A Comparison of Internal, External, and Successive Unfolding Based on Occupational Judgments

Abstract

Successive unfolding is a recently developed technique that appears to have advantages over traditional unfolding techniques. To assess its usefulness, successive, internal, and a type of external unfolding were compared with regard to the stimulus configurations recovered, the fit of the models as a whole, the fit of individual subjects, and each model's stability in a cross-validation sample. The data were obtained from judgments of similarity of and preference for occupational titles. Internal unfolding yielded degenerate solutions and was dropped from subsequent analyses. External and successive unfolding yielded interpretable dimensions, but the nature of the dimensions uncovered differed somewhat from one type of unfolding to the other. Both types of unfolding fit the data adequately for the stimulus configuration, though successive unfolding appeared to fit better. The first two dimensions of both models were well reproduced in cross-validation samples. Dimensions beyond the second were more easily cross-validated using external unfolding. Successive unfolding proved superior to external unfolding in representing preference data as indicated by the fit of ideal points. Successive unfolding appears a viable method in unfolding research. Methods for deriving joint spatial representations of individuals and objects have been available since Coombs' (1950) technique effectively &dquo;un-folded&dquo; preference rankings into a joint scale containing both persons and objects. An individual's preferences corresponded to the rank order of the distances from that individual to the objects. Thus, the most preferred objects should be the closest to the individual. Coombs' original model assumed a single underlying dimension, and served both as method and criterion, that is, a set of data would or would not unfold. Bennett and Hays (1960) extended unfolding into multiple dimensions, and Carroll (1972) developed a hierarchy of multidimensional unfolding models relating level of preference to various functions of distance. Several recent books on multidimensional scaling (MDS) treat unfolding models in some detail (Coxon, 1982;

External Variables as Points in Smallest Space Analysis: a Theoretical, Mathematical and Computer-Based Contribution

“…External unfolding (Brannick andHahn 1985, Rogers andYoung 1981) was developed to resolve inherent computational problems found in internal unfolding which tend to cause the structure of the data to degenerate. In external unfolding, a stimulus space is obtained, based on paired comparison similarity judgments or some other matrix of similarities, or dissimilarities, using any of the well-known MDS (multi-dimensional scaling) techniques.…”