Yield in mt treatment 1 248 325 30 345 394 treatment

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Yield (in MT) Treatment 1 2.48 3.25 3.87 3.6 4.0 3.0 3.45 3.94 Treatment 2 3.88 2.87 3.27 2.8 2.84 3.1 3.5 2.37 Treatment 3 2.69 2.88 3.4 3.17 3.44 3.15 2.46 2.85 Solution The sequence of orderings and their ranks for the data are given as
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127 15.3.4. RANKING FOR THE KRUSKAL WALLIS TEST Score 2.37 2.46 2.48 2.69 2.8 2.84 2.85 2.87 2.88 Rank 1 2 3 4 5 6 7 8 9 Sample 2 3 1 3 2 2 3 2 3 Score 3.0 3.1 3.15 3.17 3.25 3.27 3.4 3.44 3.45 Rank 10 11 12 13 14 15 16 17 18 Sample 1 2 3 3 1 2 3 3 1 Score 3.5 3.6 3.87 3.88 3.94 4.0 Rank 19 20 21 22 23 24 Sample 2 1 1 2 1 1 We obtain the following totals; 1 r = sum of the ranks of elements of sample 1=133 2 r = sum of the ranks of the elements of sample 2 = 87 3 r = sum of the ranks of the elements of sample 3 = 80 Using the Kruskal Wallis test to ascertain the validity of the null hypothesis that all the three samples belong to the same population, we have 2 1 1 12 3( 1) ( 1) k i i r H n n n n 2 2 2 12 133 87 80 3 25 24(24 1) 8 8 8 79.145 75 4.145 H H   The critical value for H from 2 distribution for = 0.05 with d.f.=2 is 5.99. Thus, the null hypothesis is not rejected; i.e., all the three treatments have similar yields. Example - 4 Use Kruskal-Wallis test to determine whether there is a significant difference in the following populations. Use 0.05 level of significance Population 1 : 17 19 27 20 35 40 Population 2 : 28 36 33 22 27 Population 3 : 37 30 39 42 28 25 31
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128 Solution Three populations are considered for study so k=3 and n=18. The observations in three populations are combined and ranked. The smallest value is given rank 1 , as shown bellow Population 1 Population 2 Population 3 Value Rank Value Rank Value Rank 17 19 20 27 35 40 1 2 3 6.3 13 17 22 27 28 33 36 4 6.5 8.5 12 14 25 28 30 31 37 39 42 5 8.5 10 11 15 16 18 1 6 n 1 42.5 r 2 5 n 2 45 r 3 7 n 3 83.5 r Suppose 0 : H Three population are identical 1 2 3 1 1 2 3 . ., : i e H The Kruskal- Wallis test statistic is 2 1 2 2 2 12 3 1 1 42.5 45 83.5 12 3 19 18(18 1) 6 5 7 0.035 301.4 405 996.03 57 2.572 k i i i r H n n n n H H H Since computed value of ( 2.572) H is less than table value of 2 5.99 at d.f= k-1 =2 and 0.05 , the null hypothesis is accepted and conclude that three populations are identical. 15.4. REVISION POINTS 1) Mann-whitney U test is based on median and ranking of the data and is equivalent to wilcoxon rank-sum test. 2) Kruskal-walis one way analysis of variance is applied to populations from which the samples drawn are not normally distributed with equal variances, or when the data for analysis consists only of ranks.
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129 15.5. INTEXT QUESTIONS 1) Explicate Mann-Whitney U test for testing the identicalness of two populations. 2) Which non-parametric tests are substitutes for the analysis of variance ?
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