The Marginal Likelihood of two-way tables and Ecological Inference
Antonio Forcina
TL;DR
The paper tackles identifiability of association structure in two-way contingency tables using marginal data. It shows that for a single $R\times C$ table, the marginal likelihood attains local maxima on extreme tables with log-odds diverging to $+\infty$, making the conditional row distributions unidentifiable from margins alone. In the second part, it extends the framework to a collection of tables that share the same row-conditional structure and introduces an efficient Fisher scoring algorithm to maximize the exact multinomial marginal likelihood, supported by a small simulation study comparing with established ecological inference methods. The analysis clarifies the geometry of the feasible table set (Fréchet class), leverages the extended hypergeometric distribution for conditional expectations, and discusses computational challenges for practical ecological inference.
Abstract
The paper derives new results on the marginal likelihood of a two-way table which clarify the conditions under which Ecological inference is possible and lead to an efficient algorithm for maximizing the exact multinomial likelihood. The first part generalizes the work of Placket(1977} on the marginal likelihood of a 2 x 2 table to a general R x C table. In doing so, new conceptual tools are introduced and new insights on the geometry of the collection of tables having fixed row and column margins and the extended hypergeometric distribution are derived. In the second part, when observations on the row and the column marginal distributions are available for a collection of two-way tables sharing the same association structure, an efficient Fisher scoring algorithm for maximizing the exact likelihood under multinomial sampling is introduced and a small simulation study is used to compare the performance of the proposed method with two well established ones.