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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