Statistical hypothesis testing of pattern heterogeneity via differences in underlying distributions across multiple contingency tables. Five tests are included: the comparative chi-squared test (Song et al. 2014) <doi:10.1093/nar/gku086> (Zhang et al. 2015) <doi:10.1093/nar/gkv358>, the Sharma-Song test (Sharma et al. 2021) <doi:10.1093/bioinformatics/btab240>, the heterogeneity test, the marginal-change test (Sharma et al. 2020) <doi:10.1145/3388440.3412485>, and the strength test (Sharma et al. 2020) <doi:10.1145/3388440.3412485>. Under the null hypothesis that row and column variables are statistically independent and joint distributions are equal, their test statistics all follow an asymptotically chi-squared distribution. A comprehensive type analysis categorizes the relation among the contingency tables into type null, 0, 1, and 2 (Sharma et al. 2020) <doi:10.1145/3388440.3412485>. They can identify heterogeneous patterns that differ in either the first order (marginal) or the second order (differential departure from independence). Second-order differences reveal more fundamental changes than first-order differences across heterogeneous patterns.
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