Statistical relationships between independent and dependent variables are often referred to as effects. So the difference between the average concentration score in the quiet condition and the average concentration score in the noisy condition can be called “the effect of noise level on concentration.” Likewise, the difference between the average intelligence rating in the smile condition and the average intelligence rating in the no-smile condition can be called “the effect of smiling on intelligence ratings.” So whenever we manipulate an independent variable, we are interested in its effect on the dependent variable.
In a factorial design, a main effect is the overall effect of one independent variable. In an experiment in which both the type of psychotherapy (cognitive vs. behavioral) and the duration of psychotherapy (short vs. long) are independent variables, there is one main effect of type and another main effect of duration. The main effect of type is the difference between the average score for the cognitive group and the average score for the behavioral group … ignoring duration. That is, short-duration subjects and long-duration subjects are combined together in computing these averages. The main effect of duration is the difference between the average score for the short-duration group and the average score for the long-duration group … this time ignoring type. Cognitive-therapy subjects and behavioral-therapy subjects are combined together in computing these averages.
Main Effects in Design Tables
Imagine that the dependent variable in the psychotherapy study is some measure of improvement, with higher scores indicating more improvement. Let us assume for simplicity that there are only three subjects per condition, so we can write their improvement scores directly into the design table (in place of the sample sizes; see below.) To see the main effect of the type of psychotherapy, all we need to do is compare the average score in the behavioral condition with the average score in the cognitive condition. We compute both of these averages across subjects in the long and short duration conditions. We see that the six subjects in the cognitive conditions scored three points higher on average than the six subjects in the behavioral conditions. This is the main effect of the type of psychotherapy. To see the main effect of the duration of psychotherapy, we compare the average score in the short condition with the average score in the long condition, now computing these averages across subjects in the cognitive and behavioral conditions. We see that the six subjects in the long conditions scored three points higher on average than the six subjects in the short conditions. This is the main effect of the duration of psychotherapy.
Main Effects in Graphs
Below are the same results plotted in the form of a bar graph, which is probably the best way to display them. The main effect of type is indicated by the fact that the two cognitive bars are higher on average than the two behavioral bars. The main effect of duration is indicated by the fact that the two long-duration (dark) bars are higher on average than the two short-duration (light) bars.
Below are some results (for a different experiment) presented in the form of a line graph. It is a line graph because the independent variable on the x-axis is quantitative with a small number of values. Note that if you were to “average” the two lines, the result would be a line sloping downward, showing a negative effect of noise on concentration. This is the main effect of noise because it combines the extroverts and the introverts. To see the main effect of extroversion, note that extroverts tended to have higher scores regardless of the noise level. The extrovert line is above the introvert line.
An interaction is a special kind of effect that can be observed in factorial experiments. You have an interaction whenever the effect of one independent variable depends on the level of the other. This is actually a fairly easy idea. Here are some examples.
1) If cognitive psychotherapy is better than behavioral psychotherapy when the therapy is short but not when the therapy is long, then there is an interaction between type and duration of therapy. 2) If the negative effect of noise level on concentration is greater for introverts than for extroverts, then there is an interaction between these two independent variables. 3) If the boost in intelligence judgments due to smiling is greater for male stimulus persons than for female stimulus persons, then there is an interaction between smiling and sex. 4) If drawing a smiley face on checks increases tips for female servers but not for male servers, then there is an interaction between drawing smiley faces (or not) and sex of the server.
There is no interaction when the effect of one variable is essentially the same regardless of the level of the other. One example we have seen is that a smile boosts perceptions of a stimulus person about the same amount regardless of whether that person is attractive or unattractive.
A particularly interesting kind of interaction is a crossover interaction. In a crossover interaction, the effect of one independent variable is not only different across levels of the second independent variable, it actually reverses. Imagine, for example, that introverts’ concentration levels started high and then dropped as the noise level increased, but extroverts’ concentration level started low and then increased as the noise level increased. This would be a crossover interaction. (Why do you think it is called that?)
The Relationship Between Main Effects and Interactions
In a 2x2 factorial experiment, there are two main effects (one for each independent variable) and one interaction (the one between the two independent variables). It gets more complicated with more independent variables. An experiment with three independent variables would have three main effects (again, one for each independent variable) and four interactions. There is an interaction between IV’s 1 and 2, 2 and 3, and 1 and 3. There is also something called a three-way interaction, which has to do with whether the interaction between two variables depends on the level of the third. But do not worry about this. Until you understand simple factorial designs, there is no way you will understand more complicated ones.
The important point here is that these effects are all independent of each other. A 2x2 factorial experiment might result in no main effects and no interaction, one main effect and no interaction, two main effects and no interaction, no main effects and an interaction, one main effect and an interaction, or two main effects and an interaction. Your job now is to get good at looking at results presented in a design table or (more importantly) a graph, and interpreting what happened in terms of main effects and interactions.