factorial analysis of variance

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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

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13

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Difficult

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Factor B: Arousal

Fac

tor

A:

Tas

k D

iffi

cult

y

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Partitioning Variance

Low Medium High

Easy

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…

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Difficult

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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.

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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

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3

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7

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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

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1

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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