Chapter Fourteen The Two-Way Analysis of Variance

Chapter Fourteen

The Two-Way Analysis of Variance

Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 2

New Statistical Notation

1. The two-way ANOVA is the parametric inferential procedure performed when an experiment contains two independent variables

2. When both factors involve independent samples, we perform the two-way, between-subjects ANOVA

3. When both factors involve related samples, we perform the two-way, within-subjects ANOVA

4. When one factor is tested using independent samples and the other factor using related samples, we perform the two-way, mixed-design ANOVA


Understanding the Two-Way ANOVA


Factorial Designs

• When we combine all levels of one factor with all levels of the other factor, this produces a complete factorial design

• When all levels of the two factors are not combined, this produces an incomplete factorial design


Overview of the Two-Way Between-Subjects ANOVA


Assumptions of the Two-WayBetween-Subjects ANOVA

1.Each cell contains an independent sample

2.The dependent variable measures interval or ratio scores that are approximately normally distributed

3.The populations have homogenous variance


Main Effects

• The main effect of a factor is the effect that changing the levels of that factor has on dependent variable scores while ignoring all other factors in the study

• We collapse across a factor. Collapsing across a factor means averaging together all scores from all levels of that factor.


Interaction Effects

• The interaction of two factors is called a two-way interaction

• The two-way interaction effect is the influence on scores that results from combining the levels of factor A with the levels of factor B

• When you look for the interaction effect, you compare the cell means. When you look for a main effect, you compare the level means.


Interaction Effect

• An interaction effect is present when the relationship between one factor and the dependent scores change with, or depends on, the level of the other factor that is present

• A two-way interaction effect indicates that the influence that one factor has on scores depends on which level of the other factor is present


Summary Table of a Two-way ANOVA


Computing the Two-Way ANOVA


N

XXSS

2tot2

tottot

)(

Computing Fobt

1.Compute the total sum of squares (SStot)


N

X

nSS

2tot

2

A

)(

columntheinscoresof

)columntheinscoresofsum(

Computing Fobt

2.Compute the sum of squares between groups for column factor A (SSA)


N

X

nSS

2tot

2

B

)(

rowtheinscoresof

)rowtheinscoresofsum(

Computing Fobt

3.Compute the sum of squares between groups for row factor B (SSB)


N

X

nSS

2tot

2

bn

)(

celltheinscoresof

)celltheinscoresofsum(

Computing Fobt

4.Compute the overall sum of squares between groups (SSbn)


BAbnBA x SSSSSSSS

Computing Fobt

5.Compute the sum of squares between groups for the interaction (SSA x B)


bntotwn SSSSSS

Computing Fobt

6.Compute the sum of squares within groups (SSwn)


Computing Fobt

7. Compute the degrees of freedom1. The degrees of freedom between groups for

factor A is kA - 1

2. The degrees of freedom between groups for factor B is kB - 1

3. The degrees of freedom between groups for the interaction is (dfA)(dfB)

4. The degrees of freedom within groups equals N – kAxB

5. The degrees of freedom total equals N - 1


8. Compute the mean squares

1.

2.

3.

B

BB df

SSMS

A

AA df

SSMS

BA x

BA x BA x df

SSMS

wn

wnwn df

SSMS

Computing Fobt

4.


9. Compute Fobt

1.

2.

3.

wn

AA MS

MSF

wn

BB MS

MSF

wn

BA x BA x MS

MSF

Computing Fobt


Interpreting the Two-Way Experiment


Graphing the Effects

• To graph main effects, plot the dependent variable along the Y axis and the levels of a factor along the X axis

• To graph interaction effects, plot the dependent variable along the Y axis. Place the levels of one factor along the X axis, and show the second factor by drawing a separate line connecting the means for each level of that factor.


Graphs Showing Main Effects


Graph of Cell Means, Showing the Interaction


Two Graphs Showing When an Interaction Is and Is Not Present


Performing Post Hoc Comparisons

• Perform post hoc comparisons on the level means from significant main effect using Tukey’s HSD

• Perform Tukey’s HSD for the interaction using only unconfounded comparisons– A confounded comparison occurs when

two cells differ along more than one factor– An unconfounded comparison occurs

when two cells differ along only one factor


tot

2 effecttheforgroupsbetweensquaresofsum

SS

Describing the Effect Size

• Compute eta squared to describe effect size. That is, the proportion of variance in dependent scores that is accounted for by a manipulation.


Xtn

MSXt

n

MS

)()( crit

wncrit

wn

Confidence Interval

• The computational formula for the confidence interval for a single is

Factor A

Group A1 Group A2 Group A3

Factor B

Group B1

14 14 10 13 11 15

13 10 12 11 14 13

Group B2

17 18 10 12 14 12

19 16 11 10 14 15

Example

• Using the following data set, conduct a two-way ANOVA. Use = 0.05


500.14824

3184362

)( 22tot2

tottot

N

XXSS

750.6424

318

8

108

8

89

8

121

)(

columntheinscoresof

)columntheinscoresofsum(

2222

2tot

2

A

N

X

nSS

Example


500.1324

318

12

168

12

150

)(

rowtheinscoresof

)rowtheinscoresofsum(

222

2tot

2

B

N

X

nSS

Example


500.111

24

318

4

55

4

43

4

70

4

53

4

46

4

51

)(

celltheinscoresof

)celltheinscoresofsum(

2222222

2tot

2

bn

N

X

nSS

Example


25.33500.13750.64500.111BAbnBA x

SSSSSSSS

00.37500.111500.148bntotwn

SSSSSS

Example


Example

• dfA = 3 - 1 = 2

• dfB = 2 - 1 = 1

• dfA X B = (2)(1) = 2

• dfwn = 24 - 6 = 18


500.131

500.13

B

BB

df

SSMS

375.322

750.64

A

AA

df

SSMS

625.162

25.33

BA x

BA x BA x

df

SSMS

056.218

000.37

wn

wnwn

df

SSMS

Example


747.15056.2

375.32

wn

AA

MS

MSF

566.6056.2

500.13

wn

BB

MS

MSF

086.8056.2

625.16

wn

BA x BA x

MS

MSF

Example


Example

• Fobt for 2 and 18 degrees of freedom is 3.55

• Fobt for 1 and 18 degrees of freedom is 4.41

• The main effect for Factor A is significant

• The main effect for Factor B is significant

• The interaction term (A X B) is significant

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Chapter Fourteen The Two-Way Analysis of Variance