Analysis of Variance (F-ratio Test) Two Way Classification

Analysis of Variance (F-ratio Test) Two Way Classification

1. ANALYSIS OF VARIANCE (F-RATIO TEST) TWO WAY CLASSIFICATION

2. This test is designed for more than two groups of objects studies to see if each group is affected by two different experimental conditions.

3. This test is used when the number of observation in the subclasses are equal.

4. Formulas for Two Way ANOVA For Equal and Proportionate Entries in the Subclasses Source of Variation Sum of Squares Degrees of F r e e d om Mean Square F- value Row R – 1 Column C – 1 Interaction (R–1)(C–1) Within cells RC (n-1) Total nRC – 1

5. Formula for Mean Square

6. Example An agricultural experiment was conducted to compare the yields of three varieties of rice applied by two types of fertilizer. The following table represents the yield in grams using eight plots.

7. Types of Fertilizer Varieties of Rice V 1 V 2 V 3 t 1 26 14 41 82 36 87 41 16 26 86 39 99 28 29 19 45 59 126 92 31 59 37 27 104 t 2 51 35 39 114 42 133 96 36 104 92 92 124 97 28 130 87 156 68 22 76 122 64 144 142

8. Hypothesis There is no difference in the yields of the three varieties of rice. The two types of fertilizer does not significantly affect the yields meaning the yield is not dependent of the type of fertilizer used. There is no significant interaction between the variety of rice and the types of fertilizer used.

9. Summary of the Data (Sum of all entries in each cell) Types of F e r t i l i z er Varieties of Rice Total V 1 V 2 V 3 t 1 277 395 577 1249 t 2 441 752 901 2094 Total 718 1147 1478 3343

10. Sum of Squares Computations 

11. Source of Variation Sum of S qua r es Degree of Freedom Mean S qua r e F-Value Rows 14, 875.52 1 14,875.52 14.64 Columns 18,150.04 2 9,075.02 8.93 Interaction 1,332.04 2 666.02 0.656 Within Cells 42,667.38 42 1,015.89 TOTAL 77,024.98 47

12. Interpretations For the different varieties of rice, we have Fc=8.93 with 2 df associated with the numerator and 42 df with the denominator. The values required for significance at 5% and 1% levels are 3.22 and 5.15, respectively. We conclude that the different varieties of rice differ significantly in their yields. For the different types of fertilizer, we have Fr=14.64 with 1 df associated with the numerator and 42 df with the denominator. The value required for the significance at 5% and 1% levels are 4.072 & 7.287 respectively. We conclude therefore that the different types of fertilizer affect significantly the yields of rice. 3. For significant interaction, we have Frc=0.656 which is lower than the table value. Therefore hypothesis number three is accepted.

13. This method is applied in two way ANOVA where the number of observations in the subclasses or cell frequency is unequal. The data is to be adjusted by the method of unweight mean. This method is in effect the analysis of variance applied to the means of the subclasses. The sum of the squares for rows, columns, and interaction are then adjusted using the harmonic mean.

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15. Formulas for Two Way ANOVA For Unequal Frequency in the Subclasses Source of Variation Sum of Squares Degrees of F r e e d om Mean Square F- value Row R – 1 Column C – 1 Interaction (R–1)(C–1) Within cells N-RC

16. Example The following t able shows of f a ct i tiou s data for a tw o w a y cla s sification experiment wit h two le v els o f one factor and three levels of the other factor.

17. C 1 C 2 C 3 R 1 7 6 6 2 8 12 17 19 16 17 13 14 4 3 16 21 10 24 22 R 2 23 22 11 26 9 16 14 26 15 14 27 17 9 18 26 13 31 18 31 42 20

18. C 1 C 2 C 3 R 1 N=6 T=28 N=8 T=1 3 9 N=5 T=70 R 2 N=6 T=112 N=7 T=136 N=8 T=180 Summary of the Data

19. Computation for Harmonic Mean: 

20. Means of each cell and other Computations of the data C 1 C 2 C 3 Total R 1 4.67 17.38 14 36.05 R 2 18.67 19.43 22.5 60.6 TOTAL 23.34 36.81 36.5 96.65

21. Sum of Squares Computations 

22. Source of Variation Sum of S qua r es Degree of Freedom Mean S qua r e F-Value Rows 650.92 1 650.92 13.97 Columns 383.1 2 191.55 4.11 Interaction 231.85 2 115.93 2.49 Within Cells 1,584.25 34 46.60 Interpretation: In this factitious data, the row effect is significant at .01 level of significance, the column effect is also significant at .05 level while the interaction effect is not significant.

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