Two way anova balanced design

1. Factorial Analysis
Two Way ANOVA (Balanced Design)
Presented by : Eng. Waleed Alzaghal
You Tube Channel : Waleed Alzaghal

2. A researcher wanted to study the effect of applying two new teaching
techniques on the achievement of students in three classes (Math, Science and
English). He randomly assigned 10 students in each of the 6 groups, and after
finishing the sessions, he recorded the students’ scores. The details are shown in
below table. State if there is any significant effect of the teaching techniques and
classes on the student achievements. (alpha = 0.05)

3. 1) State the Hypotheses
Ways : Ho : µ (Way A) = µ (Way B)
Ha : µ ( Way A) ≠ µ (Way B)
Classes : Ho : µ (Math) = µ (Science) = µ (English)
Ha : At least one mean is not equal to other
Interaction : Ho : there is no interaction between the teaching ways and the classes.
Ha : There is an interaction between the teaching ways and the classes.

4. 2) Calculate Degree of Freedom (DF)
DF way = number of ways – 1 = (a – 1) = 2 – 1 = 1
DF class = number of classes - 1 = (b – 1) = 3 – 1 = 2
DF way x class = ( a -1 ) ( b – 1 ) = (1) (2) = 2
DF Error = (a)(b) (r – 1) = (2)(3)(10 – 1) = (6)(9) = 54

5. Source of
Variation
Degree of Freedom
(DF)
Sum of Squares
(SS)
Mean Squares
(MS)
F stat. F critical
Way 1
Class 2
Way x Class 2
Error 54
Total 59

6. 3) Calculate the Correction Factor (CF)
CF =
∑ 𝑥 2
𝑁
=
𝑇2
60
=
7+8+5+ …….+5+6 2
60
CF =
417 2
60
=
173,889
60
CF = 2,898.15‬

7. 4) Calculate SS for Ways of Teaching ( SS way )
SS way =
∑ ∑ 𝑎ì
2
(𝑏)(𝑛)
- CF SS way =
2192
+ 1982
3 (10)
- 2,898.15
SS way =
47,961 + 39,204‬
30
- 2,898.15 SS way =
87,165‬
30
- 2,898.15
SS way = 2,905.5 - 2,898.15 SS way = 7.35

8. 5) Calculate SS for Class ( SS class )
SS class =
∑ ∑ 𝑏𝑖 2
(𝑎)(𝑛)
- CF SS class =
1372
+1482 +1322
2 (10)
- 2,898.15
SS class =
18,769+ 21,904+17,424‬
20
- 2,898.15 SS class =
58,097
20
- 2,898.15
SS class = 2,904.85 - 2,898.15 SS class = 6.7‬

9. 6) Calculate SS for Interaction (SS way x class )
SS (way x class ) = SS (subtotals way x class) – SS way – SS class
SS (subtotals way x class) =
∑ ∑ 𝑎𝑖 𝑏𝑖 2
(𝑛)
- CF
SS (subtotals way x class) =
692+832+672+682+652+652
10
– 2,898.15 = 23.15
SS (way x class ) = 23.15 – 7.35 – 6.7
SS (ways x class ) = 9.1

10. 7) Calculate Sum Square Total (SS total)
SS total = ∑ y2 – CF
∑y2 = 72 + 82 + 52 + 92 + ……. + 52 + 62
∑y2 = 3003
SS total = 3003 - 2,898.15
SS total = 104.85‬

11. 8) Calculate Sum Square Error ( SS Error)
SS total = SS way +SS class + SS (way x class ) + SS Error
SS Error = SS total - SS way - SS class - SS (way x class )
SS Error = 104.85 – 7.35 – 6.7 – 9.1
SS Error = 81.7

12. 9) Calculate Mean Squares ( MS)
MS way = SS way / DF way = 7.35 / 1 = 7.35
MS class = SS class / DF class = 6.7 / 2 = 3.35
MS way x class = SS way x class / DF way x class = 9.1 / 2 = 4.55
MS Error = SS Error / DF Error = 81.7 / 54 = 81.7 / 54 = 1.51
Source of
Variation
Degree of Freedom
(DF)
Sum of Squares
(SS)
Mean Squares
(MS)
F stat. F critical
Way 1 7.35
Class 2 6.7
Way x Class 2 9.1
Error 54 81.7
Total 59 104.85

13. 10) Calculate F stat.
F way = MS way / MS Error = 7.35 / 1.51 = 4.87
F class = MS class / MS Error = 3.35 / 1.51 = 2.22
F ways x class = MS ways x class / MS Error = 4.55 / 1.51 = 3.01
Source of
Variation
Degree of Freedom
(DF)
Sum of Squares
(SS)
Mean Squares
(MS)
F stat. F critical
Way 1 7.35 7.35
Class 2 6.7 3.35
Way x Class 2 9.1 4.55
Error 54 81.7 1.51
Total 59 104.85

14. 11) Find Critical Values of fcrit.
F way = F (0.05, 1 , 54)
F class = F (0.05, 2 , 54)
F way x class = F (0.05, 2 , 54)
Source of
Variation
Degree of Freedom
(DF)
Sum of Squares
(SS)
Mean Squares
(MS)
F stat. F critical
Way 1 7.35 7.35 4.87
Class 2 6.7 3.35 2.22
Way x Class 2 9.1 4.55 3.01
Error 54 81.7 1.51
Total 59 104.85

15. F way = F (0.05, 1 , 54) = 3.92
F class = F (0.05, 2 , 54) = 3.17

16. 12) Complete the ANOVA table
Source of
Variation
Degree of Freedom
(DF)
Sum of Squares
(SS)
Mean Squares
(MS)
F stat. F critical
Way 1 7.35 7.35 4.87*
3.92
Class 2 6.7 3.35 2.22 3.17
Way x Class 2 9.1 4.55 3.01 3.17
Error 54 81.7 1.51
Total 59 104.85

17. 13) Draw conclusion
Way : As f calculated (4.87) is bigger than f critical (3.92) at 0.05 level of significance,
we reject Ho which means that the means are not same.
Class : As f calculated (2.22) is less than f critical (3.17) at 0.05 level of significance,
we fail to reject Ho which means no effect for classes.
Interaction (Ways x Class) : As the f calculated (3.01) is less than f critical at 0.05 level
of significance, we fail to reject Ho which means there is no interaction effect.
Source of
Variation
Degree of Freedom
(DF)
Sum of Squares
(SS)
Mean Squares
(MS)
F stat. F critical
Way 1 7.35 7.35 4.87*
3.92
Class 2 6.7 3.35 2.22 3.17
Way x Class 2 9.1 4.55 3.01 3.17
Error 54 81.7 1.51
Total 59 104.85