Factorial Designs

More than one Independent Variable:

•Each IV is referred to as a Factor

•All Levels of Each IV represented in the Other IV

A Two-Way ANOVA

Factor A has 3 Levels

Factor BHas 2Levels

Each Cell is a COMBINATIONOf Treatments for a Group ofSubjects

A Two-Way ANOVA

Marginal MeansFor Factor B; doThey differ

Marginal Means for Factor A; do they differ?

Marginal Means average across the Levels of the OTHER Factor

A Two-Way ANOVA

A Two-Way ANOVA tells you:1. What a One-Way ANOVA would find out about Factor A2. What a One-Way ANOVA would find out about Factor B3. If there is an Interaction between Factor A and Factor B

A Two-Way Interaction

An Interaction is the effect which one IV has on the effect whichThe other IV has on the DV

Is the Difference between subjects who got Treatment B1 and B2The Same irrespective of whether they got Treatment A1, A2, or A3?

The Other Side of the Same Coin:

Are the Differences among subjects who got Treatments A1, A2 and A2The Same irrespective of whether they got Treatment B1 or B2?

Main Effects & Interactions

SPORTS

Yes .. No

Mea

n A

GG

RE

SS

ION

70

60

50

40

30

20

10

SEX

Female

Male

IV1: SportsIV2: GenderDV: Aggression

IV1: Main EffectIV2: Main EffectInteraction: None

Main Effect: AveragedAcross levels of theOther IV

Is the impact of sports on aggression differentFor Males and females?

M

FN

Y

Main Effects & Interactions

IV1: SportsIV2: GenderDV: Aggression

IV1: Main EffectIV2: Main EffectInteraction: Yes

Main Effect: AveragedAcross levels of theOther IV

Is the impact of sports on aggression differentFor Males and females?

M

FN

Y

SPORTS

Yes .. No

Mea

n A

GG

RE

SS

ION

100

80

60

40

20

0

SEX

Female

Male

M

FN

Y

Main Effects & Interactions

IV1: SportsIV2: GenderDV: Aggression

IV1: Main EffectIV2: Main EffectInteraction: Yes

Main Effect: AveragedAcross levels of theOther IV

Is the impact of sports on aggression differentFor Males and females?

M

FN

YM

FN

Y

SPORTS

Yes .. No

Mea

n A

GG

RE

SS

ION

100

80

60

40

20

0

SEX

Avg

Female

Male

F

M

Main Effects & Interactions

IV1: SportsIV2: GenderDV: Aggression

IV1: No Main EffectIV2: Main EffectInteraction: Yes

Main Effect: AveragedAcross levels of theOther IV

Is the impact of sports on aggression differentFor Males and females?

M

FN

YM

FN

Y

F

M

SPORTS

Yes .. No

Mea

n A

GG

RE

SS

ION

80

70

60

50

40

30

20

SEX

Avg

Female

Male

M

F

Main Effects & Interactions

IV1: SportsIV2: GenderDV: Aggression

IV1: No Main EffectIV2: No Main EffectInteraction: Yes

Main Effect: AveragedAcross levels of theOther IV

Is the impact of sports on aggression differentFor Males and females?

M

FN

YM

FN

Y

F

MM

F

SPORTS

Yes .. No

Mea

n A

ggre

ssio

n

80

70

60

50

40

30

20

SEX

Avg

Female

Male

No Interaction

Main EffectDifferences Same for B1 and B2And Same as Marginal Means

Main EffectDifferences sameFor A1, A2, & A3And Same asMarginal Means

Yummy Interaction

B1 and B2 Subjects change differently (across Factor A) fromOne another and from the Marginal Means

A1, A2, & A3 SubjectsChange differently(across Factor b) from oneAnother and fromMarginal Means

In my opinion: Interactions are more interesting and more importantThan main effects

Explaining the Relationship Between the IV and DV

Does Sports (IV1) affect Aggression (DV)?Yes but more so in malesYes but only in malesYes but oppositely in males and females (IV2)

If you have to qualify the relationship with a “But,” then youHave an Interaction.

Interactions

Statistical Symbols

Σ Sumά Type I Errorβ Type II Errorμ Population Meanρ Population Correlationσ Population Standard Deviation

Interaction

InteractionsArms & Legs Not Parallel

SPORTS

Yes .. No

Mea

n A

GG

RE

SS

ION

100

80

60

40

20

0

SEX

Female

Male

SPORTS

Yes .. No

Mea

n A

GG

RE

SS

ION

100

80

60

40

20

0

SEX

Avg

Female

Male

SPORTS

Yes .. No

Mea

n A

GG

RE

SS

ION

80

70

60

50

40

30

20

SEX

Avg

Female

Male

SPORTS

Yes .. No

Mea

n A

ggre

ssio

n

80

70

60

50

40

30

20

SEX

Avg

Female

Male

Yes But more so in males

Yes But only in males

Yes But in different directions

Yes But in different directions, from different directions

Parvulus te Tergum

The Structure of the ANOVAPartitioning the Total Sum of Squared Deviations

From the Grand Mean

If you must run your reaction time study at 3 different times of day:1. Counter Balance2. Use Time as a Second IV to pull Main Effect and Interaction

Variance (SS) out of Error Term

D.V.: Reaction Time

SS Main Effect A

SS Main Effect B

SS Interaction

AxB

SS Error

Variation W/I Drug Time Combination

E.G., Drug

E.G., Time of Day

Partitioning the Sums of SquaresInto 4 Parts

Variation of cell meansFrom Grand Mean(SS_Between Cell)

Variation of IndividualsFrom their cell means(SS-Within Cell)

Variation of Individuals fromThe Grand Mean

Sum to SSCell

Every Subjects’ Score is composed of these 4 parts

Do It!Step 1:Calculate SS-Total &SS-Error

Xi X-XBarA1B1 dev sq X-GM dev sqGroup A1B1 12 1 1 2.583333333 6.673611mean= 11 0 0 1.583333333 2.50694411 10 -1 1 0.583333333 0.340278

X-XBarA1B2

Group A1B2 12 1.66666667 2.777778 2.583333333 6.673611mean= 10 -0.3333333 0.111111 0.583333333 0.34027810.333333 9 -1.3333333 1.777778 -0.416666667 0.173611

X-XBarA2B1 dev sq X-GM dev sq

Group A2B1 8 0.66666667 0.444444 -1.416666667 2.006944mean= 7 -0.3333333 0.111111 -2.416666667 5.8402787.3333333 7 -0.3333333 0.111111 -2.416666667 5.840278

X-XBarA2B2 dev sq X-GM dev sq

Group A2B2 10 1 1 0.583333333 0.340278mean= 9 0 0 -0.416666667 0.1736119 8 -1 1 -1.416666667 2.006944

GM= 9.416667 9.333333 32.91667SSError SSTotal

Step 2: Calculate SS Main Effects For A & B

A1 B112 1211 1110 1012 810 79 7

Mean A1= 10.66667 9.166667MeanA1-GM= 1.25 -0.25

dev sq= 1.5625 0.0625

A2 B28 127 107 9

10 109 98 8

Mean A2= 8.166667 9.666667MeanA2-GM= -1.25 0.25

dev sq= 1.5625 0.0625

18.75 0.75SSA SSB

Step 3: Calculate SS Interaction

SSAxB= "f21-i23-j23-d21= 4.083333SSAxB= "o9-i23-j23= 4.083333

SSTot-SSA-SSB-SSError

OrSSCell-SSA-SSB

Step 4: Calculate Degrees of Freedom

DF Tot= N-1 11DF A= "2-1 1DF B= "2-1 1

DF AxB= (2-1)*(2-1) 1DFError= "11-3 8

Step 5: Calculate Mean SquaresDivide SS by df

DF Tot= N-1 11DF A= "2-1 1 MS A= 18.75DF B= "2-1 1 MS B= 0.75

DF AxB= (2-1)*(2-1) 1 MS AxB= 4.083333333DFError= "11-3 8 MSError= 1.166666667

Step 5: Calculate F-ValuesDivide MS by MSError

FMS A= 18.75 16.07 p<0.01MS B= 0.75 0.64 p>0.05

MS AxB= 4.083333333 3.50 p>0.05MSError= 1.166666667

Decision

If Interaction is non-significant:•Interpret Each Main Effect as if it came from a One-Way ANOVA•Do Tukey Post Hoc HSD test for every Significant IV with more

than 2 Levels

If Interaction is Significant:Do a Simple Effects ANOVA on Each IV

For EVERY Level of the other IVA 3x2 design would require 5 One-way ANOVAs

Post Hoc Tests for Each Significant IV (If No Interaction)

•X-Bars are the Marginal Means•Nt is the number of scores going into the Marginal Mean•Nt must be same size for both Marginal Means