Research Methodology - Dr. Krishan K. Pandey
Research Methodology - Dr. Krishan K. Pandey Research Methodology - Dr. Krishan K. Pandey
Testing of Hypotheses-II 289 fact, equally good, (alternatively the probability that the particular result or worse for B group would occur) be worked out. This should be done keeping in view the probability principles. For the given case, the probability that Group A has the particular result or a better one, given the null hypothesis that the two programmes are equally good, is as follows: Pr. of Group A doing as well or better = Pr. (5 passing and 1 failing) + Pr. (6 passing and 0 failing) = 8 5 4 1 C × C 12 C 6 + 8 4 C6× C 12 C 224 28 = + = 024 . + 003 . = 027 . 924 924 Alternatively, we can work out as under: Pr. of Group B doing as well or worse = Pr. (3 passing and 3 failing) + Pr. (2 passing and 4 failing) = 8 4 C3× C 12 C 6 3 + 8 6 4 0 C2× C 12 C 6 4 224 28 = + = 024 . + 003 . = 027 . 924 924 Now we have to compare this calculated probability with the significance level of 5% or 0.05 already specified by the management. If we do so, we notice that the calculated value is greater than 0.05 and hence, we must accept the null hypothesis. This means that at a significance level of 5% the result obtained in the above table is not significant. Hence, we can infer that both training programmes are equally good. This test (Fisher-Irwin test), illustrated above, is applicable for those situations where the observed result for each item in the sample can be classified into one of the two mutually exclusive categories. For instance, in the given example the worker’s performance was classified as fail or pass and accordingly numbers failed and passed in each group were obtained. But supposing the score of each worker is also given and we only apply the Fisher-Irwin test as above, then certainly we are discarding the useful information concerning how well a worker scored. This in fact is the limitation of the Fisher-Irwin test which can be removed if we apply some other test, say, Wilcoxon test as stated in the pages that follow. 3. McNemer Test McNemer test is one of the important nonparametric tests often used when the data happen to be nominal and relate to two related samples. As such this test is specially useful with before-after measurement of the same subjects. The experiment is designed for the use of this test in such a way that the subjects initially are divided into equal groups as to their favourable and unfavourable views about, say, any system. After some treatment, the same number of subjects are asked to express their views about the given system whether they favour it or do not favour it. Through McNemer test we in fact try to judge the significance of any observed change in views of the same subjects before
290 Research Methodology and after the treatment by setting up a table in the following form in respect of the first and second set of responses: Table 12.3 Before treatment After treatment Do not favour Favour Favour A B Do not favour C D Since A + D indicates change in people’s responses (B + C shows no change in responses), the expectation under null hypothesis H 0 is that (A + D)/2 cases change in one direction and the same proportion in other direction. The test statistic under McNemer Test is worked out as under (as it uses the under-mentioned transformation of Chi-square test): χ 2 = c A − D − 1h bA + Dg 2 with d.f. = 1 The minus 1 in the above equation is a correction for continuity as the Chi-square test happens to be a continuous distribution, whereas the observed data represent a discrete distribution. We illustrate this test by an example given below: Illustration 3 In a certain before-after experiment the responses obtained from 1000 respondents, when classified, gave the following information: Before treatment After treatment Unfavourable Favourable Response Response Favourable response 200 =A 300 =B Unfavourable response 400 = C 100 = D Test at 5% level of significance, whether there has been a significant change in people’s attitude before and after the concerning experiment. Solution: In the given question we have nominal data and the study involves before-after measurements of the two related samples, we can use appropriately the McNemer test. We take the null hypothesis (H 0 ) that there has been no change in people’s attitude before and after the experiment. This, in other words, means that the probability of favourable response before and unfavourable response after is equal to the probability of unfavourable response before and favourable response after i.e., H 0 : P(A) = P (D) We can test this hypothesis against the alternative hypothesis (H a ) viz., H a : P (A) ≠ P (D)
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290 <strong>Research</strong> <strong>Methodology</strong><br />
and after the treatment by setting up a table in the following form in respect of the first and second<br />
set of responses:<br />
Table 12.3<br />
Before treatment After treatment<br />
Do not favour Favour<br />
Favour A B<br />
Do not favour C D<br />
Since A + D indicates change in people’s responses (B + C shows no change in responses), the<br />
expectation under null hypothesis H 0 is that (A + D)/2 cases change in one direction and the same<br />
proportion in other direction. The test statistic under McNemer Test is worked out as under (as it<br />
uses the under-mentioned transformation of Chi-square test):<br />
χ 2<br />
=<br />
c A − D − 1h<br />
bA + Dg<br />
2<br />
with d.f. = 1<br />
The minus 1 in the above equation is a correction for continuity as the Chi-square test happens to be<br />
a continuous distribution, whereas the observed data represent a discrete distribution. We illustrate<br />
this test by an example given below:<br />
Illustration 3<br />
In a certain before-after experiment the responses obtained from 1000 respondents, when classified,<br />
gave the following information:<br />
Before treatment After treatment<br />
Unfavourable Favourable<br />
Response Response<br />
Favourable response 200 =A 300 =B<br />
Unfavourable response 400 = C 100 = D<br />
Test at 5% level of significance, whether there has been a significant change in people’s attitude<br />
before and after the concerning experiment.<br />
Solution: In the given question we have nominal data and the study involves before-after<br />
measurements of the two related samples, we can use appropriately the McNemer test.<br />
We take the null hypothesis (H 0 ) that there has been no change in people’s attitude before and<br />
after the experiment. This, in other words, means that the probability of favourable response before<br />
and unfavourable response after is equal to the probability of unfavourable response before and<br />
favourable response after i.e.,<br />
H 0 : P(A) = P (D)<br />
We can test this hypothesis against the alternative hypothesis (H a ) viz.,<br />
H a : P (A) ≠ P (D)
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