The Marginal Likelihood of two-way tables and Ecological Inference

TL;DR

This paper develops new theoretical results on the marginal likelihood of two-way tables, providing conditions for ecological inference and an efficient algorithm for likelihood maximization, extending previous work to general tables.

Contribution

It generalizes Placket's 1977 work to R x C tables, introduces new geometric tools, and proposes an efficient Fisher scoring algorithm for likelihood maximization.

Findings

New insights into the geometry of fixed-margin tables

An efficient algorithm for likelihood maximization

Simulation results comparing methods

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.