Graphics Reference
In-Depth Information
Suppose dissimilarity data have been collected on a set of n objects or individu-
als, where there is a value of dissimilarity measured for each pair. he dissimilarity
measure used might be a subjective judgement made by a judge, where for example
a teacher subjectively scores the strength of friendship between pairs of pupils in her
class, or, as an alternative, more objective, measure, she might count the number of
contacts madeinadaybetween eachpairofpupils.Inother situations thedissimilar-
ity measure might be based on a data matrix. he general aim of multidimensional
scaling is to find a configuration of points in a space, usually Euclidean, where each
point represents one of the objects or individuals, and the distances between pairs
of points in the configuration match as well as possible the original dissimilarities
between the pairs of objects or individuals. Such configurations can be found us-
ing metric and non-metric scaling, which are covered in Sects.
and
. A number
of other techniques are covered by the umbrella title of multidimensional scaling
(MDS), and here the techniques of Procrustes analysis, unidimensional scaling, in-
dividual differences scaling, correspondence analysis and reciprocal averaging are
briefly introduced and illustrated with pertinent data sets.
Much of the initial impetus and theory of MDS was developed by mathematical
psychologistswhopublishedmanyoftheirfindingsinthejournalPsychometrika.Al-
though its roots are in the behavioural sciences, MDS has now become more widely
popular and has been employed in a wide variety of areas of application. his popu-
larity isreflectedbyitsinclusion inmany computer-based statistical packages. Topics
on the subject include those by Borg and Groenen (
), Cox and Cox (
) and
Young (
).
Proximity Data
3.1
Proximity means nearness in whatever space is under consideration. he “nearness”
of objects, individuals or stimuli needs defining prior to any analysis. In some situa-
tions, such as with simple Euclidean distance, this is straightforward. here are two
typesofbasicmeasureofproximity,similarity anddissimilarity, withthesebeingem-
ployed to indicate how similar or dissimilar objects are. he similarity/dissimilarity
measured between two objects is a real function, resulting in similarity s
rs
or dis-
similarity δ
rs
between the rth and sth objects. Usually all measures are taken to be
non-negative. he dissimilarity of an object with itself is taken to be zero, while the
similarity ofanobjectwithitselfisthemaximumsimilarity possible,withsimilarities
usually scaled so that the maximum similarity is unity. hechoice of proximity mea-
sure will depend on the problem under consideration. Sometimes the measure be-
tween two individuals isnotbasedonany underlying observations andistotally sub-
jectiveaswiththeteacherscoringfriendshipbetweenpupils.Inothersituations,sim-
ilarities (dissimilarities) are constructed froma data matrix forthe objects. hese are
then called similarity (dissimilarity) coe
cients. Several authors, for example Cor-
mack (
), Jardine and Sibson (
), Anderberg (
), Sneath and Sokal (
),
DidayandSimon(
),Mardiaetal.(
),Gordon(
),Hubalek(
),Gower
