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When X and Y are ordinal with counts {n_{ij}}:

a. Explain why the (^{n}_{ 2}) pairs of observations partition into C + D + T_{X} + T_{Y} â€“ T_{XY}, where T_{X} = âˆ‘n_{i} + (n_{i+} â€“1)/2 pairs are tied on X, T_{Y} pairs are tied on Y, and T_{XY} pairs are tied on X and Y.

b.

Explain why d is the difference between the proportions of concordant and discordant pairs out of those pairs united on X (Somers 1962). (For 2 Ã— 2 tables, d equals the difference of proportions, and tau-b equals the correlation between X and Y.)

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