Slides to accompany Weathington, Cunningham & Pittenger (2010),

Chapter 13: Between-Subjects Factorial Designs

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Objectives

• Two-variable design

• GLM and two-variable design

• Advantages of 2-variable design

• Main effects

• Interactions

• Designing a two-variable study

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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?

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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

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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

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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

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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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