a general non-linear index of association between two continuous rank-order variables

Non-linear dependence between two continuous variables has been given but little consideration among statisticians to this day, and no correlation index has been contrived, apart from the semi-categorized eta square coefficient in the anova context. Here, a non-parametric, rank-based approach is implemented, giving rise to two coefficients, RY, which measures the non-linear (and non-monotonic) variation of the Y series concomitant to the X series, and RXY, a symmetrised measure of the non-linear correspondence between the two series. The gist of the approach resides in the postulate that, if the series are related in any manner, numerically consecutive values of one variable should be linked to values of the other variable having reduced mutual differences. RY and RXY are presented here, with their first moments and sets of exact and approximate critical values, and they are the distribution-free counterparts of coefficients A and AS (Laurencelle, 2012) formerly presented for the normal parametric context.