Correlation, Prediction, and Explanation
The slogan that correlation does not imply causation is universally known and routinely misapplied. It is often taken to mean that statistical evidence is worthless for causal questions, which is false — randomised trials are statistical evidence, and epidemiology established that smoking causes cancer without a single controlled human experiment. The accurate statement is narrower and more useful: correlation constrains causal structure without determining it, and closing the gap requires assumptions that data alone cannot supply.
This page collects the practical consequences of the folder's argument: the ways correlations mislead, why predictive success does not confer causal understanding, and what statistical evidence can establish.
How correlations arise without direct causation
A correlation between and has a limited menu of explanations, and enumerating them is the first step in any causal assessment.
- causes — the hypothesis of interest.
- causes — reverse causation. Poverty correlates with poor health in both directions.
- A common cause — confounding.
- Selection — the sample was chosen in a way that induces the association.
- Chance — particularly with small samples or many comparisons.
- Artifact — shared measurement error, shared denominators, or definitional overlap.
Distinguishing these is the whole problem, and it cannot be done from the joint distribution of and alone.
Selection and collider bias
The most treacherous case, because conditioning on the wrong variable creates an association that does not exist in the population, and the resulting correlation is entirely real within the sample.
If is a common effect of and — a collider — then conditioning on makes and dependent even when they are independent. The mechanism is simple: given that the effect occurred, one cause being absent makes the other more likely.
Berkson's paradox is the medical instance. Among hospital patients, two unrelated diseases appear negatively correlated, because either disease alone can cause hospitalisation — so a patient with one is less likely to also have the other than a member of the general population. The correlation is genuine in the hospital and absent in the population, and the hospital sample is selected on a collider.
The pattern is pervasive: restaurant quality and location among surviving restaurants; talent and attractiveness among celebrities; test scores and interview performance among admitted students. In each case the sample is conditioned on a common effect, and the induced correlation is an artifact of selection.
This is why "controlling for more variables" is not a conservative strategy. Adding a collider to a regression introduces bias where none existed. As the Simpson discussion showed, whether to condition depends on causal role, and the data do not reveal it.
Prediction without understanding
Modern machine learning makes vivid a distinction that was always available: a model can predict superbly while encoding no causal structure at all.
A model predicting hospital mortality may rely on the fact that patients receiving palliative care die more often. The correlation is strong and stable, and the model will perform well on data drawn from the same regime. But intervening on the predictor — withholding palliative care — would not reduce mortality, and might increase it.
The general principle: predictive models exploit any stable association, causal or not, and they are optimised for exactly that. Purely predictive accuracy is therefore no evidence of causal correctness.
The distinction matters for three practical reasons.
- Intervention. Acting on a variable changes the distribution, which is and not . A model trained on observational data estimates the latter.
- Transportability. A model relying on a confounded association fails when transferred to a population where the confounding differs. Causal relationships are more stable across populations than correlations precisely because they reflect mechanisms, which is one of the better arguments for caring about causal structure at all.
- Distribution shift. When a predictor is itself acted upon — because the model's use changes behaviour — the association it relies on can disappear. Goodhart's law is this phenomenon: a measure that becomes a target ceases to be a good measure, because the correlation between measure and underlying quality was not causal in the direction assumed.
What statistical evidence can establish
The negative points should not obscure how much is achievable, since the sceptical slogan is more often used to dismiss good evidence than to catch bad.
Randomisation severs the edges from confounders into the treatment, making assignment independent of everything prior. The observed association then estimates the causal effect, and this is why controlled trials are the standard. The assumptions are not nil — compliance, blinding, no interference between units — but they are far weaker than in observational work.
Observational inference with explicit assumptions. Given a causal graph and the back-door criterion, effects can be identified from observational data. The assumptions are strong and usually untestable, but they are stateable, which allows disagreement to be located precisely. Natural experiments, instrumental variables, and regression discontinuity exploit situations where something approximating randomisation occurs naturally.
Convergent evidence. The case that smoking causes lung cancer was made without human experiments, by combining dose–response relationships, temporal ordering, animal experiments, biological mechanism, consistency across populations with different confounding structures, and the implausibility of any common cause accounting for the magnitude of the association. The Bradford Hill considerations systematise this, and the argument's strength comes from the fact that no single confounder could explain all the strands at once. This is the practical answer to the sceptic: not one decisive statistic, but a structure of evidence whose alternative explanations have been eliminated piecemeal.
Elimination of alternatives. Fisher's suggestion that a genetic common cause explained the smoking–cancer association was a serious hypothesis, and it was refuted by evidence — twin studies, dose–response, the effect of cessation — not by assertion. That is how causal inference from observational data proceeds: by making the alternatives explicit and testing them.
Explanation
The final connection back to chance and explanation. A correlation does not explain, even when it predicts perfectly. The barometer's fall predicts the storm and explains nothing, because the explanatory relation runs through the atmospheric pressure.
This supports a general moral about the relationship between the two folders. Probabilistic dependence is evidence about causal and explanatory structure but is not identical to it, and the same holds for chance: a chance is evidence about the mechanism without being a description of one. In both cases the probability is a symmetric relation and what it is evidence for is asymmetric, which is why the extra structure — intervention, mechanism, direction of explanation — has to be supplied from outside the calculus.
Where this sits
This completes the causation folder. Its argument runs: probability raising fails as an analysis in both directions; screening off is the principle that makes causal inference possible and is empirically false in the quantum case; Simpson's paradox shows the same statistics can require opposite analyses; causal models supply the missing structure through intervention; and the practical upshot is that statistical evidence establishes causal conclusions only in conjunction with causal assumptions — which can nonetheless be made explicit, tested, and sometimes eliminated.
The remaining folders take up the standing puzzles where these tools are stress-tested, the scientific cases where the interpretation of probability has the most at stake, and the historical and methodological context.