© Dec 2006 Paul Cooijmans
(It is assumed in this article that the reader already knows what a correlation is.)
It is often said, typically in a warning manner, that a correlation does not imply a causal relation. Provided no dishonest manipulation or fraud takes place in selecting the data on which the correlation is based however, this well-meant advice is wrong.
Any correlation implies a causal relation, with a probability inversely proportional to its significance.
For clarity: A correlation as meant here is not zero, as the value of zero denotes the ABSENCE of correlation. The significance of a correlation is the probability of that correlation resulting from mere coincidence if the true correlation were zero.
In other words, the significance is the probability that the correlation does NOT imply a causal relation, and [1 - significance] is the probability that it does.
Then to the nature and direction of the causality; If a correlation exists between A and B, at least one of the following explanations applies:
It is of course the latter explanation, the common cause, that is often overlooked, and that may be what some people are really trying to say with "Correlation does not imply causality". But a common cause is a cause nevertheless, so there is causality after all. There can be no correlation without causality, except for by chance as reflected in the correlation's significance.
Exactly which variable is a cause, and through which mechanism it works, can be found out and proven by studying all intercorrelations between a broader set of variables, by logical thinking, or by other non-statistical methods like experiment.