Interpreting stress in multidimensional scaling

Abstract

Several authors have studied the stress associated with a configuration of points obtained from non-metric multidimensional scaling. Most studies involve the simulation of a configuration of points in a Euclidean space from which dissimilarities are derived usually in the form of distances between points together with associated errors. The dissimilarities are then used to attempt to recover the initial configuration of points using multidimensional scaling. The resulting stress for the configuration is then related to dimension of the space, parameters of the associated errors, etc. This paper uses similar methods to these previous ones but by introducing uniform errors instead of the usual normal errors interesting results occur. It is found that a linear relationship occurs between stress and the parameter of the uniform errors and indeed this linear relationship is approximately invariant to the number of points in the configuration and to the spatial pattern of the configuration. This leads directly to an interpretation of stress in terms of the parameter and allows stress values to be compared directly across configurations.

Keywords:

• Non-metric multidimensional scaling
• spatial pattern
• stress

^∗Requests for reprints should be sent to Dr M. A. A. Cox, Department of Engineering Mathematics, The University, Newcastle-upon-Tyne, NEl 7RU, UK.

^∗Requests for reprints should be sent to Dr M. A. A. Cox, Department of Engineering Mathematics, The University, Newcastle-upon-Tyne, NEl 7RU, UK.

Notes

^∗Requests for reprints should be sent to Dr M. A. A. Cox, Department of Engineering Mathematics, The University, Newcastle-upon-Tyne, NEl 7RU, UK.