ANOVA%20With%20More%20Than%20One%20IV

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ANOVA With More Than One IV
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2-way ANOVA

• So far, 1-Way ANOVA, but can have 2 or more IVs.
  IVs aka Factors.
• Example Study aids for exam
• IV 1 workbook or not
• IV 2 1 cup of coffee or not

Workbook (Factor A) Workbook (Factor A)
Caffeine (Factor B) No Yes
Yes Caffeine only Both
No Neither (Control) Workbook only
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Main Effects
N30 per cell Workbook (Factor A) Workbook (Factor A) Row Means
Caffeine (Factor B) No Yes
Yes Caff 80 SD5 Both 85 SD5 82.5
No Control 75 SD5 Book 80 SD5 77.5
Col Means 77.5 82.5 80
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Main Effects and Interactions

• Main effects seen by row and column means Slopes
  and breaks.
• Interactions seen by lack of parallel lines.
• Interactions are a main reason to use multiple IVs

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Single Main Effect for B
(Coffee only)
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Single Main Effect for A
(Workbook only)
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Two Main Effects Both A B
Both workbook and coffee
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Interaction (1)
Interactions take many forms all show lack of
parallel lines.
Coffee has no effect without the workbook.
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Interaction (2)
People with workbook do better without coffee
people without workbook do better with coffee.
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Interaction (3)
Coffee always helps, but it helps more if you use
workbook.
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Labeling Factorial Designs

• Levels each IV is referred to by its number of
  levels, e.g., 2X2, 3X2, 4X3 designs. Two by two
  factorial ANOVA.

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Example Factorial Design (1)

• Effects of fatigue and alcohol consumption on
  driving performance.
• Fatigue
• Rested (8 hrs sleep then awake 4 hrs)
• Fatigued (24 hrs no sleep)
• Alcohol consumption
• None (control)
• 2 beers
• Blood alcohol .08

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Cells of the Design
Alcohol (Factor A) Alcohol (Factor A) Alcohol (Factor A)
Fatigue (Factor B) None 2 beers .08
Tired Cell 1 Cell 2 Cell 3
Rested Cell 4 Cell 5Rested, 2 beers, Porsche 911 Cell 6
DV closed course driving performance ratings
from instructors.
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Factorial Example Results
Main Effects? Interactions?
Both main effects and the interaction appear
significant.
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ANOVA Summary Table
Two Factor, Between Subjects Design
Source SS Df MS F
A SSA a-1 SSA/dfA MSA/MSError
B SSB b-1 SSB/dfB MSB/MSError
AxB SSAxB (a-1)(b-1) SSAxB/dfAxB MSAxB/MSError
Error SSError ab(n-1)or N-ab SSError/dfError
Total SSTotal N-1
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Review

• In a 3 X 3 ANOVA
• How many IVs are there?
• How many df does factor A have
• How many df does the interaction have

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Test

• We can see the main effect for a variable if we
  examine means of the dependent variable while
  ________
• Considering the joint effects of both variables
• Examining a single value of a second factor
• Examining each cell
• Ignoring the other variable

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Test

• In two-way ANOVA, the term interaction means
• Both IVs have an impact on the DV
• The effect of one IV depends on the value of the
  other IV
• The on IV has no effect unless the other IV has a
  certain value
• There is a crossover a graph of two lines shows
  an X.