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Implications of Interaction
1. Main effects, alone, will not fully describe the results. 2. Each factor (or IV) must be interpreted in terms of the factor(s) with which it interacts. 3. Analysis of findings, when an interaction is present, will focus on individual treatment means rather than on overall factor (IV) means.
4. Interaction indicates moderation.
Interactions are Non-Additive Relationships Between Factors
1. Additive: When presence of one factor changes the expression of another factor consistently, across all levels.
2. Non-Additive: When the presence of one factor changes the expression of another factor differently, at different levels.
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5
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North South
Democrat
Republican
0
5
10
15
20
25
30
35
North South
Democrat
Republican
Ordinal and Disordinal Interactions
0
10
20
30
40
50
B1 B2
A1
A2
0
10
20
30
40
50
B1 B2
A1
A2
XXXX Interaction
YYY Interaction
Ordinal and Disordinal Interactions
0
10
20
30
40
50
B1 B2
A1
A2
0
10
20
30
40
50
B1 B2
A1
A2
Ordinal Interaction
Disordinal Interaction
Birth Order and Gender Effects on Ratings of Help Seeker
0
1
2
3
4
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First Born Last Born
Rat
ing Male
Female
Birth Order Main Effect: NO
Gender Main Effect: NO
Interaction: NO
Birth Order and Gender Effects on Ratings of Help Seeker
0
1
2
3
4
5
First Born Last Born
Rat
ing Male
Female
Birth Order Main Effect: YES
Gender Main Effect: NO
Interaction: NO
Birth Order and Gender Effects on Ratings of Help Seeker
0
1
2
3
4
5
First Born Last Born
Rat
ing Male
Female
Birth Order Main Effect: NO
Gender Main Effect: YES
Interaction: N0
Birth Order and Gender Effects on Ratings of Help Seeker
0
1
2
3
4
5
First Born Last Born
Rat
ing Male
Female
Birth Order Main Effect: YES
Gender Main Effect: YES
Interaction: NO
Birth Order and Gender Effects on Ratings of Help Seeker
0
1
2
3
4
5
First Born Last Born
Rat
ing Male
Female
Birth Order Main Effect: NO
Gender Main Effect: NO
Interaction: YES
Birth Order and Gender Effects on Ratings of Help Seeker
123
456
First Born Last Born
Rat
ing Male
Female
Birth Order Main Effect: YES
Gender Main Effect: NO
Interaction: YES
Birth Order and Gender Effects on Ratings of Help Seeker
0
1
2
3
4
5
First Born Last Born
Rat
ing Male
Female
Birth Order Main Effect: NO
Gender Main Effect: YES
Interaction: YES
Birth Order and Gender Effects on Ratings of Help Seeker
0123456
First Born Last Born
Rat
ing Male
Female
Birth Order Main Effect: YES
Gender Main Effect: YES
Interaction: YES
Birth Order and Gender Effects on Ratings of Help Seeker
01234567
Male Female
Ratin
g First BornLast Born
Development of ANOVA Analytic Components
1. Individual scores Condition (cell) sums 2. Condition sums Condition means 3. Cond. means – ind. scores Deviations Deviations2
4. Deviations2 Sums of squares (SS between, SS within)
5. Sum Sqrs / df Mean squares (Between and Within) 6. MS Between F Ratio MS Within F (X, Y df) Probability of null (p) p Accept null, or accept alt.
Birth Order and Ratings of “Activity” Deviation Scores
AS Total Between Within (AS – T) = (A – T) + (AS –A)
1.33 (-2.97) = (-1.17) + (-1.80) 2.00 (-2.30) = (-1.17) + (-1.13) 3.33 (-0.97) = (-1.17) + ( 0.20) 4.33 (0.03) = (-1.17) + ( 1.20) 4.67 (0.37) = (-1.17) + ( 1.54)
Level a1: Oldest Child
Level a2: Youngest Child4.33 (0.03) = (1.17) + (-1.14) 5.00 (0.07) = (1.17) + (-0.47) 5.33 (1.03) = (1.17) + (-0.14) 5.67 (1.37) = (1.17) + ( 0.20) 7.00 (2.70) = (1.17) + ( 1.53)
Sum: (0) = (0) + (0)
Mean scores: Oldest = 3.13 Youngest = 5.47 Total = 4.30
Sum of Squared Deviations
Total Sum of Squares = Sum of Squared between-group deviations + Sum of Squared within-group deviations
SSTotal = SSBetween + SSWithin
Computing Sums of Squares from Deviation Scores
Birth Order and Activity Ratings (continued)
SS = Sum of squared diffs, AKA “sum of squares”
SST = Sum of squares., total (all subjects)
SSA = Sum of squares, between groups (treatment)
SSs/A = Sum of squares, within groups (error)
SST = (2.97)2 + (2.30)2 + … + (1.37)2 + (2.70)2 = 25.88
SSA = (-1.17)2 + (-1.17)2 + … + (1.17)2 + (1.17)2 = 13.61
SSs/A = (-1.80)2 + (-1.13)2 + … + (0.20)2 + (1.53)2 = 12.27Total (SSA + SSs/A) = 25.88
Variance
Code Calculation Meaning
Mean Square Between Groups
MSA SSA
dfA
Between groups variance
Mean Square Within Groups
MSS/A SSS/A
dfS/A
Within groups variance
Variance
Code Calculation Data Result
Mean Square Between Groups
MSA SSA
dfA
13.61 1
13.61
Mean Square Within Groups
MSS/A SSS/A
dfS/A
12.278
1.53
Mean Squares Calculations
F Ratio Computation
F = 13.611.51
= 8.78
F = MSA = Between Group Variance
MSS/A Within Group Variance
Conceptual Approach to Two Way ANOVA
SS total = SS between groups + SS within groups
Oneway ANOVA
SS between groups =
Factor A and its levels (e.g., birth order; older/younger) Twoway ANOVA
SS between groups =
Factor A and its levels (e.g., birth order; older/younger)
XXXX
YYYY
Conceptual Approach to Two Way ANOVA
SS total = SS between groups + SS within groups
Oneway ANOVA
SS between groups =
Factor A and its levels (e.g., birth order; older/younger) Twoway ANOVA
SS between groups =
Factor A and its levels (e.g., birth order; older/younger)
Factor B and its levels (e.g., gender; male / female)
The interaction between Factors A and B (e.g., how ratingsof help seeker are jointly affected by birth order and gender)
Total Mean (4.32)
Distributions of All Four ConditionsBirth Order and Gender Effects on Ratings of
Help Seeker
01234567
Male Female
Ratin
g First BornLast Born
Total Mean (4.32)
Gender Effect (collapsing across birth order)Birth Order and Gender Effects on Ratings of
Help Seeker
01234567
Male Female
Ratin
g First BornLast Born
Total Mean (4.32)
Birth Order Effect (collapsing across gender)Birth Order and Gender Effects on Ratings of
Help Seeker
01234567
Male Female
Ratin
g First BornLast Born
Understanding Effects of Individual Treatment Groups
How much can the variance of any particular treatment group be explained by:Factor AFactor B
The interaction of Factors A and B
Quantification of AB Effects
AB - T = (A effect) + (B effect) + ?????
AB - T = (A - T) + (B - T) + (AB - A - B + T)
(AB - A - B + T) = ??? AKA "????"
(AB - T) - (? - T) - (? - T) = Interaction
Error Term in Two-Way ANOVA
Error = (ABS - AB)
Understanding Effects of Individual Treatment Groups
How much can the variance of any particular treatment group be explained by:Factor AFactor B
The interaction of Factors A and B
Quantification of AB Effects
AB - T = (A effect) + (B effect) + (A x B Interaction)
AB - T = (A - T) + (B - T) + (AB - A - B + T)
(AB - A - B + T) = Interaction AKA "residual"
(AB - T) - (A - T) - (B - T) = Interaction
Error Term in Two-Way ANOVA
Error = (ABS - AB)
Deviation of an Individual Score in Two Way ANOVA
ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bj + T) + (ABSijk – ABij)
Ind. score
Total Mean
??? Effect ??? Effect ??? Effect ??? (w’n Effect)
Deviation of an Individual Score in Two Way ANOVA
ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bj + T) + (ABSijk – ABij)
Ind. score
Total Mean
Factor A Effect
Factor B Effect
Interaction AXB Effect
Error (w’n Effect)
Degrees of Freedom in 2-Way ANOVA
Between Groups
Factor A df A = a - 1 2 – 1 = 1
Factor B df B = b – 1 2 – 1 = 1
Interaction Effect
Factor A X Factor B dfA X B = (a –1) (b – 1) (2-1) x (2-1) = 1
Error Effect
Subject Variance df s/AB = ab(s – 1)
df s/AB = n - ab 20 – (2 x 2) = 16
Total Effect
Variance for All Factors df Total = abs – 1
df Total = n – 1 20 – 1 = 19
Conceptualizing Degrees of Freedom (df) in Factorial ANOVA
Factor A
Factor B a1 a2 Sum
b1 # X B1
b2 X X X
Sum A1 X T
Factor A = Birth Order
Factor B = Gender # = Known quantity
Conceptualizing Degrees of Freedom (df) in Factorial ANOVA
Birth Order
Gender Youngest Oldest Sum
Males
Sum
Females
4.50
5.50
9.00
11.00
4.50
5.50
20.0010.0010.00
NOTE: “Fictional sums” for demonstration.
Conceptualizing Degrees of Freedom (df) in Factorial ANOVA
Factor A
Factor B a1 a2 a3 Sum
b1 # # X B1
b2 # # X B2
b3 X X X X
Sum A1 A2 X T
A, B, T, # = free to vary; X = determined by #s
Once # are established, Xs are known
Analysis of Variance Summary Table:
Two Factor (Two Way) ANOVA
A SSA a - 1 SSA
dfA
MSA
MSS/AB
B SSB b - 1 SSb
dfb
MSB
MSS/AB
A X B SSA X B (a - 1)(b - 1) SSAB
dfA X B
MSA X B
MSS/AB
Within(S/AB)
SSS/A ab (s- 1) SSS/AB
dfS/AB
Total SST abs - 1
Source of Variation Sum of Squares df Mean Square F Ratio
(SS) (MS)
Effect of Multi-Factorial Design on Significance Levels
MeanMen
MeanWomen
Sum of Sqrs.
Betw'n
dt Betw'n
MSBetw'n
Sum of Sqrs. Within
df Within
MS Within
F p
One Way
4.78 3.58 3.42 1 3.42 22.45 8 2.81 1.22 .30
Two Way
4.78 3.58 3.42 1 3.42 5.09 6 .85 4.03 .09
ONEWAY ANOVA AND GENDER MAIN EFFECT
Source Sum of Squares
df Mean Square
F Sig.
Gender 3.42 1 3.42 1.22 .34
Error 22.45 8 2.81
Source Sum of Squares
df Mean Square
F Sig.
Gender 3.42 1 3.42 4.03 .09
Birth Order 16.02 1 16.02 18.87 .005
Interaction 3.75 1 3.75 4.42 .08
Error 5.09 6 0.85
Total 9
TWOWAY ANOVA AND GENDER MAIN EFFECT
Oneway F: 3.42 = 1.22 Twoway F: 3.42 = 4.42 2.81 .85