**
Glossary**

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

*
Terms concerning Data**
Terms concerning perceptual spaces**
Terms concerning MDS computer programs*

**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 X_{A}, Y_{A},
Z_{A})

and stimulus B (coordinates X_{B}
, Y_{B} , Z_{B} ) is:

[(X_{A} - X_{B} )^{2} + (Y_{A}
- Y_{B})^{2} + (Z_{A} - Z_{B})^{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:

[(X_{A} - X_{B} )^{3} + (Y_{A}
- Y_{B})^{3} + (Z_{A} - Z_{B})^{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.

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

Cummins - 2001