Multidimensional ranking tests

Textbooks of nonparametric statistics propose a few methods only for the analysis of bi- or multivariate semiquantitative data. Section 5.1 has shown that there actually exist many numerical approaches for analysing multidimensional data, corresponding to all levels of precision (Table 5.1). These methods, which include most of those described in this book, belong to nonparametric statistics in a general sense, because they do not focus on the parameters of the data distributions. Within the specific realm of ranking tests, however, the only statistical techniques available for

Table 5.3 Numerical example. Perfect rank correlation between descriptors yj and y2.

Objects Ranks of objects on the two descriptors

(observation units) yj y2

x5 33

multidimensional semiquantitative data are two rank correlation coefficients (Spearman r and Kendall t), which both quantify the relationship between two descriptors, and the coefficient of concordance (Kendall W), which assesses the relationship among several descriptors. These are described in some detail in the present section.

Spearman The Spearman r statistic, also called p (rho), is based on the idea that two corr. coeff. descriptors yj and y2 carry the same information if the largest object on yj also has the highest rank on y2, and so on for all other objects. Two descriptors are said to be in perfect correlation when the ranks of all object are the same on both descriptors, as in the numerical example of Table 5.3. If, however, object xj which has rank 5 on yj had rank 2 on y2, it would be natural to use the difference between these ranks dj = (yjj - ^2) = (5 - 2) = 3 as a measure of the difference between the two descriptors, for this object. For the whole set of objects, differences di are squared before summing them, in order to prevent differences with opposite signs from cancelling each other out.

The expression for the Spearman r may be derived from the general formula of correlation coefficients (Kendall, !948):

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