Statistical entailments and the Galois lattice
White
DR
SOCIAL NETWORKS
18 (3): 201-215 AUG 1996
Abstract:
Statistical entailment
analysis (White, 1984; White and McCann, 1988 Social Structures: Form and
Behaviour in Social Life) (Cambridge University Press) 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 (Boolean) 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 (the specific model of
independence derives from a distribution of randomly permuted entries in the
columns of the data matrix marginals, i.e. keeping univariate marginals fixed).
This expands on the initial design of entailment analysis (White, 1984) 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 (ignoring exceptions), 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 (tendencies towards further statistical
constraints on Boolean patterns that occur in the data) that can be used to
simplify (by approximation) the lattice of empirical patterns. As demonstrated
with studies of dual orderings of material possessions (possessions stratify
people; people possessions), the statistical interpretability of discrete
structure lattices is enhanced by using the results of entailment analysis for
cansensus-simplification of statistically strong or significant implicational
relations.
Addresses:
White DR, UNIV CALIF IRVINE,INST MATH BEHAV
SCI,IRVINE,CA 92717