Abstract
Information identities derived from entropy and relative entropy can be useful in statistical inference. For discrete data analyses, a recent study by the authors showed that the fundamental likelihood structure with categorical variables can be expressed in different yet equivalent information decompositions in terms of relative entropy. This clarifies an essential difference between the classical analysis of variance and the analysis of discrete data, revealing a fallacy in the analysis of hierarchical loglinear models. The discussion here is focused on the likelihood information of a three-way contingency table, without loss of generality. A classical three-way categorical data example is examined to illustrate the findings.
Highlights
The analysis of contingency tables with multi-way classifications originates from the historical development of statistical inference with 2 × 2 tables
Since partitions of chi-squares are closely related to the likelihood ratio tests (Wilks, 1935), maximum likelihood estimation of association and interaction significantly influenced the early studies of multi-way contingency tables (Roy and Kastenbaum, 1956)
It is noteworthy that the likelihood ratio test statistic (2.4), being the sample analog of I(X; Y ), is an average of empirical log-likelihood; and the sample version of (2.2) does not include a constant term as the loglinear model does with standard ANOVA
Summary
The analysis of contingency tables with multi-way classifications originates from the historical development of statistical inference with 2 × 2 tables. In related research in biostatistics, Cochran (1954), Woolf (1955) and Mantel and Haenszel (1959) developed chi-square tests for no association between two variables across levels of the third variable These early studies led to further analyses of three-way tables, which include estimating the common odds ratio, testing zero interaction and testing no association across strata, for. Since partitions of chi-squares are closely related to the likelihood ratio tests (Wilks, 1935), maximum likelihood estimation of association and interaction significantly influenced the early studies of multi-way contingency tables (Roy and Kastenbaum, 1956). A basic likelihood information approach to categorical data analysis, coined linear information models, will provide a remedy, and this will be discussed in a forthcoming study
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