1. Two way ANOVA
Dr. Anshul Singh Thapa

2. Two way ANOVA
Till now we have learned how the effect of one independent or one type of treatment was studied on single dependent variable.
Now when we want to study the effect of two independent variables on a single dependent variable. Further suppose our aim is to study the independent effects of the independent variables as well as their combined or joint effect on the dependent variable. For example:
The independent effect of one training on selected behavior
The independent effect of other training on selected behavior
The interactional effect of first and second training i.e., 1 x 2 on selected behavior.
In two way analysis of variance, usually the two independent variables are taken simultaneously. It has two main effects and one interactional or joint effect. In such situation we have to use analysis of variance in two way i.e., vertically as well as horizontally.

3. Level of Socio Economic Status
For example
Suppose we are interested in studying the intelligence i.e., level of boys and girls studying in M.P.Ed first in relation to their level of socio economic status in such conditions we have 3 x 2 design
Gender
Groups
Level of Socio Economic Status
High
Average
Low
Total
Boys
MHB
MAB
MLB MB
Girls
MHG
MAG
MLG MG MH MA ML M
In this table:
M is mean of intelligence scores.
MHB, MAB and MLB : Mean of intelligence scores of boys belonging to different levels of SES
MHG, MAG and MLG : Mean of intelligence scores of girls belonging to different levels of SES
MH, MA and ML : Mean of intelligence scores of students belonging to different levels of SES
MB, MG : Mean of intelligence scores of boys and girls

4. Games X1
Judo X2
Badminton X3
Tennis X4
Squash
Gender
Male 17 24 15 12 20 18 28 21 23 22
Female 13 14 16 10 83 70 91 62

5. Steps in calculation of ‘F’ Two way ANOVA
Step 1 – Correction factor Step 2 – Sum of Square of Total SST Step 3 – Sum of Square between the Group SSA Step 4 – Sum of Square between the groups (Gender) SSAS Step 5 – Sum of Square among the groups (Game) SSAG Step 6 – Sum of Square between the interaction (Gender x Game) SSAS x AG Step 7 – Sum of Square within the Group SSW Step 8 – Mean Sum of Squares between the groups MSSA Step 9 – Mean Sum of Squares between the groups (Gender) MSSAS Step 10 – Mean Sum of Squares between the groups (Game) MSSAG Step 11 – Mean Sum of Squares between the groups (Gender X Game) MSSAS x AG Step 12 – Mean Sum of Squares within the groups MSSW Step 13 – Summary of ANOVA

6. Games X1
Judo X2
Badminton X3
Tennis X4
Squash
Gender
Male 17 24 15 12 20 18 28 21 23 22 88
117 87 78
R1= 370
Female 13 14 16 10 83 70 91 62
R1= 306
C = 171
C = 187
C = 178
C = 140
G = 676

7. Step 1 – Calculation of Correction factor
Calculation of Correction Factor (CF) G2 N CF =

8. Games X1
Judo X2
Badminton X3
Tennis X4
Squash
Gender
Male 17 24 15 12 20 18 28 21 23 22 88
117 87 78
Female 13 14 16 10 83 70 91 62

9. Games X1
Judo
X12 X2
Badminton
X22 X3
Tennis
X32 X4
Squash
X42
Gender
Male 17
289 24
576 15
225 12
144 20
400 18
324 28
784 21
441 23
529 22
484 88
117 87 78
Female 13
169 14
196 16
256 10
100 83
3073 70
3783 91
3322 62
2014

10. Games X1
Judo
X12 X2
Badminton
X22 X3
Tennis
X32 X4
Squash
X42
Gender
Male 17
289 24
576 15
225 12
144 20
400 18
324 28
784 21
441 23
529 22
484 88
117 87 78
Female 13
169 14
196 16
256 10
100 83
3073 70
3783 91
3322 62
2014
X12 + X22 + X32 + X42 = 12192

11. Step 2 – Calculation of Sum of Square of Total SST
Calculation of Raw Some of Square (RSS)
(RSS) = X12 + X22 + X32 + X42 = 12192
SST = = 12192
= – CF
= –
= 767.6

12. Games X1
Judo X2
Badminton X3
Tennis X4
Squash
Gender
Male 17 24 15 12 20 18 28 21 23 22
(88)2 /5
(117)2/5
(87)2/5
(78)2/5
Female 13 14 16 10
(83)2/5
(70)2/5
(91)2/5
(62)2/5

13. Step 3 – Calculation of Sum of Square of Total SSA
=7744/ / / / / / / /5 –
= –
= –
= 375.6

14. Games X1
Judo X2
Badminton X3
Tennis X4
Squash
Gender
Male 17 24 15 12
R = 370 20 18 28 21 23 22 88
117 87 78
Female 13 14
R = 306 16 10 83 70 91 62

15. Step 4 – Calculation of Sum of Square between the groups (Gender) SSAS
SSAS = (ΣR1)2/ Rn + (ΣR2)2/ Rn - CF
= (370)2/ 20 + (306)2/20 –
= –
= – =

16. Games X1
Judo X2
Badminton X3
Tennis X4
Squash
Gender
Male 17 24 15 12 20 18 28 21 23 22 88
117 87 78
Female 13 14 16 10 83 70 91 62

17. Games X1
Judo X2
Badminton X3
Tennis X4
Squash
Gender
Male 17 24 15 12 20 18 28 21 23 22 88
117 87 78
Female 13 14 16 10 83 70 91 62

18. Games X1
Judo X2
Badminton X3
Tennis X4
Squash
Gender
Male 17 24 15 12 20 18 28 21 23 22 88
117 87 78
Female 13 14 16 10 83 70 91 62
171
187
178
140

19. Step 5 – Calculation of Sum of Square among the groups (Game) SSAG
SSAG = (ΣC1)2/ Cn + (ΣC2)2/ Cn + (ΣC3)2/ Cn + (ΣC4)2/Cn - CF
= (171)2/10 + (187)2/10 + (178)2/10 + (140)2/10 –
= –
= –
= 125

20. Calculation of Sum of Square within the Group SSW SSW = SST – SSA
Step 6 – Calculation of Sum of Square between the interaction (Gender x Game) SSAS x AG
Calculation of Sum of Square between the interaction (Gender x Game) SSAS x AG SSAS x AG = SSA– SSAS – SSAG = – – 125 =
Step 7 – Calculation of Sum of Square within the Group SSW
Calculation of Sum of Square within the Group SSW
SSW = SST – SSA
= – 375.6
= 392
Step 8 – Calculation of Mean Sum of Square among the groups (MSSA)
Calculation of Mean Sum of Square among the groups (MSSA)
MSSA = SSA/ k – 1
= 375.6/ 7
= 53.67
Step 9 – Calculation of Mean Sum of Square between the groups (Gender) (MSSAS)
Calculation of Mean Sum of Square between the groups (Gender) (MSSAS)
MSSAS = SSAS/ k1 – 1
= / 1 =

21. Step 12 – Calculation of Mean Sum of Square within the group MSSW
Step 10 – Calculation of Mean Sum of Square among the group (Games) MSSAG
Calculation of Mean Sum of Square among the groups (Games) (MSSAG)
MSSAG = SSAG/ k2 – 1
= 125/3
= 41.67
Step 11 – Calculation of Mean Sum of Square between the group (Gender X Game) MSSAG X AS
Calculation of Mean Sum of Square between the group (Gender X Game) (MSSAG X AS)
MSSAG X AS = SSAG X AS/ (k1 – 1) (k2 – 1)
= / 3
= 49.40
Step 12 – Calculation of Mean Sum of Square within the group MSSW
Calculation of Mean Sum of Square within the group MSSW
MSSW = SSW/ N – k
= 392/32
= 12.25

22. *Significant at 0.05 level of Significance
Step 13 – ANOVA Summary
Source of Variance df
Sum of Square
Mean Sum of Square
F – value
(calculated value)
(table value)
Among the groups 7
Between the Groups (Gender) 1
Among the groups (Game) 3
Interaction (Gender x Game)
Within the Group 32
*Significant at 0.05 level of Significance
Degree of Freedom:
Among the Group = k – 1
Between the Groups SSAS = k1 – 1
Between the group SSAG = k2 - 1
Within the group = N – k
k = 8 (Number of groups)
n = 5 (Number of subjects in each group)
N = 40 [n x k or n1+n2+n3 (Total number of scores in an experiment)]

23. *Significant at 0.05 level of Significance
Step 8 – ANOVA Summary
Source of Variance df
Sum of Square
Mean Sum of Square
F – value
(calculated value)
(table value)
Among the groups 7
375.6
Between the Groups (Gender) 1
102.40
Among the groups (Game) 3
125
Interaction (Gender x Game)
148.20
Within the Group 32
392
*Significant at 0.05 level of Significance
Degree of Freedom:
Among the Group = k – 1
Between the Groups SSAS = k1 – 1
Between the group SSAG = k2 - 1
Within the group = N – k
k = 8 (Number of groups)
n = 5 (Number of subjects in each group)
N = 40 [n x k or n1+n2+n3 (Total number of scores in an experiment)]

24. *Significant at 0.05 level of Significance
Step 8 – ANOVA Summary
Source of Variance df
Sum of Square
Mean Sum of Square
F – value
(calculated value)
(table value)
Among the groups 7
375.6
53.65
Between the Groups (Gender) 1
102.40
Among the groups (Game) 3
125
41.66
Interaction (Gender x Game)
148.20
49.40
Within the Group 32
392
12.25
*Significant at 0.05 level of Significance
Degree of Freedom:
Among the Group = k – 1
Between the Groups SSAS = k1 – 1
Between the group SSAG = k2 - 1
Within the group = N – k
k = 8 (Number of groups)
n = 5 (Number of subjects in each group)
N = 40 [n x k or n1+n2+n3 (Total number of scores in an experiment)]

25. *Significant at 0.05 level of Significance
Step 8 – ANOVA Summary
Source of Variance df
Sum of Square
Mean Sum of Square
F – value
(calculated value)
(table value)
Among the groups 7
375.6
53.65
4.37*
Between the Groups (Gender) 1
102.40
8.35*
Among the groups (Game) 3
125
41.66
3.40*
Interaction (Gender x Game)
148.20
49.40
4.03*
Within the Group 32
392
12.25
*Significant at 0.05 level of Significance
Degree of Freedom:
Among the Group = k – 1
Between the Groups SSAS = k1 – 1
Between the group SSAG = k2 - 1
Within the group = N – k
k = 8 (Number of groups)
n = 5 (Number of subjects in each group)
N = 40 [n x k or n1+n2+n3 (Total number of scores in an experiment)]

26. *Significant at 0.05 level of Significance
Step 8 – ANOVA Summary
Source of Variance df
Sum of Square
Mean Sum of Square
F – value
(calculated value)
(table value)
Among the groups 7
375.6
53.65
4.37*
2.25 (F.057,32)
Between the Groups (Gender) 1
102.40
8.35*
4.15 (F.052,32)
Among the groups (Game) 3
125
41.66
3.40*
2.90 (F.053,32)
Interaction (Gender x Game)
148.20
49.40
4.03*
Within the Group 32
392
12.25
*Significant at 0.05 level of Significance
Degree of Freedom:
Among the Group = k – 1
Between the Groups SSAS = k1 – 1
Between the group SSAG = k2 - 1
Within the group = N – k
k = 8 (Number of groups)
n = 5 (Number of subjects in each group)
N = 40 [n x k or n1+n2+n3 (Total number of scores in an experiment)]

27. Since computed values of F for row, column and interaction are greater than their corresponding tabulated values, therefore null hypothesis for row, column and interaction may be rejected at .05 level of significance.