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I heard a story a long time ago, and I don’t remember who told it to me and I’m probably getting all the details wrong, but I wanted to try to recreate the story from memory because it illustrates one of the perils of blind reliance on statistical models to identify “important” variables.
A statistician was asked to analyze some data about an industrial process and there were about a dozen or so independent variables that affect the outcome. So the statistician did some sort of stepwise regression or R-squared calculation and came up with an ordering for all the independent variables. The most important variable was the one with largest correlation or the first variable entered in the stepwise model (I’m not sure which, but the point is the same either way). The second most important variable was the one with the second largest correlation or the second variable entered in the stepwise model.
The statistician reviewed each variable in order starting with the most important variable. It was rather dull, of course, until the statistician got to the bottom of the list. He proclaimed “and the least important variable is the amount of water in the raw material.”
At this point the engineers in the room burst into laughter. It turns out that water was the most important variable. If you had even a small amount of water in the raw material, the entire production process would explode. The engineers spent a huge amount of effort to keep the water down to a level that was barely measurable.
If a variable has very little variability in it by design, you cannot expect to see a large correlation. This is sometimes called a restriction of range problem. The SAT test for college applicants has an upper bound of 2400 and for some high end Universities, they may end up admitting only students scoring 2350 or higher. That’s a very narrow range, and if it turns out that the SAT scores at this place are a poor predictor of future performance (like GPAs or graduation rates), that may be more a function of the very narrow range of students that were admitted than anything else.
Now whenever I hear a story like this, I think of the preachers quote “There but for the grace of God go I” that has been attributed to John Bradford. I bet I’ve said stuff even stupider than what this statisticians is supposed to have said. It is just dumb luck, or God is looking out for me, or something else that keeps me from the one being publicly humiliated.
How do you avoid saying something so stupid that everyone laughs at you? Well, the obvious answer is to talk to one of the engineers first and show them what you are going to say. Better for that one Engineer to laugh at you in private than having a whole room of Engineers laugh at you in public.
What do you do if you recognize that have a restriction of range problem? Well, first, drop the correlations in favor of a regression model. A linear regression model is not perfect, but it is a lot better than a correlation coefficient in this situaiton. Second, draw lots of graphs. Third, talk to the experts. Fourth, disclose the restriction of range as a possible limitation to your findings. A restriction of range means that you may be trying to extrapolate beyond the range of your data (your graphs will help show this) and this type of extrapolation will often require making untestable assumptions.
And if anyone knows the source of this story or can point me to a reference, I would be forever in your debt.