Statistics for the Social Sciences

1
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

3

• Statistical analysis follows design

• The factorial (between groups) ANOVA

4
Factorial experiments
B1 B2 B3
A1
A2

• 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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Example
Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level
Low B1 Low B1 Medium B2 Medium B2 High B3 High B3
FactorA Task Difficulty A1 Easy 3 1 1 6 4 2 5 9 7 7 7 9 11 10 8
FactorA Task Difficulty A2 Difficult 3 0 0 2 0 3 8 3 3 3 0 0 0 5 0
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Example
Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level
Low B1 Low B1 Medium B2 Medium B2 High B3 High B3
FactorA Task Difficulty A1 Easy 3 1 1 6 4 2 5 9 7 7 7 9 11 10 8
FactorA Task Difficulty A2 Difficult 3 0 0 2 0 3 8 3 3 3 0 0 0 5 0
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Example
Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level
Low B1 Low B1 Medium B2 Medium B2 High B3 High B3
FactorA Task Difficulty A1 Easy 3 1 1 6 4 2 5 9 7 7 7 9 11 10 8
FactorA Task Difficulty A2 Difficult 3 0 0 2 0 3 8 3 3 3 0 0 0 5 0
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Example
Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level Factor B Arousal Level
Low B1 Low B1 Medium B2 Medium B2 High B3 High B3
FactorA Task Difficulty A1 Easy 3 1 1 6 4 2 5 9 7 7 7 9 11 10 8
FactorA Task Difficulty A2 Difficult 3 0 0 2 0 3 8 3 3 3 0 0 0 5 0
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Example ANOVA table
Source SS df MS F
Between
A B AB 120 60 60 1 2 2 120 30 30 27.7 6.9 6.9
Within Total 104 344 24 4.33
v
v
v
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Factorial ANOVA in SPSS

• What we covered today is a completely between
  groups Factorial ANOVA
• Enter your observations in one column, use
  separate columns to code the levels of each
  factor
• Analyze -gt General Linear Model -gt Univariate
• Enter your dependent variable (your observations)
• Enter each of your factors (IVs)
• Output
• Ignore the corrected model, intercept, total
  (for now)
• F for each main effect and interaction

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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