factorial analysis of variance
DESCRIPTION
Factorial Analysis of Variance. 46-511 Between Groups Fixed Effects Designs. Two-Way ANOVA Example: (Yerkes – Dodson Law). Factor B: Arousal. Factor A: Task Difficulty. Partitioning Variance. Factor B: Arousal. Variation among means on A represent effect of A. Factor A: Task Difficulty. - PowerPoint PPT PresentationTRANSCRIPT
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Factorial Analysis of Variance
46-511
Between Groups Fixed Effects Designs
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Two-Way ANOVA Example:(Yerkes – Dodson Law)
Low Medium High
Easy
3
1
1
6
4
5
5
9
7
7
9
9
13
6
8
Difficult
0
2
0
0
3
3
8
3
3
3
0
0
0
5
0
Factor B: Arousal
Fac
tor
A:
Tas
k D
iffi
cult
y
3
Partitioning Variance
Low Medium High
Easy
3
…
4
5
…
7
9
…
8
Difficult
0
…
3
3
…
3
0
…
0
Factor B: Arousal
Fac
tor
A:
Tas
k D
iffi
cult
y
Variation among people treated the same = error
Variation among means on A represent effect of A
Variation among means on B represents effect of B
Leftover variation = interaction
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Partitioning Variance: Interaction
Low Medium High Total
Easy 3.00 6.00 9.00 6.00
Difficult 1.00 4.00 1.00 2.00
Total 2.00 3.00 5.00 4.00
Factor B: Arousal
Fac
tor
A:
Tas
k D
iffi
cult
y
Dependence of means on levels of both A & B represents the effect of an interaction.
5
0
1
2
3
4
5
6
7
8
9
10
Low Medium High
Easy
Difficult
Or Graphically…
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In words
Types of Effects vs. 1-way Main Effect for A Main Effect for B Interaction (A x B)
Structural Model: XIJK = μ++++IJK
Partitioning Variance/Sums of Squares
First, total variance:
Between Groups:
Thus Total is:
BG A B AXBSS SS SS SS TOT BG WSS SS SS
TOT A B AXB WSS SS SS SS SS
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Sums of Squares Between
___2( )ijBGSS n AB G
Definitional Formula
Computational Formula
2 2
BG
T GSS
n N
Variation of cell means around grand mean, weighted by n.
Computational formulae:
•More accurate for hand calculation
•Easier to work
•Less intuitive
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Sums of Squares A
Definitional Formula
Computational Formula
_ _2( )iASS nq A G
2 2ROW
AROW
T GSS
n N
Variation of row means around grand mean, weighted by n times the number of levels of B, or q.
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Sums of Squares B
Definitional Formula
Computational Formula
_ _2( )jBSS np B G
2 2COL
BCOL
T GSS
n N
Variation of column means around grand mean, weighted by n times the number of levels of A, or p.
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Sums of Squares AxB
Definitional Formula
Computational Formula
2( )ij jAxB iSS n AB A B G
A B Between A BSS SS SS SS
SSAxB = Variation of cell means around grand mean, that cannot be accounted for by effects of A or B alone.
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Sums of Squares Within (Error)
Definitional Formula
Computational Formula
2( )ijW ijkSS X AB
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W ijk
TSS X
n
SSW = Variation of individual scores around cell mean.
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Numerical Example
Effect of Task Difficulty and Anxiety Level on Performance Low Medium High Marginals for B 3 2 9 1 5 9 Easy 1 9 13 T 90 6 7 6 Mean 6 4 7 8 SS 162 T 15 30 45 Mean 3 6 9
SS 18 28 26
0 3 0 2 8 0 Difficult 0 3 0 T 30 0 3 5 Mean 2 3 3 0 SS 78 T 5 20 5 Mean 1 4 1
SS 8 20 20
Marginal for A T 20 50 50 Mean 2 5 5 SS 36 58 206
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Degrees of Freedom
df between = k – 1; or, (kA x kB – 1)
df A = kA – 1
df B = kB – 1
df A x B = dfbetween – dfA – dfB
dfW = k(n-1)
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Source Table
Source SS df MS F
Between
A
B
AxB
Within
Total
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More Digression on Interactions
Ways to talk about interactions Scores on the DV depend upon levels of both
A and B The effect of A is moderated by B The effect of B is moderated by A There is a multiplicative effect for A and B
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More Digresions (cont’d)No effect whatsoever…
Interaction Effect: Cell & Marginal Means
A: Task Difficulty Low Medium High TotalsEasy 4 4 4 4Hard 4 4 4 4Totals 4 4 4 4
Deviations: cell mean - row mean - column mean + grand mean
Task Difficulty Low Medium HighEasy 0 0 0Hard 0 0 0
Interaction Sum of Squares: 0Main Effect for A 0Main Effect for B 0
B: Anxiety
Anxiety
No Significant Effects
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Main effects for A and B…
Interaction Effect: Cell & Marginal Means
A: Task Difficulty Low Medium High TotalsEasy 3 6 9 6Hard 1 4 7 4Totals 2 5 8 5
Deviations: cell mean - row mean - column mean + grand mean
Task Difficulty Low Medium HighEasy 0 0 0Hard 0 0 0
Interaction Sum of Squares: 0Main Effect for A 30Main Effect for B 180
Only Main Effects Significant
B: Anxiety
Anxiety
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Graphically…
2-Way ANOVA Anxiety by Task Difficulty: Main Effects, No Interaction
0
1
2
3
4
5
6
7
8
9
10
Low Medium High
Anxiety Level
Per
form
ance
Easy
Hard
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Interaction significant also…
Interaction Effect: Cell & Marginal Means
A: Task Difficulty Low Medium High TotalsEasy 3 6 9 6Hard 1 4 1 2Totals 2 5 5 4
Deviations: cell mean - row mean - column mean + grand mean
Task Difficulty Low Medium HighEasy -1 -1 2Hard 1 1 -2
Interaction Sum of Squares: 60Main Effect for A 120Main Effect for B 60
Anxiety
Significant Interaction
B: Anxiety
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Graphically…
2-Way ANOVA Anxiety by Task Difficulty: Main Effects AND Interaction
0
1
2
3
4
5
6
7
8
9
10
Low Medium High
Anxiety Level
Per
form
an
ce
Easy
Hard
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Further Analyses on Main Effects
Contrasts
Planned Comparisons
Post-Hoc Methods
In the presence of a significant interaction
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Further Analyses on Interaction
What it means
Simple (Main) Effects
Contrasts
Partial Interactions
Contrasts
Simple Comparisons / Post-Hoc Methods
How to get q
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Simple Main Effects Analysis
Low Medium High Total
Easy 3.00 6.00 9.00 6.00
Difficult 1.00 4.00 1.00 2.00
Total 2.00 3.00 5.00 4.00
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Simple Main Effects
2 2_ij j jJ
j
T at r TSS
n n
Sum of Squares Formula:
F Ratio: jc
W
MSF
MS
df = dfj,dfw:
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Partial Interaction Analysis
Low Medium High Total
Easy 3.00 6.00 9.00 6.00
Difficult 1.00 4.00 1.00 2.00
Total 2.00 3.00 5.00 4.00
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In Class Exercise
Drug B1 (no dose) B2 (low dose) B3 (high dose)
A1 (no dose) 57568
910
765
48459
A2 (low dose) 55468
78456
54397
A3 (high dose) 89786
1211151310
1817192020
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Based on two pieces of information
Descriptive Statistics
Dependent Variable: anxiety
6.2000 1.30384 5
7.4000 2.07364 5
6.0000 2.34521 5
6.5333 1.92230 15
5.6000 1.51658 5
6.0000 1.58114 5
5.6000 2.40832 5
5.7333 1.75119 15
7.6000 1.14018 5
12.2000 1.92354 5
18.8000 1.30384 5
12.8667 4.95504 15
6.4667 1.50555 15
8.5333 3.24844 15
10.1333 6.63181 15
8.3778 4.51406 45
FactorB1.00
2.00
3.00
Total
1.00
2.00
3.00
Total
1.00
2.00
3.00
Total
1.00
2.00
3.00
Total
FactorA1.00
2.00
3.00
Total
Mean Std. Deviation N
1)
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Compute simple main effects
Tests of Between-Subjects Effects
Dependent Variable: anxiety
781.378b 8 97.672 30.523 .000 .872 244.181 1.000
3158.422 1 3158.422 987.007 .000 .965 987.007 1.000
458.178 2 229.089 71.590 .000 .799 143.181 1.000
101.378 2 50.689 15.840 .000 .468 31.681 .999
221.822 4 55.456 17.330 .000 .658 69.319 1.000
115.200 36 3.200
4055.000 45
896.578 44
SourceCorrected Model
Intercept
FactorA
FactorB
FactorA * FactorB
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
Partial EtaSquared
Noncent.Parameter
ObservedPower
a
Computed using alpha = .05a.
R Squared = .872 (Adjusted R Squared = .843)b.
2)
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3-Way ANOVA
Effects A B C A x B A x C B x C A x B x C
A Vague Example DV = Treatment
Outcome Factor A: Gender Factor B: Age (14 or 17) Factor C: Treatment
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Results
Tests of Between-Subjects Effects
Dependent Variable: SCORE
221.556b 11 20.141 2.224 .049 .505 24.466 .804
2635.111 1 2635.111 290.994 .000 .924 290.994 1.000
.111 1 .111 .012 .913 .001 .012 .051
36.000 1 36.000 3.975 .058 .142 3.975 .482
24.889 2 12.444 1.374 .272 .103 2.748 .266
.111 1 .111 .012 .913 .001 .012 .051
80.889 2 40.444 4.466 .022 .271 8.933 .710
4.667 2 2.333 .258 .775 .021 .515 .086
74.889 2 37.444 4.135 .029 .256 8.270 .674
217.333 24 9.056
3074.000 36
438.889 35
SourceCorrected Model
Intercept
SEX
AGE
TREAT
SEX * AGE
SEX * TREAT
AGE * TREAT
SEX * AGE * TREAT
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
Partial EtaSquared
Noncent.Parameter
ObservedPower
a
Computed using alpha = .05a.
R Squared = .505 (Adjusted R Squared = .278)b.
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Significant Two-Way Interaction
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Significant Three-Way Interaction
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Other Stuff
Higher order models (4-way, 5-way, etc.) Unequal Cell Sizes and SS Type Use of contrast coefficients Short-Cuts using SPSS Custom Models in SPSS Observed Power