about how regression to the mean at least partially explains the “curse” of building extravagant headquarters (which often signals an impending downturn in the performance of a company). Whenever we come up with a detailed causal explanation for something that is simply a statistical effect we are caught up in what is known as the narrative fallacy. As Kahneman documents so well in his book Thinking Fast and Thinking Slow, the brain has long evolved to want to see causal connections and tell stories about them which is why we are so prone to this fallacy.

Regression to the mean can be explained by considering that skill and performance are imperfectly correlated due to the role of luck. Imagine for a moment the converse: a hypothetical sport in which outcomes are strictly based on skill every single encounter. Such a sport would be incredibly boring because skills don’t change very quickly and so for many encounters the results would be pre-determined. There would also be no regression to the mean in such a sport. If your team had a great season last season it will have a great season again next season, because all of last season’s success was due to skill, none of it due to luck.

As it turns out imperfect correlation is also at the heart of another powerful narrative fallacy: inverse correlation at the extremes. Because imperfect correlation is everywhere (perfect correlation being rare), this narrative fallacy is pervasive. How does it work? Here is a graph of two imperfectly correlated variables

Now ask yourself the following question: why is it that countries with the highest GDP per capita do not have the highest well being? And watch your brain immediately go into narrative overdrive. “Super high GDP per capita means people are working all the time, so they have less time for their friends and family.” Voila. A perfect explanation. Done and moving on.

Except you have just committed a case of the narrative fallacy. Whenever two variables are imperfectly correlated there is negative correlation at the extremes and it simply must be that way. The only time that’s not the case is when the two variables are perfectly correlated. To see this better visually consider the following version of the chart

Now look at the points to the right of the green line. Those are countries with extremely high GDP per capita. You can easily see that these countries are unlikely to have the highest well being -- the only way they could would be if well being and GDP per capita were perfectly correlated. This is also true of course at the other extreme, when you look at points to the left of the blue line. These are countries with the lowest GDP per capita. Again they are unlikely to have the lowest well being because the only way that could happen is with perfect correlation.

Since correlation is symmetric you could go through the same exercise drawing horizontal lines and looking only at countries with the highest well being (or the lowest well being). And since this shape is the same for all imperfectly correlated variables (just remove the axis labels) this has absolutely nothing to do with the measures being GDP per capita and well being but applies to any two imperfectly correlated variables. Kahneman gives the example of the intelligence of spouses -- which can make for a very controversial dinner party conversation (given that it is imperfectly correlated). People will come up with all sorts of amazing explanations for that one!