Abstract

Statistical entailment analysis pp. 380-404) aims first at a rigorous evaluation of null hypotheses of statistical independence as a potential source of binary data structure, and second at constructing a discrete structure model of those statistical interactions that remain when the null hypothesis is rejected for particular subsets of variables. Signal detection theory, rather than a conventional significance level, is used to specify optimal cutoffs given an ordering of ratios of actual to expected across levels of exception and relevance. Bivariate entailment analysis is generalized here to improve its utility for use in lattice approximation. Generalized statistical entailment analysis describes Boolean patterns in a set of data in terms of those that occur with greater frequency than expected by chance according to a model of complete statistical independence. This expands on the initial design of entailment analysis to deal with partial orders of quasi-implication in pairs or chains of dichotomous variables, supported by statistical evidence of departure from bivariate independence and conformity to the rules of transitivity. Statistical approximations simplify a lattice representation of discrete structure by forcing quasi-implications, for example, but they also provide information about those implications in the lattice that represent statistically significant tendencies. Given a lattice representing the discrete structure of a raw data matrix, the findings of entailment analysis describe additional structural regularities that can be used to simplify the lattice of empirical patterns. As demonstrated with studies of dual orderings of material possessions, the statistical interpretability of discrete structure lattices is enhanced by using the results of entailment analysis for consensus-simplification of statistically strong or significant implicational relations.