Small Contingency Tables with Large Gaps

  title={Small Contingency Tables with Large Gaps},
  author={Seth Sullivant},
  journal={SIAM J. Discret. Math.},
  • S. Sullivant
  • Published 3 May 2004
  • Mathematics
  • SIAM J. Discret. Math.
We construct examples of contingency tables on n binary random variables where the gap between the linear programming lower/upper bound and the true integer lower/upper bounds on cell entries is exponentially large. These examples provide evidence that linear programming may not be an effective heuristic for detecting disclosures when releasing margins of multiway tables. 
On the occurrence of large gaps in small contingency tables
Examples of small contingency tables on binary random variables with large integer programming gaps on the lower bounds of cell entries were constructed by Sullivant. We argue here that the margins
Cell Bounds in Two-Way Contingency Tables Based on Conditional Frequencies
This paper derives the closed-form solutions for the linear relaxation bounds on cell counts of a two-way contingency table given observed conditional probabilities and compute the corresponding sharp integer bounds via integer programming and shows that there can be large differences in the width of these bounds.
Cell Bounds in k-way Tables Given Conditional Frequencies
This work considers bounds on cell counts for k-way tables when observed conditional probabilities and total sample size are released and compute sharp integer bounds using integer programming and demonstrates that, in some cases, they can be unacceptably narrow.
Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation
Contingency tables have provided a fertile ground for the growth of algebraic statistics. In this paper we briefly outline some features of this work and point to open research problems. We focus on
Integrality Gaps of Integer Knapsack Problems
In a randomised setting, it is shown that the integrality gap of a “typical” knapsack problem is drastically smaller than the integralities gap that occurs in a worst case scenario.
Algebraic and Geometric Methods in Statistics: The generalised shuttle algorithm
The Generalized Shuttle Algorithm for computing integer bounds of multi-way contingency tables induced by arbitrary linear constraints on cell counts is described and illustrated how the algorithm can be used to compute exact p-values of goodness-of-fit tests in exact conditional inference.
Partial Information Releases for Confidential Contingency Table Entries: Present and Future Research Efforts
The main focus is the partial information release strategy, through which agencies can release relevant marginal and conditional information along with the sample size instead of a full contingency table.
Algebraic statistics
This tutorial will consist of a detailed study of two examples where the algebra/statistics connection has proven especially useful: in the study of phylogenetic models and in the analysis of contingency tables.
Rapid Mixing and Markov Bases
It is shown that under a dilation of the underlying polytope, these random walks do not mix rapidly when a fixed Markov based is used, and this phenomenon does not disappear after adding more moves to the Markov basis.


Computing the integer programming gap
We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is
Controlled rounding for tables with subtotals
It is shown that some forms of tables with subtotals always have a controlled rounding solution, while other table structures cannot be guaranteed such a solution under “zero-restrictedness”.
Graphical models in R
This paper presents Graphical Models for Complex Stochastic Systems, a meta-modelling framework for graphical models of complex systems that combines Gaussian Graphical models, Mixed Interaction Models, and Log-Linear Models.
Markov Bases of Binary Graph Models
The topological invariant of a graph given by the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables is described and the algebraic degree of the model when the underlying graph is a forest.
Disclosure Detection in Multivariate Categorical Databases: Auditing Confidentiality Protection Through Two New Matrix Operators
This article addresses inferential disclosure of confidential views in multidimensional categorical databases and demonstrates that any structural, so data-value-independent method for detecting disclosure can fail.
Gröbner bases and convex polytopes
Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The
An algorithm to calculate the lower and uppoer bounds of the elements of an array given its marginals
  • 1999
An algorithm to calculate the lower and uppoer bounds of the elements of an array given its marginals
  • Statistical Data Protection Proceedings, Eurostat
  • 1999
An algorithm to calculate the lower and uppoer bounds of the elements of an array given its marginals, in Statistical Data Protection
  • 1999