F = Variance (differences) between sample means

Variance (differences) expected from sampling error

No Audience AudienceX = 5 X = 7

Self-Esteem

Low HighX = 9 X = 3

No Audienc

eAudienc

e

High

Low

No Audienc

eAudienc

e

High X = 3 X = 3

Low X = 7 X = 11

No Audienc

eAudienc

e

High X = 3 X = 3

Low X = 7 X = 11

X = 5 X = 7

No Audienc

eAudienc

e

High X = 3 X = 3 X = 3

Low X = 7 X = 11 X = 9

X = 5 X = 7

High Low

10

8

6

4

2

Mean

Num

ber

of E

rror

s

No Audience

Audience

Single Independent VariableDependent Variable

Single Factor - One-way ANOVA

Now:

Two Independent Variables Dependent Variable

Two Factor ANOVA

Factor B (class size)18-student

class24-student

class30-student

class

Factor A

(program)

Program I

Scores for n = 15

subjects taught by program I in a class

of 18

Scores for n = 15

subjects taught by program I in a class

of 24

Scores for n = 15

subjects taught by program I in a class

of 30

Program II

Scores for n = 15

subjects taught by program II in a class

of 18

Scores for n = 15

subjects taught by program II in a class

of 24

Scores for n = 15

subjects taught by program II in a class

of 30

Two-factor analysis of variance permits us to

test:1. Mean difference between the

2 teaching programs

2. Mean differences between the 3 class sizes

3. Combinations of teaching program and class size

Three separate hypothesis tests in

one ANOVAThree F-ratios:

F = Variance (differences) between sample means

Variance (differences) expected from sampling error

18-student class

24-student class

30-student class

Program I X = 85 X = 77 X = 75 X = 79

Program II X = 75 X = 67 X = 65 X = 69

X = 80 X = 72 X = 70

Factor A Teaching Program

€

Ho : μ A1 = μ A2

€

H1 : μ A1 ≠ μ A2

(Teaching program has no effect on math scores)(Teaching program has an effect on math scores)

F =

Variance (differences) between treatment means

for Factor AVariance (differences) expected from sampling

error

18-student class

24-student class

30-student class

Program I X = 85 X = 77 X = 75 X = 79

Program II X = 75 X = 67 X = 65 X = 69

X = 80 X = 72 X = 70

Factor B Class Size

€

Ho : μ B1 = μ B2 = μ B3

€

H1 :

(Class size has no effect on math scores)(Class size has an effect on math scores)

F =

Variance (differences) between treatment means

for Factor BVariance (differences) expected from sampling

error

At least one population mean is different

18-student class

24-student class

30-student class

Program I X = 85 X = 77 X = 75 X = 79

Program II X = 75 X = 67 X = 65 X = 69

X = 80 X = 72 X = 70

18-student class

24-student class

30-student class

Program I X = 80 X = 77 X = 80 X = 79

Program II X = 80 X = 67 X = 60 X = 69

X = 80 X = 72 X = 70

18

Students

9085807570

Mean

mat

h te

st s

core

s

24 Students

30 Students

6560

Program I

Program II

18 Students

24 Students

30 Students

9085807570

Mean

mat

h te

st s

core

s

6560

Program I

Program II

Thus two-factor AVOVA composed of 3 distinct

hypothesis tests:1. The main effect of A (called the A-

effect)

2. The main effect of B (called the B-

effect)

3. The interaction (called the AxB

interaction)

(a) Data showing a main effect for factor A, but no B-effect and no

interaction

B1 B2

A1 20 20 A1 mean = 20

A2 10 10 A2 mean = 10

B1 mean = 15

B2 mean = 15

No difference

10-point difference

(b) Data showing a main effects for both factor A and factor B, but

no interaction

B1 B2

A1 10 30 A1 mean = 20

A2 20 40 A2 mean = 30

B1 mean = 15

B2 mean = 35

20-point difference

10-point difference

(c) Data showing no main effect for either factor A or factor B,

but an interaction

B1 B2

A1 10 20 A1 mean = 15

A2 20 10 A2 mean = 15

B1 mean = 15

B2 mean = 15

No difference

No difference

Interaction

H0 : There is no interaction between factors A and B.

H1 : There is an interaction between factors A and B.

OR

H0 : The effect of factor A does not depend on the levels of factor B (and B does not depend on A).H1 : The effect of one factor does depend on the levels of

the other factor (and B does not depend on A).

F = Variance (differences) not explained by main effects

Variance (differences) expected from sampling error

Between Treatment Variability1. Treatment (factor A, B, or AxB

interaction)2. Individual differences (difference

of SS in each treatment condition)3. Experimental error

Variability Within Treatments (Chance)1. Individual differences2. Experimental error

F = Treatment effect + Individual Differences + Experimental Error

Individual Differences + Experimental Error

Treatment (Cell) Combinations

Factor BLevel B1 Level B2 Level B3

Factor A

Level A1

Treatment (cell)A1B1

Treatment (cell)A1B2

Treatment (cell)A1B3

Level A2

Treatment (cell)A2B1

Treatment (cell)A2B2

Treatment (cell)A2B3

Total Variability

Within-treatments Variability

Between-treatments Variability

Factor AVariability

Factor BVariability

InteractionVariability

Stage 2

Stage 1

Factor B

B1 B2 B3

Factor A

A1

16111

AB = 10SS = 20

771146

AB = 35SS = 26

31164

AB = 15SS = 18

A1 = 60

A2

03755

AB = 20SS = 28

00050

AB = 5SS = 20

02003

AB = 5SS = 8

A2 = 30

B1 = 30 B2 = 40 B3 = 20N = 30G = 90

∑x2 = 520

Factor B

B1 B2 B3

Factor A

A1

16111

AB = 10SS = 20

771146

AB = 35SS = 26

31164

AB = 15SS = 18

A1 = 60

A2

03755

AB = 20SS = 28

00050

AB = 5SS = 20

02003

AB = 5SS = 8

A2 = 30

B1 = 30 B2 = 40 B3 = 20N = 30G = 90

∑x2 = 520

€

X = 2

€

X = 7

€

X = 3

€

X = 4

€

X =1

€

X =1

2 - Factor Data Table w/ Cell Means & Marginal Means

Factor B

B1 B2 B3

Factor A

A1

16111

AB = 10SS = 20

771146

AB = 35SS = 26

31164

AB = 15SS = 18

A1 = 60

A2

03755

AB = 20SS = 28

00050

AB = 5SS = 20

02003

AB = 5SS = 8

A2 = 30

B1 = 30 B2 = 40 B3 = 20N = 30G = 90

∑x2 = 520

€

X = 2

€

X = 7

€

X = 3

€

X = 4

€

X =1

€

X =1

€

X A1 = 4

€

X A2 = 2

€

X B1 = 2

€

X B3 = 2

€

X B2 = 4

Breakdown of Variability Sources

and Formulas

InteractionSS found by subtraction

= 80

Stage 2

Stage 1

Total

€

SS = X 2∑ −G2

N = 250

Between Treatments

€

SS = AB2

n∑ −G

2

N =130

Within Treatments

€

SS = SSeach cell∑ =120

Factor A

€

SS = A2

bn∑ −G

2

N = 30

Factor B

€

SS = B2

an∑ −G

2

N = 20

Breakdown of Degrees of Freedom and

Formulas

Interactiondf = dfA x dfB

= 2

Stage 2

Stage 1

Totaldf = N - 1

= 29

Between Treatmentsdf = ab - 1

= 5

Within Treatmentsdf = N - ab

= 24

Factor Adf = a - 1

= 1

Factor Bdf = b - 1

= 2

Breakdown of both Degrees of Freedom and

SS Formulas

InteractionSS is found bysubtraction

df = (a - 1)(b - 1)

Stage 2

Stage 1

Total

df = N - 1

Between Treatments

df = ab - 1

Within Treatments

df = N - ab

Factor A

df = a - 1

Factor B

df = b - 1

€

SS = X 2∑ −G2

N

€

SS = AB2

n∑ −G

2

N

€

SS = SSeach cell∑

€

SS = A2

bn∑ −G

2

N

€

SS = B2

an∑ −G

2

N

0 1 2 3 4 5 6

0 1 2 3 4 5 6

4.26

3.40

Distribution of F-ratiosdf = 2.24

Distribution of F-ratiosdf = 1.24

Source SS df MS F p < .05

Between Treatments 130 5

Factor A (program)

30 1 30 F(1,24)= 6.0

√

Factor B (class size)

20 2 10 F(1,24)= 2.0

n.s.

AxB Interaction

80 2 40 F(1,24)= 8.0

√

Within Treatments 120 24 5

Total 250 29

Plot of the mean scores of Factor A and

B

7

6

5

4

3

2

1

0B1 B2 B3

Factor B

Mean

sco

re

A2

A1

Schacter (1968)Obesity and Eating Behavior

• Hypothesis: obese individuals do not respond to internal biological signals of hunger

Variables• 2 Independent Variables or Factors– Weight (obese vs. normal)– Fullness (full stomach vs. empty stomach)

• Dependent Variable– Number of crackers eaten by each subject

Factor B (fullness)

Empty Full

Factor A

(weight)

Normal n = 20 n = 20

Obese n = 20 n = 20

Factor B (fullness)

EmptyStomach

Full Stomach

Factor A

(weight)

Normal

n = 20X = 22

AB = 440SS = 1540

n = 20X = 15

AB = 300SS = 1270

A1 = 740

Obese

n = 20X = 17

AB = 340SS = 1320

n = 20X = 18

AB = 360SS = 1266

A2 = 700

B1 = 780 B2 = 660

G = 1440x2 = 31,836N = 80

Source SS df MS F p < .05

Between Treatments

520 3

Factor A (weight)

20 1 20 F(1,76)= 0.28

n.s.

Factor B (fullness)

180 1 180 F(1,76)= 2.54

n.s.

AxB Interaction

320 1 320 F(1,76)= 4.51

√

Within Treatments 5396 76 71

Total 5916 79

Plot of mean number of crackers eaten for

each group

Empty Full

Normal 22 15

Obese 17 18

Mean Number of Crackers Eaten

14151617181920212223

Mean Number of Crackers

Eaten

EmptyStomach

FullStomach

Obese

Normal

Graph of Schacter 1

Mean Number of Crackers Eaten as Function of Weight anf Fullness

0

5

10

15

20

25

Empty FullFullness

Mean Number of Crackers Eaten

NormalObese

Empty FullNormal 22 15Obese 17 18

Schacter Graph 2

Mean Number of Crackers Eaten as a Function of Weight and Fullness

0

5

10

15

20

25

Normal ObeseWeight

Mean Number of Crackers Eaten

EmptyFull

Normal ObeseEmpty 22 17Full 15 18

The means and standard deviations are presented in Table 1. The two-factor analysis of variance showed no significant main effect for the weight factor, F(1,76) = 0.28, p > .05; no significant main effect for the fullness factor, F(1,76) = 2.54, p > .05; but the interaction between weight and fullness was significant, F(1,76) = 4.51, p < .05.Mean number of crackers eaten in each

treatment conditionFullness

Empty Stomach

Full Stomach

WeightNormal

M = 22.0SD = 9.00

M = 15.0SD = 8.18

ObeseM = 17.0SD = 8.34

M = 18.0SD = 8.16

TABLE 1

Treatment 1

Treatment 2

3 53 41 32 51 Males 37 Females 77 75 86 95 9

T1 = 40 T2 = 60SS1 = 48 SS2 = 48

Data table for Treatment 1 vs. 2 & M

vs. FFactor A (Treatment)

Treatment 1 Treatment 2

Factor B

(Sex)

Males

33121

AB = 10SS = 4

54353

AB = 20SS = 4

B1 = 30

Females

77565

AB = 30SS = 4

77899

AB = 40SS = 4

B2 = 70

A1 = 40 A2 = 60

Assumptions for the 2-factor ANOVA:

(Independent Measures)1. Observations within each sample are

independent

2. Populations from which the samples are drawn are normal

3. Populations from which the samples are selected must have equal variances (homogeneity of variance)