Introduction to Factorial ANOVA Designs. Factorial Anova With factorial Anova we have more than one...

Introduction to Factorial ANOVA

Designs

Factorial Anova With factorial Anova we have more than

one independent variable The terms 2-way, 3-way etc. refer to how

many IVs there are in the analysis The following will discuss 2-way design but

may extended to more complex designs. The analysis of interactions constitutes the

focal point of factorial design

Recall the one-way Anova Total variability comes from:

Differences between groups Differences within groups

S S Between groups S S with in groups

S S total

Factorial Anova With factorial designs we have additional

sources of variability to consider Main effects

Mean differences among the levels of a particular factor

Interaction Differences among cell means not attributable

to main effects When the effect of one factor is influenced by

the levels of another

Partition of Variability

Total variability

Between-treatments var. Within-treatments var.

Factor Avariability

Factor Bvariability

Interactionvariability

Example: Arousal, task difficulty and performance

Yerkes-Dodson

Descriptive Statistics

Dependent Variable: Score

1.0000 2.23607 5

1.0000 1.41421 5

4.0000 2.23607 5

2.0000 2.36039 15

9.0000 2.54951 5

3.0000 2.12132 5

6.0000 2.64575 5

6.0000 3.40168 15

5.0000 4.78423 10

2.0000 2.00000 10

5.0000 2.53859 10

4.0000 3.52332 30

Arousalhi

lo

med

Total

hi

lo

med

Total

hi

lo

med

Total

Difficultydifficult

easy

Total

Mean Std. Deviation N

Example SStotal = ∑(X – grand mean)2

SStotal = 360 Df = N – 1 = 29

SSb/t =∑n(cell means – grand mean)2

= 5(3-4)2 + … 5(1-4)2 SSb/t =240 Df= K# of cells – 1 = 5

SSw/in = ∑(X – respective cell means)2 or SStotal- SSb/t

SSw/in = 120 Df = N-K = 24

Sums of Squares Between SSDifficulty = n(row means –grand mean)2

= 15(6-4)2 + 15(2-4)2 = 120 Df = # of rows (levels) – 1 = 1

SSArousal = n(col means – grand mean)2

= 10(2-4)2 + 10(5-4)2 + 10(5-4)2 = 60 Df = # of columns (levels) – 1 = 2

SSDXA = SSb/t - SSDifficulty - SSArousal = 240 - 120 - 60 = 60 Df = dfb/t - dfDifficulty – dfArousal = 5-1-2 = 2

Or dfdiff X dfarous

Output Mean squares and F-statistics are

calculated as before

Tests of Between-Subjects Effects

Dependent Variable: Score

240.000a 5 48.000 9.600 .000 .667

120.000 1 120.000 24.000 .000 .500

60.000 2 30.000 6.000 .008 .333

60.000 2 30.000 6.000 .008 .333

120.000 24 5.000

360.000 29

SourceB/t groups

Difficulty

Arousal

Difficulty * Arousal

Error

Total

Type IIISum ofSquares df Mean Square F Sig.

Partial EtaSquared

R Squared = .667 (Adjusted R Squared = .597)a.

Initial Interpretation Significant main effects of task difficulty

and arousal level, as well as a significant interaction

Difficulty Better performance for easy items

Arousal Low worst

Interaction Easy better in general but much more so with high

arousal

Eta-squared is given as the effect size for B/t groups (SSeffect/SStotal)

Partial eta-squared is given for the remaining factors: SSeffect/(SSeffect + SSerror)

End result: significance w/ large effect sizes

Graphical display of interactions Two ways to display previous results

lo med hi

Arousal

0.00

2.00

4.00

6.00

8.00

10.00

Mea

n S

core

Difficulty

difficult

easy

easy difficult

Difficulty

0.00

2.00

4.00

6.00

8.00

10.00

Mea

n S

core

Arousal

hi

lo

med

Graphical display of interactions What are we looking for? Do the lines behave similarly (are parallel)

or not? Does the effect of one factor depend on

the level of the other factor?

No interaction Interaction

The general linear model Recall for the general one-way anova

Where: μ = grand mean = effect of Treatment A (μa – μ)

ε = within cell error

So a person’s score is a function of the grand mean, the treatment mean, and within cell error

( )ijk i k ijY

Population main effect associated with the treatment Aj (first factor): jj

Population main effect associated with treatment Bk (second factor): kk

The interaction is defined as , the joint effect of treatment levels j and k (interaction of and ) so the linear model is:

ijkjkkjijk ey )(

jk)(

Each person’s score is a function of the grand mean, the treatment means, and their interaction (plus w/in cell error).

Effects for 2-way

The general linear model The interaction is a residual:

Plugging in and leads to:

kjjkjk )(

kjjkjk)(

Partitioning of total sum of squares Squaring yields

Interaction sum of squares can be obtained as remainder

TR A B ABSS SS SS SS 2

** ***

2* * ***

2* ** * * ***

( )

( )

( )

A ii

B jj

AB ij i ji j

SS n Y Y

SS n Y Y

SS n Y Y Y Y

AB TR A BSS SS SS SS

Partitioning of total sum of squares SSA: factor A sum of squares measures the

variability of the estimated factor A level means The more variable they are, the bigger will be SSA

Likewise for SSB

SSAB is the AB interaction sum of squares and measures the variability of the estimated interactions

( )ijk i j k ijijY

0 1 2Treatment A. :

pH

0 1 2Treatment B. :

qH

0 11 12Interaction. :

pqH

Statistical Hypothesis:

Statistical Model:

GLM Factorial ANOVA

The interaction null is that the cell means do not differ significantly (from the grand mean) outside of the main effects present, i.e. that this residual effect is zero

Interpretation: sig main fx and interaction Note that with a significant interaction, the

main effects are understood only in terms of that interaction

In other words, they cannot stand alone as an explanation and must be qualified by the interaction’s interpretation

Some take issue with even talking about the main effects, but noting them initially may make the interaction easier for others to understand when you get to it

Interpretation: sig main fx and interaction However, interpretation depends on common

sense, and should adhere to theoretical considerations Plot your results in different ways

If main effects are meaningful, then it makes sense to talk about them, whether or not an interaction is statistically significant or not E.g. note that there is a gender effect but w/ interaction

we now see that it is only for level(s) X of Factor B To help you interpret results, test simple effects

Is simple effect of A significant within specific levels of B?

Is simple effect of B significant within specific levels of A?

Simple effects Analysis of the effects of one factor at one

level of the other factor Some possibilities from previous example

Arousal for easy items (or hard items) Difficulty for high arousal condition (or medium

or low)

Simple effects SSarousal for easy items = 5(3-6)2 + 5(6-6)2 + 5(9-6)2 = 90 SSarousal for difficult items = 5(1-2)2 + 5(4-2)2 + 5(1-2)2 = 30

SSdifficulty at lo = 5(3-2)2 + 5(1-2)2 = 10 SSdifficulty at med = 5(6-5)2 + 5(4-5)2 = 10 SSdifficulty at hi = 5(9-5)2 + 5(1-5)2 = 160

Simple effects Note that the simple effect represents a

partitioning of SSmain effect and SSinteraction NOT JUST THE INTERACTION!!

From Anova table: SSarousal + SSarousal by difficulty = 60 + 60 = 120

SSarousal for easy items = 90 SSarousal for difficult items = 30 90 + 30 = 120

SSdifficulty + SSarousal by difficulty = 120 + 60 = 180 SSdifficulty at lo = 10 SSdifficulty at med = 10 SSdifficulty at hi = 160 10 + 10 + 160 = 180

Pulling it off in SPSS

Paste!

Pulling it off in SPSS Add

/EMMEANS = tables(a*b)compare(a) /EMMEANS = tables(a*b)compare(b)

Pulling it off in SPSS Output

Univariate Tests

Dependent Variable: Score

30.000 2 15.000 3.000 .069 .200

120.000 24 5.000

90.000 2 45.000 9.000 .001 .429

120.000 24 5.000

Contrast

Error

Contrast

Error

Difficultydifficult

easy

Sum ofSquares df Mean Square F Sig.

Partial EtaSquared

Each F tests the simple effects of Arousal within each level combination of the other effects shown.These tests are based on the linearly independent pairwise comparisons among the estimated marginalmeans.

Univariate Tests

Dependent Variable: Score

160.000 1 160.000 32.000 .000 .571

120.000 24 5.000

10.000 1 10.000 2.000 .170 .077

120.000 24 5.000

10.000 1 10.000 2.000 .170 .077

120.000 24 5.000

Contrast

Error

Contrast

Error

Contrast

Error

Arousalhi

lo

med

Sum ofSquares df Mean Square F Sig.

Partial EtaSquared

Each F tests the simple effects of Difficulty within each level combination of the other effects shown.These tests are based on the linearly independent pairwise comparisons among the estimatedmarginal means.

Test for simple fx with no sig interaction? What if there was no significant

interaction, do I still test for simple effects?

Maybe, but more on that later A significant simple effect suggests

that at least one of the slopes across levels is significantly different than zero

However, one would not conclude that the interaction is ‘close enough’ just because there was a significant simple effect

The nonsig interaction suggests that the slope seen is not statistically different from the other(s) under consideration.

1.00 2.00

VAR00001

0

1

2

3

4

Mea

n V

AR

0000

2

VAR00003

1.00

2.00

Multiple comparisons and contrasts For main effects multiple comparisons and

contrasts can be conducted as would be normally

One would have all the same considerations for choosing a particular method of post hoc analysis or weights for contrast analysis

Multiple comparisons and contrasts With interactions post hocs can be run

comparing individual cell means The problem is that it rarely makes

theoretical sense to compare many of the pairs of means under consideration

Contrasts for interactions We may have a specific result to

look for with regard to our interaction

For example, we may think based on past research moderate arousal should result in optimal performance for difficult items

We would assign contrast weights to reflect this hypothesis

Pulling it off in SPSS

Analyze General Linear Model Univariate

Select Dependent Variable and Specify Fixed and/or Random Factor(s) (Treatment Groups and or Patient Characteristic(s), Treatment Sites, etc.)

Paste Launches Syntax Window

Add /LMATRIX command lines

AllRUN

/LMATRIX ‘<Title for 1st Contrast>’<Specify Weights for 1st Contrast>;

‘<Title for 2nd Contrast>’<Specify Weights for 2nd Contrast>;

…

‘<Title for Final Contrast>’<Specify Weights for Final Contrast>

/LMATRIX Command

For this 3 X 2 design the weights will order as follows:

A1B1 A1B2 A2B1 A2B2 A3B1 A3B2

Note for this example, SPSS is analyzing categories in alphabetical order

Arousal hi lo med Task Diff Easy

In other words Hi:Difficult Hi:Easy Lo:Difficult … Med:Easy

Test Results

Dependent Variable: Score

30.000 1 30.000 6.000 .022 .200

120.000 24 5.000

SourceContrast

Error

Sum ofSquares df Mean Square F Sig.

Partial EtaSquared

As alluded to previously it is possible to have:

Sig overall F Sig contrast Nonsig posthoc

Nonsig F Nonsig contrast

e.g. 1 & 3 VS. 2 Sig posthoc

1 vs. 2 sig1.00 2.00 3.00

VAR00001

0

1

2

3

4

5

Mea

n V

AR

0000

4

A different model ☺

If cognitive anxiety is low, then the performance effects of physiological arousal will be low; but if it is high, the effects will be large and sudden.