I M.Com.

1. TWO WAY ANOVA
Dr. R. MUTHUKRISHNAVENI
SAIVA BHANU KSHATRIYA COLLEGE, ARUPPUKOTTAI

2. Two Way ANOVA
• A two-way ANOVA is an extension of the one-way ANOVA (analysis of variances) that reveals
the results of two independent variables on a dependent variable.
• A two-way ANOVA test is a statistical technique that analyzes the effect of the independent
variables on the expected outcome along with their relationship to the outcome itself.

3. ANOVA
Set two hypothesis(for row and column) H0: µ1= µ2= µ3=…. µk or H1: µ1≠ µ2 ≠ µ3 ≠ …. µk
Calculate variance between the rows(SSR) and columns(SSC)
Calculate variance Residual(SSE)
Sequence of
Variance
Sum of
Squares
Degrees of
freedom
Mean of Square Ratio of F
Between columns SSC V1 = c – 1
MSC =
𝑆𝑆𝐶
𝑐 −1
F =
𝑀𝑆𝐶
𝑀𝑆𝐸
Between Rows SSR V2 = r – 1
MSR =
𝑆𝑆𝑅
𝑟−1
F =
𝑀𝑆𝑅
𝑀𝑆𝐸
Residual or Error SSE V3 =(c – 1) (r – 1)
MSE =
𝑆𝑆𝐶
(𝑐−1)(𝑟−1_
Total SST n – 1

4. Illustration
• A tea company appoints four salesmen - A,B,C and D and observes their sales in three seasons –
summer, winter and monsoon. The figures(in lakhs) are given in the following table
• (i) Do the salesman significantly differ in performance?
• (ii) Is there significant difference between the seasons?
Seasons Salesmen Season Total
A B C D
Summer 36 36 21 35 128
Winter 28 29 31 32 120
Monsoon 26 28 29 29 112
Salesmen total 90 93 81 96 360

5. Solution
• H0 1: There is no significant difference in performance of salesmen
• H0 2: There is no significant difference in seasons sales
• Find salesman’s sales average, season’s sales average and grand average
Seasons Salesmen Season Total Average =
Total/4A B C D
Summer 36 36 21 35 128 32
Winter 28 29 31 32 120 30
Monsoon 26 28 29 29 112 28
Salesmen total 90 93 81 96 360 32+30+28
3
= 30
Average = total/3 30 31 27 32 30+31+27+32
4
= 30
360
12
= 30
Grand
average

6. • Sum of Square between the salesmen
• SSC = 0+3+27+12 = 42
• MSC =
𝑆𝑆𝐶
𝐶−1
=
42
4−1
=14
Salesmansales
A
(30 -30)2
B
(30 -31)2
C
(27 -30)2
D
(32 -30)2
0 1 9 4
0 1 9 4
0 1 9 4
Total 0 3 27 12

7. • Sum of Square between the seasons
• SSC = 16+0+10 = 32
• MSC =
𝑆𝑆𝑅
𝑟−1
=
32
3−1
=16
Seasons sales Total
Summer
(32 -30)2
4 4 4 4 16
Winter
(30 -30)2
0 0 0 0 0
Monsoon
(28 -30)2
4 4 4 4 16

8. • Residual
• SSE = 136
• MSE =
𝑆𝑆𝐸
(𝑐−1)(𝑟−1)
=
136
3 𝑥 2
= 22. 67
Seasons Salesmen Season
A – 30 B – 31 C – 27 D – 32
(Summer – 32) (4)6 = 24 (4)5 = 20 (11)6 = 66 (3)3 = 9 119
(Winter – 30) (-2)-2 = 4 (-1)-2 = 2 (1)4 = 4 (2)0 =0 10
(Monsoon – 28) (-2)-4 = 8 (0)-3 =0 (1)2 = 2 (1)-3 = -3 7
Salesmen total 36 22 72 6 136

9. • Total sum of square
(Seasons – 30)2 (Salesmen – 30)2 Season Total
A B C D
Summer 36 36 81 25 178
Winter 4 1 1 4 10
Monsoon 16 4 1 1 22
Salesmen total 56 41 83 30 210

10. • Inference
• H0 - 1 There is no significant difference in performance of salesmen
• Table value for V1 = 3 and V3 = 6 at 5% level of significance is 4.76. CV < TV the hypothesis is
accepted
• H0 – 2 There is no significant difference in seasons sales
• Table value for V2 = 2 and V3 = 6 at 5% level of significance is 5.14. CV < TV the hypothesis is
accepted
Sequence of
Variance
Sum of
Squares
Degrees of
freedom
Mean of
Square
Ratio of F
Between columns 42 3 14
F =
𝑀𝑆𝐶
𝑀𝑆𝐸
=
14
22.67
= 1.619
Between Rows 32 2 16
F =
𝑀𝑆𝑅
𝑀𝑆𝐸
=
16
22.67
= 1.417
Residual or Error 136 6 22.67
Total 210 11