Two Way ANOVA.pptx

In statistics, the two-way analysis of variance is an extension of the one-way ANOVA that examines the influence of two different categorical independent variables on one continuous dependent variable.

1. Two-Way ANOVA Calculation Research Methodology (Module Code :AG 4016) GROUP NO : 02 1

2. Two way ANOVA is a extension of the one way ANOVA in terms of the second factor in to the analysis. This is an extension the one factor situation to take account of a second factor are often determined by grouping of subjects or units used in the investigation. One way ANOVA is independent only one factor but two way ANOVA independent two factor in experiment. A blocking factors are included in Two way ANOVA ,which is places subjects or units into homogeneous groups called Blocks. So it self called a Randomized Block Design (RCBD) Introduction of Two Way ANOVA 2

3. - Two way ANOVA table 3

4. Introduction of ANOVA table - Two way - Anova Table Source df SS MSS F P Between row Between column Within Error Total SS = Sum of squares df = Degrees of Freedom MSS = Mean Sum of Square F = F- ratio P = P - Value 4

5. Experiment 1 A farmer applied three types of fertilizers on 4 separate plots for his cultivation. The Figure on Yield per acre are tabulated below. . 01 -Selected data set Table 1 –Yield of field with inorganic fertilizer Fertilizer Yield A B C D Nitrogen 6 4 8 6 Phosphorus 7 6 6 9 Potassium 8 5 10 9 5

6. Calculation part 6

7. 02 - FIND df Value Source of variation df SS Mss = F ratio P value Between Row (Fertilizer) 2 Between Column (Plots) 3 Within Error 6 Total 11 Source of variation df SS F ratio P value Between Row (Fertilizer) 2 Between Column (Plots) 3 Within Error 6 Total 11 d f of Between row = No. of rows – 1 = 3 -1 = 2 df of Between c olumn = No. of Column – 1 = 4 – 1 = 3 df of Total = Total No. of Elements -1 = 12 – 1= 11 df of Within Error = Total – ( df of Between row + df of Between column ) = 11- (2 + 3) = 6 Fertilizer Yield A B C D Nitrogen 6 4 8 6 Phosphorus 7 6 6 9 Potassium 8 5 10 9 7

8. 03- FIND SS Value Fertilizer Yield Total A B C D Nitrogen 6 4 8 6 24 Phosphorus 7 6 6 9 28 Potassium 8 5 10 9 32 Total 21 15 24 24 84 8

9. Corection Factor C=   = = = 588   03-FIND SS Value Fertilizer Yield Total A B C D Nitrogen 6 4 8 6 24 Phosphorus 7 6 6 9 28 Potassium 8 5 10 9 32 Total 21 15 24 24 84 9

10. 03-FIND SS Value SS Between Rows (Fertilizers) - = - = 596 – 588 = 8   SS Between Columns (Plots ) - = - = 606 – 588 = 18   SS Total value = - = 624 – 588 = 36   Fertilizer Plots Yield Total A B C D Nitrogen 6 4 8 6 24 Phosphorus 7 6 6 9 28 Potassium 8 5 10 9 32 Total 21 15 24 24 84 10

11. Two Way Anova Value Source of variation df SS MSS = F ratio P value Between Row (Fertilizer) 2 8 Between Column (Plots) 3 18 Within Error 6 10 Total 12 36 Source of variation df SS F ratio P value Between Row (Fertilizer) 2 8 Between Column (Plots) 3 18 Within Error 6 10 Total 12 36 Within Error = Total SS – ( SS of Between Row + SS of Between Column ) = 36 – (8 +18) = 10 11

12. 04-FIND MSS Value Source of variation df SS MSS = F ratio P value Between Row (Fertilizer) 2 8 4 Between Column (Plots) 3 18 6 Within Error 6 10 1.6667 Total 12 36 Source of variation df SS F ratio P value Between Row (Fertilizer) 2 8 4 Between Column (Plots) 3 18 6 Within Error 6 10 1.6667 Total 12 36 MSS of Between Row = = = 4   MSS of Between Column = = = 6   MSS of Within Error = = = 1.66667   12 MSS =  

13. Two Way Anova Value Source of variation df SS MSS = F ratio P value Between Row (Fertilizer) 2 8 4 Between Column (Plots) 3 18 6 Within Error 6 10 1.6667 Total 12 36 Source of variation df SS F ratio P value Between Row (Fertilizer) 2 8 4 Between Column (Plots) 3 18 6 Within Error 6 10 1.6667 Total 12 36 13

14. 05-Find F Value F ratio of Bw (R) = =   F ratio of Bw (C) = =   Bw (R) = Between row Bw (C) = Between column Source of variation df SS MSS = F ratio P value Between Row (Fertilizer) 2 8 4 2.4000 Between Column (Plots) 3 18 6 3.6001 Within Error 6 10 1.6667 Total 12 36 Source of variation df SS F ratio P value Between Row (Fertilizer) 2 8 4 2.4000 Between Column (Plots) 3 18 6 3.6001 Within Error 6 10 1.6667 Total 12 36 14

15. 15 Two Way Anova Value Source of variation df SS MSS = F ratio P value Between Row (Fertilizer) 2 8 4 Between Column (Plots) 3 18 6 Within Error 6 10 1.6667 Total 12 36 Source of variation df SS F ratio P value Between Row (Fertilizer) 2 8 4 Between Column (Plots) 3 18 6 Within Error 6 10 1.6667 Total 12 36

16. 06-Find P value Source of variation df ss MSS = F ratio P value Between Row (Fertilizer) 2 8 4 = 2.4009 F (2,6) Between Column (Plots) 3 18 6 = 3.6001 F ( 3,6) Within Error 6 10 1.6667 Total 12 36 Source of variation df ss F ratio P value Between Row (Fertilizer) 2 8 4 F (2,6) Between Column (Plots) 3 18 6 F ( 3,6) Within Error 6 10 1.6667 Total 12 36 df of Between Row MSS value =2 df of Within Error MSS value =6 df of Between Column MSS value =3 df of Within Error MSS value =6 16

17. 06-Find P value For Fertilizer (Row ) Degrees of freedom in the denumerator Degrees of freedom in the numerator Accept H 5.14 Critical Region reject H F 2.4009 17 F (2,6 ) = 5.14

18. Accept H 4.76 Critical Region reject H F 3.6001 For Plots (Column ) Degrees of freedom in the denumerator Degrees of freedom in the numerator 06-Find P value 18 F ( 3,6 ) = 4.76

19. Accept H 4.76 Critical Region reject H F 3.6001 Accept H 5.14 Critical Region reject H F 2.4009 Interpretation Since 2.4009  5.14 , H is not rejected or fail to reject H at 5% significance level . Interpretation Since 3.6001  4.76 , H is not rejected or fail to reject H at 5% significance level . For Fertilizer (Row ) For Plots (Column ) 19

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