On the central role of Somers' D

Somers' D and Kendall's tau-a are parameters behind rank or nonparametric statistics, interpreted as differences between proportions. Given two bivariate data pairs (X1, Y1) and (X2, Y2), Kendall’s tau-a parameter tau-XY is the difference between the probability that the two X–Y pairs are concordant and the probability that the two X–Y pairs are discordant, and Somers' D parameter DYX is the difference between the corresponding conditional probabilities, given that the X-values are ordered. The somersd package computes confidence intervals for both parameters. The Stata 9 version of somersd uses Mata to increase computing speed and greatly extends the definition of Somers' D, allowing the X and/or Y variables to be left- or right-censored and allowing multiple versions of Somers' D for multiple sampling schemes for the X–Y pairs. In particular, we may define stratified versions of Somers' D, in which we compare only X–Y pairs from the same stratum. The strata may be defined by grouping a Rubin–Rosenbaum propensity score, based on the values of multiple confounders for an association between exposure variable X and an outcome variable Y . Therefore, rank statistics can have not only confidence intervals but also confounder-adjusted confidence intervals. Usually, we either estimate DYX as a measure of the effect of X on Y , or we estimate DXY as a measure of the performance of X as a predictor of Y, compared with other predictors. Alternative rank-based measures of the effect of X on Y include the Hodges–Lehmann median difference and the Theil–Sen median slope, both of which are defined in terms of Somers' D.