1. ANOVA II (Part 2)
Class 18

2. 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.

3. Ordinal and Disordinal Interactions

4. Eyeballing Interactions and Main Effects*
Dem
GOP * * X X X
North
South X
Dem
GOP * * * X X
North
South

5. Birth Order Main Effect:
Gender Main Effect:
Interaction: NO NO NO

6. Birth Order Main Effect:
Gender Main Effect:
Interaction:
YES NO NO

7. Birth Order Main Effect:
Gender Main Effect:
Interaction: NO
YES NO

8. Birth Order Main Effect:
Gender Main Effect:
Interaction:
YES
YES NO

9. Birth Order Main Effect:
Gender Main Effect:
Interaction: NO NO
YES

10. Birth Order Main Effect:
Gender Main Effect:
Interaction:
YES NO
YES

11. Birth Order Main Effect:
Gender Main Effect:
Interaction: NO
YES
YES

12. Birth Order Main Effect:
Gender Main Effect:
Interaction:
YES
YES
YES

13. To What Degree Does a Person Who Discloses Personal Problems Appear "Active"?
(3)
(2)
(2)
(3)
Birth Order Means
Main Effects are? Interaction is? Simple effects are?
Diff. betwen males & females, youngest/oldest
How birth order effect is moderated by gender
Youngest females v. Oldest females, for example
Note: Condition ns in parentheses

15. ANOVA: A MACHINE FOR SEPARATING TREATMENT EFFECTS (T) FROM ERROR (E)

16. ABS Matrix (Treatment Combinations)
Design and Notation for Two-Factor Design
Experimental Design
Factor A
Factor A: Birth Order
a1 = oldest a2 = youngest
Factor B: Gender
b1 = male b2 = female
Factor B a a n
b s = 3 s = 2
b s = 2 s = 3 5 5 n 5 5
Total n = 10
ABS Matrix (Treatment Combinations)
ab ab ab ab22
ABS ABS ABS211 ABS221
ABS ABS ABS ABS222
ABS ABS223
AB Matrix
Levels of Factor A
a a Marginal Sum
b AB AB B1
b AB AB B2
A A T
Levels of Factor B
Marginal sum

17. Conceptual Approach to Two Way ANOVA
 SS total = SS between groups + SS within groups
One-way ANOVA
SS between groups =
Factor A and its levels, e.g., birth order: level 1 = older
level 2 = younger
Two-way ANOVA
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 ratings
of help seeker are jointly affected by birth order and gender)

18. Distributions of All Four Conditions
Total Mean (4.32)

19. Gender Effect (collapsing across birth order)
Total Mean (4.32)

20. Birth Order Effect (collapsing across gender)
Total Mean (4.32)

21. Interaction: Gender * Birth Order
Total Mean (4.32)

22. Understanding Effects of Individual Treatment Groups
How much can the variance of any particular treatment group be explained by:
Factor A
Factor B
The interaction of Factors A and B
Quantification of AB Interaction
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)

24. Deviation of an Individual Score in Two Way ANOVA
ABSijk – T = (Ai – T) + (Bj – T) + (ABij – Aij – Bij + T) + (ABSijk – ABij)
Total Mean
Ind. score
Factor A Effect
Factor B Effect
Interaction AXB Effect
Error (w’n Effect)
(Birth Order)
(Gender)
(Birth * Gender)

25. Variance for All Factors
Degrees of Freedom in
2-Way ANOVA
Between Groups
Factor A
(Birth Order)
df A = a - 1
2 – 1 = 1
Factor B
(Gender)
df B = b – 1
2 – 1 = 1
Interaction Effect
Factor A X Factor B
(Birth X Gender)
dfA X B = (a –1) (b – 1)
(2-1) x (2-1) = 1
Error Effect
Subject Variance
df s/AB = n - ab
10 – (2 x 2) = 6
Total Effect
Variance for All Factors
df Total = n – 1
10 – 1 = 9

26. Conceptualizing Degrees of Freedom (df) in Factorial ANOVA
Birth Order
Gender Youngest Oldest Sum
9.00
Males
4.50
4.50
11.00
Females
5.50
5.50
Sum
10.00
10.00
20.00
df A = a - 1
df B = b – 1
dfA X B = (a –1) (b – 1)
NOTE: “Fictional sums” for demonstration.

27. Conceptualizing Degrees of Freedom(df)
For Conditions and Factors in Factorial ANOVA
Factor A
Factor B a1 a2 a3 Sum
b1 # # X #
b # # X #
b3 X X X X
Sum # # X T
# = free to vary; T has been computed
X = determined by #s
Once # are established, Xs are known
df Formulas:
Factor A =
(Σa – 1)
Factor B =
(Σb – 1)
A X B =
(Σa – 1) *
(Σb – 1)

28. Analysis of Variance Summary Table: Two Factor (Two Way) ANOVA
Source of Variation Sum of Squares df Mean Square F Ratio
(SS) (MS) A
SSA
a - 1
dfA
MSA
MSS/AB B
SSB
b - 1
SSb
dfb
MSB
A X B
SSA X B
(a - 1)(b - 1)
SSAB
dfA X B
MSA X B
Within
(S/AB)
SSS/A
ab (s- 1)
SSS/AB
dfS/AB
Total
SST
abs - 1
MSA =
MSB =
MSAB =
MSS/AB =

29. F Ratios for 2-Way ANOVA

30. Effect of Multi-Factorial Design on
Significance Levels:
Gender Main Effects
Mean
Men
Women
Sum of Sqrs.
Betw'n dt MS
Within df
MS Within F p
One Way
4.78
3.58
3.42 1
22.45 8
2.81
1.22
.30
Two Way
5.09 6
.85
4.03
.09

31. Source Sum of Squares df Mean Square F Sig. Source Sum of Squares df
ONEWAY ANOVA AND GENDER MAIN EFFECT
Source
Sum of Squares df
Mean Square F
Sig.
Gender
3.42 1
1.22
.34
Error
22.45 8
2.81
TWO-WAY ANOVA AND GENDER MAIN EFFECT
Source
Sum of Squares df
Mean Square F
Sig.
Gender
3.42 1
4.03
.09
Birth Order
16.02
18.87
.005
Interaction
3.75
4.42
.08
Error
5.09 6
0.85
Total 9
Oneway F: = Twoway F: =