Interaction effects occur when two or more variables taken together behave differently than would be expected by simply considering the sum of their individual effects. To take a simple example, while inflating a single tyre on a car (while the others remain flat) will have only a small positive effect on vehicle speed, inflating all the tyres at once will dramatically increase the speed, far more than the sum of the individual increases from each tyre considered singly. Essentially this occurs because the different tyres interact with each other to collectively produce effects not found by examining a single tyre by itself.

Interaction effects are very common when considering the risk factors for developing particular diseases. Often, having only one risk factor (such as certain genetic predispositions or a poor diet) will somewhat increase one’s risk, but having two or more risk factors will dramatically increase the risk by far more than the sum of each individual factor. Discrimination is another area were interaction effects are common. It may be the case, for instance, that women and black people both receive lower wages on average, but black women earn considerably less again even compared with the sum of the average deficit for woman and black people combined. This difference between the expected wage deficit for being black plus the deficit for being a women compared to the the actual wage deficit for being a black women would be the *interaction effect* of the ‘black’ and ‘female’ variables. Statistical analysis that do not consider interaction effects are likely to make incorrect inferences about the relationship between multiple variables.

Simpson’s paradox is a particularly extreme example of interaction effects. It occurs when associations between two variables that are found when examining particular groups disappear or are even reversed when those groups are combined. A famous case of Simpson’s paradox occurred with regard to admissions rates to graduate programs in the University of California, Berkeley in 1973. While it was found that women had overall much lower rates of acceptance into graduate programs than men, no individual department showed this bias (if anything the bias went the other way) when examined by themselves. The cause of this paradox was an interaction effect – women tended to apply to more competitive departments than men, and thus had an overall lower admission rate, even though within each individual competitive they had slightly higher admission rates than their male counterparts.

**Further Reading**

Finding interactions: a graphical approach to teaching how to detect interactions between variables

Interpreting interactions in regression: a more advanced discussion focusing on regression analysis

Simpson’s paradox: an excellent and very clear introduction to Simpson’s paradox with graphical illustrations