The document discusses the analysis of variance (ANOVA) methodology for two-way classifications, focusing on the impacts of different experimental conditions on multiple groups. It provides a comprehensive example involving agricultural experiments comparing rice yields affected by various fertilizers, detailing hypotheses, sum of squares computations, and interpretations of F-values for significance testing. Additionally, it addresses the methodology for two-way ANOVA with unequal cell frequencies, demonstrating the use of harmonic means and interpretations of results.

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
Freedom
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
V1 V2 V3
t1
26 14 41 82 36 87
41 16 26 86 39 99
28 29 19 45 59 126
92 31 59
37
27 104
t2
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
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.

9. 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

10. Sum of Squares Computations


11. Source 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

12. Interpretations
1.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.

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.

14. 

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

16. 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.

17. C1 C2 C3
R1
7
6
6
2
8
12
17
19
16
17
13
14
4 3 16 21 10
24 22
R2
23 22 11 26 9 16
14 26 15 14 27 17
9 18 26 13 31 18
31 42 20

18. C1 C2 C3
R1
N=6
T=28
N=8
T=139
N=5
T=70
R2
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
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

21. Sum of Squares Computations


22. Source 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.

23. THANK YOU
