Factorial ANOVA allows analysis of experiments with two or more independent variables. There are different types of factorial designs depending on whether data comes from different or same respondents. A two-way ANOVA breaks down variability in a dependent variable due to two independent variables and their interaction. An example shows sales data collected from respondents of different ethnicities and genders. The total variability is broken down into portions explained by each independent variable and their interaction.

1. FACTORIAL
ANOVA

2. Factorial
Design
When an experiment has two or more
independent variables
n-Way
ANOVA
When there are two or more independent
variables in the ANOVA
Typesof
Factorial
Design
Independent
Factorial Design
Repeated
Measures
Factorial Design
Mixed Design
Two or More
Number of IV Data from
Two or More
Two or More
Different
Respondents
Same
Respondents
Some from
same / other
from different

3. Example: Two way ANOVA
Independent Variable
Ethnicity Gender
Dependent Variable
Sales
1
2
3
4
5
Respondents

4. Breakup of Variability
through 2 Way ANOVA
SST (Total Variability)
SSM
(Variation Explained by Model)
SSE
(Unexplained
Variation)
SSA
(Variation
Explained by
Variable A)
SSB
(Variation
Explained by
Variable B)
SSAxB
(Variation
Explained by
the interaction
of A&B)

5. SST
SSM SSE
SSTR1 SSTR2 SS Interaction

6. Independent Variable
Ethnicity Gender
Dependent Variable
Sales
1
2
5
6
11
Respondents
Chinese
Indian
Male
Female
100
12
Chinese
Malay
Malay
Indian
Male
Female
Male
Female
110
3
4
Chinese Male
Female
105
Chinese 112
7
8
Malay
Malay
Male
Female
9 Indian
10 Indian
Male
Female
90
100
95
105
80
90
84
94

7. Independent
Variable
Ethnicity Gender
Dependent
Variable
Sales
1
2
5
6
11
Respondents
Chinese
Indian
Male
Female
100
12
Chinese
Malay
Malay
Indian
Male
Female
Male
Female
110
3
4
Chinese Male
Female
105
Chinese 112
7
8
Malay
Malay
Male
Female
9 Indian
10 Indian
Male
Female
90
100
95
105
80
90
84
94

8. SST = 100.8 x 11 =1108.8

9. SSM = 2{(82-97.08)2+(92-97.08)2 +(92.5-
97.08)2+(102.5-97.08)2 +(102.5-97.08)2 +(111-97.08)2 = 1063.1
SSF1 = 4(87-97.08)2+4(97.5-97.08)2 +4(106.75-97.08)2
SSF2 = 6(92.33-97.08)2+6(101.83-97.08)2
= 781.16
= 270.75
SSInteraction = SSM – SSF1 – SSF2 = 1.5

10. SSE = 8+8+12.5+12.5+12.5+2 =43

11. SST
SSM SSE
SSTR1 SSTR2 SS Interaction
1108.8
1063.1 43
781.16 270.75 1.5

16. Sales
Mean Count Grand Mean Variation
Malay
Male 526.22 9 541.15 2005.546944
Female 522.14 7 541.15 2528.900357
Indian
Male 510.00 9 541.15 8732.9025
Female 502.43 7 541.15 10495.44321
Chinees
Male 592.00 8 541.15 20685.78
Female 592.63 8 541.15 21197.405
Model Variation 65645.97802
Sales
Mean Count G Mean Variation
Malay 524.44 16 541.15 4466.69
Indian 506.69 16 541.15 18998.02
Chinees 592.31 16 541.15 41888.44
Variation Because of Ethnicity 65353.16
Sales
Mean Count G Mean Variation
Male 540.85 26 541.15 2.33
Female 541.50 22 541.15 2.75
Variation Because of Gender 5.09
Variation because of Interaction 287.71
SM = SA +SB + SSAxB 65645 = 65353 +5 + 288

17. What is interaction?
The combined effect of two or more variables
on dependent variables
Interaction