Multidimensional scaling (MDS) is a mathematical tool that uses proximities
between objects, subjects or stimuli to produce a spatial representation of
these items. The proximities are defined as
any set of numbers that express the amount of
similarity or dissimilarity between pairs of objects, subjects or stimuli.
In contrast to the techniques considered so far,
MDS does not start from the raw multivariate data matrix ,
but from a
dissimilarity or distance matrix,
, with
the elements
and
respectively. Hence, the underlying dimensionality of the data under
investigation is in general not known.
MDS is a data reduction technique because it is concerned with the
problem of finding a set of points in low dimension that represents the
``configuration'' of data in high dimension. The ``configuration'' in high
dimension is represented by the distance or dissimilarity matrix .
MDS-techniques are often used to understand how people perceive and evaluate certain signals and information. For instance, political scientists use MDS techniques to understand why political candidates are perceived by voters as being similar or dissimilar. Psychologists use MDS to understand the perceptions and evaluations of speech, colors and personality traits, among other things. Last but not least, in marketing researchers use MDS techniques to shed light on the way consumers evaluate brands and to assess the relationship between product attributes.
In short, the primary purpose of all MDS-techniques is to uncover
structural relations or patterns in the data and to represent it in a
simple geometrical model or picture.
One of the aims is to determine the dimension of the model
(the goal is a low-dimensional, easily interpretable model)
by finding the -dimensional space in which there is maximum correspondence
between the observed proximities and the distances between points
measured on a metric scale.
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Multidimensional scaling based on proximities is usually referred to as metric MDS, whereas the more popular nonmetric MDS is used when the proximities are measured on an ordinal scale.
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What does the ranking describe? The answer is given by Figure 15.4 which shows the correlation between the MDS projection and the variables. Apparently, the first MDS direction is highly correlated with service(-), value(-), design(-), sportiness(-), safety(-) and price(+). We can interpret the first direction as the price direction since a bad mark in price (``high price'') obviously corresponds with a good mark, say, in sportiness (``very sportive''). The second MDS direction is highly positively correlated with practicability. We observe from this data an almost orthogonal relationship between price and practicability.
In MDS a map is constructed in Euclidean space that
corresponds to given distances.
Which solution can we expect? The solution is
determined only up to rotation, reflection and shifts.
In general, if
with coordinates
for
represents a MDS solution in
dimensions, then
with an orthogonal matrix
and a shift vector
also represents a MDS solution.
A comparison of Figure 15.1 and Figure 15.2
illustrates this fact.
Solution methods that use only the rank order of the distances are
termed nonmetric methods of MDS.
Methods aimed at finding
the points directly from a distance matrix like the one in
the Table 15.2 are called metric methods.