how to solve Two way ANOVA

1. Two way ANOVA
द्वि मार्गी प्रसरण विश्लेषण

2. Two way ANOVA
A two-way ANOVA tests the effect of
two independent variables on a
dependent variable.A two-way
ANOVA test analyzes the effect of the
independent variables on the
expected outcome along with their
relationship to the outcome itself.
By using ANOVA, a researcher is able
to determine whether the variability
of the outcomes is due to chance or
to the factors in the analysis.
एक दो-तरफा एनोिा एक आश्रित चर
पर दो स्ितंत्र चर क
े प्रभाि का
परीक्षण करता है। एक दो-तरफा
एनोिा परीक्षण अपेक्षक्षत पररणाम पर
स्ितंत्र चर क
े प्रभाि का विश्लेषण
करता है, साथ ही पररणाम क
े साथ
उनक
े संबंध का भी विश्लेषण करता
है।
एनोिा का उपयोर्ग करक
े , एक
शोधकताा यह ननधााररत करने में
सक्षम होता है कक पररणामों की
पररितानशीलता संयोर्ग क
े कारण है
या विश्लेषण क
े कारकों क
े कारण है।

3. Two way ANOVA
As the name suggests, in two-way ANOVA
reference station, the larger group or total
group is divided on two grounds, e.g. if the
effectiveness of three teaching methods is to be
compared for boys and girls separately, Then
the researcher has to make a total of 6 groups.
There will be three groups of boys, each of
whom will be taught by one method of teaching
and three groups of girls, each of whom will be
taught by one method of teaching, then it is
clear that the larger group consisting of six
groups grouped together on the basis of two
variables – gender and method of teaching.
Hence there is a TWO WAY partition which can
be represented as follows
जैसा कक नाम से ही स्पष्ट है कक टू-िे एनोिा
सन्दभा स्टेशन में, बडे समूह या क
ु ल समूह को
दो आधारों पर विभाजजत ककया जाता है,
जैसे यदद तीन शशक्षण विश्रधयों की प्रभािशीलता
की तुलना लडकों और लडककयों क
े शलए अलर्ग-
अलर्ग करनी हो ,तब शोधकताा को क
ु ल 6
समूह बनाने होते हैं। तीन समूह बालकों क
े होंर्गे,
जजनमें से प्रत्येक को एक-एक शशक्षण पद्धनत से
पढाया जाएर्गा और तीन समूह लडककयों क
े होंर्गे,
जजनमें से प्रत्येक को एक-एक शशक्षण पद्धनत से
पढाया जाएर्गा, तो स्पष्ट है कक छह समूहों को
शमलाकर बने बडे समूह को दो चरों -शलंर्गभेद
तथा शशक्षण विश्रध क
े आधार पर एक साथ
बांटा र्गया है। अतः TWO WAY विभाजन है जजसे
ननम्न ढंर्ग से प्रस्तुत ककया जा सकता है

4. VATIABL
E
TEACHING METHOD
Level Lecture
Method
A1
Text book Method
A2
Participation
Method
A3
Total
FACTOR
ES
B
Boys
B1
8
4
5
7
6
2
3
5
1
4
12
8
13
7
5
Girls
B2
10
7
9
8
6
8
10
3
9
5
19
10
15
17
14
Total

5. Steps of two way ANOVA
1. Correction factor(संशोधन पद) (C)
2. Total sum of square( क
ु ल िर्गा योर्ग) ( SST)
3. sum square between column( स्तंभ का मध्यम िर्गा योर्ग)(SSB
COLUMN))
4. sum of square between Rows( पंजतत का मध्यिर्गा योर्ग)(SSB R)
5. sum of square of cell( कोषों का िर्गा योर्ग)(SS CELL)
6. sum of square between interaction (SSB INT)
7. with in sum of square(आंतररक िर्गा योर्ग)(SSCO)
8. preparation of analysis of variation table( प्रसरण ताशलका का
ननमााण करना)
9. degree of freedom( मुतता ंंश) (df)
10. variance estimates( प्रसरण अनुमान )
11. F-Ratio( ऍफ अनुपात)
12. interpretation (व्याख्या)

6. VARIABLE TEACHING METHOD
Level Lecture Method
A1
Text book Method
A2
Participati
on
Method
A3
Total
XA1 X2 A1 XA2 X2 A2 XA3 X2 A3 X X2
FACTORES
B
Boys
B1
8
4
5
7
6
64
16
25
49
36
2
3
5
1
4
4
9
25
1
16
12
8
13
7
5
144
64
169
49
25
∑ 30 190 15 55 45 451 90 696
Girls
B2
10
7
9
8
6
100
49
81
64
36
8
10
3
9
5
64
100
9
81
25
19
10
15
17
14
361
100
225
289
196
∑ 40 330 35 279 75 1171 150 1780
Total 70 520 50 334 120 1622 240 2476

7. VA
RIA
BLE
TEACHING METHOD
Leve
l
Lecture
Method
A1
Text book
Method
A2
Participation
Method
A3
Tota
l
XA
1
X2 A1 XA2 X2 A2 XA3 X2 A3 X X2
FAC
TOR
ES
B
Boys
B1
8
4
5
7
6
64
16
25
49
36
2
3
5
1
4
4
9
25
1
16
12
8
13
7
5
144
64
169
49
25
∑ 30 190 15 55 45 451 90 696
Girls
B2
10
7
9
8
6
100
49
81
64
36
8
10
3
9
5
64
100
9
81
25
19
10
15
17
14
361
100
225
289
196
∑ 40 330 35 279 75 1171 150 1780
Tota
l
70 520 50 334 120 1622 240 2476
1. Correction Factor (C)
𝑐 =
σ 𝑥 2
𝑁
𝑐 =
240 2
30
𝑐 =
57600
30
𝑐 = 1920

8. VA
RIA
BLE
TEACHING METHOD
Leve
l
Lecture
Method
A1
Text book
Method
A2
Participation
Method
A3
Tota
l
XA1 X2 A1 XA2 X2 A2 XA3 X2 A3 X X2
FAC
TOR
ES
B
Boys
B1
8
4
5
7
6
64
16
25
49
36
2
3
5
1
4
4
9
25
1
16
12
8
13
7
5
144
64
169
49
25
∑ 30 190 15 55 45 451 90 696
Girls
B2
10
7
9
8
6
100
49
81
64
36
8
10
3
9
5
64
100
9
81
25
19
10
15
17
14
361
100
225
289
196
∑ 40 330 35 279 75 1171 150 1780
Total 70 520 50 334 120 1622 240 2476
2. . Total Sum Of Square
(SST)
SS𝑇 = 𝛴𝑋𝑖
2
− 𝐶
SS𝑇 = 2476 − 1920
SS𝑇 = 556

9. VA
RIA
BLE
TEACHING METHOD
Leve
l
Lecture
Method
A1
Text book
Method
A2
Participation
Method
A3
Tota
l
XA
1
X2 A1 XA2 X2 A2 XA3 X2 A3 X X2
FAC
TOR
ES
B
Boys
B1
8
4
5
7
6
64
16
25
49
36
2
3
5
1
4
4
9
25
1
16
12
8
13
7
5
144
64
169
49
25
∑ 30 190 15 55 45 451 90 696
Girls
B2
10
7
9
8
6
100
49
81
64
36
8
10
3
9
5
64
100
9
81
25
19
10
15
17
14
361
100
225
289
196
∑ 40 330 35 279 75 1171 150 1780
Tota
l
70 520 50 334 120 1622 240 2476
3. sum square between
column (SSB CO)
𝑆𝑆 𝐶𝑂𝐿 =
σ 𝑐1
2
+
𝑛1
σ⋅ 𝑐2
2
𝑛2
+ σ 𝑐3
2
𝑛3
− 𝑐
𝑆𝑆 𝐶𝑂𝐿 =
70 2 +
10
50 2
10
+1202
10
− C
𝑆𝑆 𝐶𝑂𝐿 = 490 + 250 + 1440 − 1920
𝑆𝑆 𝐶𝑂𝐿 = 260

10. VA
RIA
BLE
TEACHING METHOD
Leve
l
Lecture
Method
A1
Text book
Method
A2
Participation
Method
A3
Tota
l
XA
1
X2 A1 XA2 X2 A2 XA3 X2 A3 X X2
FAC
TOR
ES
B
Boys
B1
8
4
5
7
6
64
16
25
49
36
2
3
5
1
4
4
9
25
1
16
12
8
13
7
5
144
64
169
49
25
∑ 30 190 15 55 45 451 90 696
Girls
B2
10
7
9
8
6
100
49
81
64
36
8
10
3
9
5
64
100
9
81
25
19
10
15
17
14
361
100
225
289
196
∑ 40 330 35 279 75 1171 150 1780
Tota
l
70 520 50 334 120 1622 240 2476
4. sum of square between
Rows(SSB RO)
𝑆𝑆 𝑅𝑂𝑊 =
σ 𝑅1
2
+
𝑛1
σ⋅ 𝑅2
2
𝑛2
+ σ 𝑅3
2
𝑛3
… … − 𝑐
𝑆𝑆 𝑅𝑂𝑊 =
90 2 +
15
1502
15
… … − 1920
𝑆𝑆 𝑅𝑂𝑊 = 540 + 1500 − 1920
𝑆𝑆 𝑅𝑂𝑊 = 120

11. VA
RIA
BLE
TEACHING METHOD
Leve
l
Lecture
Method
A1
Text book
Method
A2
Participation
Method
A3
Tota
l
XA
1
X2 A1 XA2 X2 A2 XA3 X2 A3 X X2
FAC
TOR
ES
B
Boys
B1
8
4
5
7
6
64
16
25
49
36
2
3
5
1
4
4
9
25
1
16
12
8
13
7
5
144
64
169
49
25
∑ 30 190 15 55 45 451 90 696
Girls
B2
10
7
9
8
6
100
49
81
64
36
8
10
3
9
5
64
100
9
81
25
19
10
15
17
14
361
100
225
289
196
∑ 40 330 35 279 75 1171 150 1780
Tota
l
70 520 50 334 120 1622 240 2476
5. sum of square of cell
𝒔𝒔𝑪=
෍𝒙𝑨𝑩
𝟐
𝑵𝑨𝑩
−𝑪
𝑆𝑆 𝐶𝐸𝐿𝐿 =
𝐵 σ 𝑋1A
2 +
𝑛1
σ⋅ 𝑋2A
2
𝑛2
+ σ 𝑋3A
2
𝑛3
+
𝐺
σ 𝑋1A
2
+
𝑛1
σ⋅ 𝑋2A
2
𝑛2
+ σ 𝑋3𝐴
2
𝑛3
𝑆𝑆 𝐶𝐸𝐿𝐿 =
𝐵 30 2 +
5
15 2
5
+ 45 2
5
+
𝐺
40 2
+
5
352
5
+ 75 2
5
𝑆𝑆 𝐶𝐸𝐿𝐿 = 400

12. VA
RIA
BLE
TEACHING METHOD
Leve
l
Lecture
Method
A1
Text book
Method
A2
Participation
Method
A3
Tota
l
XA
1
X2 A1 XA2 X2 A2 XA3 X2 A3 X X2
FAC
TOR
ES
B
Boys
B1
8
4
5
7
6
64
16
25
49
36
2
3
5
1
4
4
9
25
1
16
12
8
13
7
5
144
64
169
49
25
∑ 30 190 15 55 45 451 90 696
Girls
B2
10
7
9
8
6
100
49
81
64
36
8
10
3
9
5
64
100
9
81
25
19
10
15
17
14
361
100
225
289
196
∑ 40 330 35 279 75 1171 150 1780
Tota
l
70 520 50 334 120 1622 240 2476
6. sum of square between
interaction (SSB INT)
𝑆𝑆𝐼𝑁𝑇 = 𝑠𝑠𝑐𝑒𝑙𝑙 − 𝑠𝑠𝑟𝑜𝑤 − 𝑠𝑠𝑐𝑜𝑙
𝑆𝑆𝐼𝑁𝑇 = 400 − 120 − 260
𝑆𝑆𝐼𝑁𝑇 = 20

13. VA
RIA
BLE
TEACHING METHOD
Leve
l
Lecture
Method
A1
Text book
Method
A2
Participation
Method
A3
Tota
l
XA
1
X2 A1 XA2 X2 A2 XA3 X2 A3 X X2
FAC
TOR
ES
B
Boys
B1
8
4
5
7
6
64
16
25
49
36
2
3
5
1
4
4
9
25
1
16
12
8
13
7
5
144
64
169
49
25
∑ 30 190 15 55 45 451 90 696
Girls
B2
10
7
9
8
6
100
49
81
64
36
8
10
3
9
5
64
100
9
81
25
19
10
15
17
14
361
100
225
289
196
∑ 40 330 35 279 75 1171 150 1780
Tota
l
70 520 50 334 120 1622 240 2476
7. with in sum of square(आंतरिक वर्ग
योर्)(SSCO)
𝑆𝑆𝑊 = 𝑠𝑠𝑇 − 𝑠𝑠𝑐𝑒𝑙𝑙
𝑆𝑆𝑊 = 556 − 400
𝑆𝑆𝑊 = 156

14. 8. preparation of analysis of variation table
Source of
Variation
df SS MSS
=SS/df
F RATIO
IV A 3 – 1 =2 𝟐𝟔𝟎 260/2
=130 130/6.5
=20
IV B 2 – 1=1 𝟏𝟐𝟎
120/1
=120 120/6.5
=18.46
Interaction
AXB
( 3-1) ( 2 –
1) = 2
𝟐𝟎 20/2=10 10/6.5
=1.54
Cell 6- 1=5 400
Within 30 – 6 = 24 156 156/24
=6.5
Total 30 – 1=29 556 556/29
=19.17
Source of
Variation
df SS MSS F
RATIO
IV
A(METHO
D)
Column-1
( C-1)
𝒔𝒔𝒄𝒐 M𝒔𝒔𝑨 F1
IV
B(GENDER
)
Row – 1
( R – 1)
𝒔𝒔𝒓𝒐𝒘 M𝒔𝒔𝑩 F2
Interaction
AXB
( C-1) ( R –
1)
𝒔𝒔𝒊𝒏𝒕 𝑴𝒔𝒔𝒊𝒏𝒕F3
Cell No. of Cell
- 1
𝒔𝒔𝒄𝒆𝒍𝒍
Within N – no. of
cell
𝒔𝒔𝒘 𝑴𝒔𝒔𝒘
Total N - 1 SST