Somers' D: A common currency for associations

Somers' D(Y|X) is an asymmetric measure of ordinal association between two variables Y and X, on a scale from –1 to 1. It is defined as the difference between the conditional probabilities of concordance and discordance between two randomly-sampled (X,Y)-pairs, given that the two X-values are ordered. The somersd package enables the user to estimate Somers' D for a wide range of sampling schemes, allowing clustering and/or sampling-probability weighting and/or restriction to comparisons within strata. Somers' D has the useful feature that a larger D(Y|X) cannot be secondary to a smaller D(W|X) with the same sign, enabling us to make scientific statements that the first ordinal association cannot be caused by the second. An important practical example, especially for public-health scientists, is the case where Y is an outcome, X an exposure, and W a propensity score. However, an audience accustomed to other measures of association may be culture-shocked, if we present associations measured using Somers’ D. Fortunately, under some commonly-used models, Somers' D is related monotonically to an alternative association measure, which may be more clearly related to the practical question of how much good we can do. These relationships are nearly linear (or log-linear) over the range of Somers' D values from –0.5 to 0.5. We present examples with X and Y binary, with X binary and Y a survival time, with X binary and Y conditionally Normal, and with X and Y bivariate Normal. Somers' D can therefore be used as a common currency for comparing a wide range of associations between variables, not limited to a particular model.