15.1 Two causes at once
A factorial design crosses two or more independent variables so that every combination of their levels is run. Each independent variable is a factor. Each value of a factor is a level, as before. A study with two factors, each with two levels, is a 2 × 2 design. It has four cells. If the factors are drug (present, absent) and instruction (given, withheld), the cells are drug-plus-instruction, drug alone, instruction alone, and neither. The point of the crossing is that the design can say more than two separate experiments. It can say whether the effect of one factor depends on the level of the other.
A main effect is the effect of one factor averaging over the levels of the other factors. The main effect of the drug compares everyone who received it with everyone who did not, pooling across instruction. A main effect can be real and still be a poor summary, because the average can hide a reversal in one cell. The sentence “there was a main effect of the drug” is true as an average and misleading as a picture of every row. Factorial results are read from the cells, not from the main effects alone.
An interaction occurs when the effect of one factor changes in size or direction across the levels of another factor. The drug helps when the instruction is withheld and does nothing when the instruction is given. That is an interaction. The instruction’s effect is not a single number. It depends on the drug. Interactions are why factorial designs exist. A theory that says “this works, especially when…” or “this works only if…” is an interaction hypothesis, and a one-factor experiment cannot test it. Cozby and Bates treat the ability to state an interaction in a plain sentence as the skill, ahead of any particular test (Cozby & Bates, 2020).
15.2 How to say what the cells did
The cells of a 2 × 2 are four means. A reader describes them by comparing simple effects: the effect of factor A at one level of factor B, then at the other level. If those two simple effects differ, there is an interaction. If they are the same size, there is no interaction, and the main effects tell the story. “People in the drug condition scored higher” is a main-effect sentence. “The drug raised scores only among people who did not receive the instruction” is an interaction sentence. Both can be numerically accurate. They are different claims, and a paper that reports only the first has buried the second.
A graph of the four means makes the interaction visible. One factor is on the horizontal axis. The other factor is two lines. Parallel lines are the picture of no interaction. Lines that diverge, converge, or cross are the picture of an interaction. The picture is not a substitute for the means. It is a check that the sentence matches them. A crossing interaction is the dramatic case: the effect reverses. A treatment that helps one group and harms another is not “a modest main effect.” It is a reason to stop averaging.
Higher-order factorials, a 2 × 2 × 2 and beyond, add further interactions, including a three-way interaction in which a two-way interaction itself depends on a third factor. Those designs are legitimate and they are easy to run past the point of interpretability. Each added factor multiplies cells, and empty or tiny cells cannot support the sentence the design promised. A methods judgment says: cross the factors the theory actually needs, and fill the cells. A demonstration that eight factors can be named is not a design.
| Instruction given | Instruction withheld | |
|---|---|---|
| Drug present | Cell: both | Cell: drug only |
| Drug absent | Cell: instruction only | Cell: neither |
15.3 Mixing assigned causes with characteristics of people
An IV × PV design crosses a manipulated independent variable with a participant variable. The instruction is assigned. Expertise is measured. The interaction asks whether the instruction works differently for novices and experts. The manipulated factor can still support a causal sentence inside each level of the participant variable, if assignment was random within those levels. The participant variable itself remains a measured predictor. “Expertise moderated the effect” is the careful sentence. “Expertise caused the instruction to work” smuggles a cause the design did not assign.
A mixed design combines a between-subjects factor with a repeated-measures factor. People are assigned to one level of the first factor and experience every level of the second. The order problems of repeated measures apply to the repeated factor only. Mixed designs match questions like “does this training change the slope across a set of trials?” Training is between people. Trials are within people. The interaction, training by trial, is often the result that matters, and it should be described as a difference in slopes rather than as two disconnected main effects.
Testing these patterns uses factorial analysis of variance and related models. That machinery, including how an interaction is partitioned from main effects in a significance test, is Statistics for Psychology, the next course on this map. This chapter stops at the design and the sentence. A student who can draw the cells, say the main effects without hiding an interaction, and refuse to call a participant variable a manipulation, has the methods content. The test that puts a p value on the interaction is the further course, and it will not repair a design whose cells were never filled.
References
- Cozby, P. C., & Bates, S. C. (2020). Methods in behavioral research (14th ed.). McGraw-Hill.
