Correlation vs. Causation

DOES LUNG CANCER MAKE YOU SMOKE CIGARETTES?

And what about when two variables are correlated? What does that really mean?

To make this simple, let’s start with the simplest kind of variable, a binary variable with only two possible values. Oftentimes a binary variable is the answer to a yes-or-no question: “Are you married?” “Do you smoke?” “Are you now, or have you ever been, a member of the Communist Party?”

When you’re comparing two binary variables, correlation takes on a particularly simple form. To say that marital status and smoking status are negatively correlated, for example, is simply to say that married people are less likely than the average person to smoke. Or, to put it another way, smokers are less likely than the average person to be married. It’s worth taking a moment to persuade yourself that those two things are indeed the same! The first statement can be written as an inequality

married smokers / all married people < all smokers / all people

and the second as

married smokers / all smokers < all married people / all people

If you multiply each inequality on both sides by its common denominator you can see that the two statements are different ways of saying the same thing:

(married smokers) × (all people) < (all smokers) × (all married people)

In the same way, if smoking and marriage were positively correlated, it would mean that married people were more likely than average to smoke and smokers more likely than average to be married.

One problem presents itself immediately. In a finite sample, the observed proportion of smokers among married people will seldom be exactly the same as the proportion in the sample as a whole. A small sample correlation can appear even when variables are independent in the population. Sexual orientation, U.S. citizenship, or first-initial-in-the-last-half-of-the-alphabet might likewise show some sample association with smoking. These observations do not by themselves establish meaningful population relationships. The discussion of null hypotheses in chapter 7 raises a related question: how much does an exact zero, rather than a small effect, matter?

To throw up our hands and say, “Everything is correlated with everything else!” would be fairly uninformative. We do not report every observed correlation. Reports often select findings considered “strong enough” because they pass a statistical significance test. But statistical significance is not the same as a large or practically important relationship. A test result may prompt a statistician to say, “Something must be going on.” Its meaning still depends on the test’s assumptions, the study design, and how the finding was selected.

But what? Here we come to the really sticky part. Marriage is negatively correlated with smoking; that’s a fact. A typical way to express that fact is to say

“If you’re a smoker, you’re less likely to be married.”

But one small change makes the meaning very different:

“If you were a smoker, you’d be less likely to be married.”

It seems strange that changing the sentence from the indicative to the subjunctive mood can change what it says so drastically. But the first sentence is merely a statement about what is the case. The second concerns a much more delicate question: What would be the case if we changed something about the world? The first sentence expresses a correlation; the second suggests a causation. As we’ve already mentioned, the two are not the same. That smokers are less frequently married than others doesn’t mean that quitting smoking will summon up your future spouse. The mathematical account of correlation has been pretty much fixed in place since the work of Galton and Pearson a century ago. Putting the idea of causation on a firm mathematical footing has been much more elusive.*

There’s something slippery about our understanding of correlation and causation. Your intuition tends to grasp it quite firmly in some circumstances but lose its grip in others. When we say that HDL is correlated with a lower risk of heart attack, we’re making a factual statement: “If you’ve got a higher level of HDL cholesterol, you’re less likely to have a heart attack.” It’s hard not to think that the HDL is doing something—that the molecules in question are literally causing your cardiovascular health to improve, say, by “scrubbing” lipidic cruft off your arterial walls. If that were so—if the mere presence of a lot of HDL were working to your benefit—then it would be reasonable to expect any HDL-increasing intervention to reduce your risk of heart attack.

But it might be that HDL and heart attack are correlated for a different reason; say, that some other factor, one we haven’t measured, tends both to increase HDL and decrease the risk of cardiovascular events. If that’s the case, raising HDL alone need not reduce heart attacks. The effect would depend on which causal pathways an intervention changes.

For a financial analogy, imagine that Tim and Sara both hold Honda stock. Their fortunes may rise together because of that shared holding, without either person’s fund causing the other’s gains.

A biomarker used in place of a clinical outcome is called a surrogate endpoint. An association between the biomarker and the outcome does not by itself show that changing the biomarker will improve the outcome.