Exact partially conditional binomial analysis for multinomial data in 2 × 2 tables

Starting with Barnard (1945, 1947), many papers have shown that exact unconditional tests outperform Fisher’s Exact Test in 2 × 2 tables with independent binomial data. Less has been published about unconditional tests with multinomial data. However, in many multinomial 2 × 2 analyses, a binomial-like comparison of proportions is of interest rather than inference in terms of odds ratios. Thus, this paper proposes using a partially conditional binomial analysis with data that are actually multinomially distributed. This partially conditional analysis, conditioning on the row totals and then using the unconditional binomial analysis, is more powerful than the fully conditional Fisher’s Exact Test, has good power comparable to the fully unconditional multinomial analysis, and provides exact confidence intervals for the difference of proportions. Also, the partially conditional binomial analysis requires considerably less computation than the fully unconditional analysis.