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Slides to accompany Weathington, Cunningham & Pittenger (2010),
Chapter 13: Between-Subjects Factorial Designs
1
Objectives
• Two-variable design
• GLM and two-variable design
• Advantages of 2-variable design
• Main effects
• Interactions
• Designing a two-variable study
2
Two-Variable Design
• Relationship between two IV and a DV
– How much does each IV influence DV?
– How much do the IVs together influence DV?
3
Figure 13.1
Total Variation
Between-groups variation
Within-groups variation
SINGLE-VARIABLE DESIGN & ANOVA
TWO-VARIABLE DESIGN & ANOVA
Total Variation
Between-groups variation
Within-groups variation
Effects of Variable A
Effects of Variable B
Joint effects of A x B
4
GLM and Two-Variable Design
• Single-variable,
Xij = µ + αj + εij
• Now,
Xij = µ + αj + βk + αβjk + εijk
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Advantages of 2-Variable Design• Efficiency
– Fewer people, more power to examine more questions simultaneously
– See Table 13.1
• Can consider interaction of variables
– Influence of variable combinations
• Increased power
– W-g variance < in one-group design6
A Bit More on Interactions
• Pattern of results unexplainable by a single IV by itself
– Compare Figure 13.2 with 13.3
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Figures 13.2 & 13.3
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Variables, Levels, Cells
• Factorial design = study with independent groups for each possible combination of levels of the IV
– e.g., A x B, 2 x 2, 3 x 4
• Can have more than 2 variables (A x B x C)
– Here we consider A x B
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Example• From text, “Reaction to Product
Endorsement”
• DV = Willingness to buy
• IV A = source credibility (high vs. low)
• IV B = type of review (strong, ambiguous, and weak)
• 2 x 3 factorial design (Figure 13.4)
• Interaction of A x B10
Main Effects
• Effect of one IV on the DV, holding the other IV constant
• Special form of b-g variance
• Two-factor design has two main effects
– Fig. 13.8 and 13.9(a) = significant findings
– Fig. 13.9(b) = nonsignificant findings11
Figure 13.8
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Figure 13.9
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More on Main Effects
• Main effect = additive effect
– Figure 13.10
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More on Interactions
• Interaction = interplay between two variables
– Figures 13.11 and 13.12
• When you have a significant interaction, interpret mean differences carefully
15
Figure 13.11
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Figure 13.12
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Designing a Factorial Study
• Each participant in only one IV combo condition
• At least 2 levels of each IV
– Sometimes more levels is better
• Best to have a DV with an interval or ratio scale (easier than nominal/ordinal)
• Try for equal n across each tx condition
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Estimating Sample Size
• Can be accomplished with power analysis
• See the appropriate table in Appendix B
– Effect size estimate, f
– Desired power
– Three F-ratios in a two-factor design: A, B, AxB
• Plan for sample size needed for weakest effect
– Formula for estimating n’ is Equation 13.2
19
Interpreting Interactions
• Residual = effect of interaction after removing influence of the main effects
Δij = Mij – Mai – Mbj + Moverall
– If interaction not statistically significant then residual (Δij) will be
close to 0
– Stronger interactions lead to larger residuals in multiple treatment conditions
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Interpreting Interactions
• Residuals represent the effects of the interaction on the DV that are not explained by the individual main effects alone
• When no interaction is present, the residuals for each treatment condition will be close to or equal to 0
• Table 13.7 and Figure 13.14 illustrate
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Table 13.7Variable A
Credibility of Source
Variable BReview type
Lowa1
Higha2
Row means
Strongb1
M11 = 6.60Δ11 = M11 –Ma1 –Mb1 + Moverall
Δ11 = 6.60 – 5.70 – 6.25 + 5.667 Δ11 = 0.32
M21 = 5.9Δ21 = M21 –Ma2 –Mb2 + Moverall
Δ21 = 5.90 – 5.63 – 6.25 + 5.667 Δ21 = -0.32
Mb1 = 6.25
Ambiguousb2
M12 = 5.40Δ12 = M12 –Ma1 –Mb2 + Moverall
Δ12 = 5.40 – 5.70 – 6.70 + 5.667 Δ12 = -1.33
M22 = 8.00Δ22 = M22 –Ma2 –Mb2 + Moverall
Δ22 = 8.00 – 5.63 – 6.70 + 5.667 Δ22 = 1.33
Mb2 = 6.70
Weakb3
M13 = 5.10Δ13 = M13 –Ma1 –Mb3 + Moverall
Δ13 = 5.10 – 5.70 – 4.05 + 5.667 Δ13 = 1.02
M23 = 3.00Δ23 = M23 –Ma2 –Mb3 + Moverall
Δ23 = 3.00 – 5.63 – 4.05 + 5.667 Δ23 = -1.02
Mb3 = 4.05
Column means
Ma1 = 5.70 Ma2 = 5.63 MOverall = 5.67
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Figure 13.14
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What is Next?
• **instructor to provide details
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