Further Examples Related to Correlations Between Variables and Ranks

Chunming Zhang

The American Statistician, 2020, vol. 75, issue 2, 226-229

Abstract: Rank statistics {R1,…,Rn} of actual variates {X1,…,Xn} play an important role in university undergraduate nonparametric statistics courses. This article derives explicit expressions of the correlation coefficients between Xi and Rj for not only i = j but also i≠j, for iid continuous variables X1,…,Xn with a distribution function FX(·) of X and n≥2: (a) ρXi,Ri=n−1n+1 ρX,FX(X)∈(0,n−1n+1] for any i, revealing that the correlation can be as close to one as expected, while may also unexpectedly decrease approaching zero for other distributions of X; (b) ρXi,Rj=−1n−1ρXi,Ri∈[−1n2−1,0) for any i≠j, inferring a negligible negative association with ranks from other data; (c) the partial correlation coefficient between Xi and Ri on Xj for any i≠j equals ρ(Xi,Ri)·Xj=ρXi,Ri/1−ρXj,Ri2∈(ρXi,Ri,n−1n2−2], invariably exceeding ρXi,Ri. Implications of the results necessitate more relevant interpretation of ranks in sharing information of data.

Date: 2020
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DOI: 10.1080/00031305.2020.1831956

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