On properties of multi-dimensional statistical tables

Statistical data often can be conveniently organized in tabular form for display and analysis. Counts are nonnegative integers, and often magnitude data take nonnegative integer values. Two-dimensional tables enjoy mathematical proper-ties on which important statistical methods depend, e.g., for stratified sampling, imputation, disclosure limitation, and sampling and fitting log-linear models to contingency tables. We demonstrate that many desirable mathematical properties, and consequently their associated statistical methods, are not extendible in all cases to three or higher dimensions. We demonstrate that ill-behaved examples are ubiquitous, abundant and consequently not mathematical anomalies. To address these shortcomings, we provide necessary and sufficient conditions and an empirical test for the existence of an n-dimensional table with prescribed (n - 1)-dimensional marginal totals (feasibility) and a characterization of n-dimensional tables with prescribed (n - 1)-dimensional marginals for which continuous bounds on integer-valued entries exist and are integer (integrality). (C) 2002 Elsevier B.V. All rights reserved.