FACTORIAL DESIGNS

What is a Factorial Design?

• A design containing two or more independent variables (factors), with all combinations of levels of factors measured.

Levels and Conditions

• A level is a value of a factor. Each factor has two or more levels.

• A condition is a combination of levels of different factors.

Factor A

Factor B

Level 1 Level 2

Level 1

Level 2

Level 3

condition condition condition

condition condition condition

Types of Factorial Designs

• between subjects• within subjects• mixed

Between Subjects

B

1

2

A1 2

Subjects1-10

Subjects11-20

Subjects31-40

Subjects21-30

Within Subjects

B

1

2

A1 2

Subjects1-40

Subjects1-40

Subjects1-40

Subjects1-40

Mixed (A Between, B Within)

B

1

2

A1 2

Subjects1-20

Subjects1-20

Subjects21-40

Subjects21-40

Notation for Factorials• The number of numbers tells you how

many IV’s.• The numbers tell you how many levels.• A factorial with two IV’s that each have

two levels is a 2 x 2 factorial.

Notation for Factorials

2x2 2x33x4How many i.v.’s?How many d.v.’s?How many conditions?

Why Are Factorials Useful?

• Reduce amount of non-systematic variance• Ability to measure interaction– Many behaviors are affected by interactions– Main effects can be misleading without

considering the interaction

Drug

Therapy

1 2

1

2

40 60

60 40

What is a Main Effect?

• The overall effect of one IV, averaging over the levels of the other IV.

• If the means of the levels (marginal means) are different, there is a main effect.

• On a graph of means, marginal means can be estimated visually.

A

B

1 2

1

2

40 40

60 60

40

60

50 50

A

B

1 2

1

2

3

20

30

20

30

50

60

What is an Interaction?

• The effect of one IV changes depending on the level of the other IV.

• If the simple effects are different, there is an interaction.

What is an Interaction?

• A simple effect is the difference in means between levels of an IV for just one level of another IV.

• On a graph, non-parallel lines indicate an interaction.

A

B

1 2

1

2

40 60

60 80

+20

+20

A

B

1 2

1

2

40 60

60 100

+20

+40

A

B

1 2

1

2

3

20

30

20

30

50

90

Drug

Therapy

1 2

1

2

40 60

60 40