A factorial design includes two or more factors studied at once.
This allows us to test not only the effect of each factor separately, but also whether the factors interact.
Example: 2 × 2 Design
- Factor A: Teaching method (Lecture, Online)
- Factor B: Time of day (Morning, Afternoon)
This design has 4 groups (2 levels of A × 2 levels of B).
We can test:
- The main effect of Factor A (method).
- The main effect of Factor B (time).
- The interaction between method and time.
The ANOVA Partition
For a 2 × 2 design:
- Main effect A: $$df_A = a - 1$$
- Main effect B: $$df_B = b - 1$$
- Interaction A × B: $$df_{A \times B} = (a - 1)(b - 1)$$
- Error (within): $$df_{\text{within}} = N - ab$$
Where $$a$$ = levels of Factor A, $$b$$ = levels of Factor B, $$N$$ = total number of observations.
Interaction
An interaction occurs when the effect of one factor depends on the level of the other factor.
- If lines in a plot are parallel, there is no interaction.
- If lines cross or diverge, there is an interaction.
Example
Suppose means are:
- Lecture: Morning = 70, Afternoon = 90
- Online: Morning = 80, Afternoon = 80
Here:
- Main effect of method: Online > Lecture overall
- Main effect of time: Afternoon > Morning overall
- Interaction: Lecture scores rise with time, Online scores stay flat → non-parallel lines.
Visuals
Figure L7.1 — Factorial Layout (2 × 2). A 2 × 2 grid: Method × Time.
Figure L7.2 — Interaction Plot. Lecture line slopes upward, Online line flat. Caption: “Lines not parallel = interaction.”
Figure L7.3 — ANOVA Summary Table for 2 × 2 design. Source | SS | df | MS | F | p.
Why This Matters
Factorial designs let us test more than one factor at a time.
They are efficient and powerful, and the concept of interaction is central in science.
Two-way ANOVA is the foundation for more complex designs, including repeated measures and mixed ANOVA.
Practice self-test quiz
In the space below, please find practice problems and self-test quizzes. For full access, please signup free.