Weighted Chi Square: an Extension of the Kappa Method

IN the usual analysis of contingency tables by means of x2, it is frequently the case that the investigator has a priori expectations that a specified few of the cells will have larger than chance pro-portions (or frequencies), while he has no particular hypotheses with regard to the remaining cells. Or, if his theory is stronger, he may be able to articulate his hypotheses about the outcome for each cell with greater refinement, distinguishing chance and larger and smaller than chance expectations in varying degrees. In either case, neither the measure of association he computes (if any) nor the x2 test he performs takes into account in any way his a priori hypoth-eses about the contingency table. They merely index the degree and significance of the collective departure from chance, in all directions and degrees indiscriminately, of the values he observes in the cells when he has collected and organized his data. This article presents a very general method for the study of m-way tables of proportions or frequencies (where m is one or more) in which the investigator’s a priori hypotheses about the cells are expressed numerically and used as weights. These weights are then used in ~ an index of hypothesized association, and also in a test of its significance, weighted x2 (xw2), which thus utilizes as relevant information the investigator’s hypotheses (anticipations, hunches, theory). 1 The material on which this article is based was presented at the Oxford University meeting of the European Society for Multivariate Experimental