Fibers of multi-way contingency tables given conditionals: relation to marginals, cell bounds and Markov bases

Abstract

A reference set, or a fiber, of a contingency table is the space of all realizations of the table under a given set of constraints such as marginal totals. Understanding the geometry of this space is a key problem in algebraic statistics, important for conducting exact conditional inference, calculating cell bounds, imputing missing cell values, and assessing the risk of disclosure of sensitive information.Motivated primarily by disclosure limitation problems where constraints can come from summary statistics other than the margins, in this paper we study the space F T of all possible multi-way contingency tables for a given sample size and set of observed conditional frequencies. We show that this space can be decomposed according to different possible marginals, which, in turn, are encoded by the solution set of a linear Diophantine equation. We characterize the difference between two fibers: F T and the space of tables for a given set of corresponding marginal totals. In particular, we solve a generalization of an open problem posed by Dobra et al. (2008). Our decomposition of F T has two important consequences: (1) we derive new cell bounds, some including connections to Directed Acyclic Graphs, and (2) we describe a structure for the Markov bases for the space F T that leads to a simplified calculation of Markov bases in this particular setting.AMS 2000 subject classifications: 13P10, 62B05, 62H17, 62P25.

An Intuitive Formulation and Solution of the Exact Cell-Bounding Problem for Contingency Tables of Conditional Frequencies

“…Let ν Note that this theorem appears in a somewhat different form in the paper by Slavković et al [22]. It shows that we can replace the 2J problems (1)-(5) for cells ı in rowī with an equivalent pair of simpler problems for the entire rowī.…”

“…The details are algorithm-specific and beyond the scope of the present work; a forthcoming paper of the authors presents such an algorithm. Algebraic table-enumeration concepts can also be used (e.g., [22]). …”

“…As we prepared this paper for submission, we became aware of a recent unpublished manuscript [22] that examines the same problem as this paper through the lens of algebraic statistics. Their core result agrees with ours, namely, that the heart of the matter lies in the Diophantine equation that constitutes the principal constraint in our knapsack formulation.…”

“…Both their paper and ours recognize that this provides a characterization of the case in which all cell counts are disclosed, but the papers largely diverge from there. Slavković et al [22] use algebraic methods and number theory to count the number of possible tables for a given set of conditionals, and then suggest using a full enumeration of those tables to obtain cell bounds from unrounded conditionals. Using Markov bases and graph theory, they provide interesting results on how several tables might be combined to produce bounds and they also address some issues pertaining to inference.…”

Conditional inference given partial information in contingency tables using Markov bases

“…In general, the Markov bases methodology and other tools from algebraic statistics can help in three related problems that are relevant for evaluating disclosure risk and data utility: (1) characterizing conditions under which released fragmentary data will uniquely identify the original table, thus leading to full disclosure of the original table; (2) computing sharp bounds on cell counts with small entries; and (3) counting the number of feasible tables consistent with released data. For further discussion on these problems, see Refs 30, 3, 31, 32, and references therein. Chapter 14 of Ref 5 also relates the results on Markov basis for preserving marginals to another commonly used disclosure control technique of data swapping.…”