Dealing with a common problem in Social science: A simplified rank‐biserial coefficient of correlation based on the U statistic

Abstract

and Macalester College, MinnesotaIt has long been recognized that a common problem in describing ecological or social psychological relationships can involve variables one of which is ranked, the other dichotomous. According to Whitfield (1947), and valid even a t a time of justified multivariate emphasis, '. . . in applied or field work the relation of psychological "measurement" and an external criterion nearly always appears in this form.' Subsequent to Whitfield's own modification of Kendall's tau, several other rank-biserial coefficients have been defined. In particular, Brogden (1949), Cureton (1956) and Glass (1966) have written expressions based on Spearman-type approaches. An interesting aspect came to light when Stanley (1968) reemphasized their essential equality and, moreover, proved them to be identical to a point-biserial r corrected by its maximum theoretical value.Empirical studies to date d o not show any very prominent use (or awareness) of the available coefficients. A contemporary neglect of correlation in favor of significance statistics, on which Bakan (1966) and others have commented with occasional dismay, may have contributed to this development. But also -with the probable exception of Glass' rb coefficient -several of the measures are either not very easily computed under field conditions or the computational apparatus seems out of proportion vis-l-vis the kinds of small samples with which many of us are so familiar. Be this as i t may, we feel that simplified solutions will make the techniques more accessible and may in fact promote a better understanding of the problem area itself. In this connection we note, first, that numerous significance tests utilizing ranked data exhaust the same maximal information (cf. Lienert, 1962). Second, many comparisons of central tendency, especially in the case of small N, are currently performed using the Wilcoxon T or the Mann-Whitney U-tests.Given the above rationale a rank correlation, which we propose to call rc, can be defined as rL. = 1 ---where Pu is the theoretical mean for the sample sizes in question, with U as usually defined being the smaller of the two cumulative countsU max ---2 -2 obtained via the Mann-Whitney procedure. Since in turn, pu =

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