Two Factor ANOVA for Independent Measures

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Two Factor ANOVA for Independent Measures

• Completely randomized (between-groups) design
• A x B Design (Two-Way ANOVA)
• Factor A with a1, a2, . . .ai levels
• Factor B with b1, b2, . . .bj levels
• Main effects
• Interaction
• Simple effects and contrasts

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2 x 2 Design Example

• Effects of motivation and task difficulty on the
  time it takes children to solve a problem
• Factor A Level of Motivation (High vs. Low)
• Factor B Task Difficulty (Hard vs. Easy)

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Assignment 5 1

• Source of variation SS df MS
  F
• _________________________________________________
  
• Factor A 11.9596 1 11.9596
  5.66
• Factor B 3.6258 1 3.6258
  1.72
• A x B Interaction 4.7841 1
  4.7841 2.26
• Within treatments 42.2784 20
  2.1139
• _________________________________________________
• Total 62.6479 23

How many groups? How many subjects per group?
Are any additional analyses needed?
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2 x 2 Design Example
Factor A Level of Motivation Low
High
Row Means
Factor B Hard Difficulty Easy
Column Means
39 39
49 29
What is the difference 49 29 attributable to?
Is there a main effect of Factor B?
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2 x 2 Design Example
Factor A Level of Motivation Low
High
Row Means
Factor B Hard Difficulty Easy
Column Means
39 39
49 29
What is an interaction?
Do these data show an interaction?
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2 x 2 Design Example

• Factorial designs Graphical results
• Easier to see main effects and interactions

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Is there a main effect of Factor B?
Is there an interaction?
What happened in this experiment?
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Is there a main effect of Factor A?
What happened in this experiment?
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What happened in this experiment?
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What happened in this experiment?
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What happened in this experiment?
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What happened in this experiment?
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B
A
What happened in this experiment?
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A
What happened in this experiment?
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What happened in this experiment?
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What happened in this experiment?
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What happened in this experiment?
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What happened in this experiment?
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a1 vs. a2 vs. a3
b1 vs. b2
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b1 vs. b2
a1 vs. a2 vs. a3
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Assignment 5 2

• Source of variation SS df MS
  F
• _________________________________________________
  
• Factor A 11.9596 3 3.9865
  3.02
• Factor B 5.5226 1 5.5226
  4.18
• A x B Interaction 4.7841 3
  1.5947 1.21
• Within treatments 42.2784 32
  1.3212
• _________________________________________________
• Total 64.5447 23

How many groups? How many subjects per group?
Are any additional analyses needed?
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Assume main effect of B is significant neither
main effect of A nor A x B interaction significant
What additional analysis is needed?
Interpretation of results?
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Assume main effect of A, main effect of B, no
interaction.
What additional analysis is needed?
Interpretation of results?
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Significant main effect of B no main effect of
A no interaction
What additional analyses are needed?
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Main effects of both A and B no interaction
What additional analyses are needed?
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Assume main effect of A, main effect of B, and A
x B interaction are all significant
What additional analysis is needed?
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Simple Effects
Simple effect of A _at_ b1 Simple effect of A _at_ b2
Means a1b1 40, a2b1 20, a1b2 80, a2b2 30
Which means are used to test for simple effect of
A _at_ b1, and A _at_ b2
Method of calculation same as planned comparisons
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Simple Effects and Interpretation of Results
Both simple effect of A _at_ b1 and A _at_ b2 are
significant. Interpretation?
Participants solved problems faster under high
motivation than under low motivation and this
effect of motivation was larger for the hard task
than for the easy task.
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Simple Effects and Interpretation of Results
Only simple effect of A _at_ b2 is significant.
Interpretation?
Participants solved problems faster under high
motivation than under low motivation when the
task was hard, but level of motivation had no
affect when the task was easy.
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Simple Effects
Simple effect of B _at_ a1 Simple effect of B _at_ a2
Means a1b1 40, a2b1 20, a1b2 80, a2b2 30
Which means are used to test for simple effect of
B _at_ a1 and B _at_ a2
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Simple Effects and Interpretation of Results
Both simple effect of B _at_ a1 and B _at_ a2 are
significant. Interpretation?
Participants solved problems more slowly when the
task was hard than when it was easy and this
effect of task difficulty was larger under low
motivation.
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Simple Effects and Interpretation of Results
Only simple effect of B _at_ a1 is significant.
Interpretation?
Participants solved problems faster on the easy
task than on the hard task when motivation was
low, but task difficulty made no difference when
motivation was high.
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Interpreting Factorial ANOVA
Source F p Factor A
4.87 .041 Factor B 1.25 ns A x B
6.28 .029
Is there a need for any additional analyses?
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Simple Effects
Simple effect of A _at_ b1 Simple effect of A _at_ b2
Means a1b1 30, a2b1 32, a1b2 50, a2b2 10
Which means are used to test for simple effect of
A _at_ b1 and A _at_ b2 ?
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Simple Effects
Only simple effect of A _at_ b2 is significant.
Interpretation?
Participants solved problems faster under high
motivation than under low motivation when the
task was difficult, but level of motivation had
no affect when the task was easy.
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Simple Effects
Simple effect of B _at_ a1 Simple effect of B _at_ a2
Means a1b1 30, a2b1 32, a1b2 50, a2b2 10
Which means are used to test for simple effect of
B _at_ a1 and B _at_ a2 ?
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Simple Effects
Both simple effect of B _at_ a1 and B _at_ a2 are
significant. Interpretation?
Participants solved problems faster on the easy
task than on the hard task when motivation was
low, but under high motivation they solved
problems faster when the task was difficult than
when it was easy.
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Interpreting Factorial ANOVA
Source F p Factor A
.47 Factor B 10.29 .01 A x B
.32 ns
Are any additional analyses required?
Main comparisons on marginal means for Factor B
(planned or post-hoc)
Why not simple effects?
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Interpreting Factorial ANOVA
Source F p Factor A
8.42 .01 Factor B 7.16 .01 A x
B .32 ns
What additional comparisons are needed?
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Interpreting Factorial ANOVA
Source F p Factor A
57.77 .000 Factor B 431.87 .000 A x B
59.25 .000
What additional tests are needed?
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Simple Effects
Simple effect of A _at_ b1 Simple effect of A _at_
b2 Simple effect of A _at_ b3 or Simple effect of B
_at_ a1 Simple effect of B _at_ a2 Simple effect of B _at_
a3
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Simple Effects
Simple effect of A _at_ b1 F 111.09, p lt .001
Simple effect of A _at_ b2 F 63.69, p lt .001
Simple effect of A _at_ b3 F 1.48, ns
Method of calculation MSbg from one-way ANOVA
divided by MS within treatments (MS error) from
factorial ANOVA
Are any additional analyses needed?
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Simple Effects
Simple effect of B _at_ a1 F 200.81
Simple effect of B _at_ a2 F 63.69
Simple effect of B _at_ a3 F 285.87
Are any additional analyses needed?
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Main effect of B, no main effect of A, A x B
interaction
What additional analyses are needed?
What pairs of means are involved in those
comparisons?