PYB350 Chap.6 Factorial ANOVA: Main Effects and Interactions
Factorial ANOVA: Main Effects and Interactions
A factorial design crosses two or more independent variables so every combination forms a cell. This chapter explains how a two-way between-groups ANOVA separates the main effect of each factor, tested on marginal means, from their interaction, tested against the cell means.
It walks through the handout's formulas for the main-effect sums of squares, the interaction by subtraction, the degrees of freedom and the shared within-cells error term, shows how to read an interaction from a line graph, and covers the data layout the assignment expects.
The worked example computes the three sums of squares and F tests from cell means, and the drill page asks you to read main effects and interactions straight from small tables, including a crossover pattern that leaves the margins flat.
What this chapter covers
- 01
Fully crossed factors and counting cells
- 02
Marginal means and main effects
- 03
Interactions and non-parallel lines
- 04
Sums of squares for A, B and A by B
- 05
One error term for three F tests
- 06
Setting up the data file with a participant ID
- 07
Spotting crossover interactions that leave the margins flat
Worked example · free
Main effects and the interaction in the feedback study
- 1Grand mean 6.6; marginal means easy 7.8, hard 5.4, immediate 7.4, delayed 5.8.
- 2SS difficulty = 6 times 2 times (1.44 + 1.44) = 34.56; SS feedback = 6 times 2 times (0.64 + 0.64) = 15.36.
- 2SS cells = 6 times (1.96 + 1.00 + 0.04 + 6.76) = 58.56, so SS interaction = 58.56 minus 34.56 minus 15.36 = 8.64.
- 1MS within = 36 / 20 = 1.8, giving F(1, 20) = 19.20, 8.53 and 4.80, all above the critical 4.35.
Key terms
- Factorial design
- A design in which every level of one independent variable is combined with every level of the other.
- Cell mean
- The mean score of the participants who received one particular combination of factor levels.
- Main effect
- The average effect of one factor, ignoring the other, tested by comparing marginal means.
- Orthogonal effects
- Effects that are independent of one another, as the two main effects and the interaction are when cell sizes are equal.
- Participant ID column
- A typed-in identifier for each case, kept so the original order is not lost if the file is sorted.
Factorial ANOVA: Main Effects and Interactions FAQ
How many cells does a 3 by 4 design have?
Twelve. A two-way A by B design has a times b cells, so a 2 by 2 has four and a 3 by 4 has twelve, while adding a third two-level factor to a 2 by 2 gives a 2 by 2 by 2 design with eight cells. The cell count also sets the error degrees of freedom, N minus ab.
Can a main effect and an interaction both be significant?
Yes. The two main effects and the interaction are orthogonal with equal cell sizes, so any combination of significant and non-significant results is possible. When the interaction is significant, the main effects are still reported but may mislead, because each averages over an effect that changes from one level of the other factor to the next.
Why include a participant ID column instead of using row numbers?
Because sorting the file by any variable reorders the rows, and once the original order is lost it cannot be recovered. The lecture recommends typing the IDs in yourself. The written assignment asks for exactly this: a formatted data file with value labels for the categorical variables and the participant ID from the data cards.
What is the difference between marginal means and cell means?
Cell means are the averages of each combination of factor levels, one per cell. Marginal means average across the levels of the other factor, giving one mean per level of the factor you are looking at. Main effects compare marginal means; the interaction and its follow-up simple effects work with cell means, because they ask how one factor behaves at each level of the other.
Exam move
Draw every 2 by 2 as a small table with margins, fill in the marginal means and the grand mean first, then sketch the line graph; most interaction questions are answered by that sketch before any F is read. Recompute one interaction by both routes, by subtraction and by the contrast, until the two answers agree without effort; the check catches most arithmetic slips.
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