Error structure and identification condition in maximum likelihood nonmetric multidimensional scaling

The author addresses two previously unresolved issues in maximum likelihood estimation (MLE) for multidimensional scaling (MDS). First, a theoretically consistent error model for nonmetric MLDMS is proposed. In particular, theoretical arguments are given that the disturbance should be multiplicative with distance when a stochastic choice model is used on rank-ordered similarity data. This assumption implies that the systematic component of similarity in the rank order is a logarithmic function of distances between stimuli. Second, a problem with identification condition of the maximum likelihood estimators is raised. The author provides a set of constraints that guarantees the identification in MLE, and produces more desirable asymptotic confidence regions that are parameter independent. An example using perception of business schools illustrates these ideas and demonstrates the computational tractability of the MLE approach to MDS.