Mixtures of (Constrained) Ultrametric Trees

• Published in: Psychometrika Vol. 63; no. 4; pp. 419 - 443
• Main Authors: Wedel, Michel, DeSarbo, Wayne S.
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.12.1998; Psychometric Society, etc
• Subjects: Biological and medical sciences; Estimation (Mathematics); Fundamental and applied biological sciences. Psychology; Latent Class Analysis; Maximum Likelihood Statistics; Psychology. Psychoanalysis. Psychiatry; Psychology. Psychophysiology; Psychometrics. Statistics. Methodology; Statistics. Mathematics; Stimuli; Statistical method; Parameter estimation; Tree(graph); Hierarchical classification; Latent variable model; Maximum likelihood; Mixture
• ISSN: 0033-3123; 1860-0980
• DOI: 10.1007/BF02294863

This paper presents a new methodology concerned with the estimation of ultrametric trees calibrated on subjects' pairwise proximity judgments of stimuli, capturing subject heterogeneity using a finite mixture formulation. We assume that a number of unobserved classes of subjects exist, each having a different ultrametric tree structure underlying the pairwise proximity judgments. A new likelihood based estimation methodology is presented for those finite mixtures of ultrametric trees, that accommodates ultrametric as well as other external constraints. Various assumptions on the correlation of the error of the dissimilarities are accommodated. The performance of the method to recover known ultrametric tree structures is investigated on synthetic data. An empirical application to published data from Schiffman, Reynolds, and Young (1981) is provided. The ability to deal with external constraints on the tree-topology is demonstrated, and a comparison with an alternative clustering based method is made.