Singularities in the analysis of binomial data

Arrays of zeros and ones in two or more dimensions may be analysed in terms of scales (Guttman, 1944), the idea being to permute the rows, columns, etc. of the array so that it becomes as monotonic as possible, according to some criterion, in each dimension. Arrays of relative frequencies are also sometimes analysed in this way, especially if most or many of the entries are zero or one. More commonly, however, frequency data are fitted by transformed additive models which decompose the binomial probabilities underlying the data into separate row, column, etc., effects (Cox, 1970; Haberman, 1974). In this paper we Bhow that by suitably formulating the notion of a scale, and by using Haberman's extension of the maximum likelihood criterion to allow singular estimates, the scaling and parametric approaches to the analysis of frequency data are made to coincide. We work in two dimensions, but our definitions, theorem and proofs carry over in obvious ways to higher dimensions. Let P be an m x n array of binomial probabilities where each ptj independently generates rti successes out of N{i > 0 trials. We assume that where/is a strictly increasing function on [ — oo, +oo] whose range is the entire interval [0,1], and which assures the strict concavity of the likelihood function, e.g. the cumulative normal or the logistic transform. Unlike untransformed linear models where use of least squares assures finite parameter estimates, transformed models for fitting binomial frequencies admit divergent maximum likelihood estimates for ft, the o4 and the by Various sets of necessary and sufficient conditions for finite estimates have been studied by Haberman (1974, 1977), Wedderburn (1976) and others. Our main result here is a theorem giving necessary and sufficient conditions, in terms of the scalability of the data matrix rjN, for obtaining the various possible patterns of singularities in the parameter estimates. These singularities are interpretable in a well-defined way, even when the model apparently involves the addition of two infinite quantities of opposite sign.