# Interactions

An effect of interaction occurs when a relation between (at least) two variables is modified by (at least one) other variable. In other words, the strength or the sign (direction) of a relation between (at least) two variables is different depending on the value (level) of some other variable(s). (The term interaction was first used by Fisher, 1926). Note that the term "modified" in this context does not imply causality but represents a simple fact that depending on what subset of observations [regarding the "modifier" variable(s)] you are looking at, the relation between the other variables will be different.

For example, imagine that we have a sample of highly achievement-oriented students and another of achievement "avoiders." We now create two random halves in each sample, and give one half of each sample a challenging test, the other an easy test. We measure how hard the students work on the test. The means of this (fictitious) study are as follows:

 Achievement-oriented Achievement-avoiders Challenging Test 10 5 Easy Test 5 10

How can we summarize these results? Is it appropriate to conclude that 1) challenging tests make students work harder, and 2) achievement-oriented students work harder than achievement-avoiders? None of these statements captures the essence of this clearly systematic pattern of means. The appropriate way to summarize the result would be to say that challenging tests make only achievement-oriented students work harder, while easy tests make only achievement-avoiders work harder. In other words, the relation between the type of test and effort is positive in one group but negative in the other group. Thus, the type of achievement orientation and test difficulty interact in their effect on effort; specifically, this is an example of a two-way interaction between achievement orientation and test difficulty. (Note that statements 1 and 2 above would describe so-called main effects.)