Assessing implications between genotypic and phenotypic variables through lattice analysis

A previous paper assessed a "Molecular Mapping of Twenty-Four Features of Down Syndrome on Chromosome 21" (Delabar et al., 1993), by analyzing the genotypes/phenotypes of patients suffering from partial trisomy. The mapping was defined through implications--each feature was mapped to the conjunction of cytogenetic bands that were shared by all patients having that feature. In the present paper, we extend that approach to determine how far those implications depart from defining equivalences. Finding equivalences is important. Local equivalences permit a genetic characterization of a feature. And if global equivalences held for all features, that set of bands would be sufficient to characterize the various phenotypes observed in individuals with partial trisomy 21. To extend the earlier approach, we examine the structure of equivalences as well as the structure of implications. We examine both conjunctions of bands and conjunctions of features. The use of Galois lattices permits simultaneous evaluation of both kinds of structures. Each Galois lattice is labeled with a basis (minimal generating set) of implications going from conjunctions of features into bands and those going from conjunctions of bands into features. Analysis reveals that about half of the conjunctions of bands that characterize the genetic structure embody equivalences. This allows us to improve the genetic description of features and to specify minimal sets of questions that need to be investigated to make the global genetic description more precise.