Markov bases of three-way tables are arbitrarily complicated

We show the following two universality statements on the entry-ranges and Markov bases of spaces of 3-way contingency tables with fixed 2-margins: (1) For any finite set D of nonnegative integers, there are r, c, and 2-margins 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 2-margins 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 1-margined tables in any dimension and any format and with 2-margined (r, c, h)-tables with both c, h fixed. These results have implications to confidential statistical data disclosure control. Specifically, they demonstrate that the entry-range of 2-margined 3-tables can contain arbitrary gaps, suggesting that even ...

• Type: Journal
• Year: 2006
• Where: JSC
• Authors: Jesús A. De Loera, Shmuel Onn
• Keywords: 3-way Contingency Tables, JSC 2006, Markov Bases, Universality Statements