anthony greene1 advanced anova 2-way anova complex factorial designs i.the factorial design...
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Anthony Greene 1
Advanced ANOVA 2-Way ANOVA
Complex Factorial Designs
I. The Factorial Design
II. Partitioning The Variance For Multiple Effects
III. Independent Main Effects of Factor A and Factor B
IV. Interactions
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The Source Table• Keeps track of all data in complex ANOVA
designs
• Source of SS, df, and Variance (MS)– Partitioning the
SS, df and MS– All variability is attributable to
effect differences or error (all unexplained differences)
Total Variability
Effect Variability
(MS Between)
ErrorVariability
(MS Within)
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Partitioning of Variability for Two-Way ANOVA
Total Variability
Effect Variability
(MS Between)
Error Variability
(MS Within)
Factor A Variability
Stage 1 {{Stage 2 Factor B
VariabilityInteraction Variability
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Source Table for 1-Way ANOVA
Effect VariabilityError Variability
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2-Way ANOVA
• Used when two variables (any number of levels) are crossed in a factorial design
• Factorial design allows the simultaneous manipulation of variables
A1 A2 A3 A4
B1 A1•B1 A2 • B1 A3 • B1 A4 • B1
B2 A1 • B2 A2 • B2 A3 • B2 A4• B2
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2-Way ANOVA
For Example: Consider two treatments for mood disorders
1.This design allows us to consider multiple variables2.Importantly, it allows us to understand Interactions among variables
Placebo Prozac Zanex Bourbon
Depression A1•B1 A2 • B1 A3 • B1 A4 • B1
Anxiety A1 • B2 A2 • B2 A3 • B2 A4 • B2
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2-Way ANOVA
Hypothetical Data:
1.You can see that the effects of the drug depend upon the disorder
2.This is referred to as an Interaction
Placebo Prozac Zanex Bourbon
Depression -2.3 0.2 -1.1 -3.2
Anxiety -2.0 -0.1 1.3 -1.6
-3.5
-3.0
-2.5-2.0
-1.5
-1.0
-0.5
0.0
0.51.0
1.5
2.0
Placebo Prozac Zanex Bourbon
DepressionAnxiety
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Example of a 2-way ANOVA: Main Effect A
Daytime Heart rate
Nighttime Heart rate
No-Meditation 75 62
Mediation 74 63
60
65
70
75
80
Daytime Nightime
No Meditation
Meditation
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Example of a 2-way ANOVA: Main Effect B
Daytime Heart rate
Nighttime Heart rate
No-Meditation 75 74
Mediation 64 63
60
65
70
75
80
Daytime Nightime
No Meditation
Meditation
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Example of a 2-way ANOVA: Main Effect A & B
Daytime Heart rate
Nighttime Heart rate
No-Meditation 80 71
Mediation 71 60
60
65
70
75
80
85
Daytime Nightime
No Meditation
Meditation
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Example of a 2-way ANOVA: InteractionDaytime Heart rate
Nighttime Heart rate
No-Meditation 75 62
Mediation 65 63
60
65
70
75
80
Daytime Nightime
No Meditation
Meditation
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Partitioning of Variability for Two-Way ANOVA
Total Variability
Effect Variability
(MS Between)
Error Variability
(MS Within)
Factor A Variability
Stage 1 {{Stage 2 Factor B
VariabilityInteraction Variability
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Partitioning of Variability for Two-Way ANOVA
Total Variability
Effect Variability
(MS Between)
Error Variability
(MS Within)
Factor A Variability
Stage 1 {{Stage 2 Factor B
VariabilityInteraction Variability
Numerator for Omnibus F-ratio
Denominator for all F-ratios
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Partitioning of Variability for Two-Way ANOVA
Total Variability
Effect Variability
(MS Between)
Error Variability
(MS Within)
Factor A Variability
Stage 1 {{Stage 2 Factor B
VariabilityInteraction Variability
Numerator for Factor A F-ratio
Denominator for F-ratio
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Partitioning of Variability for Two-Way ANOVA
Total Variability
Effect Variability
(MS Between)
Error Variability
(MS Within)
Factor A Variability
Stage 1 {{Stage 2 Factor B
VariabilityInteraction Variability
Numerator for Factor B F-ratio
Denominator for F-ratio
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Partitioning of Variability for Two-Way ANOVA
Total Variability
Effect Variability
(MS Between)
Error Variability
(MS Within)
Factor A Variability
Stage 1 {{Stage 2 Factor B
VariabilityInteraction Variability
Numerator for Interaction F-ratio
Denominator for F-ratio
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2 Main Types of Interactions
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Simple Effects of An Interaction
0102030405060708090
100
B1 B2 B3 B4
A1A2A3
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Simple Effects of An Interaction
0102030405060708090
100
B1 B2 B3 B4
A1
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Simple Effects of An Interaction
0102030405060708090
100
B1 B2 B3 B4
A2
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Simple Effects of An Interaction
0102030405060708090
100
B1 B2 B3 B4
A3
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Simple Effects of An Interaction
0102030405060708090
100
B1 B2 B3 B4
A1A2A3
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Simple Effects of An Interaction
010
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4050
6070
8090
100
B1
A1A2A3
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Simple Effects of An Interaction
010
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8090
100
B2
A1A2A3
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Simple Effects of An Interaction
010
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8090
100
B3
A1A2A3
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Simple Effects of An Interaction
010
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8090
100
B4
A1A2A3
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How To Make the Computations
A1 A2
B1 153
374
B2 254
324
A1 A2 RowTot
B1 TSS
TSS
TB1
B2 TSS
TSS
TB2
Col
Tot.
TA1 TA2
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A1 A2 RowTotal
B1 TSS
TSS
TB1
B2 TSS
TSS
TB2
Col
Total
TA1 TA2
BAbtwAXB
BAbtwAXB
BB
BB
AA
AA
dfdfdfdf
SSSSSSSS
dfN
G
n
TSS
dfN
G
n
TSS
1-B) of levels of(number ,
1-A) of levels of(number ,
22
22
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Higher Level ANOVAN-Way ANOVA: Any number of factorial variables may be crossed; for example, if you wanted to assess the effects of sleep deprivation:
1. Hours of sleep per night: 4, 5, 6, 7, 8
2. Age: 20-30, 30-40, 40-50, 50-60, 60-70
3. Gender: M, F
You would need fifty samples
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Higher Level ANOVA
Mixed ANOVA: Any number of between subjects and repeated measures variables may be crossedFor example, if you wanted to assess the effects of sleep deprivation using sleep per night as the repeated measure:1. Hours of sleep per night: 4, 5, 6, 7, 82. Age: 20-30, 30-40, 40-50, 50-60, 60-703. Gender: M, FYou would need 10 samples
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How to Do a Mixed Factorial DesignTotal
Variability
Effect Variability
(MS Between)
MS Within
Individual Variability
ErrorVariability
Stage 1 {{Stage 2 Factor A
VariabilityInteractionVariability
Factor BVariability
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Two-Way ANOVAAn experimenter wants to assess the simultaneous effects of having breakfast and enough sleep on academic performance. Factor A is a breakfast vs. no breakfast condition. Factor B is three sleep conditions: 4 hours, 6 hours & 8 hours of sleep. Each condition has 5 subjects.
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Two-Way ANOVA
Source d.f. SS MS F
Between 60
Main A 5
Main B
A x B 30
Within 2
Total
Factor A has 2 levels, Factor B has 3 levels, and n = 5 (i.e., six conditions are required and each has five subjects). Fill in the missing values.
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Two-Way ANOVA
Source d.f. SS MS F
Between 5 60
Main A 1 5
Main B 2
A x B 30
Within 2
Total
First the obvious: The degrees freedom for A and B are the number of levels minus 1 . The degrees freedom Between is the number of conditions (6 = 2x3) minus 1.
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Two-Way ANOVA
Source d.f. SS MS F
Between 5 60
Main A 1 5
Main B 2
A x B 2 30
Within 2
Total
The interaction (AxB) is then computed: d.f.Between = d.f.A + d.f.B + d.f.AxB. OR d.f.AxB = d.f.A d.f.B
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Two-Way ANOVA
Source d.f. SS MS F
Between 5 60
Main A 1 5
Main B 2
A x B 2 30
Within 24 2
Total 29
d.f.Within= Σd.f. each cell
d.f.Total = N-1 = 29. d.f.Total= d.f.Between+ d.f.Within
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Two-Way ANOVA
Source d.f. SS MS F
Between 5 60 12
Main A 1 5
Main B 2
A x B 2 30
Within 24 2
Total 29
Now you can compute MSBetween by dividing SS by d.f.
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Two-Way ANOVA
Source d.f. SS MS F
Between 5 60 12
Main A 1 10 10 5
Main B 2
A x B 2 30
Within 24 2
Total 29
You can compute MSA by remembering that FA= MSA MSWithin, so 5 = ?/2. SSA is then found by remembering that MS = SS df,so 10 = ?/1
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Two-Way ANOVA
Source d.f. SS MS F
Between 5 60 12
Main A 1 10 10 5
Main B 2 20 10
A x B 2 30 15
Within 24 2
Total 29
Now SSB is computed by SSA + SSB + SSAxB = SSBetween
MSB = SSB/dfB and MSAxB = SSAxB/dfAxB
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Two-Way ANOVA
Source d.f. SS MS F
Between 5 60 12
Main A 1 10 10 5
Main B 2 20 10
A x B 2 30 15
Within 24 48 2
Total 29
MSWithin=SSWithin/dfWithin, solve for SS.
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Two-Way ANOVA
Source d.f. SS MS F
Between 5 60 12 6
Main A 1 10 10 5
Main B 2 20 10 5
A x B 2 30 15 7.5
Within 24 48 2
Total 29
Now Solve for the missing F’s (Between, B, AxB). F=MS/MSWithin
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Two-Way ANOVAAn experimenter is interested in the effects of efficacy on self-esteem. She theorizes that lack of efficacy will result in lower self-esteem. She also wants to find out if there is a different effect for females than for males. She conducts an experiment on a sample of college students, half male and half female. She then puts them through one of three experimental conditions: no efficacy, moderate efficacy, and high efficacy. Then she measures level of self-esteem. Her results are below. Conduct a two-way ANOVA. Report all significant findings with α= 0.05.
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Data
No Moderate High Efficacy Efficacy Efficacy
1 4 7Males 3 8 8 0 7 10Females 2 10 16
5 7 134 8 15
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No Moderate HighEfficacy Efficacy Efficacy
1 T=4 4 T=19 7 T=25Males 3 SS=4.6 8 SS=8.6 8 SS=4.7
0 7 10 Tm= 48Females 2 T=11 10 T=25 16 T=44
5 SS=4.6 7 SS=4.7 13 SS=4.7 Tf= 804 8 15Tne=15 Tme=44 The=69
n=3k=6N=18G=128∑x2=1260
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SSbetween
SSbtw = ∑T2/n – G2/N
SSbtw = (42 + 192 + 252 + 112 + 252 + 442)/3 –282/18
SSbtw = 1228-910.2=317.8
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SSsex, SSefficacy, SSinteractionSSsex = ∑T2sex/nsex – G2/N
SSsex = (482 + 802)/9 – 910.2
SSsex = 56.9
SSefficacy= ∑T2e/ne– G2/N
SSefficacy = (152 + 442 + 692)/6 – 910.2
SSefficacy = 243.47
SSinteraction = SSbetween – SSsex – SSefficacy
SSinteraction = 317.8-56.9-243.47
SSinteraction = 17.43
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SSwithin and SStotal
SSwithin = ∑SS
SSwithin=4.6+8.6+4.7+4.6+4.7+4.7=31.9
SStotal = ∑x2 – (∑x)2/N
SStotal = 1260 – 910.2
SStotal = 349.8
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Degrees Freedom
dfbtw = cells – 1 = k-1
dfsex = rows - 1
dfeff = columns - 1
dfint = dfbtw – dfsex - dfeff
dfwin = Σdfeach cell = dftot-dfbtw
dftot = N-1 = nk-1
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Degrees Freedom
dfbtw = cells – 1 = k-1 = 5
dfsex = rows – 1 = 1
dfeff = columns – 1 = 2
dfint = dfbtw – dfsex – dfeff = dfsex dfeff = 2
dfwin = Σdfeach cell = dftot-dfbtw = 12
dftot = N-1= nk-1= 17
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Source TableSource SS df MS F Fcrit
Between 317.8
Sex 56.9
Efficacy 243.5
Int. 17. 4
Within 31. 9
Total 349.8
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Source TableSource SS df MS F Fcrit
Between 317.8 5
Sex 56.9 1
Efficacy 243.5 2
Int. 17. 4 2
Within 31. 9 12
Total 349.8 17
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Source TableSource SS df MS F Fcrit
Between 317.8 5 63.6
Sex 56.9 1 56.9
Efficacy 243.5 2 121.7
Int. 17. 4 2 8.7
Within 31. 9 12 2.7
Total 349.8 17
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Source TableSource SS df MS F Fcrit
Between 317.8 5 63.6 23.5 F(5,12)=3.11
Sex 56.9 1 56.9 21.4 F(1,12)=4.75
Efficacy 243.5 2 121.7 45.8 F(2,12)=3.88
Int. 17. 4 2 8.7 3.3 F(2,12)=3.88
Within 31. 9 12 2.7
Total 349.8 17
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Source TableSource SS df MS F Fcrit
Between 317.8 5 63.6 23.5 F(5,12)=3.11
Sex 56.9 1 56.9 21.4 F(1,12)=4.75
Efficacy 243.5 2 121.7 45.8 F(2,12)=3.88
Int. 17. 4 2 8.7 3.3 F(2,12)=3.88
Within 31. 9 12 2.7
Total 349.8 17
1 main effect for sex2 main effect for efficacy 3 no significant interaction