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Glossary

The following MDS Terminology has been adapted
from Schiffman, REynolds, and Young (1981)

Terms Concerning Data:

Object.  A thing or event, for example, an apple.

Stimulus.  A perceived object, for example, a tasted apple.

Attribute.  A perceived characteristic of a stimulus, for example, sweet.

Proximity.  A number that shows the amount of similarity or difference between a pair of stimuli.  Input
data values are often referred to as proximities.

Data Matrix.  Arrangement of the data into a table where the first column shows the proximities
between stimulus 1 and each of the succeeding stimuli, the second column shows the
proximities between stimulus 2 and succeeding stimuli, and so on.

The total matrix is said to be square.  It is also symmetric if the proximities in the
upper-half matrix are equal to the proximities in the lower-half matrix.
MDS procedures often assume this is so.

Measurement Level (of Data).
Nominal.  Objects are sorted into groups only, such as males and females.  This is the weakest or lowest
measurement level.  It is also referred to as categorical.

Ordinal.  Objects are arranged in rank order of magnitude.  For example, bus-van-car is the ranking of
these vehicles in terms of size.  At the ordinal level of measurement, there is no indication of whether
the size difference between bus and van is more or less than the size of the difference between van
and car.

Interval.  Objects are placed on a scale such that the magnitude of the differences between objects is
shown by the scale.  However, interval scales do not have true zeros.  The Fahrenheit temperature
scale is an interval scale.  The difference between 20ºF and 50ºF is the same as the difference
between 50ºF and 80ºF.  However, it is not true to say that 80ºF is four times as hot as 20ºF.

Ratio.  Objects are placed on a scale such that the position along the scale represents the absolute
magnitude of the attribute.  This is the most stringent level of measurement.  Mass and velocity are
examples of ratio scales.  A car traveling at 60 m.p.h. is traveling twice as fast as a car traveling at
30 m.p.h.

Terms Concerning Perceptual Spaces

Point.  A position in a space that is an abstract representation of a stimulus.

Dimension.  A characteristic that serves to define a point in a space; an axis through the space.

Space.  The set of all potential points defined by a set of dimensions.

Configuration.  A particular organization of a set of points, that is, a map.

Direction.  A vector through a space that relates to an attribute.  A vector is a quantity which possesses
both magnitude and direction.

Orthogonal.  This means perpendicular to.  Most MDS spaces are developed with orthogonal axes.

Subject Weights.  Numbers showing the relative importance a subject attaches to each of the stimulus
dimensions when making his or her similarity judgements.  Sometimes they are also referred to as
dimensional saliences.

Subject Space.  A map showing the vectors of subject weights.  The length of the weight vector in
INDSCAL and ALSCAL indicates how much of the subject's data (or some transformation of that
data) is explained by the model.  The subject space has as many dimensions as the stimulus space.

Euclidean Distance.  This is the distance that corresponds to everyday experience.  The distance
between two stimuli can be calculated from their coordinates according to the Pythagorean formula.
In a three-dimensional map, for example, the distance between stimulus A (coordinates XA, YA, ZA)
and stimulus B (coordinates XB , YB , ZB ) is:

[(XA - XB )2  + (YA  - YB)2 + (ZA - ZB)2 ] 1/2.

Minkowski Distance.  This is a generalization of Euclidean distance.  The interstimulus distances,
instead of being given by the square root of the sum of the squares of the coordinate differences, are
given by the rth root of the sum of the rth powers of the absolute coordinate differences.  In the
example above, for the Minkowski  r of 3, the distance between A and B is:

[(XA - XB )3  + (YA  - YB)3 + (ZA - ZB)3 ] 1/3.

City Block Distance.  This is a Minkowski distance with  r = 1.  The distance between two stimuli
roughly corresponds to "walking" halfway round a city block contained in a regular grid of streets.
Interstimulus distances for the city block metric, Euclidean space, and a space with
Minkowski metric with r.

The meaning of minkowski r greater than 2 is complicated.  A large Minkowski r emphasizes the
dimension on which two stimuli are most different.  Gregson's (1965) suggestion that it is a measure
of the importance a subject places on stimulus dimensions is more straightforwardly handled by
subject weights (INDSCAL, ALSCAL).  While the city-block metric (r = 1) is conceptually attractive
for stimuli judged with more than one sense (taste and texture, for example), Euclidean solutions have
been found to be adequate in practice.

Terms concerning MDS Computer Programs

MDS.  Multidimensional Scaling.

Algorithm.  The mathematical procedure used to solve the problem.  The word becomes tied to the
computer program based on the procedure:  "The INDSCAL algorithm allows for individual
perceptual differences." Or it is dropped entirely:  "INDSCAL allows for individual perceptual
differences."

Metric.  The type of measuring system.  The word is used very widely in different contexts which can
be confusing.  It is common in MDS to refer to metric and nonmetric solutions for the stimulus space.
The distances in metric solutions preserve (as far as possible) the original similarity data in a linear
fashion.  The distances in nonmetric solutions preserve only the rank order of the original similarity
data.  The actual computation of the coordinates of the stimulus space is, of course, a metric
(numerical) operation.  Monotone transformations of the original similarity data
provide the bridge between rank order and distances in the stimulus space.

Transformations.  Nonmetric MDS programs apply monotone transformations to the original data to
allow performance of arithmetric operations on the rank orders of proximities.  A monotone
transformation need only maintain the rank order of the proximities. The logarithm function is an
example of a monotone transformation.

Disparities.  Monotonic transformations of the data which are much like distances (usually in a
least-squares sense) as possible.

Stress.  A particular measure that shows how far the data depart from the model.  There are several
stress formulas available in the various algorithms.

Shepard Diagram.  Also scattergram or scatter diagram.  A plot comparing the distances derived by
MDS and the transformed data (disparities) with the original data values or proximities.

 Back Home Onward

Life loves to be taken by the lapel and be told:
" I am with you kid. let's go."
Maya Angelou

Cummins - 2001