In this paper, hierarchical and non-hierarchical tree structures are proposed as models of similarity data. Trees are viewed as intermediate between multidimensional scaling and simple clustering. Procedures are discussed for fitting both types of trees to data. The concept of multiple tree structures shows great promise for analyzing more complex data. Hybrid models in which multiple trees and other discrete structures are combined with continuous dimensions are discussed. Examples of the use of multiple tree structures and hybrid models are given. Extensions to the analysis of individual differences are suggested.

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configuration X relative to the proximity data P and the loss of X relative to a pseudo-data matrix R. The matrix R is set up such that the side constraints to be imposed on X’s distances are expressed by the relations among R’s numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solution X. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matrices P and R. Further generalizations are introduced by substituting R by a set of R matrices, Ri, i = 1, …r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.