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We adapt Higher Criticism (HC) to the comparison of two frequency tableswhich may -- or may not -- exhibit moderate differences between the tables insome unknown, relatively small subset out of a large number of categories. Ouranalysis of the power of the proposed HC test quantifies the rarity and size ofassumed differences and applies moderate deviations-analysis to determine theasymptotic powerfulness/powerlessness of our proposed HC procedure.  Our analysis considers the null hypothesis of no difference in underlyinggenerative model against a rare/weak perturbation alternative, in which thefrequencies of $N^{1-\beta}$ out of the $N$ categories are perturbed by $r(\logN)/2n$ in the Hellinger distance; here $n$ is the size of each sample. Ourproposed Higher Criticism (HC) test for this setting uses P-values obtainedfrom $N$ exact binomial tests. We characterize the asymptotic performance ofthe HC-based test in terms of the rarity parameter $\beta$ and the perturbationintensity parameter $r$. Specifically, we derive a region in the$(\beta,r)$-plane where the test asymptotically has maximal power, while havingasymptotically no power outside this region. Our analysis distinguishes betweencases in which the counts in both tables are low, versus cases in which countsare high, corresponding to the cases of sparse and dense frequency tables. Thephase transition curve of HC in the high-counts regime matches formally thecurve delivered by HC in a two-sample normal means model.

Higher Criticism to Compare Two Large Frequency Tables, with sensitivity to Possible Rare and Weak Differences