Introduction

Though multidimensional scaling techniques require only nonmetric (ordinal) inputs, metric (ratio scale) data typically results. This transformation is made possible by reducing the number of constraints used to describe the data.

In the absence of transformation ordered relationships among n elements of a stimulus set can be depicted in a space of n -1 dimensions. For example, the distance between any two objects can be respresented by a straight line in unidimensional space. Similarly the relationships among three objects in three-dimensional space can be depicted by several triangles in two-dimensional space.

As the number of elements n contained in a stimulus set becomes large, the number of nonmetric, ordered relationships (nonmetric constraints) increases by approximately the square of the number of elements:

number of ordered relationships = n · (n - 1) / 2

Notwithstanding, the number of constraints (order pairs) required for a complete metric specification of the same n elements increases only linearly by the number of elements - namely, n.

By way of example, ten elements require 45 = 10 · (10 - 1) / 2 ordered relationships for complete specification in nonmetric space, but only 20 coordinates in two-dimensional metric space -- namely, (n · k) = 20 where the n (the number of elements) equals 10 and k (the number of dimensions) equals 2.

Paired City Distances - Plotting geographical distances using ordered pairs
mds (number and interpretation of dimensions)

A useful way of evaluating any analytical procedure is to test it against a problem with a known solution.
In an experiment performed by Marshall G. Greenberg (1969) respondents were asked to rank order the distances between pairs of large, well-known cities in the United States.

The number of cities was 15, and the number of pairs required to describe the relationships among the 15 cities was 105 = 15 · (15 - 1) / 2. The pair of cities separated by the largest distance in the minds of the respondents was assigned the number 1, and the pair of cities separated by the shortest distance was assigned the number 105. All other city-pairs were assigned some unique number between 1 and 105. From these 105 observations metric data were computed, averaged, and plotted in two-dimensional space. The coordinate axes (dimensions) of this space were the geographical directions north-south and east-west. The relative locations of the cities plotted on the perceptual map closely ressembled the locations of these cities on a standard geographical map.

Of course, for most multidimensional scaling problems the number and interpretation of dimensions are far less obvious than those in the experiment just described. Nevertheless, the power of multidimensional scaling should be clear from the above example.

statsitical modelling (mds)

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statistical modelling (diagnostics)