1. Factorial Analysis of variance

2. Factorial Analysis of variance

3. Orthogonal
Is the property that every level of one factor is present in the experiment in combination with every level of the other factor

4. Effects of season (winter/spring and summer/fall) and adult density (8,15, 30 y 45 animales/225 cm2) on egg production of Siphonaria diemenensis

5. Spring-30
Summer- 15
Summer- 45
Summer- 30
Spring-8
Spring-15
Summer- 15
Summer- 8
Spring-15
Summer- 8
Spring-45
Summer- 45
Spring-45
Summer- 45
Spring-30
Summer- 30
Spring-8
Spring-45
Spring-15
Summer- 15
Spring-8
Summer- 8
Summer- 30
Spring-30

6. Two single-factor design
animales/225 cm2
Winter/Spring
Summer Fall
Density treatment 8 15 30 45

7. Null hypothesis No effects of treatment A No effects of treatment B
No effects of the interaction

8. Data
Density 8 15 30 45
Spring
Summer
2.875
2.125
2.600
0.867
2.230
1.267
1.400
0.711
2.625
1.50
1.866
0.933
1.466
0.467
1.022
0.356
1.750
1.875
2.066
1.733
1.00
0.700
1.177

9. Consider the entire analysis as though it were a single factor factorial experiment with ab experimental treatments
Source
Sum of Squares
d f AB
Σia Σjb Σjn (Xij-X)2
ab-1
Residual
Σia Σjb Σjn (Xijk-Xij)2
ab(n-1)
Total
Σia Σjb Σkn(Xijk-X)2
abn-1

10. Now start again and ignore any differences among the data that might be due to factor B. Equivalent to a single-factor analysis of variance of means of the levels of factor A (with a treatments each replicated bn times)
Source
Sum of Squares
d f A
Σia Σjb Σjn (Xi-X)2
a-1
Residual
Σia Σjb Σjn (Xijk-Xij)2
a(bn-1)
Total
Σia Σjb Σkn(Xijk-X)2
abn-1
A symmetrical argument can be made to analyze the data as a single factor B

11. The remaining differences can be identified empirically as
Not all the differences among means of the ab combinations of treatments have been accounted for by the two single factor analyses
The remaining differences can be identified empirically as
SS among all treatments- SS factor A- SS factor B=
Σia Σjb(Xij-Xi-Xj+X)2

12. Expected mean squares for test of null hypothesis for two factorial analysis (A fixed, B fixed)
Source df
Sum of squares
Mean square
Expected mean square
F ratio
Factor A
a-1
Factor B
b-1
Interaction AxB
(a-1)(b-1)
Within groups
ab (n-1)
Total
abn-1

13. Source
Type III Sum of Squares df
Mean Square F
Sig.
DENSITY
5.284 3
1.761
9.669
.001
SEASON
3.250 1
17.842
DENSITY * SEASON
.165
.055
.301
.824
Error
2.915 16
.182
Total
11.614 23

15. Expected mean squares for test of null hypothesis for two factorial analysis (A random, B random)
Source df
Sum of squares
Mean square
Expected mean square
F ratio
Factor A
a-1
Factor B
b-1
Interaction AxB
(a-1)(b-1)
Within groups
ab (n-1)
Total
abn-1