From the marginal likelihood of a two-way table to ecological inference

The paper extends in two directions the work of \cite{Plackett77} who studied how, in a $2\times 2$ table, the likelihood of the column totals depends on the odds ratio. First, we study the marginal likelihood of a single $R\times C$ frequency table when only the marginal frequencies are observed and then consider a collection of, say, $s$ $R\times C$ tables, where only the row and column totals can be observed, which is the basic framework which in applications of Ecological Inference. In the simpler context, we derive the likelihood equations and show that the likelihood has a collection of local maxima which, after a suitable rearrangement of the row and column categories, exhibit the strongest positive association compatible with the marginals, a kind of paradox, considering that the available data are so poor. Next, we derive the likelihood equations for the marginal likelihood of a collection of tow-way tables, under the assumption that they share the same row conditional distributions and derive a necessary condition for the information matrix to be well defined. We also describe a Fisher-scoring algorithm for maximizing the marginal likelihood which, however, can be used only if the number of available replications reaches a given threshold.