Distribution of the Spearman’s Rank Correlation Coefficient
Methods based not on observed values of random variables but on ranks corresponding to these values are more and more used in statistical research. On the ground of research in the rank application theory we can formulate the thesis that statistical methods based on ranks allow to solve majority of problems of statistical practice. The object of our consideration is the Spearman’s rank correlation coefficient which is often applied to empirical research. The aim of this work ist to show recurrent formula of the distribution of the Spearman’s rank correlation coefficient which allows to construct the tables of its basic quantiles. Exact distribution of the Spearman’s rank correlation coefficient was tabled first by Owen (1962), and next for n = 12(1)16 by Otten (1973). Zar (1972) used approximation based on the Pearson’s distribution of the 2nd type and showed critical values for n = 4(1)50(2)100 and α = 0.50,0.20,0.10,0.05,0.02,0.01,0.005,0.002,0.001. It can be noticed that 16 values (given by Zar (1972)) out of 45, differ from the values given by Otten (1973). Apart from the methods of the Spearman’s rank correlation coefficient approximation, it is worth considering the possibility of finding the numerical effective algorithm for calculation the direct distribution. To construct tables published so far (compare Otten (1973)), an algorithm generating all or a half of permutations of n-element set was used. In spite of the development of computer calculations techniques this method seems to be useless in case of n much greater than 20.
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