FACTORIAL ANOVA

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FACTORIAL ANOVA
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Overview of Factorial ANOVA

• Factorial Designs
• Types of Effects
• Assumptions
• Analyzing the Variance
• Regression Equation
• Fixed and Random Effects

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FACTORIAL DESIGNS

• All combinations of levels of two or more
  independent variables (factors) are measured

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Types of Factorials

• Between subjects (independent)
• Within subjects (related)
• Mixed

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Between Subjects
A
1
2
Subjects 1-10
Subjects 21-30
1
B
Subjects 11-20
Subjects 31-40
2
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Within Subjects
A
1
2
Subjects 1-40
Subjects 1-40
1
B
Subjects 1-40
Subjects 1-40
2
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Mixed (A Between, B Within)
A
1
2
Subjects 1-20
Subjects 21-40
1
B
Subjects 1-20
Subjects 21-40
2
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TYPES OF EFFECTS

• A main effect is the overall effect of each IV
  by itself, averaging over the levels of any other
  IVs
• An interaction occurs when the effects of one
  factor change depending on the level of another
  factor

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Simple Effects

• An interaction can be understood as a difference
  in simple effects
• A simple effect is the effect of one factor on
  only one level of another factor
• If the simple effects differ, there is an
  interaction

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70
60
50
B2
40
d.v.
30
B1
20
10
0
1
2
A
16
70
B2
60
50
40
B1
d.v.
30
20
10
0
1
2
A
17
B2
70
60
50
40
d.v.
30
B1
20
10
0
1
2
A
18
70
60
B2
50
40
d.v.
30
20
B1
10
0
1
2
A
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ASSUMPTIONS

• Interval/ratio data
• Normal distribution or N at least 30
• Independent observations
• Homogeneity of variance
• Proportional or equal cell sizes

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ANALYZING THE VARIANCE

• Total Variance Model Residual
• Model Variance is further divided into
• Factor A
• Factor B
• A x B interaction

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Comparing Variance

• F-test for each main effect and for the
  interaction
• Each F-test compares variance for the effect to
  Residual variance

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REGRESSION EQUATION

• bo is mean of base group
• b1 is the main effect of factor A
• b2 is the main effect of factor B
• b3 is the A x B interaction

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FIXED VS. RANDOM EFFECTS

• Fixed Factor only the levels of interest are
  selected for the factor, and there is no intent
  to generalize to other levels
• Random Factor the levels are selected at random
  from the possible levels, and there is an intent
  to generalize to other levels

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APA Format Example

• The two-way between subjects ANOVA showed a
  significant main effect of customer type,
  F(1,1482) 5.04, p .025, partial h2 .00, a
  non-significant main effect of industry type,
  F(2,1482) 0.70, p .497, partial h2 .00, and
  a significant interaction, F(2,1482) 3.12, p
  .044, partial h2 .00.