multi-dimensional scaling

multi-dimensional scaling Often, a set of data (for example attitude items) cannot be represented in one dimension, such as in a unidimensional scale or factor analysis. The items may then be modified or selected, so that they can be so represented (as in item analysis and scale construction); or, alternatively, a representation can be sought in a space of two or more dimensions. The purpose of multi-dimensional scaling (MDS) is to seek as good a representation of the data as possible in as few dimensions as possible.

In the simplest case (non-metric Euclidean distance MDS), data are thought of as giving information on the similarity or dissimilarity between pairs of objects: for example, positive correlations can be interpreted as similarities, such that the higher the value the more similar are the variables. The purpose of MDS is to represent each variable as a point in a low-dimensional space so that the distance between the points accurately reflects the relative size of the data similarities and dissimilarities. Non-metric (or ordinal) MDS preserves only the rank order of the data; metric (linear or power) MDS preserves the quantitative information. Computer programs implementing MDS usually seek such a solution iteratively; that is, from a preliminary guess successively brought into closer conformity to the data, in a cycle of improvements.

MDS is a very useful and general family of procedures (a General Distance Model) appropriate to a wide variety of data, including (for example) two-way correlation matrices, rectangular individual by variable matrices, and three-way stacks of data; to different models (distance, scalar-products or factor, weighted distance); and different levels of measurement (such as monotonic/non-metric, linear/metric, and power transformations). It has been used successfully on a wide range of sociological and psychological data (see J. B. Kruskal and and M. Wish , Multidimensional Scaling, 1978
).