Cell Bounds in Two-Way Contingency Tables Based on Conditional Frequencies

Smucker, Byran J.; Slavković, Aleksandra

doi:10.1007/978-3-540-87471-3_6

Cited by 6 publications

(26 citation statements)

References 11 publications

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“…The earlier papers cited above have noted a lack of such fundamental insight. In particular, we address several conjectures/questions raised by Smucker et al [26] concerning patterns they observed in the examples they studied. We illustrate our basic principles with numerous examples, worked in their entirety.…”

Section: Overviewmentioning

confidence: 99%

“…Subsequent rounding then provides additional protection against disclosure. Although some earlier works such as Smucker et al [26] have made preliminary efforts in this direction by obtaining relaxed bounds, no method has been presented for calculating the sharpest possible bounds from rounded conditionals. A forthcoming paper focuses precisely on such calculations, extending the results presented here.…”

Section: Overviewmentioning

confidence: 99%

“…This has implications for disclosure risk because the apparent difficulty of exactly solving such large problems has been cited as evidence that tables of conditionals might not pose as much risk as their tight integer bounds would suggest [26]. Furthermore, table redesign, through aggregation over or within variables, is one of the key tools available to the data owner for protecting privacy [15].…”

Section: Overviewmentioning

confidence: 99%

“…As noted by the referees, this can have serious effects on data utility with regard to inference involving small cell counts and (say) the ability to distinguish them from structural or observed zero counts. In the current paper, we again follow Smucker et al [26] and note that the assumption of unrounded conditionals may be viewed as a worst-case scenario that aids the data owner in determining disclosure risk when designing a table for release. Subsequent rounding then provides additional protection against disclosure.…”

Section: Overviewmentioning

confidence: 99%

“…Slavković [23] and Fienberg and Slavković [13] first examined this issue, and more recently Smucker and Slavković [25] and Smucker et al [26] studied integer programming formulations to determine bounds on cells given conditional probabilities and tabular sample size, as well as easier-to-solve linear programming relaxations. However, for large tables the integer programs may not solve in a reasonable amount of time, and the linear programming relaxations give bounds that are much wider than the sharp integer bounds.…”

Section: Overviewmentioning

confidence: 99%

See 4 more Smart Citations

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

Wright

Smucker

2014

JPC

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Recent research has raised questions about the structure and calculation of bounds on the underlying cell counts for a contingency table released in the form of conditional probabilities. This problem has implications for statistical disclosure limitation. We elucidate the mathematical structure of the problem in fairly elementary terms, under the assumption that the unrounded conditionals and sample size are known. To do so, we reformulate a standard integer programming approach as a knapsack problem, show that this provides many insights into the problem, and provide illustrations in the context of several datasets. In particular, we demonstrate that the tightest bounds are much easier to calculate than previously believed, and we also identify circumstances in which disclosure is either guaranteed or unlikely to occur.

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Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

See 3 more Smart Citations

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

Wright

Smucker

2014

JPC

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Conditional inference given partial information in contingency tables using Markov bases

Karwa

Slavković

2013

WIREs Computational Stats

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In this article, we review a Markov chain Monte Carlo (MCMC) algorithm for performing conditional inference in contingency tables in the presence of partial information using Markov bases, a key tool arising from the area known as algebraic statistics. We review applications of this algorithm to the problems of conditional exact tests, ecological inference, and disclosure limitation and illustrate how these problems fall naturally in the setting of inference with partial information. We also discuss some issues associated with computing Markov bases which are needed as an input to the algorithm. WIREs Comput Stat 2013, 5:207–218. doi: 10.1002/wics.1256 This article is categorized under: Data: Types and Structure > Traditional Statistical Data

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

2014

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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.

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Cell Bounds in Two-Way Contingency Tables Based on Conditional Frequencies

Cited by 6 publications

References 11 publications

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

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

Conditional inference given partial information in contingency tables using Markov bases

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

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