Robustness and Small-Sample Properties of the Estimators of Probabilistic Multidimensional Scaling (PROSCAL)

Abstract

Probabilistic multidimensional scaling (MDS) models, in comparison with the deterministic ones, offer several theoretical advantages that are likely to appeal to marketing researchers. These advantages are based on asymptotic sampling theory assuming that an explicated error distribution holds true. Whether such models are perceived as practical and useful by marketing researchers, however, will depend partially on the robustness and small-sample properties of their estimators. The error underlying the judgments may very well violate the distributional assumptions, and the typical sample size in marketing applications is not likely to meet the asymptotic requirements. The authors examine the robustness and small-sample properties of PROSCAL, a probabilistic MDS algorithm based on a multivariate generalization of Thurstone's pair comparison model. The results of two extensive simulations experimentally manipulating number of subjects, number of stimuli, and type of error distribution and its variance suggest that PROSCAL estimators are robust to violations of the assumption of normality of the error term within the manipulated ranges of the parameter values. Also, goodness of recovery and dimensionality tests are fairly accurate for sample sizes as small as nine.