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Statistica Sinica 17(2007), 1371-1393





CONTINGENCY TABLES OF NETWORK TYPE:

MODELS, MARKOV BASIS AND APPLICATIONS


Lawrence H. Cox


National Center for Health Statistics


Abstract: Contingency tables are a staple of quantitative science. Statistical science has paid continuing attention to developing theory, analytical methods, and models for contingency tables that in turn require theoretically verifiable, computationally efficient algorithms. Despite recent advances, there remain theoretical and computational obstacles to developing such algorithms, in some cases for tables with relatively few cells or in low dimensions. We define and investigate a class of multi-dimensional contingency tables -- tables of network type -- that overcome these limitations and enjoy strong theoretical properties and efficient computational algorithms. We demonstrate that tables in this class are abundant and familiar, including 2-dimensional tables, the Rasch model, log-linear models involving summation over mostly dichotomous variables, and tables of these types subject to structural zeroes. We describe ways to collapse non-network tables into network tables. We construct a Markov basis for tables of network type based on moves involving only coefficients $-1$, 0, $+1$. We provide theoretical models and efficient algorithms for solving three important statistical problems over tables of network type -- sampling contingency tables subject to specified marginals, imputation and analysis for item and unit nonresponse subject to edit constraints, and obtaining exact bounds on entries in partially specified tables for purposes including statistical disclosure limitation. We relate our results to the De Loera-Onn formulation of all integer linear programs as slim 3-dimensional contingency tables.



Key words and phrases: Data confidentiality, exact bounds, imputation, integer optimization, log-linear model, Markov chain Monte Carlo, nonresponse, probability sampling.

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