Two way ANOVA

Dr. Anshul Singh Thapa

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. Then we should use Two Way ANOVA. The Two Way ANOVA studies the following:
• 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.

Steps in calculation of ‘F’�Two way ANOVA

Step 1 – Correction factor

Step 2 – Sum of Square of Total SS_T

Step 3 – Sum of Square among the group SS_A

Step 4 – Sum of Square between the A group (Gender) SS_AS

Step 5 – Sum of Square among the B group (Game) SS_AG

Step 6 – Sum of Square of interaction (Gender x Game) SS_{S x G}

Step 7 – Sum of Square within the Groups SS_W

Step 8 – Mean Sum of Squares among the groups MSS_A

Step 9 – Mean Sum of Squares between the A groups (Gender) MSS_AS

Step 10 – Mean Sum of Squares among the B groups (Game) MSS_AG

Step 11 – Mean Sum of Squares interaction (Gender X Game) MSS_{S x G}

Step 12 – Mean Sum of Squares within the groups MSS_W

Step 13 – Summary of ANOVA

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Steps in calculation of ‘F’�Two way ANOVA

Step 1 – Correction factor

Step 2 – Sum of Square of Total SS_T

Step 3 – Sum of Square among the group SS_A

Step 4 – Sum of Square between the A group (Gender) SS_AS

Step 5 – Sum of Square among the B group (Game) SS_AG

Step 6 – Sum of Square of interaction (Gender x Game) SS_{S x G}

Step 7 – Sum of Square within the Groups SS_W

Step 8 – Mean Sum of Squares among the groups MSS_A

Step 9 – Mean Sum of Squares between the A groups (Gender) MSS_AS

Step 10 – Mean Sum of Squares among the B groups (Game) MSS_AG

Step 11 – Mean Sum of Squares interaction (Gender X Game) MSS_{S x G}

Step 12 – Mean Sum of Squares within the groups MSS_W

Step 13 – Summary of ANOVA

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Step 1 – Calculation of Correction factor

Step 1 – Calculation of Correction factor

Calculation of Correction Factor (CF)

G²

N

676²

40

CF = 11424.40

Step 2 – Calculation of Sum of Square of Total SS_T

�Step 2 – Calculation of Sum of Square of Total SS_T �

Calculation of Raw Some of Square (RSS)

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(RSS) = X₁² + X₂² + X₃² + X₄² = 12192

SS_T = 3073 + 3783 + 3322 + 2014 = 12192

SS_T = 12192 – CF

SS_T = 12192 – 11424.4

SS_T = 767.6

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Step 3 – Calculation of Sum of Square among the groups SS_A

Step 3 – Calculation of Sum of Square among the groups SS_A

Calculation of Sum of Square of Among the group SS_A

SS_A = (88)²/5 + (117)²/5 + (87)²/5 + (78)²/5 + (83)²/5 + (70)²/5 + (91)²/5 + (62)²/5 – 11424.4

SS_A = 7744/5 + 13689/5 + 7569/5 + 6084/5 + 6889/5 + 4900/5 + 8281/5 + 3844/5 – 11424.4

SS_A = 1548.8 + 2737.8 + 1513.8 + 1216.8 + 1377.8 + 980 + 1656.2 + 768.8 – 11424.4

SS_A = 11800 – 11424.4

SS_A = 375.6

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Step 4 – Calculation of Sum of Square between the groups (Gender) SS_AS

Step 4 – Calculation of Sum of Square between the groups (Gender) SS_AS

Calculation of Sum of Square between the groups (Gender) SS_AS

SS_AS = (ΣR₁)²/ R_n + (ΣR₂)²/ R_n - CF

SS_AS = (370)²/ 20 + (306)²/20 – 11424.4

SS_AS = 6845 + 4681.4 – 11424.4

SS_AS = 11526.8 – 11424.4

SS_AS = 102.40

Step 5 – Calculation of Sum of Square among the groups (Game) SS_AG

Step 5 – Calculation of Sum of Square among the groups (Game) SS_AG

Calculation of Sum of Square among the groups (Game) SS_AG

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SS_AG = (ΣC₁)²/ C_n + (ΣC₂)²/ C_n + (ΣC₃)²/ C_n + (ΣC₄)²/C_n - CF

SS_AG = (171)²/10 + (187)²/10 + (178)²/10 + (140)²/10 – 11424.4

SS_AG = 2924.1 + 3496.9 + 3168.4 + 1960 – 11424.4

SS_AG = 11549.4 – 11424.4

SS_AG = 125

Step 6 – Calculation of Sum of Square among the interaction (Gender x Game) SS_{AS x AG}

Calculation of Sum of Square between the interaction (Gender x Game) SS_{AS x AG}

SS_{AS x AG} = SS_A– SS_AS – SS_AG

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SS_{AS x AG} = 375.6 – 102.40 – 125

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SS_{AS x AG} = 148.20

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Step 7 – Calculation of Sum of Square within the Group SS_W

Calculation of Sum of Square within the Group SS_W

SS_W = SS_T – SS_A

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SS_W = 767.60 – 375.6

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SS_W = 392

Step 8 – Calculation of Mean Sum of Square among the groups (MSS_A)

Calculation of Mean Sum of Square among the groups (MSS_A)

MSS_A = SS_A/ k – 1

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MSS_A = 375.6/ 7

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MSS_A = 53.67

Step 9 – Calculation of Mean Sum of Square between the groups (Gender) (MSS_AS)

Calculation of Mean Sum of Square between the groups (Gender) (MSS_AS)

MSS_AS = SS_AS/ k₁ – 1

MSS_AS = 102.40/ 1

MSS_AS = 102.40

Step 10 – Calculation of Mean Sum of Square among the group (Games) MSS_AG

Calculation of Mean Sum of Square among the groups (Games) (MSS_AG)

MSS_AG = SS_AG/ k₂ – 1

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MSS_AG = 125/3

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MSS_AG = 41.67

Step 11 – Calculation of Mean Sum of Square among the group (Gender X Game) MSS_{AG X AS}

Calculation of Mean Sum of Square among the group (Gender X Game) (MSS_{AG X AS})

MSS_{AG X AS} = SS_{AG X AS}/ (k₁ – 1) (k₂ – 1)

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MSS_{AG X AS} = 148.20/ 3

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MSS_{AG X AS} = 49.40

Step 12 – Calculation of Mean Sum of Square within the group MSS_W

Calculation of Mean Sum of Square within the group MSS_W

MSS_W = SS_W/ N – k

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MSS_W = 392/32

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MSS_W = 12.25

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

Step 13 – ANOVA Summary

^*Significant at 0.05 level of Significance

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Step 13 – ANOVA Summary

^*Significant at 0.05 level of Significance

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Degree of Freedom:

Among the Group = k – 1

Between the Groups SS_AS = k₁ – 1

Between the group SS_AG = k₂ - 1

Within the group = N – k

Step 13 – ANOVA Summary

^*Significant at 0.05 level of Significance

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Degree of Freedom:

Among the Group = k – 1

Between the Groups SS_AS = k₁ – 1

Between the group SS_AG = 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)]

Step 13 – ANOVA Summary

^*Significant at 0.05 level of Significance

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Degree of Freedom:

Among the Group = k – 1

Between the Groups SS_AS = k₁ – 1

Between the group SS_AG = 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)]

• 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.

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