Multidimensional Scaling

 

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Introduction

Given a set of n points on a map, it is fairly straightforward to find the distances among the points and construct an n p matrix of inter-point distances. But the inverse problem of locating the points, given an n p matrix of inter-point distances is not so straightforward. Multidimensional scaling (MDS) attempts to solve such an inverse problem. This type of problem is quite important in social and behavioral research, scientometrics, marketing research, etc. In these problems, the primary goal is to detect the underlying dimensions of a set of variables.

Essentially, the purpose of multidimensional scaling (MDS) is to provide a visual representation of the pattern of proximities (i.e., similarities or distances) among a set of objects. MDS plots the objects on a map such that objects that are very similar to each other are placed near each other on the map, and objects that are very different from each other, are placed far away from each other on the map.

Multidimensional scaling can be considered as an alternative to factor analysis. In general, the goal of these techniques is to detect meaningful underlying dimensions that allow the researcher to explain observed similarities or dissimilarities (distances) between the investigated objects. However, these two techniques are fundamentally different in terms of methodology. Factor analysis requires that the underlying data are distributed as multivariate normal, and that the relationships are linear. MDS imposes no such restrictions. Moreover, MDS can be applied to any kind of distances or similarities, whereas factor analysis requires us to first compute a correlation or covariance matrix. Factor analysis tends to extract more factors (dimensions) than MDS; as a result, MDS often yields more readily interpretable solutions.

Multidimensional scaling is a mathematical procedure by means of which information contained in a data set can be represented by points in a space. MDS finds a set of vectors in a p-dimensional space such that the matrix of Euclidean distances among them corresponds as closely as possible to some function of the input matrix (i.e. matrix of similarities or dissimilarities) according to a criterion function called stress. Stress is a measure of the lack of correspondence between the distances among points implied by MDS map and the input matrix.

MDS is primarily concerned with the representation of objects as a configuration of points, usually in two - dimensional maps, in such a way that maximizes the fit between the proximity measure of each pair of variables (or objects) and the distances between all of them in the map. The points located in the graphs reproduce distances between each pair, controlled by the distance of each variable (object) with each one of the remaining variables (objects). The points that are close to each other in the map indicate relationships between the pairs (i.e., cohesion) as well as similarity of behavior with respect to the remaining variables or objects (i.e., structural equivalence).

There are two types of multidimensional scaling models: Metric and Non-metric.

  1. Multidimensional scaling which is based on measured proximities is called metric multidimensional scaling.
  2. Multidimensional scaling, which is based on proximities based on judgments, is called non-metric multidimensional scaling.

If the measured proximities are rank ordered, then their scaling would be non-metric.

In metric MDS, the spatial representation attempts to preserve the distances among the objects, whereas in non-metric MDS the spatial representation attempts to preserve the rank order among the dissimilarities.

IDAMS module Mdscal performs non-metric multidimensional scaling.