Concordance between two linear orders: The Spearman and Kendall coefficients revisited

This paper discusses the two classic measures of concordance between two linear orders L and L′, the Kendall tau and the Spearman rho, equivalently, the Kendall and Spearman distances between such orders. We give an expression for ρ(L,L′)?τ(L,L′) as a function of the parameters of the partial order L∪L′, which allows the determination of extremal values for this difference and an investigation of when tau and rho are equal. This expression for ρ(L,L′)?τ(L,L′) is derived from a relation between the Kendall and Spearman distances between linear orders that is equivalent to both the Guilbaud (1980) formula linking rho, tau, and a third coefficient sigma, and Daniels’s (1950) inequality. We also prove an apparently new monotonicity property of rho. In the conclusion we point out possible extensions and add general historical comments.