Advanced ANOVA

Anthony Greene 1

Advanced ANOVA 2-Way ANOVA

Complex Factorial Designs

I. The Factorial DesignII. Partitioning The Variance For

Multiple EffectsIII. Independent Main Effects of Factor

A and Factor BIV. 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 toeffect 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 ANOVAFor 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


Depression -2.3 0.2 -1.1 -3.2

Anxiety -2.0 -0.1 1.3 -1.6

-3.5-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.0


DepressionAnxiety

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Example of a 2-way ANOVA: Main Effect A

Daytime Heart rate

Nighttime Heart rate

No-Meditation 75 62Mediation 74 63

60

65

70

75

80

Daytime Nightime

No MeditationMeditation

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Example of a 2-way ANOVA: Main Effect B

Daytime Heart rate



60

65

70

75

80

Daytime Nightime


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Example of a 2-way ANOVA: Main Effect A & B

Daytime Heart rate



60

65

70

75

80

85

Daytime Nightime


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Example of a 2-way ANOVA: InteractionDaytime Heart rate



60

65

70

75

80

Daytime Nightime


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Total Variability

Effect Variability

(MS Between)

Error Variability

(MS Within)




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Total Variability

Effect Variability

(MS Between)

Error Variability

(MS Within)




Numerator for Omnibus F-ratio

Denominator for all F-ratios

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Total Variability

Effect Variability

(MS Between)

Error Variability

(MS Within)




Numerator for Factor A F-ratio

Denominator for F-ratio

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Total Variability

Effect Variability

(MS Between)

Error Variability

(MS Within)




Numerator for Factor B F-ratio


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Total Variability

Effect Variability

(MS Between)

Error Variability

(MS Within)




Numerator for Interaction 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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0102030405060708090

100

B1 B2 B3 B4

A1

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0102030405060708090

100

B1 B2 B3 B4

A2

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0102030405060708090

100

B1 B2 B3 B4

A3

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0102030405060708090

100

B1 B2 B3 B4

A1A2A3

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0102030405060708090

100

B1

A1A2A3

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0102030405060708090

100

B2

A1A2A3

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0102030405060708090

100

B3

A1A2A3

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0102030405060708090

100

B4

A1A2A3

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+

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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

ColTot.

TA1 TA2

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A1 A2 RowTotal

B1 TSS

TSS

TB1

B2 TSS

TSS

TB2

ColTotal

TA1 TA2

BAbtwAXB

BAbtwAXB

BB

BB

AA

AA

dfdfdfdfSSSSSSSS

dfNG

nTSS

dfNG

nTSS

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, 82. Age: 20-30, 30-40, 40-50, 50-60, 60-703. Gender: M, FYou 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 FBetween 60 Main A 5 Main B A x B 30Within 2Total

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 FBetween 5 60 Main A 1 5 Main B 2 A x B 30Within 2Total

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 FBetween 5 60 Main A 1 5 Main B 2 A x B 2 30Within 2Total

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 FBetween 5 60 Main A 1 5 Main B 2 A x B 2 30Within 24 2Total 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 FBetween 5 60 12 Main A 1 5 Main B 2 A x B 2 30Within 24 2Total 29

Now you can compute MSBetween by dividing SS by d.f.

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Two-Way ANOVA

Source d.f. SS MS FBetween 5 60 12 Main A 1 10 10 5 Main B 2 A x B 2 30Within 24 2Total 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 FBetween 5 60 12 Main A 1 10 10 5 Main B 2 20 10 A x B 2 30 15Within 24 2Total 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 FBetween 5 60 12 Main A 1 10 10 5 Main B 2 20 10 A x B 2 30 15Within 24 48 2Total 29

MSWithin=SSWithin/dfWithin, solve for SS.

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Two-Way ANOVA

Source d.f. SS MS FBetween 5 60 12 6 Main A 1 10 10 5 Main B 2 20 10 5 A x B 2 30 15 7.5Within 24 48 2Total 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/NSSsex = (482 + 802)/9 – 910.2SSsex = 56.9

SSefficacy= ∑T2e/ne– G2/NSSefficacy = (152 + 442 + 692)/6 – 910.2SSefficacy = 243.47

SSinteraction = SSbetween – SSsex – SSefficacy

SSinteraction = 317.8-56.9-243.47SSinteraction = 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. 9Total 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 8.7Within 31. 9 12 2.7Total 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)=3.88Within 31. 9 12 2.7Total 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)=3.88Within 31. 9 12 2.7Total 349.8 17

1 main effect for sex2 main effect for efficacy 3 no significant interaction

Advanced ANOVA

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