A note on the graphical representation of multivariate binary data

Various ordination methods for mapping n units characterized by v binary variables are in common use in which the distance between points Pi and Pj, representing units i and j, approximates some function (a similarity coefficient) of (aij ,bij, cij, dij), the usual cell-counts in a 2 × 2 table. Ordination generally requires (n - 1) dimensions to represent the distances exactly, but the quantities bij - cij can always be represented in one dimension. This leads to a simple graphical extension of ordination that helps with interpretation, reveals discrepancies, screens clustering possibilities and permits the recovery of approximations to all the (a,b,c,d)-values. Two examples illustrate the technique.