There are three questions the researcher need consider in a 2 x 2 factorial design. (1) Is there a significant main effect for Factor A? (2) Is there a significant main effect for Factor B? (3) Is there a significant interaction between Factor A and Factor B? Significance can be determined statistically through an analysis called the F test. (A discussion of the F test is beyond the scope of this tutorial; the reader is encouraged to consult any textbook in basic statistics for an overview of this topic.)
Based on the answers to these three questions, there are eight possible outcomes, as presented in Table 6 below (adapted from Cozby, 1977). Take some time now to review them by clicking on each one. When you do, you will be linked to hypothetical data illustrative of that outcome, first displayed in a table and then plotted on graphs. While examining these displays, think about the similarities and differences among the outcomes.
| OUTCOME | Factor A Main Effect? | Factor B Main Effect? | A x B Interaction? |
|---|---|---|---|
| Outcome 1 | No | No | No |
| Outcome 2 | Yes | No | No |
| Outcome 3 | No | Yes | No |
| Outcome 4 | Yes | Yes | No |
| Outcome 5 | Yes | Yes | Yes |
| Outcome 6 | Yes | No | Yes |
| Outcome 7 | No | Yes | Yes |
| Outcome 8 | No | No | Yes |
The data you reviewed for each outcome are idealized; the data from a real experiment would probably be somewhat more ambiguous, thus the need for a statistical analysis. You may have noted at least a couple of important points. First, graphically, parallel lines mean no interaction between Factor A and Factor B (Outcomes 1-4). In other words, the nature of the relationship across the two levels of Factor A is the same at the two levels of Factor B (or, alternatively, the nature of the relationship across the two levels of Factor B is the same at the two levels of Factor A). Returning to our example, we would say the drug (Factor A) similarly affects memory when no task description is provided (B1) and when the task is described as "hard" (B2).
Second, an interaction between Factor A and Factor B is possible when there are two main effects (Outcome 5), one main effect (Outcome 6 and Outcome 7), or even no main effects (Outcome 8). Think about this latter outcome in terms of our example. Focusing on main effects only would lead the researcher to believe that neither the drug nor the task description were effective independent variables, that is, neither was particularly relevant with respect to memory. Additional consideration to the interaction reveals that nothing could be further from the truth. The drug (A) greatly increased the number of errors when the task was described as "hard" (line B2) and greatly decreased the number of errors when no task description was provided (line B1). In other words, the effect of the drug was completely opposite at the two levels of the task description. This particular type of interaction has been called antagonistic (McBurney, 2004).