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)
2

3. -
3

4. -
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..
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. 6

7. Source of variation df SS Mss =
𝒔𝒔
𝒅𝒇
F ratio P value
Between Row
(Fertilizer)
2
Between Column
(Plots)
3
Within Error 6
Total 11
df of Between row = No. of rows – 1
= 3 -1 = 2
df of Between column = 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. 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=
𝑮𝟐
𝑵
=
(𝟖𝟒)𝟐
𝟏𝟐
=
𝟕𝟎𝟓𝟔
𝟏𝟐
= 588
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. SS Between Rows (Fertilizers)
(𝟐𝟒)𝟐
𝟒
+
(𝟐𝟖)𝟐
𝟒
+
(𝟑𝟐)𝟐
𝟒
-
(𝟖𝟒)𝟐
𝟏𝟐
=
𝟓𝟏𝟔
𝟒
+
𝟕𝟖𝟒
𝟒
+
𝟏𝟎𝟐𝟒
𝟒
-
𝟕𝟎𝟓𝟔
𝟏𝟐
= 596 – 588
= 8
SS Between Columns (Plots)
(𝟐𝟏)𝟐
𝟑
+
(𝟏𝟓)𝟐
𝟑
+
(𝟐𝟒)𝟐
𝟑
+
(𝟐𝟒)𝟐
𝟑
-
(𝟖𝟒)𝟐
𝟏𝟐
=
𝟒𝟒𝟏
𝟑
+
𝟐𝟐𝟓
𝟑
+
𝟓𝟕𝟔
𝟑
+
𝟓𝟕𝟔
𝟑
-
𝟕𝟎𝟓𝟔
𝟏𝟐
= 606 – 588
= 18
SS Total value
= 𝟔𝟐 + 𝟒𝟐 + 𝟖𝟐 + 𝟔𝟐 + 𝟕𝟐 + 𝟔𝟐 + 𝟔𝟐 + 𝟗𝟐 + 𝟖𝟐 + 𝟓𝟐 + 𝟏𝟎𝟐 + 𝟗𝟐 -
(𝟖𝟒)𝟐
𝟏𝟐
= 624 – 588
= 36
Fertilizer
Plots
Yield Tot
al
A B C D
Nitrogen 6 4 8 6 24
Phospho
rus
7 6 6 9 28
Potassiu
m
8 5 10 9 32
Total 21 15 24 24 84
10

11. Source of variation df SS MSS =
𝒔𝒔
𝒅𝒇
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. Source of
variation
df SS MSS =
𝒔𝒔
𝒅𝒇
F
rati
o
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 =
𝐒𝐒
𝐝𝐟
=
𝟖
𝟐
= 4
MSS of Between Column =
𝑺𝑺
𝒅𝒇
=
𝟏𝟖
𝟑
= 6
MSS of Within Error =
𝑺𝑺
𝒅𝒇
=
𝟏𝟎
𝟔
= 1.66667
12
MSS =
𝒔𝒔
𝒅𝒇

13. Source of variation df SS MSS =
𝒔𝒔
𝒅𝒇
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. F ratio of Bw (R) =
MSS (Bw) row
MSS (𝐖) 𝐫𝐨𝐰
=
𝟒
𝟏.𝟔𝟔𝟔𝟔𝟕
= 𝟐. 𝟒𝟎𝟎𝟎
F ratio of Bw (C) =
MSS (Bw) 𝒄𝒐𝒍𝒖𝒎𝒏
MSS (𝐖) 𝒄𝒐𝒍𝒖𝒎𝒏
=
𝟔
𝟏.𝟔𝟔𝟔𝟔𝟕
= 𝟑. 𝟔𝟎𝟎𝟏
Bw (R) = Between row
Bw (C) = Between column
Source of
variation
df SS MSS =
𝒔𝒔
𝒅𝒇
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
Source of variation df SS MSS =
𝒔𝒔
𝒅𝒇
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. Source of
variation
df ss
MSS =
𝒔𝒔
𝒅𝒇
F ratio
P
value
Between Row
(Fertilizer)
2 8 4
𝟒
𝟏.𝟔𝟔𝟔𝟔𝟕
=
2.4009
F (2,6)
Between
Column
(Plots)
3
18 6
𝟔
𝟏.𝟔𝟔𝟔𝟔𝟕
=
3.6001
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. For Fertilizer (Row )
Degrees
of
freedom
in
the
denumerator
Degrees of freedom in the numerator
Accept H0
5.14
Critical
Region
reject H0
F
2.4009
17
F (2,6) = 5.14

18. Accept H0
4.76
Critical
Region
reject H0
F
3.6001
For Plots (Column )
Degrees
of
freedom
in
the
denumerator
Degrees of freedom in the numerator
18
F ( 3,6) = 4.76

19. Accept
H0
4.76
Critical
Region
reject H0
F
3.6001
Accept
H0
5.14
Critical
Region
reject H0
F
2.4009
Interpretation
• Since 2.4009  5.14 , H0 is
not rejected or fail to reject
H0 at 5% significance level .
Interpretation
• Since 3.6001  4.76 , H0 is not
rejected or fail to reject H0 at
5% significance level .
For Fertilizer (Row ) For Plots (Column )
19

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