A Generalisation of Emerson’s recurrence formulae and the Gray-Williams index

Abstract When analysing the association between the ordered categorical variables of a contingency table, orthogonal polynomials derived from the recurrence formulae of Emerson (1968, Biometrics , 24: 695 - 701) have been extensively used. The calculation of such polynomials is somewhat limited because they reflect only the univariate structure of each variable. This paper proposes a new generalisation of Emerson’s recurrence formulae that reflects bivariate and, more generally, multivariate, association structures for the construction of orthogonal polynomials for multi-way contingency tables. We shall demonstrate the utility of this generalisation by giving special attention to the Gray-Williams index, a three-way variant of the Goodman-Kruskal tau index.