Correlation and causation

You have probably heard people warn you, "Correlation does not imply causation." This is a reminder that when you are sampling natural variation in two variables, there is also natural variation in a lot of possible confounding variables that could cause the association between A and B. So if you see a significant association between A and B, it doesn't necessarily mean that variation in A causes variation in B; there may be some other variable, C, that affects both of them. For example, let's say you went to an elementary school, found 100 random students, measured how long it took them to tie their shoes, and measured the length of their thumbs. I'm pretty sure you'd find a strong association between the two variables, with longer thumbs associated with shorter shoe-tying times. I'm sure you could come up with a clever, sophisticated biomechanical explanation for why having longer thumbs causes children to tie their shoes faster, complete with force vectors and moment angles and equations and 3-D modeling. However, that would be silly; your sample of 100 random students has natural variation in another variable, age, and older students have bigger thumbs and take less time to tie their shoes.

So what if you make sure all your student volunteers are the same age, and you still see a significant association between shoe-tying time and thumb length; would that correlation imply causation? No, because think of why different children have different length thumbs. Some people are genetically larger than others; could the genes that affect overall size also affect fine motor skills? Maybe. Nutrition affects size, and family economics affects nutrition; could poor children have smaller thumbs due to poor nutrition, and also have slower shoe-tying times because their parents were too overworked to teach them to tie their shoes, or because they were so poor that they didn't get their first shoes until they reached school age? Maybe. I don't know, maybe some kids spend so much time sucking their thumb that the thumb actually gets longer, and having a slimy spit-covered thumb makes it harder to grip a shoelace. But there would be multiple plausible explanations for the association between thumb length and shoe-tying time, and it would be incorrect to conclude "Longer thumbs make you tie your shoes faster."

Since it's possible to think of multiple explanations for an association between two variables, does that mean you should cynically sneer "Correlation does not imply causation!" and dismiss any correlation studies of naturally occurring variation? No. For one thing, observing a correlation between two variables suggests that there's something interesting going on, something you may want to investigate further. For example, studies have shown a correlation between eating more fresh fruits and vegetables and lower blood pressure. It's possible that the correlation is because people with more money, who can afford fresh fruits and vegetables, have less stressful lives than poor people, and it's the difference in stress that affects blood pressure; it's also possible that people who are concerned about their health eat more fruits and vegetables and exercise more, and it's the exercise that affects blood pressure. But the correlation suggests that eating fruits and vegetables may reduce blood pressure. You'd want to test this hypothesis further, by looking for the correlation in samples of people with similar socioeconomic status and levels of exercise; by statistically controlling for possible confounding variables using techniques such as multiple regression; by doing animal studies; or by giving human volunteers controlled diets with different amounts of fruits and vegetables. If your initial correlation study hadn't found an association of blood pressure with fruits and vegetables, you wouldn't have a reason to do these further studies. Correlation may not imply causation, but it tells you that something interesting is going on.

In a regression study, you set the values of the independent variable, and you control or randomize all of the possible confounding variables. For example, if you are investigating the relationship between blood pressure and fruit and vegetable consumption, you might think that it's the potassium in the fruits and vegetables that lowers blood pressure. You could investigate this by getting a bunch of volunteers of the same sex, age, and socioeconomic status. You randomly choose the potassium intake for each person, give them the appropriate pills, have them take the pills for a month, then measure their blood pressure. All of the possible confounding variables are either controlled (age, sex, income) or randomized (occupation, psychological stress, exercise, diet), so if you see an association between potassium intake and blood pressure, the only possible cause would be that potassium affects blood pressure. So if you've designed your experiment correctly, regression does imply causation.