Integer estimation of inner-cell values in RxC ecological tables

Abstract

The estimation of the inner entries of a set of related RxC contingency tables when only the margins are known poses one of the most difficult problems in the field of the social sciences, present in many areas from marketing to quantitative history; being particularly prevalent in political science and sociology. With dozens of methods proposed to solve this, (almost) all of them have been devised to infer conditional proportion distributions, despite interior values of contingency tables being integers. This paper develops, within the linear programming framework, a new algorithm to output integer solutions, and assesses it using real data from more than 500 elections where actual cross-distributions of votes are known. Although the new approach is proposed with the expectation that more accurate solutions would be obtained by narrowing the search space from a continuous to a discrete simplex space, the results attained suggest that the use of a pure integer approach does not lead to more accurate solutions. The recommendation is therefore to integer-adjust decimal solutions when the focus is on counts. Interested practitioners can easily use the new models as they have been programmed in the R-package lphom.