View
229
Download
0
Category
Tags:
Preview:
Citation preview
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 3
Understanding the Two-Way ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 4
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 5
Overview of the Two-Way Between-Subjects ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 6
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 7
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.
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 8
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.
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 9
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 10
Summary Table of a Two-way ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 11
Computing the Two-Way ANOVA
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 12
N
XXSS
2tot2
tottot
)(
Computing Fobt
1.Compute the total sum of squares (SStot)
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 13
N
X
nSS
2tot
2
A
)(
columntheinscoresof
)columntheinscoresofsum(
Computing Fobt
2.Compute the sum of squares between groups for column factor A (SSA)
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 14
N
X
nSS
2tot
2
B
)(
rowtheinscoresof
)rowtheinscoresofsum(
Computing Fobt
3.Compute the sum of squares between groups for row factor B (SSB)
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 15
N
X
nSS
2tot
2
bn
)(
celltheinscoresof
)celltheinscoresofsum(
Computing Fobt
4.Compute the overall sum of squares between groups (SSbn)
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 16
BAbnBA x SSSSSSSS
Computing Fobt
5.Compute the sum of squares between groups for the interaction (SSA x B)
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 17
bntotwn SSSSSS
Computing Fobt
6.Compute the sum of squares within groups (SSwn)
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 18
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 19
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.
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 20
9. Compute Fobt
1.
2.
3.
wn
AA MS
MSF
wn
BB MS
MSF
wn
BA x BA x MS
MSF
Computing Fobt
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 21
Interpreting the Two-Way Experiment
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 22
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.
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 23
Graphs Showing Main Effects
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 24
Graph of Cell Means, Showing the Interaction
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 25
Two Graphs Showing When an Interaction Is and Is Not Present
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 26
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 27
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.
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 28
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 30
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 31
500.1324
318
12
168
12
150
)(
rowtheinscoresof
)rowtheinscoresofsum(
222
2tot
2
B
N
X
nSS
Example
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 32
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 33
25.33500.13750.64500.111BAbnBA x
SSSSSSSS
00.37500.111500.148bntotwn
SSSSSS
Example
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 34
Example
• dfA = 3 - 1 = 2
• dfB = 2 - 1 = 1
• dfA X B = (2)(1) = 2
• dfwn = 24 - 6 = 18
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 35
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 36
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
Copyright © Houghton Mifflin Company. All rights reserved. Chapter 14 - 37
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
Recommended