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ANALYSIS OF VARIANCE(F-RATIO TEST)
TWO WAY CLASSIFICATION
ANOVA – TWO WAY CLASSIFICATION
This test is designed for more than two groups of objects studies to see if each group is affected by two different experimental conditions.
Equal and Proportionate Entries in the Subclasses
This test is used when the number of observation in the subclasses are equal.
Formulas for Two Way ANOVAFor Equal and Proportionate Entries in the
SubclassesSource of Variation
Sum of Squares Degrees of
Freedom
Mean Square
F-value
Row R – 1
Column C – 1
Interaction (R–1)(C–1) c
Within cells RC (n-1)
Total nRC – 1
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.
Types of Fertilizer
Varieties of Rice
V1 V2 V3
t126 1441 1628 2992 31
41 8242 8643 4559 37
36 8739 99
59 12627 104
t251 3596 3697 2822 76
39 114104 92130 87122 64
42 13392 124
156 68144 142
Hypothesis
1. There is no difference in the yields of the three varieties of rice.
2. The two types of fertilizer does not significantly affect the yields meaning the yield is not dependent of the type of fertilizer used.
3. There is no significant interaction between the variety of rice and the types of fertilizer used.
Summary of the Data (Sum of all entries in each cell)
Types of Fertilizer
Varieties of Rice Total
V1 V2 V3
t1 277 395 577 1249
t2 441 752 901 2094
Total 718 1147 1478 3343
= 5,944,837 c= 4,015,617 = 309,851
Sum of Squares Computations
ANALYSIS OF VARIANCESource of Variation
Sum of Squares
Degree of Freedom
Mean Square
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
Interpretations1. 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.
2. 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.
Unequal Frequency in the Subclasses
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.
Consider a two-factor experiment with R levels of one factor and C levels of the other. Denote the cell frequency by Nrc.
Formula for Harmonic Mean:
Formulas for Two Way ANOVAFor Unequal Frequency in the Subclasses
Source of Variation
Sum of Squares Degrees of
Freedom
Mean Square
F-value
Row R – 1
Column C – 1
Interaction (R–1)(C–1) c
Within cells N-RC
Example
The following table shows of factitious data for a two way classification experiment with two levels of one factor and three levels of the other factor.
C1 C2 C3
R17 68 24 3
8 1712 1916 2124 22
16 1317 14
10
R223 2224 2625 18
11 2615 1426 13
31
9 1627 1731 1842 20
C1 C2 C3
R1N=6T=28
N=8T=139
N=5T=70
R2N=6
T=112N=7
T=136N=8
T=180
Summary of the Data
Computation for Harmonic Mean:
Means of each cell and other Computations of the data
C1 C2 C3 Total
R1 4.67 17.38 14 36.05
R2 18.67 19.43 22.5 60.6
TOTAL 23.34 36.81 36.5 96.65= 1,657.32 c= 1,752.22
= 13,893
Sum of Squares Computations
6.48 (1,752.22 – 1,657.32-1,615.99+1,556.87)=231.85
= 1,584.25
ANALYSIS OF VARIANCESource of Variation
Sum of Squares
Degree of Freedom
Mean Square
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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