Characterizations of continuous adequate objective functions for ordinal or interval scaled data

Abstract

Objective functions (goodness criteria which have to be optimized) that are considered, for instance, in cluster analysis, factor analysis, (linear) structural equation modeling, (linear) regression, multidimensional scaling, choice theory, and utility theory, must be adequate, i.e. carefully adapted to the structure of the observed data. Adequateness of an objective function means, in our terminology, that (certain) transformations of its arguments, e.g. changes in the unit measures of the quantities involved, do not influence the solutions of the optimization procedure. In this paper we concentrate our attention on affine strictly increasing transformations and we also incorporate the case of continuous adequate objective functions accordingly. The characterization of adequate dissimilarity coefficients for interval scaled data shows the appropriateness of the concept of adequateness that is developed in this paper.