While studying the software, we made two additional observations: Its size- O(n) memory allocation (where n is the sum of all contingency table entries) could be avoided by reordering the calculation; it is only necessary to track a few partial sums.Since likelihoods decay super-geometrically as one moves away from the most probable table, only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $O(\sqrt {n})$ \end{document}O(n) of the likelihoods can meaningfully impact the partial sums; the sum of the remaining terms is too small to consistently affect even the 10th significant digit in the final p-value. By terminating the calculation when all the partial sums stop changing (due to the newest term being too tiny to be tracked by IEEE-754 double-precision numbers), computational complexity is reduced from O(n) to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $O(\sqrt {n})$ \end{document}O(n) with no loss of precision. See Figure 1 for an example.Figure 12 × 2 contingency table log-frequencies. This is a plot of relative frequencies of 2 × 2 contingency tables with top row sum 1000, left column
