Research Methodology - Dr. Krishan K. Pandey
Research Methodology - Dr. Krishan K. Pandey Research Methodology - Dr. Krishan K. Pandey
Analysis of Variance and Co-variance 269 Table 11.6: Computations for Two-way Anova (in design with repeated values) 187 × 187 Step (i) T = 187, n = 18, thus, the correction factor = = 18 1942. 72 Step (ii) Total SS = [(14) 2 + (15) 2 + (12) 2 + (11) 2 + (10) 2 + (11) 2 + (10) 2 +(9) 2 + (7) 2 + (8) 2 + (11) 2 + (11) 2 + (11) 2 L Mb g O 18 NM P QP + (11) 2 + (10) 2 + (11) 2 + (8) 2 + (7) 2 ] – 187 2 = (2019 – 1942.72) = 76.28 × × × Step (iii) SS between columns (i.e., between drugs) = + + QP − 73 73 56 56 58 58 6 6 6 L NM = 888.16 + 522.66 + 560.67 – 1942.72 = 28.77 × × × Step (iv) SS between rows (i.e., between people) = + + QP − 70 70 59 59 58 58 6 6 6 L NM O O L 2 Mb187g O NM P 18 QP L 2 Mb187g O NM P 18 QP = 816.67 + 580.16 + 560.67 – 1942.72 = 14.78 Step (v) SS within samples = (14 – 14.5) 2 + (15 – 14.5) 2 + (10 – 9.5) 2 + (9 – 9.5) 2 + (11 – 11) 2 + (11 – 11) 2 + (12 – 11.5) 2 + (11 – 11.5) 2 + (7 – 7.5) 2 + (8 – 7.5) 2 + (10 – 10.5) 2 + (11 – 10.5) 2 + (10 – 10.5) 2 + (11 – 10.5) 2 + (11 – 11) 2 + (11 – 11) 2 + (8 – 7.5) 2 + (7 – 7.5) 2 = 3.50 Step (vi) SS for interaction variation = 76.28 – [28.77 + 14.78 + 3.50] = 29.23 Table 11.7: The Anova Table Source of variation SS d.f. MS F-ratio 5% F-limit Between columns (i.e., between drugs) Between rows (i.e.,between people) 28.77 (3 – 1) = 2 14.78 (3 – 1) = 2 28. 77 14. 385 2 0389 . = 14.385 = 36.9 14. 78 7390 . 2 0389 . = 7.390 = 19.0 F (2, 9) = 4.26 F (2, 9) = 4.26 ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ Contd.
270 Research Methodology Source of variation SS d.f. MS F-ratio 5% F-limit Interaction 29.23 * 4 * 29. 23 7308 . 4 0389 . Within samples (Error) 3.50 (18 – 9) = 9 Total 76.28 (18 – 1) = 17 350 . 9 = 0.389 F (4, 9) = 3.63 * These figures are left-over figures and have been obtained by subtracting from the column total the total of all other value in the said column. Thus, interaction SS = (76.28) – (28.77 + 14.78 + 3.50) = 29.23 and interaction degrees of freedom = (17) – (2 + 2 + 9) = 4. The above table shows that all the three F-ratios are significant of 5% level which means that the drugs act differently, different groups of people are affected differently and the interaction term is significant. In fact, if the interaction term happens to be significant, it is pointless to talk about the differences between various treatments i.e., differences between drugs or differences between groups of people in the given case. Graphic method of studying interaction in a two-way design: Interaction can be studied in a two-way design with repeated measurements through graphic method also. For such a graph we shall select one of the factors to be used as the X-axis. Then we plot the averages for all the samples on the graph and connect the averages for each variety of the other factor by a distinct mark (or a coloured line). If the connecting lines do not cross over each other, then the graph indicates that there is no interaction, but if the lines do cross, they indicate definite interaction or inter-relation between the two factors. Let us draw such a graph for the data of illustration 3 of this chapter to see whether there is any interaction between the two factors viz., the drugs and the groups of people. Graph of the averages for amount of blood pressure reduction in millimeters of mercury for different drugs and different groups of people.* (Blood pressure reduction in millimeters of mercury) Y-axis Groups of People 21 19 17 15 13 11 9 7 5 X Y Z Drugs Fig. 11.1 X-axis * Alternatively, the graph can be drawn by taking different group of people on X-axis and drawing lines for various drugs through the averages. A B C
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270 <strong>Research</strong> <strong>Methodology</strong><br />
Source of variation SS d.f. MS F-ratio 5% F-limit<br />
Interaction 29.23 * 4 * 29. 23 7308 .<br />
4 0389 .<br />
Within<br />
samples<br />
(Error)<br />
3.50 (18 – 9) = 9<br />
Total 76.28 (18 – 1) = 17<br />
350 .<br />
9<br />
= 0.389<br />
F (4, 9) = 3.63<br />
* These figures are left-over figures and have been obtained by subtracting from the column total the total of all other<br />
value in the said column. Thus, interaction SS = (76.28) – (28.77 + 14.78 + 3.50) = 29.23 and interaction degrees of freedom<br />
= (17) – (2 + 2 + 9) = 4.<br />
The above table shows that all the three F-ratios are significant of 5% level which means that<br />
the drugs act differently, different groups of people are affected differently and the interaction term<br />
is significant. In fact, if the interaction term happens to be significant, it is pointless to talk about the<br />
differences between various treatments i.e., differences between drugs or differences between<br />
groups of people in the given case.<br />
Graphic method of studying interaction in a two-way design: Interaction can be studied in a<br />
two-way design with repeated measurements through graphic method also. For such a graph we<br />
shall select one of the factors to be used as the X-axis. Then we plot the averages for all the samples<br />
on the graph and connect the averages for each variety of the other factor by a distinct mark (or a<br />
coloured line). If the connecting lines do not cross over each other, then the graph indicates that there<br />
is no interaction, but if the lines do cross, they indicate definite interaction or inter-relation between<br />
the two factors. Let us draw such a graph for the data of illustration 3 of this chapter to see whether<br />
there is any interaction between the two factors viz., the drugs and the groups of people.<br />
Graph of the averages for amount of blood pressure reduction in millimeters of<br />
mercury for different drugs and different groups of people.*<br />
(Blood pressure reduction<br />
in millimeters of mercury)<br />
Y-axis Groups of<br />
People<br />
21<br />
19<br />
17<br />
15<br />
13<br />
11<br />
9<br />
7<br />
5<br />
X Y Z<br />
<strong>Dr</strong>ugs<br />
Fig. 11.1<br />
X-axis<br />
* Alternatively, the graph can be drawn by taking different group of people on X-axis and drawing lines for various drugs<br />
through the averages.<br />
A<br />
B<br />
C
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