Regression, correlation and cause and effect

Our view. We agree with the statement in part. It is right about correlation, and right that regression is the tool used when a cause-and-effect link is believed to exist; but regression by itself does not prove that link either.

Correlation: association, not cause. Correlation measures the direction and the degree of covariation between two variables: whether they move together, and how closely. It is symmetrical (the correlation of X with Y equals that of Y with X) and does not say which variable influences the other. A high correlation can arise from pure coincidence or from a third variable acting on both, so it should never be read as cause and effect by itself. The syllabus lists Correlation Vs Causation as a topic of its own for this reason.

Example. Across Indian cities in summer, sales of air-conditioners and sales of bottled cold drinks are highly correlated. Neither causes the other: both are driven by temperature.

Regression: dependence and estimation. Regression goes further than correlation. It sets up a functional relationship in which one variable is treated as dependent and the other as independent, and uses that relationship to estimate or project the dependent variable. In general the two regression lines (Y on X and X on Y) are different lines, with coefficients b_yx = r·Sy/Sx and b_xy = r·Sx/Sy. They coincide only in the special case of perfect correlation, r = ±1, and the two coefficients are equal only in special cases, for instance when X and Y have the same standard deviation. Because the lines normally differ, the analyst must decide which variable depends on which.

Example. A farm co-operative regresses wheat yield per hectare on the quantity of fertiliser applied. Agronomy tells us that fertiliser affects yield, not the reverse, so yield is the dependent variable and the regression coefficient tells the co-operative how much extra yield to expect from each extra kilogram.

Why the statement needs a qualification. The direction of cause in a regression comes from theory and knowledge of the subject, not from the arithmetic. A regression of air-conditioner sales on cold-drink sales could be fitted and would "predict" well, yet it would describe no causal link. So regression assumes a cause-and-effect (or at least a dependence) structure chosen by the analyst; it does not establish it.

Conclusion. Correlation shows only possible association; regression measures and uses a dependence relationship for estimation. Regression is the right tool when there is a sound reason to treat one variable as the cause, but proof of causation needs reasoning beyond both measures.