Research Methodology - Dr. Krishan K. Pandey
Research Methodology - Dr. Krishan K. Pandey Research Methodology - Dr. Krishan K. Pandey
Analysis of Variance and Co-variance 263 F H SS for total variance = ∑ X − X ij I K 2 i = 1, 2, 3… j = 1, 2, 3… = (6 – 5) 2 + (7 – 5) 2 + (3 – 5) 2 + (8 – 5) 2 + (5 – 5) 2 + (5 – 5) 2 + (3 – 5) 2 + (7 – 5) 2 + (5 – 5) 2 + (4 – 5) 2 + (3 – 5) 2 + (4 – 5) 2 = 1 + 4 + 4 + 9 + 0 + 0 + 4 + 4 + 0 + 1 + 4 + 1 = 32 Alternatively, it (SS for total variance) can also be worked out thus: SS for total = SS between + SS within = 8 + 24 = 32 We can now set up the ANOVA table for this problem: Table 11.2 Source of SS d.f. MS F-ratio 5% F-limit variation (from the F-table) Between sample 8 (3 – 1) = 2 8/2 = 4.00 4.00/2.67 = 1.5 F(2, 9) = 4.26 Within sample 24 (12 – 3) = 9 24/9 = 2.67 Total 32 (12 – 1) = 11 The above table shows that the calculated value of F is 1.5 which is less than the table value of 4.26 at 5% level with d.f. being v = 2 and v = 9 and hence could have arisen due to chance. This 1 2 analysis supports the null-hypothesis of no difference is sample means. We may, therefore, conclude that the difference in wheat output due to varieties is insignificant and is just a matter of chance. Solution through short-cut method: In this case we first take the total of all the individual values of n items and call it as T. T in the given case = 60 and n = 12 Hence, the correction factor = (T) 2 /n = 60 × 60/12 = 300. Now total SS, SS between and SS within can be worked out as under: 2 Total 2 bg T SS =∑Xij − i = 1, 2, 3, … n j = 1, 2, 3, …
264 Research Methodology = (6) 2 + (7) 2 + (3) 2 + (8) 2 + (5) 2 + (5) 2 + (3) 2 + (7) 2 + (5) 2 + (4) 2 + (3) 2 + (4) 2 – = 332 – 300 = 32 2 di Tj bg T SS between = ∑ − n n = F HG 24 × 24 4 I KJ F + HG 20 × 20 4 = 144 + 100 + 64 – 300 = 8 j I KJ F + HG T SS within 2 j =∑Xij −∑ n 2 F HG 16 × 16 4 60 × 60 12 I KJ F − HG I K J 60 × 60 = 332 – 308 = 24 It may be noted that we get exactly the same result as we had obtained in the case of direct method. From now onwards we can set up ANOVA table and interpret F-ratio in the same manner as we have already done under the direct method. TWO-WAY ANOVA Two-way ANOVA technique is used when the data are classified on the basis of two factors. For example, the agricultural output may be classified on the basis of different varieties of seeds and also on the basis of different varieties of fertilizers used. A business firm may have its sales data classified on the basis of different salesmen and also on the basis of sales in different regions. In a factory, the various units of a product produced during a certain period may be classified on the basis of different varieties of machines used and also on the basis of different grades of labour. Such a two-way design may have repeated measurements of each factor or may not have repeated values. The ANOVA technique is little different in case of repeated measurements where we also compute the interaction variation. We shall now explain the two-way ANOVA technique in the context of both the said designs with the help of examples. (a) ANOVA technique in context of two-way design when repeated values are not there: As we do not have repeated values, we cannot directly compute the sum of squares within samples as we had done in the case of one-way ANOVA. Therefore, we have to calculate this residual or error variation by subtraction, once we have calculated (just on the same lines as we did in the case of oneway ANOVA) the sum of squares for total variance and for variance between varieties of one treatment as also for variance between varieties of the other treatment. di j 2 12 I K J
- Page 230 and 231: Testing of Hypotheses I 213 Table 9
- Page 232 and 233: Testing of Hypotheses I 215 σ diff
- Page 234 and 235: Testing of Hypotheses I 217 Solutio
- Page 236 and 237: Testing of Hypotheses I 219 Hence t
- Page 238 and 239: Testing of Hypotheses I 221 of succ
- Page 240 and 241: Testing of Hypotheses I 223 Thus, q
- Page 242 and 243: Testing of Hypotheses I 225 the val
- Page 244 and 245: Testing of Hypotheses I 227 and 2 X
- Page 246 and 247: Testing of Hypotheses I 229 LIMITAT
- Page 248 and 249: Testing of Hypotheses I 231 20. Ten
- Page 250 and 251: Chi-square Test 233 10 Chi-Square T
- Page 252 and 253: Chi-square Test 235 S. No. 1 2 3 4
- Page 254 and 255: Chi-square Test 237 As a test of go
- Page 256 and 257: Chi-square Test 239 (i) First of al
- Page 258 and 259: Chi-square Test 241 The expected fr
- Page 260 and 261: Chi-square Test 243 Show that the s
- Page 262 and 263: Chi-square Test 245 Events or Expec
- Page 264 and 265: Chi-square Test 247 c h χ 2 N ⋅
- Page 266 and 267: Chi-square Test 249 from a number o
- Page 268 and 269: Chi-square Test 251 Questions 1. Wh
- Page 270 and 271: Chi-square Test 253 No. of boys 5 4
- Page 272 and 273: Chi-square Test 255 23. For the dat
- Page 274 and 275: Analysis of Variance and Co-varianc
- Page 276 and 277: Analysis of Variance and Co-varianc
- Page 278 and 279: Analysis of Variance and Co-varianc
- Page 282 and 283: Analysis of Variance and Co-varianc
- Page 284 and 285: Analysis of Variance and Co-varianc
- Page 286 and 287: Analysis of Variance and Co-varianc
- Page 288 and 289: Analysis of Variance and Co-varianc
- Page 290 and 291: Analysis of Variance and Co-varianc
- Page 292 and 293: Analysis of Variance and Co-varianc
- Page 294 and 295: Analysis of Variance and Co-varianc
- Page 296 and 297: Analysis of Variance and Co-varianc
- Page 298 and 299: Analysis of Variance and Co-varianc
- Page 300 and 301: Testing of Hypotheses-II 283 12 Tes
- Page 302 and 303: Testing of Hypotheses-II 285 Illust
- Page 304 and 305: Testing of Hypotheses-II 287 Table
- Page 306 and 307: Testing of Hypotheses-II 289 fact,
- Page 308 and 309: Testing of Hypotheses-II 291 The te
- Page 310 and 311: Testing of Hypotheses-II 293 Pair B
- Page 312 and 313: Testing of Hypotheses-II 295 Table
- Page 314 and 315: Testing of Hypotheses-II 297 Illust
- Page 316 and 317: Testing of Hypotheses-II 299 Soluti
- Page 318 and 319: Testing of Hypotheses-II 301 r = nu
- Page 320 and 321: Testing of Hypotheses-II 303 where
- Page 322 and 323: Testing of Hypotheses-II 305 Table
- Page 324 and 325: Testing of Hypotheses-II 307 We now
- Page 326 and 327: Testing of Hypotheses-II 309 (ii) I
- Page 328 and 329: Testing of Hypotheses-II 311 all po
264 <strong>Research</strong> <strong>Methodology</strong><br />
= (6) 2 + (7) 2 + (3) 2 + (8) 2 + (5) 2 + (5) 2 + (3) 2<br />
+ (7) 2 + (5) 2 + (4) 2 + (3) 2 + (4) 2 –<br />
= 332 – 300 = 32<br />
2<br />
di Tj<br />
bg T<br />
SS between = ∑ −<br />
n n<br />
=<br />
F<br />
HG<br />
24 × 24<br />
4<br />
I<br />
KJ F + HG<br />
20 × 20<br />
4<br />
= 144 + 100 + 64 – 300<br />
= 8<br />
j<br />
I<br />
KJ F + HG<br />
T<br />
SS within 2 j<br />
=∑Xij −∑<br />
n<br />
2<br />
F<br />
HG<br />
16 × 16<br />
4<br />
60 × 60<br />
12<br />
I<br />
KJ F − HG<br />
I K J<br />
60 × 60<br />
= 332 – 308<br />
= 24<br />
It may be noted that we get exactly the same result as we had obtained in the case of direct<br />
method. From now onwards we can set up ANOVA table and interpret F-ratio in the same manner<br />
as we have already done under the direct method.<br />
TWO-WAY ANOVA<br />
Two-way ANOVA technique is used when the data are classified on the basis of two factors. For<br />
example, the agricultural output may be classified on the basis of different varieties of seeds and also<br />
on the basis of different varieties of fertilizers used. A business firm may have its sales data classified<br />
on the basis of different salesmen and also on the basis of sales in different regions. In a factory, the<br />
various units of a product produced during a certain period may be classified on the basis of different<br />
varieties of machines used and also on the basis of different grades of labour. Such a two-way design<br />
may have repeated measurements of each factor or may not have repeated values. The ANOVA<br />
technique is little different in case of repeated measurements where we also compute the interaction<br />
variation. We shall now explain the two-way ANOVA technique in the context of both the said<br />
designs with the help of examples.<br />
(a) ANOVA technique in context of two-way design when repeated values are not there: As we<br />
do not have repeated values, we cannot directly compute the sum of squares within samples as we<br />
had done in the case of one-way ANOVA. Therefore, we have to calculate this residual or error<br />
variation by subtraction, once we have calculated (just on the same lines as we did in the case of oneway<br />
ANOVA) the sum of squares for total variance and for variance between varieties of one<br />
treatment as also for variance between varieties of the other treatment.<br />
di<br />
j<br />
2<br />
12<br />
I K J
Change language