Cell bounds in two-way contingency tables based on conditional frequencies

Statistical methods for disclosure limitation (or control) have seen coupling of tools from statistical methodologies and operations research. For the summary and release of data in the form of a contingency table some methods have focused on evaluation of bounds on cell entries in k-way tables given the sets of marginal totals, with less focus on evaluation of disclosure risk given other summaries such as conditional probabilities, that is, tables of rates derived from the observed contingency tables. Narrow intervals - especially for cells with low counts - could pose a privacy risk. In this paper we derive the closed-form solutions for the linear relaxation bounds on cell counts of a two-way contingency table given observed conditional probabilities. We also compute the corresponding sharp integer bounds via integer programming and show that there can be large differences in the width of these bounds, suggesting that using the linear relaxation is often an unacceptable shortcut to estimating the sharp bounds and the disclosure risk.