In statistics, a spurious relationship (or, sometimes, spurious correlation) is a mathematical relationship in which two occurrences have no logical connection, yet it may be implied 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.

General example

An example of a spurious relationship can be illuminated examining a city's ice cream sales. These sales are highest when the city's rate of drownings is highest. To allege that ice cream sales cause drowning 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.

Statistics

The term is commonly used in statistics and in particular in experimental research techniques. Experimental research attempts to understand and predict causal relationships (X → Y). A non-causal correlation can be spuriously created by an antecedent which causes both (W → X & Y). Intervening variables (X → W → Y), if undetected, may make indirect causation look direct. Because of this, experimentally identified correlations do not represent causal relationships unless spurious relationships can be ruled out.

In practice, three conditions must be met in order to conclude that X causes Y, directly or indirectly:

• X must precede Y
• Y must not occur when X does not occur
• Y must occur whenever X occurs

Spurious relationships can often be identified by considering whether any of these three conditions have been violated.

The final condition may be relaxed in the case of indirect causation. For example, consider a pistol duel. Two men face off and fire at each other. If one man dies as a result of the other man's shot, we can rightly conclude that the other man caused his death. However, if a doctor saves the wounded man's life (thus violating the third premise), this does not undermine causation, only direct causation. The biological damage (W) sustained from the shot (X) causes death (Y), not the shot itself, allowing medical intervention.

See a more detailed discussion at causation.

See also

• Correlation implies causation (logical fallacy)
• Joint effect

External links and references

• Burns, William C., "Spurious Correlations", 1997.