i = i i = i v i=i ^ and the second is that the sum of the all R values in the table is pn(n + 1)/2.

Coefficient W varies between 0 (no concordance) and 1 (maximum concordance).

Its significance is tested either using eq. 5.11 directly, or after transforming W into the

associated X2 statistic:

The null hypothesis (Ho) subjected to testing is that the row sums R are equal or, in other words, that the p sets of ranks (or the p semiquantitative descriptors) are

2 2 independent of one another. The X2 statistic is compared to a y„ value read in a table 2 2 2 of critical values of %2, for v = (n - 1). When X2 is smaller than the critical value xa

(i.e. probability larger than a), the null hypothesis that the row sums R are equal cannot be rejected; this leads to the conclusion that the p descriptors are independent and differ in the way they rank the n objects. On the contrary, X2 > %a (i.e. probability smaller than or equal to a) indicates good agreement among the descriptors in the way they rank the objects. Textbooks of nonparametric statistics provide modified formulae for X2, for data sets with tied observations.

Table 5.7 Numerical example. Ranks of six objects on three descriptors, y1, y2, and y3.

Objects Ranks of objects on the three descriptors Row sums

(observation units) y1 y2 y3 Ri

x1 |

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