Correlation does not imply causation — but it implies something

“Correlation does not imply causation” is the most successfully transmitted sentence in all of statistics. It is on posters. It gets deployed in comment sections within seconds. It is also, taken alone, a conversation-ender that leaves people less able to reason than a slightly longer version would.

Because correlation does imply something. It implies a short list, and knowing the list is what turns the slogan from a dismissal into a method.

The list

If two things are genuinely correlated — not a fluke of a small sample, not an artefact — then at least one of these holds:

1. A causes B.
2. B causes A. The direction is not given by the data, and reversal is a real explanation far more often than people expect.
3. Something else causes both. The classic confounder.
4. They are linked through how the data was gathered, without any causal connection in the world at all.
5. It is chance, and with enough comparisons this happens constantly.

That is close to exhaustive. So the correct response to a correlation is not “therefore nothing” — it is which of these five, and what would distinguish them? That is an answerable question, and it is where all the interesting work lives.

Reversal is underrated

Everybody reaches for the confounder and forgets to check the arrow.

Studies find that people who take more holidays are healthier. The intuitive reading is that rest improves health. The other direction is at least as plausible — people who are unwell take fewer holidays. Same correlation. Opposite intervention.

The question worth asking of any correlation is simply: if the arrow ran the other way, would the data look different? Often it would not, and if it would not, the data cannot tell you which way it runs.

The fourth one is the interesting one

Selection effects deserve more attention than they get, because they produce correlations that are entirely real in your dataset and entirely absent in the world.

The clean example is medical. Among people admitted to hospital, two unrelated diseases can appear negatively correlated — not because either protects against the other, but because having either one raises your chance of being admitted, so among the admitted, having one makes it less necessary that you have the other to explain your presence. Nothing causal happened. Conditioning on admission did it.

This is Berkson’s paradox, and the general form is worth internalising, because it turns a piece of standard advice on its head: controlling for a variable can create a spurious correlation rather than remove one. Adding controls is not automatically more rigorous. If the variable you controlled for is a common effect of both things you are studying — a collider, in the jargon — you have manufactured the association you are now reporting.

Which is why “did you adjust for X?” is not a question with a universally correct answer of yes.

Where the slogan misleads

Two ways.

It suggests correlation is worthless. It is not — it is often exactly what you need. If you want to predict rather than intervene, causation is optional. A model using shoe size to predict reading ability in children works fine, despite the obvious confounder of age, and it will keep working as long as the relationship holds. Prediction rides on association. Only intervention needs the arrow, because intervention breaks the association it was riding on: buying bigger shoes teaches nobody to read.

Keep those apart and a lot of arguments about models dissolve. “Is it causal?” is the wrong question until somebody proposes to act on it.

And it implies randomised experiments are the only escape. They are the cleanest, but a great deal of causal knowledge comes from elsewhere — natural experiments where something arbitrary assigned people to groups, discontinuities where a cutoff creates near-identical groups either side, instruments that move the cause and nothing else. Whole fields work this way and produce real answers, because randomisation is often impossible or unethical and the question does not go away.

The better sentence

Replace the slogan with the question it should have been:

Correlation raises the question of causation. Which of the five is it, and what would tell them apart?

That keeps the caution and returns the information. It also invites the useful follow-up — is anyone planning to do something with this, or only to predict — which decides whether the answer matters at all.