Transcript kxkxk Designs & Analysis

kxkxk 3-way Factorial Designs
• kxkxk 3-way designs, effects & causality
• 3-way interactions
• describing effects and checking if they are
. “descriptive” or “misleading”
• an example
3-way Factorial Designs
The simplest factorial design is a 2x2, which can be expanded in
two ways:
1) Adding conditions to one, the other, or both IVs
2) Add a 3rd IV (making a 3-way factorial design)
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Identify the three IVs in this design . . .
Specify the properties of each condition/cell of this design . . .
3-Way Factorial Designs
There are 7 omnibus F-tests involved in a 3-way factorial
• 3 main effects F-tests (one for each IV)
• 3 2-way interaction F-tests (one for each pair of IVs)
• 1 3-way interaction F-test
For the example, these F-tests are…
• main effects 1. Topic
2. Instruction Method
3. Ed. level
• 2-way interactions
1. Topic X Inst. Method
2 .Topic X Ed. Level
3. Inst. Method X Ed. Level
• 3-way interaction
Topic X Instruction Method X Ed. level
Causal Interpretations of 3-way Designs
• When can a main effect be causally interpreted ?
When the conditions of that IV are RA, Manip, Etc.
• When can a 2-way interaction be causally interpreted ?
• When MEs of both the involved IVs are causally interpretable
• When we can causally interpret the differences between causally
interpretable simple effects.
• When can a 3-way interaction be causally interpreted ?
• When MEs of all 3 IVs are causally interpretable
• When we can causally interpret the differences between simple 2way interactions which each involve causally interpretable
differences between causally interpretable simple effects.
A 2-way interaction…
• “emerges” when there are 2 IVs (categorical or quant)
• is the non-additive/joint effect of the 2 IVs
• usually describes as “when the effect of one IV is different
for different values of the 2nd IV”
Expanding this to define 3-ways…
• “emerges” when there are 3 IVS (categorical or quant)
• is non-additive/joint effect of the 3 IVs
• usually identified as the “when the 2-way interaction of 2
IVs is different for different values of the 3rd IV”
What does a 3-way interaction look
like?
Practice
• Remember that a 2-way interaction is,
“when the effect of one IV is different
for different levels of a 2nd IV”
Difficulty
1
Easy
90
=
90
Hard
40
<
70
• Extending this to a design with 3 IVs, a
3-way interaction is, “when the
interaction of two IVs is different for
different levels of a 3rd IV”
SE of Practice is different
for Easy and Hard Tasks
Familiar Task
Unfamiliar Task
Practice
Difficulty
1
Easy
90
Hard
40
10
10
Practice
Difficulty
1
10
= 90
Easy
60
< 70
< 70
Hard
20
< 60
The 2-way interaction of
Practice and Difficulty is
different for Familiar and
Unfamiliar Tasks
Remember what the 3-way F-test tells you !!!
Familiar Task
Unfamiliar Task
Practice
Difficulty
1
Easy
80
Hard
60
10
Practice
Difficulty
1
10
< 90
Easy
50
< 80
= 60
Hard
40
= 40
F(1, 152) = 4.234,
p = .0235
The simple 2-way interaction pattern of Difficulty x Practice is
the same for Familiar and Unfamiliar Tasks?!?
Where is the interaction?
1st – the significant 3-way tells you that the two simple 2-ways
are different – that’s what a 3-way interaction means !!!
2nd – notice the Easy Practice effect is smaller for Familiar than
for Unfamiliar – sometimes that’s all it takes for a sig 3-way !!!
Basic Rules for Follow-up Analyses:
#1 Any significant effect with k>2 conditions will require a
follow-up significance test to compare the condition means
of that effect to describe the pattern of that effect.
#2 Any lower-order efffect involved in a significant higher
order effect will require follow-up significance tests to
compare the means pattern of that lower-order effect with
the means pattern of the corresponding simple effects, to
determine if the means pattern of the lower-order effect is
descriptive or misleading.
Remember…
In a 3-way effect we have 6 lower-order effects
• 3  2-way interactions
For each lower-order effect we
• 3  Main effects
have to decide… Do we care???
As we discussed… There are 2 situations when we will care!
#1  if the aggregates compared in the effect represent a
target population that we care about!
#2  when the effect is descriptive, and can be used to
generalize from the aggregate to all the separate
conditions/values (in the current study/design)
If the aggregates represent meaningful populations, but the
effect is misleading  we’ll have to carefully describe the
situation – when the pattern is descriptive and misleading, etc.
If the effect is neither representative not a useful general
description, then we may not care about the effect !!!
“Descriptive” effects in a 3-way
The 3-way -- significant or not -- is always descriptive !
If the 3-way is significant, all 2-way & main effects are “suspect”
If the 3-way is significant, a 2-way is only descriptive if that 2-way
has the same pattern for each condition of the 3rd IV
If the 3-way is significant, a main effect is only descriptive if that
main effect has the same pattern for each combination of the
other two IVs
If the 3-way is non-significant, all three 2-ways are descriptive
“Descriptive” effects in a 3-way
If a 2-way is significant, the main effects of those 2 IVs are
“suspect”
If a 2-way is significant, the main effect of an IV involved in that
interaction is only descriptive if that main effect has the same
pattern for each condition of the other IV
The main effect of an IV that is not involved in any 2-way or
3-way interaction is always descriptive
kxkxk
Stimulus Type
Shape
#Practices
1
Modality
Vision
Touch
Bimodal
Texture
10
1
#Practices
10
kxkxk
Effects for this Design
3-way interaction of
x Modality x Practice
2-way interaction of Stimulus x Modality
2-way interaction of Stimulus x Practice
2-way interaction of Modality x Practice
Main effect of Stimulus
Main effect of Modality
Main effect of Practice
Simple interaction of Stimulus x Modality at dif values of Practice
Simple interaction of Stimulus x Practice at dif values of Modality
Simple interaction of Modality x Practice at dif values of Stimulus
Simple effect of Stimulus at dif combos of Modality & Practice
Simple effect of Modality at dif combos of Stimulus & Practice
Simple effect of Practice at dif combos of Modality & Practice
kxkxk
100
90
80
70
60
50
Vision
Touch
Bimodal
10 Practices
Shape
1 Practice
Texture
kxkxk
P
E
R
F
O
R
M
A
N
C
E
100
90
80
70
60
50
1
Vision
Touch
Bimodal
Practices
10
Shape
Texture
kxkxk