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Quality & Quantity

Erschienen in:

13.10.2015

The dimensionality of discrete factor analyses

verfasst von: John Levi Martin

Erschienen in: Quality & Quantity | Ausgabe 6/2016

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Abstract

This article discusses a solution to Coombs’s project of a discrete, ordinal factor analysis for dichotomous data that is structurally homologous to Galois lattice analysis and to other related algebraic approaches. It compares this approach to the better known “biorder” approach to the same problem. In contrast to the biorder approach which is NP-hard, here the set of minimal solutions can be determined with a reasonably simple coloration algorithm. The dimensionality of the resulting solution may be larger than that retrieved by the closely related biorder approach, but the underlying space may be more parsimonious in that there are fewer possible regions. In a class of reasonably important cases, the two are equivalent.

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The degenerate case in which a threshold is at −∞ will (following Coombs) here be noted as the threshold being 0, and the trait space being confined to non-negative traits. When we speak of the set of thresholds in a Coombs factorization, we will not include such degenerate cases.

It is of course true that in this case, the biorder approach is indifferent to which of the three items we treat as requiring two as opposed to one trait. Similarly indeterminacies can occur in all Boolean factorizations.

More technically, in this hypergraph H = (V ^H, E ^H) for any two vertices, v dominates w if for any edge e in E ^H that contains w, the set B _vw = (e\w) ∪ v is “non-stable,” meaning that there is no edge e* ∈ E ^H |e*⊆ B _vw.

We note that this implies that each dimension can be considered a permutation of the items; this has implications for the relation to incomparability graphs, though we do not make use of this here (though see Golumbic et al. 1983). Also note that we here exclude degenerate thresholds as discussed in note 1.

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Titel: The dimensionality of discrete factor analyses
verfasst von: John Levi Martin
Publikationsdatum: 13.10.2015
Verlag: Springer Netherlands
Erschienen in: Quality & Quantity / Ausgabe 6/2016
Print ISSN: 0033-5177
Elektronische ISSN: 1573-7845
DOI: https://doi.org/10.1007/s11135-015-0271-4

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