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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Notes
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.
References
Birkhoff, Garrett: Lattice Theory, vol. 25. American Mathematical Society Colloquium Publications, Providence (1967)
Butts, C.T., Hilgeman, Christin: Inferring potential memetic structure from cross-sectional data: an application to american religious beliefs. J. Memet. Evolut. Models Inf. Trans. 7(2), 3976–3979 (2003)
Chubb, C.: Collapsing Binary Data for Algebraic Multidimensional Representation. J. Math. Psychol. 30, 161–187 (1986)
Cogis, Olivier: On the ferrers dimension of a digraph. Discret. Math. 38, 47–52 (1982)
Coombs, C.H.: A Theory of Data. John Wiley and Sons, New York (1964)
Degenne, Alain, Lebeauz, M.-O.: Boolean analysis of questionnaire data. Soc. Netw. 18, 231–245 (1996)
Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51, 161–166 (1950)
Doignon, J.-P., Ducamp, André, Falmagne, J.-C.: On realizable biorders and the biorder dimension of a relation. J. Math. Psychol. 28, 73–109 (1984)
Doignon, J.-P., Falmagne, J.-C.: Matching relations and the dimensional structure of social choices. Math. Soc. Sci. 7, 211–229 (1984)
Doignon, J.-P., Falmagne, J.C.: A polynomial time algorithm for unidimensional unfolding representations. J. Algorithms 16, 218–233 (1994)
Duquenne, V.: Models of possessions and lattice analysis. Soc. Sci. Inform. 34, 253–267 (1995)
Duquenne, V.: On lattice approximations: syntactic aspects. Soc. Netw. 18, 189–200 (1996)
Edelman, Paul H., Saks, M.E.: Combinatorial representation and convex dimension of convex geometries. Order 5, 23–32 (1988)
Freeman, L.C., White, D.R.: using galois lattices to represent network data. Sociol. Methodol. 23, 127–146 (1993)
Ganter, B., Rudolf, W.: Conceptual scaling. In: Roberts, F. (ed.) Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, pp. 139–167. Springer, New York (1989)
Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Berlin (1999)
Golumbic, M.C., Rotem, D., Urrutia, J.: Comparability graphs and intersection graphs. Discret. Math. 43, 37–46 (1983)
Haertel, E.H., Wiley, D.E.: Representations of ability structures: implications for testing. In: Frederiksen, N., Mislevy, R., Bejar, Isaac (eds.) Test Theory for a New Generation of Tests. Erlbaum, Hillsdale (1993)
Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)
Johnson, H.M.: Some neglected principles in aptitude testing. Am. J. Psychol. 47, 159–165 (1935)
Jungnickle, D.: Graphs, Networks and Algorithms. Springer, Berlin (2005)
Koppen, M.G.M.: On finding the bidimension of a relation. J. Math. Psychol. 31, 155–178 (1987)
Leenen, I., van Mechelen, I., De Boeck, P.: A generic disjunctive/conjunctive decomposition model for n-ary relations. J. Math. Psychol. 43, 102–122 (1999)
Lovász, L.: Perfect graphs. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory 2, pp. 55–57. Academic Press, London (1983)
Magdison, J., Vermunt, J.K.: Latent class factor and cluster models, Bi-plots, and related graphical displays. Sociol. Methodol. 31, 223–264 (2001)
Martin, J.L.: Spatial processes and galois/concept lattices. Qual. Quant. 48, 961–981 (2014)
Martin, J.L., Wiley, J.A.: Algebraic representations of beliefs and attitudes II: microbelief models for dichotomous belief data. Sociol. Methodol. 30, 123–164 (2000)
Mische, A., Pattison, P.: Composing a civic arena: publics, projects, and social settings. Poetics 27, 163–194 (2000)
Mohr, J., Duquenne, Vincent: The duality of culture and practice: poverty relief in New York City, 1888–1917. Theor. Soc. 26, 305–356 (1997)
Monjardet, Bernard: The presence of lattice theory in discrete problems of mathematical social science why. Math. Soc. Sci. 46, 103–144 (2003)
Müller-Hannemann, M.: Drawing trees, series-parallel digraphs, and lattices. In: Kaufmann, M., Wagner, D. (eds.) Drawing Graphs, pp. 46–70. Springer, Berlin (2001)
Pattison, P.: Algebraic Models for Social Networks. Cambridge University Press, Cambridge (1993)
Pattison, P.: Boolean decomposition of binary matrices. Paper presented at the European Meeting of the Psychometric Society, Leiden July 4–7, (1995)
Reise, S.P.: Personality measurement issues viewed through the eyes of IRT. In: Embretson, S.E., Hershberger, S.L. (eds.) The New Rules of Measurement, pp. 219–242. Lawrence Erlbaum, New Jersey (1999)
Stern, M.: Semimodular Lattices: Theory and Applications. Cambridge University Press, Cambridge (1999)
Takane, Y., de Leeuw, Jan: On the relationship between item response theory and factor analysis of discretized variables. Psychometrika 52, 393–408 (1987)
Trotter Jr, W.T.: Graphs and partially ordered sets. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory, vol. 2, pp. 237–288. Academic Press, London (1983)
van Mechelen, I., De Beock, P., Rosenberg, Seymour: The conjunctive model of hierarchical classes. Psychometrika 60, 505–521 (1995)
White, Douglas R.: Statistical entailments and the galois lattice. Soc. Netw. 18, 201–215 (1996)
Wiley, James, Martin, John Levi: Algebraic Representations of beliefs and attitudes: partial order models. Sociol. Methodol. 1999, 113–146 (1999)
Acknowledgments
This work was supported by the Graduate School Research Committee of the University of Wisconsin, Madison, Grant #060059. I am extremely grateful for the detailed critique and comments of an anonymous reviewer that led to a tightening of the exposition and the removal of a number of missteps. I also thank Jonathan Farley for disproving one conjecture, and especially James Montgomery for Table 2 and other comments and discussions that have greatly improved the work along these lines.
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Appendix: conventional algebraic definitions
Appendix: conventional algebraic definitions
The classic treatment of lattice algebra is Birkhoff (1967 [1940]); the exposition here includes only terms needed. The reader familiar with partial orders and lattices may skip this section. A “partially ordered set” (or “poset,” for short) is a set of elements {δ1, δ2, δ3,…} and a binary relation denoted ≤, which satisfies the following three conditions:
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(i)
transitivity: δ1 ≤ δ2, δ2 ≤ δ3 implies δ1 ≤ δ3;
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(ii)
reflexivity: δ1 ≤ δ1;
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(iii)
antisymmetry: δ1 ≤ δ2 and δ2 ≤ δ1 implies δ1 = δ2.
Given a set of elements A and a relation ≤ , δ1 ∈ A is said to be the “minimum” of A if for any δ2 ∈A, δ2 ≤ δ1 implies δ2 = δ1. Note that the set of all points is a poset.
Consider a poset A, consisting of elements δ1, δ2, δ3 etc. together with a binary relation ≤ as defined in the text. The “lower bound” of a pair of elements, δ1 and δ2, in A is an element δ3, such that δ3 ≤ δ1 and δ3 ≤ δ2. Similarly, the “upper bound” of a pair of elements, δ1 and δ2, in A is an element δ3, such that δ1 ≤ δ3 and δ2 ≤ δ3. The “greatest lower bound” or “meet” of any two elements δ1 and δ2 in A, denoted δ1∧δ2, is a unique element δ3 in A such that δ3 ≤ δ1 and δ3 ≤ δ2, and there is no δ4 in A such that δ3 ≤ δ4 ≤ δ1 and δ3 ≤ δ4 ≤ δ2. Similarly, the “least upper bound” or “join” of any two elements, δ1 and δ2 in A, denoted δ1∨δ2, is a unique element δ3 in A such that δ1 ≤ δ3 and δ2 ≤ δ3, and there is no δ4 in A such that δ1 ≤ δ4 ≤ δ3 and δ2 ≤ δ4 ≤ δ3. A “lattice” is then a poset that is closed under the binary operations of meet and join; that is, for any two elements δ1 and δ2 in A, δ1∨δ2 ∈ Α, δ1∧δ2 ∈ Α. Given a set of elements {δ1, δ2, δ3,…} we may write δ1∧ δ2∧ δ3∧… as ∧{δ1, δ2, δ3,…}.
