Interaction (statistics)

In statistics, an interaction is a term in a statistical model added when the effect of two or more variables is not simply additive. Such a term reflects that the effect of one variable depends on the values of one or more other variables.

Thus, for a response "Y" and two variables "x"₁ and "x"₂ an "additive" model would be:

:$Y\; =\; ax\_1\; +\; bx\_2\; +\; mbox\{error\},!$

In contrast to this,

:$Y\; =\; ax\_1\; +\; bx\_2\; +\; c(x\_1\; imes\; x\_2)\; +\; mbox\{error\},$

is an example of a model with an "interaction" between variables "x"₁ and "x"₂ ("error" refers to the random variable whose value by which "y" differs from the expected value of "y").

Very often the interacting variables are categorical variables rather than real numbers. For example, members of a population may be classified by religion and by occupation. If one wishes to predict a person's height based only on the person's religion and occupation, a simple "additive" model, i.e., a model without interaction, would add to an overall average height an adjustment for a particular religion and another for a particular occupation. A model with interaction, unlike an additive model, could add a further adjustment for the "interaction" between that religion and that occupation. This example may cause one to suspect that the word "interaction" is something of a misnomer.

Statistically, the presence of an interaction between categorical variables is generally tested using a form of Analysis of Variance (ANOVA). If one or more of the variables is continuous in nature, however, it would typically be tested using moderated multiple regression.citation
author = Overton, R. C.
year = 2001
title = Moderated multiple regression for interactions involving categorical variables: a statistical control for heterogeneous variance across two groups
journal = Psychol Methods
volume = 6
issue = 3
pages = 218–33
doi = 10.1037/1082-989X.6.3.218
pmid = 11570229] This is so-called because a moderator is a variable that affects the strength of a relationship between two other variables.

The consequence of an "interaction" is that the effect of one variable depends on the value of another. This has implications in design of experiments as it is misleading to vary one factor at a time.

Real-world examples of interaction include:
*"Interaction" between adding sugar to coffee and stirring the coffee. Neither of the two individual variables has much effect on sweetness but a combination of the two does.
*"Interaction" between adding carbon to steel and quenching. Neither of the two individually has much effect on strength but a combination of the two has a dramatic effect.
*"Interaction" between smoking and inhaling asbestos fibres: Both raise lung carcinoma risk, but exposure to asbestos "multiplies" the cancer risk in smokers and non-smokers. Both risk factors were not shown to be "additive" – a clear indication of interaction.citation
author = Lee, P. N.
year = 2001
title = Relation between exposure to asbestos and smoking jointly and the risk of lung cancer
journal = Occupational and Environmental Medicine
volume = 58
issue = 3
pages = 145
doi = 10.1136/oem.58.3.145
pmid = 11171926]

Genichi Taguchi contended that "interactions" could be eliminated from a system by appropriate choice of response variable and transformation. However George Box and others have argued that this is not the case in general.

See also

* Main effect
* Interaction
* Interaction variable
* Design of experiments
* Factorial experiment

Bibliography

*Box, G.E.P. (1990). Do interactions matter?. "Quality Engineering" "2", 365-369.

Further reading

*Southwood, K.E., "Substantive Theory and Statistical Interaction: Five Models," "The American Journal of Sociology", 83(5) (1978), 1154-1203.

References