ANOVA With More Than One IV

2-way ANOVA

So far, 1-Way ANOVA, but can have 2 or more IVs. IVs aka Factors.

Example: Study aids for exam IV 1: workbook or not IV 2: 1 cup of coffee or not

Workbook (Factor A)

Caffeine (Factor B)

No Yes

Yes Caffeine only

Both

No Neither (Control)

Workbook only

Main Effects N=30 per cell

Workbook (Factor A) Row Means

Caffeine

(Factor B)

No Yes

Yes Caff

=80

SD=5

Both

=85

SD=5

82.5

No Control

=75

SD=5

Book

=80

SD=5

77.5

Col Means 77.5 82.5 80

X X

X X

Main Effects and Interactions

Main effects seen by row and column means; Slopes and breaks.

Interactions seen by lack of parallel lines.

Interactions are a main reason to use multiple IVs

Workbook (Factor A)

86

84

82

80

78

76

74

Mea

n R

M T

est S

core

No Yes

Without Caffeine

With Caffeine

Factor B

Single Main Effect for B

2.0 1.0

Factor A

25

20

15

10

5

0

Mea

n R

espo

nseSingle Main Effect

B=1

B=2

A

1 2

B1

2

10 10

20 20

(Coffee only)

Single Main Effect for A

2.0 1.0

Factor A

20

16

12

8

4

0

Mea

n R

espo

nse

Single Main Effect

B=1

B=2

A

1 2

B1

2

10 20

10 20

(Workbook only)

Two Main Effects; Both A & B

2.0 1.0

Factor A

35

30

25

20

15

10

5

0

Mea

n R

espo

nse

Two Main Effects

B=1

B=2

A

1 2

B1

2

10 20

20 30

Both workbook and coffee

Interaction (1)

2.0 1.0

Factor A

35

30

25

20

15

10

5

0

Mean R

esponse

Interaction 1

B=1

B=2

A

1 2

B1

2

10 20

10 30

Interactions take many forms; all show lack of parallel lines.

Coffee has no effect without the workbook.

Interaction (2)

2.0 1.0

Factor A

25

20

15

10

5

0

Mea

n R

espo

nse

Interaction 2

B=1

B=2

A

1 2

B1

2

10 20

20 10

People with workbook do better without coffee; people without workbook do better with coffee.

Interaction (3)

2.0 1.0

Factor A

40

35

30

25

20

15

10

5

0

Mea

n R

espo

nse

Interaction 3

B = 1

B = 2

Coffee always helps, but it helps more if you use workbook.

Labeling Factorial Designs

Levels – each IV is referred to by its number of levels, e.g., 2X2, 3X2, 4X3 designs. Two by two factorial ANOVA.

Example Factorial Design (1)

Effects of fatigue and alcohol consumption on driving performance.

Fatigue Rested (8 hrs sleep then awake 4 hrs) Fatigued (24 hrs no sleep)

Alcohol consumption None (control) 2 beers Blood alcohol .08 %

Cells of the Design

Alcohol (Factor A)

Fatigue (Factor B)

None 2 beers .08 %

Tired Cell 1 Cell 2 Cell 3

Rested Cell 4 Cell 5:Rested, 2 beers, Porsche 911

Cell 6

DV – closed course driving performance ratings from instructors.

Factorial Example Results

Intox2 beersnoneAlcohol Consumption

25

20

15

10

5

0

Driv

ing

Err

ors

Factorial Design

Rested

Fatigued

Main Effects?Interactions? Both main effects and the interaction

appear significant.

ANOVA Summary Table

SourceSource SSSS DfDf MSMS FF

A SSA a-1 SSA/dfA MSA/MSError

B SSB b-1 SSB/dfB MSB/MSError

AxB SSAxB (a-1)(b-1) SSAxB/dfAxB MSAxB/MSError

Error SSError ab(n-1)or

N-ab

SSError/dfError

Total SSTotal N-1

Two Factor, Between Subjects Design

Review

In a 3 X 3 ANOVA How many IVs are there? How many df does factor A have How many df does the interaction have

Test

We can see the main effect for a variable if we examine means of the dependent variable while ________

Considering the joint effects of both variables Examining a single value of a second factor Examining each cell Ignoring the other variable

Test

In two-way ANOVA, the term interaction means

Both IVs have an impact on the DV The effect of one IV depends on the value of the

other IV The on IV has no effect unless the other IV has a

certain value There is a crossover – a graph of two lines

shows an ‘X’.