Statistics for the Social Sciences

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Statistics for the Social Sciences

• Psychology 340
• Spring 2005

Factorial ANOVA
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Outline

• Basics of factorial ANOVA
• Interpretations
• Main effects
• Interactions
• Computations
• Assumptions, effect sizes, and power
• Other Factorial Designs
• More than two factors
• Within factorial ANOVAs

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• Statistical analysis follows design

• The factorial (between groups) ANOVA

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Factorial experiments

• Two or more factors
• Factors - independent variables
• Levels - the levels of your independent variables
• 2 x 3 design means two independent variables, one
  with 2 levels and one with 3 levels
• condition or groups is calculated by
  multiplying the levels, so a 2x3 design has 6
  different conditions

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Factorial experiments

• Two or more factors (cont.)
• Main effects - the effects of your independent
  variables ignoring (collapsed across) the other
  independent variables
• Interaction effects - how your independent
  variables affect each other
• Example 2x2 design, factors A and B
• Interaction
• At A1, B1 is bigger than B2
• At A2, B1 and B2 dont differ

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Results

• So there are lots of different potential
  outcomes
• A main effect of factor A
• B main effect of factor B
• AB interaction of A and B
• With 2 factors there are 8 basic possible
  patterns of results

5) A B 6) A AB 7) B AB 8) A B AB
1) No effects at all 2) A only 3) B only 4) AB
only
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2 x 2 factorial design
Whats the effect of A at B1? Whats the effect
of A at B2?

• Condition
• mean
• A1B1

Condition mean A2B1
Condition mean A1B2
Condition mean A2B2
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Examples of outcomes
45
45
60
30
Main effect of A
v
Main effect of B
X
Interaction of A x B
X
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Examples of outcomes
60
30
45
45
Main effect of A
X
Main effect of B
v
Interaction of A x B
X
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Examples of outcomes
45
45
45
45
Main effect of A
X
Main effect of B
X
Interaction of A x B
v
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Examples of outcomes
45
30
45
30
v
Main effect of A
v
Main effect of B
Interaction of A x B
v
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Factorial Designs

• Benefits of factorial ANOVA (over doing separate
  1-way ANOVA experiments)
• Interaction effects
• One should always consider the interaction
  effects before trying to interpret the main
  effects
• Adding factors decreases the variability
• Because youre controlling more of the variables
  that influence the dependent variable
• This increases the statistical Power of the
  statistical tests

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Basic Logic of the Two-Way ANOVA

• Same basic math as we used before, but now there
  are additional ways to partition the variance
• The three F ratios
• Main effect of Factor A (rows)
• Main effect of Factor B (columns)
• Interaction effect of Factors A and B

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Partitioning the variance
Total variance
Stage 1
Within groups variance
Between groups variance
Stage 2
Factor A variance
Factor B variance
Interaction variance
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Figuring a Two-Way ANOVA

• Sums of squares

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Figuring a Two-Way ANOVA

• Degrees of freedom

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Figuring a Two-Way ANOVA

• Means squares (estimated variances)

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Figuring a Two-Way ANOVA

• F-ratios

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Figuring a Two-Way ANOVA

• ANOVA table for two-way ANOVA

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Example
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Example
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Example
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Example
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Example
v
v
v
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Assumptions in Two-Way ANOVA

• Populations follow a normal curve
• Populations have equal variances
• Assumptions apply to the populations that go with
  each cell

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Effect Size in Factorial ANOVA
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Approximate Sample Size Needed in Each Cell for
80 Power (.05 significance level)
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Extensions and Special Cases of the Factorial
ANOVA

• Three-way and higher ANOVA designs
• Repeated measures ANOVA

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Factorial ANOVA in Research Articles

• A two-factor ANOVA yielded a significant main
  effect of voice, F(2, 245) 26.30, p lt .001. As
  expected, participants responded less favorably
  in the low voice condition (M 2.93) than in the
  high voice condition (M 3.58). The mean rating
  in the control condition (M 3.34) fell between
  these two extremes. Of greater importance, the
  interaction between culture and voice was also
  significant, F(2, 245) 4.11, p lt .02.