A random effects approach to unfolding models.

Unfolding is a scaling method introduced by Coombs (1950) to model the judgment of stimuli among a sample of subjects. This model provides a geometric representation in the spirit of multidimensional scaling (MDS) on data in the form of preference orders or in the form of ratings over a set of stimuli. The technique has been used in psychology and marketing research. The unfolding model attempts to find a spatial (unfolded) configuration in multidimensional space with each stimulus represented as a point in that space. What gives unfolding its defining feature is that (unlike multidimensional scaling) subjects are also represented in the same spatial configuration as the stimuli. That is, each subject is represented as having an ideal point against which each stimulus is evaluated, and that ideal point is modeled as a point in stimuli space. We extended unfolding models to random effects in the sense the subjects are treated as random effects. The advantages of considering random effects is the ability to estimate covariates effect simultaneously, as well as reduction in the number of parameters. The current methodology in the field is to estimate ideal points, then treat those ideal points as data in a subsequent analysis (e.g., a two sample t test comparing differences between means of men and women on the computed ideal points on one dimension). We deal with this problem by proposing a model that treats ideal points as random effects. This gives the feature that one can test covariates of ideals points simultaneously to the estimation of the ideal points. Degeneracy is a potential problem in unfolding analysis. It can influence the informativeness of the resulting configuration and make it difficult to draw conclusions from a particular configuration. Past research suggests several methods to avoid de generacy. However, the methods are not general and can only be successful on a case by case basis. We found that the score pattern is crucial to avoid degeneracy. We use several examples and simulations to show the important role of score pattern in degeneracy. This finding is useful not only in unfolding, but also provides solutions for nonmetric MDS degeneracy.Ph.D.Pure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/126184/2/3237986.pd