Chance News 104

Transitivity, Correlation and Causation

Theorem 1 of the article cited by Paul Alper in the previous issue, "Is the Property of Being Positively Correlated Transitive?" (The American Statistician, Vol. 55, No. 4, November, 2001), depends on the existence of non-observed independent random variables U, V, and W which cause the correlations between X=U+V, Y=W+V, and Z=W-U to be non-transitive. An interesting question is whether this relates back to the difference between causation and correlation.

The answer turns out to be no, we can get the same sort of result even in the presence of causative relationships between X, Y and Z. Here’s an example:

• X is N(0,1);
• Y = X + U, where U is N(0,1) and independent of X;
• Z = Y - 1.5*X.

The correlation coefficients between X and Y and between Y and Z are both positive but the correlation coefficient between X and Z is negative.

Stan Lipopvetsky’s follow-up letter (The American Statistician, 56:4, 341-342, 2002) hints at this but does not include an actual example.

Submitted by Emil M Friedman