Details:
Title  Markov bases of threeway tables are arbitrarily complicated  Author(s)  Jesús A., Shmuel Onn  Type  Article in Journal  Abstract  We show the following two universality statements on the entryranges and Markov bases of spaces of 3way contingency tables with fixed 2margins: (1) For any finite set D of nonnegative integers, there are r , c , and 2margins for ( r , c , 3 ) tables such that the set of values occurring in a fixed entry in all possible tables with these margins is D . (2) For any integer n vector d , there are r , c such that any Markov basis for ( r , c , 3 ) tables with fixed 2margins must contain an element whose restriction to some n entries is d . In particular, the degree and support of elements in the minimal Markov bases when r and c vary can be arbitrarily large, in striking contrast with the case for 1margined tables in any dimension and any format and with 2margined ( r , c , h ) tables with both c , h fixed. These results have implications for confidential statistical data disclosure control. Specifically, they demonstrate that the entryrange of 2margined 3tables can contain arbitrary gaps, suggesting that even if the smallest and largest possible values of an entry are far apart, the disclosure of such margins may be insecure. Thus, the behavior of sensitive data under disclosure of aggregated data is far from what has been so far believed. Our results therefore call for the reexamination of aggregation and disclosure practices and for further research on the issues exposed herein. Our constructions also provides a powerful automatic tool in constructing concrete examples, such as the possibly smallest 2margins for (6, 4, 3)tables with entryrange containing a gap.  Keywords  Contingency tables, Markov bases, Statistical data security, Sampling and random generation, Confidentiality of statistical data, Transportation polytopes  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717105001185 
Language  English  Journal  Journal of Symbolic Computation  Volume  41  Number  2  Pages  173  181  Year  2006  Note  Computational Algebraic Statistics Computational Algebraic Statistics  Edition  0  Translation 
No  Refereed 
No 
