Factorial Experiments
- “Multiple factor experiments”
- Experiments with more than one independent variable (or “factor”)
- Allow us to study the effects of more than one “cause”
Part 1: The Notation System
- 3 basic features:
o Each factor (IV) is identified by a number
o The numerical values indicate the ”levels” of each factor (e.g., “2” = 2 levels)
o Create all possible combinations of levels by “crossing” each factor (symbol = “x” =
“by”)
Each unique combination of factors and levels defines a “condition”
Number of conditions = product of levels (multiply the levels of each factor)
- Examples:
1. 2x2 factorial (simplest design)
o Two factors (IV’s)
o Each factor has two levels – if it doesn’t have 2 levels, there’s no variability
4 conditions (2x2=4)
2. 3x3 factorial
o Two factors (IV’s); each factor has 3 levels
9 conditions (3x3=9)
3. 2x3x2 factorial
o Three factors, with 2, 3, and 2 levels
12 conditions (2x3x2=12)
- Note typo on p.284 (W&M): A 3x3x3 factorial has 27 conditions, not 9
- It’s possible to have any number of factors (with any number of levels): 2x4x3x7…
- BUT, as design becomes more complex, it’s more difficult to interpret the results o Limit the number of factors/levels to address the specific questions that you want to
answer
o Science grows by examining a few causes at a time
Part 2: The “Factorial Cell Matrix” Notation System
- Assign a capital letter to each factor (A, B…)
- Assign numerical subscripts to represent the levels of each factor (e.g., A1, A2…)
o Numbers should preserve “quantity” (e.g., low = 1; high = 2)
o If categorical, numbers are arbitrary (e.g., gender: M = 1, F = 2 or F = 1, M = 2)
- Create factorial matrix that represents all combinations of factors/levels
o Each “cell” represents a condition
- Example: 2x2 Factorial
Factor A
A1 A 2
Factor B B 1 A1B1 A 2 1
B 2 A1B2 A 2 2
- Concrete Example: 2x2 factorial (p. 274)
- DV = judged “guilt” of defendant
- Factor A: facial expression
o A1 = “neutral”
o A2 = “smiling”
- Factor B: physical attractiveness
o B1 = unattractive
o B2 = attractive
- Note: This design could be expanded to include:
1. A third level of each factor. E.g.,
o neutral (A1); small smile (A2); big smile (A3) o unattractive (B1); neutral (B2); unattractive (B3)
2. More than two factors (IV’s). E.g.,
o Gender of def (C1 = male; C2 = female)
o Trust (D1 = untrustworthy; D2 = trustworthy)
o 3x3x2x2 =36 conditions
- Remember, more factors = more complexity (or more possible outcomes)
Part 3: Outcomes of Factorial Experiments
A. Main effects: allow us to examine the separate effects of each factor
- Easy to examine main effects by focusing on “marginal means” in matrix (e.g., 2x2)
o Main effect of A (column means, p. 274); table 11.4 p274 W&M
Compare A1 (averaged over B1 and B2) and
A2 (averaged over B1 and B2)
o Main effect of B (ro

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