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 €“ T_XY, where T_X = ˆ‘n_i + (n_i+ €“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 × 2 tables, d equals the difference of proportions, and tau-b equals the correlation between X and Y.)

Transcribed Image Text:

Let d = (C – D)/:) - %3D