spurious correlations

In statistics, a spurious relationship (or, sometimes, spurious correlation or spurious regression) is a mathematical relationship in which two occurrences have no causal connection, yet it may be inferred that they do, due to a certain third, unseen factor (referred to as a "confounding factor" or "lurking variable"). The spurious relationship gives an impression of a worthy link between two groups that is invalid when objectively examined.

The misleading correlation between two variables is produced through the operation of a third causal variable. In other words we find a correlation between A and B. So we have three possible relationships:

A causes B,
B causes A,
OR
C causes both A and B.

The last is a spurious correlation. In a regression model, where A is regressed on B, but C is found to be the true causal factor for B; this is called specification error. It is therefore often said that "Correlation does not imply causation".

The true causal chain may be

C => A => B

or even

A => C => B

or as illiterated above,

C => A and C => B

An example of a spurious relationship can be illuminated examining a city's ice cream sales. These sales are highest when the rate of drownings in city swimming pools is highest. To allege that ice cream sales cause drowning, or vice-versa, would be to imply a spurious relationship between the two. In reality, a heat wave may have caused both. The heat wave is an example of a hidden or unseen variable.

Another popular example is a series of Dutch statistics showing a positive correlation between the number of storks nesting in a series of springs and the number of human babies born at that time. Of course there was no causal connection; they were correlated with each other only because they were correlated with the weather nine months before the observations.