Journal of the Institute for Psychological and Physical Science, Vol. 3, No. 1, 101–107, 2011

Review

Asymmetric Multidimensional Scaling̶1. Introduction
Naohito CHINO*   The history and scope of asymmetric multidimensional scaling (abbreviated as asymmetric MDS) are
briefly discussed. First, the unidimensional sensory scaling in psychophysics is contrasted with the unidimensional psychological scaling in psychometrics. Second, the law of comparative judgment and the law of categorical judgment are introduced, which constitute the basis for asymmetric MDS. Third, several symmetric MDS’ swhich are extensions of the unidimensional scalings and theorems which underpin the symmetric MDS are described. Fourth, asymmetric MDS’s which are extensions of the symmetric MDS and a theorem which is a basis
for asymmetric MDS are discussed.
Keywords: asymmetric MDS; law of comparative judgment, law of categorical judgment; the Young-Householder theorem; the Chino-Shiraiwa theorem; likelihood ratio test; test for symmetry; Euclidian
space; Minkowski space; Hilbelt space; information criteria

1

Introduction

As is frequently the case, psychological phenomena which
we encounter in daily life are multidimensional in nature.
For example, a famous intelligence test, WAIS, uses 11
scales in order to diagnose adult intelligence. Similarly,
EPPS consists of 15 scales which diagnose personality. In
these tests, the number of dimensions is presupposed. It isalso frequently the case that such a number is not known apriori.
Multidimensional scaling, which we shall hereafter abbreviate as MDS, is a method for locating stimuli whichvary with respect to an unknown number of dimensions, ina certain dimensional psychological continuum, given a setof dissimilarities between stimuli. Here, stimuli may bephysical objects, persons, groups, nations and so on. Of
course, such a set of dissimilarities between stimuli can beof order n, where n is thestacked into a matrix form,
number of stimuli. We shall call the method symmetric
MDS if is symmetric, and asymmetric MDS if it is asymmetric. Symmetric MDS is an extension of unidimensional
scaling to be discussed below, and asymmetric MDS is anextension of symmetric MDS.
Asymmetric MDS has a long history. In fact, if we view
sensory scalings in psychophysics (e.g., Fechner, 1860;

Herrnstein & Boring, 1965; Stevens, 1951; Weber, 1834) as
the origin of the unidimensional scaling, asymmetric MDS
goes back to the 19th century. However, it may be naturaland appropriate not to view sensory scalings as the root ofasymmetric MDS, because it is usual that psychologicalscalings in psychometrics do not necessarily assume any
correspondence between the scale values to be attached tostimuli and the physical quantities which belong to them.
In this sense, asymmetric MDS may be said to go back to
the early 20th century, especially to the pioneering work ofThurstone (1927a). According to him :… we have then two criteria, one for the stimuli and one
for the discriminal processes of these stimuli. The stimulus
continuum must of course be defined in terms of some definite stimulus attribute. The discriminal continuum is a qualitative one which does not necessarily have either magnitude or intensity (Thurstone, 1927a, p. 370).
His seminal work was a watershed of unidimenisional
scalings, and it extended the applicability of scaling methods drastically, although one of the key terms in his work,
discriminal process, which means psychological value to be
attached to any object or person in general, sounds today
somewhat strange.
In a companion paper (Thurstone, 1927b) he proposed
the famous law of comparative judgment using the notion*This paper is based on the author’s works on asymmetric MDS since 1977. Requests for reprint should be sent to Naohito Chino, Dept. of Psychology, Aichi Gakuin University, Nissin-cho, Aichi, Japan, or chino@dpc.aichi-gakuin.ac.jp.

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CHINO
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hethe
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heproposedfamouslawofcomparative
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um.
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that
that
we
view
Richardson’s
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a
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one is that
X
X
cthe probability(e.g., Xwhichonadiscriminal
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continuum.
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whichalthoughmeans
j is superiorscaling
methodsprocess,drastically,
oneto
ofthisthej law,key
in his work,
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paper(Thurstone,
1927b)
heproposedthefamouslaw wordsofcomparative
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using
and
X
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stimulionofdiscriminal
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which
plays
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centralroleinhis
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X
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X j isplays
c ,j andandsoXon)
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minal
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and
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watershedofthanunidimenisionaland
it cextended
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and
XcanX
beacademic
two stimuli
onof discriminal
which
centralrole
in histheory.
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isprocess,expressed
as playsinmeeting and there exists no way of knowing(e.g.,
criminal
continuum.
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to
this
law,
the
probability
thatX{superiorj Xj cis
ntinuum.
Accordingtocomparative
this law, theprobability
that Xj > Xc (e.g.,( j is superior)2 }∫osed the famouslawofjudgment
using
∞
In
a
companion
paper(Thurstone,
1927b)
heproposedthefamouslawofcomparative
usingIt is Tucker andalthough
one
of
the
key
words
in
his
work,
discriminal
process,
which
means
the
detailsof
his work judgment
at this point.
1
1
y
−
µ
Xc (e.g.,
Xj is jcsuperior
criminal
Accordingtothisthe probability
Xjthaniscontinuum.
heavier
than
and
so
on)
is
expressedasc ) = √ that Xj >exp
dy,
(1)
=two
Plaw,
rob(X
−
P
eavier
Xtheory.
so X
on)
is
expressedasXbestimulij > X
alorroleinhis
Xcj, andc , and Let
c jc
Messick(1963)
thatclearlypoint
outthat MDS was devel2σ
2πσ
jc Let Xj and Xc be two stimuli
thenotiondiscriminal
process,which
plays a}centraljc 0 roleached
objects
personin general,
somewhatstrange.
} in his theory.
{or Xj tois anyheavierthanorXofas{sounds()∫ ∞c , and so∫on) is expressed2()
superior
the probability that Xj > Xc 1(e.g., X∞
oped by Richardson (1938), and extended by Torgerson
j is 112 y − µ
1 comparative
y−µ
}
{
> (1)
Xc (e.g.,
Xj is superioroncontinuum.
Accordingtojc−thislaw,
thethat µXjare
hurstone,1927b)
thefamouslaw
whereµproposed
isjdeﬁned
as √thediscriminalµj jc−)µ2probability. Here,andthepsychologicalvalues( judgmentdy, µj (1)=a hePdiscriminal
rob(X
>X
Pjc
∫of
cusing
c) =
dy,
exp
−
P
∞ expdiﬀerence,
jc = P rob(Xj > Xc )jc= √(1952). c
σjcµjc (1)1 jc 202πσressed as
σjc − 12 y −
2πσ
0 √
jc
dy,
(1)
=
P
rob(X
>
X
)
=
exp
P
jc
j
cin his theory.
iscontinuum
heavier
than
XLet
on)
expressed
asand arethanandXcis bestimulicess,which
aXcentral
Xj soon
corresponding
totwostimuli,
respectively
theandmean
values of thetheorem by itself
c , or}X
c , and{ plays
Although
Young
Householder’s2theσjctwo)2 jrole( a discriminal
2πσ
∫ ∞
jc 0
deﬁneddiscriminal
diﬀerence,
µ
−
µ
.
Here,
µ
and
µ
are
the
psychologicalvalues}{1asythe
−
µ
j
c
j
c
jc
djcAccording
asisthediscriminal
diﬀerence,
µ
−
µ
.
Here,
µ
and
µ
are
the
psychologicalvalues()∫ μ∞c .
jc
isascthethat
discriminal
where
does not describe
the way of determining the number of diμ −μ j cthe2dy,defined(1)
exp − tolaw,
probability
Xjj > Xdifference,
twothisdiscriminal
processes,c (e.g., 1Xj jis superior
1 y − µjc values
σjcdiscriminal diﬀerence,
2as theµj stimuli,
− µ>
Here,
µ√j andµc areexp
the0 is deﬁned
jc
c. X
dy,
(1) are embedded,
Ptwo
rob(X
) = are−Pstimuli,riminalcontinuum
correspondingtothethepsychologicalmeanvaluesofthethe=psychologicalvalueson respectively
athe
discrimiHere,mensionsofthespace in which stimuli
μtwo
μto
jc
j respectively
cand
j and
c are
tinuumcorresponding
the
and
are
mean
valuesoftheσjc
an Xc , and so on) is expressed asϵj , jc ϵj 0∼ N (0, σj2 ),2(2)
vj = µj +2πσriminalcontinuum
corresponding
to
the
two
stimuli,andarerespectively
themeanvalues
of
the
−
µ
.
Here,
µ
and
µ
are
the
psychologicalvaluesnal
continuum
corresponding
to
the
two
stimuli,andarereYoungandHouseholder(1938)refertothe paper which enc
jcriminalprocesses,{
cesses,()2 }∫ deﬁned
∞
where
µ
is
as
the
discriminal
diﬀerence,
µ
−
µ
.
Here,
µ
and
µ
are
the
psychologicalvaluesspectively
themeanvaluesofthetwodiscriminal
processes,
ables
us
to
fit
a
lower
dimensional
setof
points to a given
j
c
j
c
1 jc
1 y of−µjc
stimuli, processes,
and are respectively
the mean values
riminal2theσj2 ),
(2)
dy,
(1)
rob(X
j∼+Nϵ(0,j , σϵj),∼ N (0,j > Xc ) = √,j =ϵjµ−(2)
vj = µj + ϵjvexp
jset. This is nothing
but the work of Eckart and Young
2σjc
2πσjc 0 continuum
on a discriminal
to
2 the
=σµ
+ ϵ two, ϵstimuli,
σc2 ).are respectively(3)(2)N (0,andc ∼, ϵj ∼ Nvc(0,(2)the mean values of the
vj = µj + ϵjcorrespondingj c), c(1936),
whichisconcerned
with the method for obtaining
2 two discriminal processes,
discriminal
diﬀerence,2 µj − µ2c . Here, µj and (2)µc are the psychological values
ϵj ∼ N (0, σj ),anvcm, respectively.
× n matrix B Moreover,of rank r thatminimizesa Frobenius
Here, σj and σc are variances ofthe discriminal processes, vj andσjc inEq.
2c ∼ N (0, σ 2 ).
(3)v,c =ϵcµc∼+Nϵ(0,(3)of)
c , σϵ
=
µ
+
ϵ
v
c
c
c
µc2 jmean+ ϵj , values
ϵj ∼
N the
(0, σj2norm,), (3)|| B − A || F, given an m × n (
vj =
c ).
orresponding to the two stimuli, and are respectivelytheµjc(2)A of rank k, andmatrix(1) is the standard
deviationas Pjc = Φ σjc .
µc + ϵc , of ϵtheN (0, σc ). diﬀerence. Eq. (1) can be abbreviated
(3)
vc2 =
c ∼ discriminal
2
2
2
of
the
discriminal
processes,
Here,ainnonnegative
integer& Hanson, 1974).
r < k (e.g., Lawsonóofthe ódiscriminal
processes,
v
and
v
,
respectively.
Moreover,
σ
in
Eq.
j and
c are variances
j
c
jc
c are
eϵcandvariancesthediscriminal
processes,
v
and
v
,
respectively.
Moreover,σ
Eq.
∼ Nσ(0,σc2of).variances
(3)
c
jc
( )
Bock and Jones (1968) call itj the response
function.
)(22, respectively.
Moreover,in Eq.
is theSince
a solution
to the problem can be obtained by utilizing
v j and ofv ctheµjcóvjjc andσc aredeviation
variancesdiscriminal
processes,Eq.
vc , (1)respectively.
Moreover,Eq.
Φ(inσjce andstandard
discriminal
diﬀerence.
(1) canbe abbreviatedasµjcPjc. =σjc
µ discriminal+ ϵjof, theϵj ∼N (0, σj2 ),σEq.
(2)
vof
j =
= σΦc2 ).
deviation
thediﬀerence.
canvbeabbreviatedas∼PNjc(0,). case of a rating scale
ϵEq.
ϵccomparative(3) (abbreviated as
cesses, vj and
vc ,jstandard
respectively.
Moreover,
in(1)
Eq.
σ
c = µthec +lawc , ofjc
jcfamous singular
Later,
Torgerson(1954,
1958)
extendedjudgment
tothedeviationof the discriminal
difference.
(1)
can
the
value decomposition
µ
jc
( )
.
=
Φ
ed standard
deviationofthediscriminal
diﬀerence.
Eq.
(1)
can
be
abbreviated
as
P
jc
σ
µ
Jones
(1968)
callittheresponsefunction.
jc
jc
68)
thewithresponsebeasBock and theJones (1968)
SVD) of A as According
in Lawson and
Hanson,SVD. proposed
ence.call
Eq.it (1)canbeabbreviatedas        .
Pjc = Φ σand2 function.
a abbreviatedorderedof categoricalto thislaw,
the plays an imporjc
Here,
σfew
and
σc2 arecategories,
variancesof thediscriminal law
processes,
vj and judgment.
vc , respectively. Moreover,σjc
in Eq.
jthed, Torgerson
Jones (1968)
call
itresponsefunction.
)((1954,
1958)
extendedthelawofcomparative
judgment
to
the
case
of
a
rating
scale
call
it
the
responsefunction.
tant
role
in
the
theoryof
MDS.
2(1954, 1958)
extendedthelaw
ofiscomparative
judgment
to thea (3)ratingµc is+ the
ϵc , standardϵc ∼Nσjudgedvc =probability
to thefall
in less thanor caseequalofto
the(1)
kthscalecategory
is writtenasasPjc = Φ µjc .
that
Xj(0,
c ).
(1)
deviationofdiscriminal
diﬀerence.
Eq.
can
be
abbreviatedσ,wTorgerson(1954,
1958)
extendedthelawofcomparative
judgment
to
the
case
of
a
ratingscale
jc
Later,
Torgerson(1954,
1958)
extendedthelawof
comNowadays
theMDS
formulated
fully
by Torgerson isordered andcategories,
andproposedthelawscaleof judgment.
categorical According
According{tocomparative
judgment
to
the
case
ratingcategories,
proposedthelaw ofofacategorical
thisthethis law,)2 } the( law,to∫judgment.
0
Bock
and
Jones
(1968)
callittheresponsefunction.
parativejudgment
theof categoricala ratingscalewith aσjc
fewclassical
MDS, because Torgerson assumed thatesofthe discriminal
processes,
vj to
and
, respectively.
in1 Eq.
ccase− theτk ) law,
1 Moreover,v −called(µtow
orderedandproposed
thev≤
law
of
judgment.
the)(According
isfalljudgedtoinorPlessorto the kthiswrittenasj thistythat Xjtocategories,ofcategorical
judgment.
Accordingtothanthisdv = Φ(Zjk ),rob(Xτkequal)the=√
exp as
−
(4)
Pfall
jk =
j law,
is judgedin less
than
equal
to
the
kth
categoryis category
writtenµorderedcategories,
andproposedthelawof
categoricalthedissimilarities
inhis MDS
had to fulfill
a distance propjc
σ
2
2πσ
j−k
Later,
Torgerson(1954,
1958)
extendedthelawofcomparative
judgment
to
the
case
of
a
rating
scale
−∞
j−k
.
=
Φ
of
the
discriminal
diﬀerence.
Eq.
(1)
can
be
abbreviated
as
P
jc
}
{
σ
is judged
toisfallin less {thanor equal to(the)kth
writtenasty thatXjkth} category
jc
)2is
∫ 0(
qual
to thecategory
written∫2(µ −thatisjudgment.
Accordinglaw, the1 probability
erty
in
the
Euclidianspace,
althoughhecalledit the comX
0 1 as to this
j
τ
)
v
−
j
k
−(τkth1ττ )}
vexp−(µ}k+1th{−jproposedk ) categoryorderedcategories,
and
the
law
of
judgment.
Accordingto thislaw, thet{
thefunction.
where
of
the
and
categoryon thediscriminal
continuum,√ boundary)2categorical
dv
= Φ(Z
=
=the
), distance(4)
Pjkjresponse∫ 10value
j a≤fewjk)kk2 is
(
√
dv
=
Φ(Z
rob(X
≤ Pτkrob(X) with
=
exp
−
),
(4)
jk
judged
to
fall
in
less
than
or
equal
to
the
kth
categoryisparative
which
did
not
necessarily
satisfy
the ratio
σj−k
1 j−k −∞
12 v − (µ2πσ1 v − (µj − τ2πσj − τk )2 exp σj−kk ) j−k −∞
√
dvthe
= Φ(Z
= P rob(Xτ=)as=−tointhe),variable(4)P−jkextended
dv
=X
Φ(Z
p958)
), judgment
(4)
jZ≤
k of
jk
jkj−k
is
judged
to
fall
less
than
or
equal
to
kth
categoryiswrittenasprobability
that
the
law
comparative
case
of
a
rating
scale
(τ
−
µ
)/σ
.
Here,
σ
is
the
standard
deviation
of
the
v
in
Eq.
(4).
and
j
written
scale
level
of
measurement.
jk
k
j
j−k
σj−k
2
σj−k
2
2πσj−k −∞boundaryvalue categoryof the kth
categoryandk+1th
on{the (discriminal
continuum,} b) discarded the metric restriction of thek is thevalue
undaryof theandk+1th
category
oncategory
thediscriminal
continuum,)∫ toKruskal(1964a,2 published
0 thiss, and proposed
the kthlaw ofafter
categorical
judgment.
According
the
Ten
years
Thurstone’s
seminalwork
referredto law,
above,
two
paperswhich underpin−
τkwere
)
1
1
v
−
(µ
j
and
k+1th
categoryonthediscriminal
continuum,istheboundary
value
of
the
kth
category
andk+1th
categoryonthediscriminal
continuum,
k
√
dv
=
Φ(Z
=
P
rob(X
≤
τ
)
=
exp
−
(4)case in which the
P
=
(τ
−
µ
)/σ
.
Here,
σ
is
the
standard
deviationofthevariable
v
in
Eq.
(4).
classical
MDS
andextended
MDS to the
jk
j
k
jk ),
k
j
j−k
j−k
/σ
.
Here,
σ
is
the
standard
deviationofthevariable
v
in
Eq.
(4).
j−k
j−k
σj−k (1938) and the other Richardson
dd tofall in lessthanor equal
tocategory
is2πσwrittentheclassical
MDS
tothe
bekth
publishedlater.
One
Young and 2Householder−∞
j−k is as
deviationofthevariable
v
in
Eq.
(4).
=
(τkafter− µjThurstone’s)/σj−k . Here,σj−k isthereferredstandard
the variable
v in Eq.
(4). underpin
similarity
is measured at an ordinal level. Therefore, hisearsseminaltodeviation
above,
twoofpapers
werewhichpublishedwhich} were{ workhurstone’s
seminalreferred
to concerned
above, twopaperspublished
underpinand(is)∫ work2the0τThe(1938).
former
with
thenecessary
suﬃcient
conditionthatthe
set of numbers
djk
where
is
the
boundary
value
of
kth
category
andk+1th
categoryonthecontinuum,
toabove,
two
paperswere
publishedwhich
underpinMDS
iscalledthediscriminal
nonmetric
MDS.
It extended
the applicak
−
τ
)
v
−
(µ
1
1
j
k above, two papers were published which underpin
ears
after
Thurstone’s
seminalwork
referred
to
√
ical
MDS
to
be
publishedlater.
One
is
Young
and
Householder(1938)
and
the
other
Richardson
(4)
dv
=
Φ(Z
τ
)
=
exp
−
),
(4)
k
jk
o be publishedbetween
later. One
is
Young
and
Householder(1938)
and
the
other
Richardson
bilityofMDS
toacertainextent.
Guttman(1968)
devel2σstimuliconstitutes
the mutualof a real
set of points
EuclidianAccording to
2πσandj−kg and Householder(1938)
andRichardson
−∞
(τ
−the
µjother)/σj−kHere,
σj−kHouseholder
isdistances
the standard
deviation
ofotherthe invariablev in space.
Eq. (4).
j−k Z
jk =
k
ical MDS
toisbepublishedlater.
One
is .Youngand(1938)
andthethesetRichardson
The
formerconcerned
withthenecessary
andsuﬃcient
conditionthatofnumbersdjk
oped
anotheralgorithm
fornonmetric
MDS
calledtheis concerned thewiththe thatnecessary
andsuﬃcient
conditionthatthe setnumbersdjk
andcondition
thek+1th
setof numbers
dthis
Young-Householder
theorem,conditionis that∆above,isofpositive
semi-deﬁnite(abbreviatedhereafteras
jk theTen
yearswith
after
Thurstone’s
seminalwork
referredtotwopaperswere publishedwhich
underpinaluesuﬃcientof the kth
category
andcategoryon
discriminal
continuum,The
formerisconcerned
thenecessary
andsuﬃcient
condition
that
the
set
of
numbersdjk
where
boundary ofvalueof settheofkth
category
andSmallest
Space Analysis,
abbreviated as SSA, and Lingoes
k is the distances
stimuli constitutes
themutualreal
pointsinspace.
Euclidianspace.
Accordingtostitutes
mutualdistancesofisaconcernedreal
set ofawith
pointsin Euclidian
Accordingtoreal σset theof ispointsin Euclidian
space.
Accordingtolater.
p.s.d.).
Thecategory
lattera continuum,
model
ofissymmetric
MDS,
which
might
beand
viewedaseries”preliminary
thekclassicalMDS
topublishedYoung(1938)
the asother
RichardsonHere,standard
deviationofthevariablev One
in points
Eq.
(4).
+ the
1th
onbethediscriminaland(1973)
provided
atoprogram
composed of several verZ andj−kj k = Householder
stimuliconstitutes
mutualdistances
of
a
real
set
of
in
Euclidianspace.
Accordingg-Householder
theorem,
thisconditionisthat ∆is positive semi-deﬁnite(abbreviatedhereafter asder
thiscondition
is
that
∆
is
positivesemi-deﬁnite(abbreviatedhereafteras∆ istheorem,
positive semi-deﬁnite(abbreviatedhereafter
as
−
)
/
.
Here,
is
the
standard
deviation
of
the
(
sions
of
SSA.
μ
ó
ó
work”
of
Torgerson’sclassical
MDS,
considering
Torgerson’s
citation(Torgerson,
1952)
toRichardson’s
k
j
j
–
k
j
–
k
(1938).
Thethisistwo
concerned
withthe necessary
suﬃcient condition
thatasthe set of numbers djk
e’sseminal work
referred
toformerabove,
paperswere
publishedwhichand
underping-Householder
theorem,conditionisthat ∆
is MDS,
positivesemi-deﬁnite
hereafterThe
latterconcerned
with
aEq.
model
of MDS,
symmetric
mightasbea(abbreviatedviewedasThea ”preliminary
concerned
withamightmodel
whichmightwhich
be viewed”preliminary
variableinsymmetric(4).
applicabilityof MDSwasfurther extended
by severvof
etric
MDS,iswhichbeviewedasa ”preliminary
work.
The
reasonthat
weview
Richardson’s
work
as
a
preliminary
one
is
that
workAccording
is merelyabetweenstimuliconstitutes
the(1938)
mutualdistancesof Richardson
a realset of points
in Euclidianhis
space.
toishedlater.
One
is YoungandHouseholder
and
the
other
The
latterisconcerned
with
a
model
of
symmetric
MDS,
which
might
be
viewedasa”preliminary
Ten
years
after
Thurstone’s
seminalwork,
referredtoalresearcherswhodevelopedtheso-calledindividual
difclassical
MDS,1952)
considering
Torgerson’s
citation (Torgerson,1952) to Richardson’s
gerson’scitation(Torgerson,
to Richardson’ssTorgerson’sclassical
MDS,
considering
Torgerson’s
citation (Torgerson,1952) meeting
to Richardson’ssummary
ofand
a 15
minutespresentation
inthe
anset
academic
andthere exists(abbreviatedno way of knowingthetheYoung-Householder
theorem,
thiscondition
is
that
∆
is
positivesemi-deﬁnite
hereafter
as
ned
with
the
necessary
suﬃcient
conditionthatofnumbersdjk
above,MDS,
two papers
were publishedwhichcitation
underpin(Torgerson,
the clas- 1952)
ferences
MDS which enables us to examine differences inTorgerson’sclassical
considering
Torgerson’s
to Richardson’s
he
that
we
Richardson’s
work
asa a preliminary
one
isthatis his
work
is merely aasareasonpreliminary
oneview
is thathis
workmerelythat
weviewRichardson’s
workasthisais preliminary
oneis thathis
workmerelydetailshis
workatconcerned
point.
It
ismodelTuckerandMessick(1963)
thatamightclearly
point
out
MDS
was
sicalofMDS
to be
publishedlater.
Oneis
Young
andAccording
House-MDS,
individuals
inbesomesenses.
methodassumes in generalp.s.d.).
The
latteriswithaofsymmetric
which
viewedas thataThis”preliminary
themutualdistances
of
a
real
set
of
pointsin
Euclidianspace.
tohereasonthatthere
wepresentation
view
Richardson’s
work theas meetinga preliminary
one exists
is thatnohis
work
is merelyaicofmeeting
andexistsno wayofknowing
yinutesa 15
minutesin theanmeetingacademic
and therewayofknowingthepresentationinanacademic
andthere
existsnowayofknowingtheholder(1938)
andother
Richardson(1938).
The
formerthatinterstimulus
distancesaredefinedforeach individual.
developedby∆Richardson(1938),and considering
extendedbyTorgerson’s
Torgerson(1952).
work”presentation
Torgerson’sclassical
MDS,
citation(Torgerson,1952) to Richardson’s
conditionisofthatis positive
semi-deﬁnite(abbreviatedhereafter
as
yfem,
of
athis
15
minutesin anacademic
andthere
existspoint
no
way
of
knowingtheessick(1963)
thatclearly
point
out
that
MDSMessick
wasmeeting
…
his
work
at
this
point.
It
is
Tuckerand(1963)
thatclearly
out
that
MDS
was
,
,
,
,
where m is theisconcerned
withthenecessaryandsufficientcondition
Let
such
a
set
of
data
be
m
1
2
at this point. It is Tucker and Messick (1963) that clearly point out that MDS was
work.
The
reasonthatwhichweviewRichardson’s
asthe
a point
preliminary
oneis thathis work is merely aedwitha(1952).
modelofsymmetric
MDS,
might
bestimuli
viewedconstitutes
asworka ”preliminary
fd hiswork
atthispoint.
It
is
TuckerandMessick(1963)
thatclearly
out
that
MDS
was
betweennumberofindividuals.
that
the
set
of
numbersdorgersonj k Torgerson (1952).
by Richardson
andbyextended
by
2
rdson
(1938), and(1938),
extended
Torgerson(1952).
summaryofa 15extended
minutespresentationanacademic
meeting andFor
there
exists Tucker
no way andof knowing
mutualdistances
ofcitationa realby
setof points inin1952)
Euclidianspace.
example,Messick the(1963) proposed aalMDS,
considering
Torgerson’s(Torgerson,
to Richardson’sd byRichardson(1938),
andTorgerson(1952).
Accordingtoa theYoung-Householder
theorem,
thisMessick
View point
Analysisas PVA. In the firstdetailsof hisasworkatthis point.
Tuckerand(1963)
that ofclearlyout abbreviatedthat MDS wasview Richardson’s
work
preliminary
one
is is
that
his work
iscondimerelya Points
2 It
2semi-definite (abbreviated hereaftertion is that is positive
stage of PVA, we first rearrange elements of each of the
2 andby Richardson
extended
Torgersonthe(1952).
esentation indeveloped
an academic
meeting and(1938),
there
existsno way byof knowing
as p.s.d.). The latter is concerned with a model of symmetdata matrices into a column vector of order n (n − 1) / 2.
oint. It is Tucker
Messick
thatclearlyout thatwork”
MDS was Next, we stack m vectors into an n (n − 1) / 2 × m matrix X,
ricandMDS,
which (1963)
might beviewedas pointa “preliminary2938), and extended by Torgerson (1952).
2

─ 102 ─was further extended by several researchers who developed the so-called individm series composed of several versions of SSA.
h enables us to examine diﬀerences in individuals in some senses. This method
cability of MDS was further extended by several researchers who developed the so-called individnterstimulus distances are deﬁned for each individual. Let such a set of data berences MDS which enables us to examine diﬀerences in individuals in some senses. This method
Asymmetric Multidimensional Scalingm is the number of individuals.
in general that interstimulus distances are deﬁned for each individual. Let such a set of data be
and Messick (1963) proposed a Points of View Analysis abbreviated as PVA.
· · ·, ∆m , where m is the number of individuals.
decompose
theofinner
matrixthea YoungX t X by
, we ﬁrst rearrange
elements
each productof the data
matrices
intocolumn vectormaximum likelihood (ML) MDS called MULTISCALE. It
xample, Tucker Householder
and Messick theorems(1963) proposedaPointsView Analysis abbreviatedas PVA.
(i.e., including theofEckart-Youngassumes thatthe observed dissimilarities ä j k r obey the logt, we stack m vectors into an n(n − 1)/2 × m matrix X, decompose the inner
of
order
which
normaldistribution,that is, ln ä j k r ~ N (ln d j k , ó 2). Here,theorem),obtain aelements
best fit matrixpX̂ ofrst stage of PVA,
we ﬁrstandrearrangeof eachthedatamatrices
intoacolumnvectorp
the Young-Householder
theorems(i.e.,
includingtheEckart-Youngtheorem),is less
than m.
best into
fit matrix
is −further
into a X, decompose
subscript ther denotesn(n − 1)/2. Next,
we stackm The
vectorsan n(n1)/2 rotated× m matrix
inner the rth replication of dissimilarity judgstructure.
Each m.
of thecolumnvectorsof therotated rotatedment on the pair of stimuli, (j, k). In general, unknown pap whichis less thanThe
best ﬁtmatrixis further
rix X̂ p of torder simple
matrix X X by the Young-Householder theorems (i.e., including the Eckart-Young theorem),
matrix F p is viewed as a view point of the dissimilarity
rameters such as coordinates of stimuli etc. in ML MDS are
Each of the column vectorsof the rotated matrix F p is viewed as a view point
order
whichstage
is lessm. Theis furthersorotatedin a best ﬁt matrix
X̂ p of In
judgments.
the psecond
wethancomputesomebest
sortﬁtofmatrix estimatedas to maximize the likelihood of the data underments. At the compromise
second stagematrices
we compute
some
sort
of suchcompromise
matricessome specified distributional assumption.
…
,
=
1,
2,
,
in
a
way
that
a
p
T
a
mple structure. Each of the∑mcolumn vectors of the rotated matrix F p is viewed as a view point1element
of of F p . Takane (1978a, b) developed a maximum likelihood (ML)where ffi ia
ch a way that           Ta = mthe(i,(i,a) a)elementa isisthe
i=1 fia ∆i ,, whereissimilarity judgments. At the second stage we compute some sort of compromise matrices
Finally we matrices,administerandMDS
to each of these commethod for nonmetric MDS for a set of empirical orderings
F p.compromise∑
DS to each of thesemobtain p conﬁgurations of stimuli
1
1, 2, · · · , p in sucha way
that Tand
fia ∆i , whereoffiastimuli
is the (i, a)F pof. stimuli, which he calls MAXSCAL-1,
a =obtain
promisematrices,on element
a set of ofpairs
p configurations
i=1
m
ew points, respectively.
corresponding
the pcompromise
view points,matrices,
respectively.
utilizing ofThurstone’s
we administer MDS
to each of tothese
and obtain p conﬁgurations
stimuli law of comparative judgment. In his70) developed anothertype
the individual
diﬀerences
MDStypecalledCarrollandofChang(1970) developed
anotherof INDSCAL,
thenonmetric ML MDS, coordinates of stimuli to be recoverednding to the p view points, respectively.
individual
differences
MDSdistancecalled INDSCAL,
considering
are related to the distance d i j between stimuli i and j by the
PVA. They assumed
a weighted
Euclidianmodel, i.e.,oll and Chang (1970)
developed
anothertypeofthe
individual
diﬀerences
MDS
called INDSCAL,
Minkowski
power metric, which is called the representation
� shortfalls of PVA. They assumed a weighted Euclidian dis� rmodel.
The
tance model,i.e.,d i j is then assumed to error-perturbed to giveng shortfallsof PVA.
They assumedaweighted
Euclidiandistancemodel,i.e.,�∑2
wit (xjt − x�i = 1, 2, · · · , m,(5)rise to a psychological value ( t ) in two ways. One is thedijk = �kt ) ,
ij
� r
t=1
�∑
normalerror,
and
the
other
the
multiplicative
error. He calls2
wit (xjt − xkt ) , i = 1, 2, · · · , m,  (5)(5)
dijk = �suchaprocesstheerror
model.
Assumeherethat a new
ight corresponding to each individual
i on each dimension t. It is apparent
t=1
variable Y i j k l equals 1 whenever i j is greater than k l, andwit denotes a weightto eachindividual toi eachon eachdimension otherwiset. It is apparentaweightcorresponding
individuwhere corresponding
0. Then, Pr (Y i j k l =1) = Pr ( i j > k l) which is an
w i t denotes
3
important assumption implied by the law of comparative
al i on each dimension t. It is apparent that INDSCAL asjudgment. For the normal error, Pr ( i j > k l) = Φ (h i j k l),sumes a stimulus space which3 is common to all individuals,
where h i j k l = (d i j − d k l) / ó i j k l. Likelihood of the data canand a subject space which represents individual differences
thus be defined using the function Φ.
in relative saliences or importances in the dimensions.
Takane (1981) developed another ML MDS called
Escoufier and his colleagues (e.g., Escoufier, 1973; Lavit,
MAXSCAL-2 utilizing law of categorical judgment. As in
C., Escoufier, Y., Sabatier, R., and Traissac, P., 1994) proMAXSCAL-1, this method assumes some representation
posed a third type individual differences MDS model called
models and error models. In addition, it presupposes rethe Structuration des Tableaux A Trois Indices de la Statissponse models. That is, subjects place error-perturbed proxtique (STATIS) (Structuralization of the three-way statisticalimities in one of the M rating scale categories C1, C2, … ,table). STATIS consists of the three steps. Step 1 is called
the inter-structure analysis, which uncovers a multidimenC M. Therefore, the probability that the error-perturbed
proximityof stimuli i and j, say, i j falls in c m is given bysional inter-structure among m occasions.
Step
2
is
calledfalls in cm is given by thethe intra-structure analysis, which discloses a holistic mulpijm = pr(bm−1 < τij < bm )..(6)
tidimensional intra-structure among n stimuli. Step 3 is
Utilizingjudgment,
the law ofone
categorical
can then of the data.
called the trajectory analysis, which estimates
the trajectory
Utilizing
the law of categorical
can then judgment,
construct one
the likelihood
construct the likelihood of the data.
of each stimulus through occasions in the holistic intraTakane & Carroll (1981) proposed a third ML MDS method in the case when dissimilarity
Takane & Carroll (1981) proposed a third ML MDS
structure (e.g., Grorud, Chino, & Yoshino, 1995).
takenanbyAlterranking procedures
thewhenmethodof conditionalrankmethod insuch
the ascasedissimilarity
measuresareorderstaken or the method
Takane, Young, & de Leeuw (1977) areproposedranking procedures such as the method of conditional
nating Least Squares algorithm for individual
differences
combinations.
They call by
it MAXSCAL-4.
rankandordersor thepioneering
method of works,
triadic somecombinations.
TheysCALing method called ALSCAL, which includes
not only Takane,
After Ramsay,
Carroll’s
other researchers
have also
call it MAXSCAL-4.
some individual differences MDS methods but also thediﬀerent ML MDS methods (e.g., Groenen, 1993; Groenen, Mathar, & Heisser, 1995; MacKa
After Ramsay, Takane, and Carroll’s pioneering works,classical MDS as well as various MDS methods developed
Oh &theRaftery (2001)
criticized
ML MDS’s
Bayesiansome otherresearchers
have andalso proposed
proposed adifferent
MLMultiDimensiona
by that time including nonmetric MDS. Recently,
They compiled
MDS
methods(e.g.,
Groenen,
1993;
Groenen,Mathar,fruits of years of study on symmetric MDS
developedbyabbreviated as Bayesian MDS (BMDS). Their major criticisms on the ML MDS&are (1) justiﬁcati
Heisser, 1995; MacKay, 1989). Recently, Oh & Rafterythat time into an algorithm.
relies on asymptotic theory, and the number of parameters to be optimized over typically grow(2001) criticized ML MDS’s and proposed a Bayesian MulAlthough symmetric MDS methods developed by thethe number of objects,tiDimensionalso that the asymptotic
theory mayasnotBayesian
apply in MDS
high dimensions (CoScaling abbreviatedlate 1970’s were merely descriptive, someaspsychometricians
Theirmajormany
criticisms
the minima
ML MDSare (1)made inroads into the inferential symmetric
MDS
at that surface(BMDS).
(2) thelikelihood
will tend
to havemore of
local
when
there are more dim
justification of ML relies on asymptotic theory, and thetime. For example, Ramsay (1977, 1978, 1982) proposed aand ﬁnding a good initial estimate will be correspondingly more diﬃcult.

In their BMDS a Euclidian distance model is used and a Gaussian measurement error is as─ 103 ─ Markov Chain Monte Carlo (MCMC) algorithm is used in order tothe observed dissimilarity.

Bayesian solution of the stimuli conﬁguration. According to them, their BMDS provided a mu

ﬁt to the data than did the classical MDS and a moderately better ﬁt than did ALSCAL i

m is given by thepijm = pr(bm−1 < τij < bm ).

(6)

cycloneandofsothe
We shall hereafter abbreviate the canonical analysis of asymmetry ag the law of categorical judgment, one can then constructthemodel,likelihood
data.
NaohitoCon.
HINO(e.g.,
Chino
Shiraiwa,
although
it is sometimes
the Gower
diagram.
cyclonemodel,and soon.1993),
We shallhereafter
abbreviate called
the canonicalanalysisof asymmetry ane & Carroll (1981) proposed a third ML MDS methodin thecase& when
dissimilarity
measures

In Chino
CASK,
observeddissimilarity
datamatrix ∆ isdecomposed
into
the symmetric par(e.g.,& the
Shiraiwa,
1993),
although
it is sometimes
calledthe Gower
diagram.
n by ranking proceduresas the methodof conditionalrankordersmethodofintotriadic
number ofsuch
parameters
to be optimized
over typically
grows or the composed
the symmetric part S s and the skew-symﬁrst,
is,
∆
= is,
S s +∆
Then,
the latterpart
is decomposedskew-symmetricpartmetric
S sk
CASK, thedissimilarity
datamatrixintolatterthe symmetric
par
that
=S sk
the
as the number of objects, so that thethe Inasymptotic
the- observed
part that
Siss. +decomposed
S s k . Then,
S s k first,
tions. They call as
it fastMAXSCAL-4.

cyclonemodel,and soon.
Weshallthathereafter
abbreviate
canonical
analysis
part
isﬁrst,decomposed
ory may not apply in high dimensions (Cox,1982),(2) theto yield,is, by∆ SVD=
S sto+yield,
S sk the. Then,the latter
partofis asymmetry
decomposeda
theskew-symmetric
part
S sk
Ramsay, Takane, and Carroll’s pioneering works, someother researchers
have
also
proposed
likelihood surface will tend to have many
more
local
minit canonicalcyclonemodel,and so on.1993),
We shallhereafter
abbreviate
theanalysis(e.g.,
Chino&
Shiraiwa,
although
it issometimes
calleddiagram.
= XΓKX,, the GowerS sk(8) of asymmetry atoyield,ML MDS methods(e.g.,there
Groenen,
1993;
Groenen,Heisser,ma whenare moredimensions,andMathar,finding a&
goodini- 1995; MacKay, 1989).
(e.g.,
Shiraiwa,
1993),
although data
it is sometimes
the Gower
tdecomposed
In Chino
CASK,&theobserveddissimilarity
matrix∆ iscalled
intodiagram.
the symmetric par=ΓXΓKX, , γnsk matrix, Oh & Raftery (2001)
criticizedML
MDS’s and proposeda BayesianMultiDimensional
whereof Γorder
isScaling
andiagonalof ordersuch that Γ =tial estimate
will becorrespondingly
morewheredifficult.
Γ is a diagonal
matrixsuchSthat= diag(γ1 1 , γ2 , γ2 , · · · , (0)), and X is an or
In
CASK,
the
observeddissimilarity
datamatrix∆is
decomposed
into
the symmetricpar… (0)),is,S s and+ SX
the latterpart
is decomposed
the skew-symmetric
S sk( ﬁrst,
sk .isThen,, 1,thatan orthogonal
matrixIn theirBMDS
a Euclidian
distance model
usedMDSand aare part
diag
2, ML
2, ∆, =
ted as Bayesian MDS(BMDS).
Their
major criticismson theisΓML(1)justiﬁcation
matrixofisordern. Moreover,K1order
is a ofspecialmatrixorder 1n.
, γ1 , γ2 , γ2 , · · · , (0)), and X is an or
where
a diagonal
matrixofn such that
Γ =ofdiag(γorderGaussian measurement error is assumed
the observed partofSordern. Moreover,n. part is decomposedthat is, ∆K=isSas special+ S sk .matrix
Then,ofthe
lattertheinskew-symmetrictoyield,
sk ﬁrst,asymptotic theory, and the number of parameters tomatrix
beAs
optimized
overtypically
grows
fast matrixaS
metric,and therefore, any
Gower(1977)
pointsAsout,elements
of Sout,of ordern. Moreover,K
isthe
aasspecial
of do
order
n.form ofsk
Gower(1977)
pointsthe not
elements
dissimilarity. Markov Chain Monte Carlo (MCMC)
algos k do not
to yield,
S sk = XΓKX t ,umber of objects,rithm
so thattheasymptotic
theory
may
not
apply
in
high
dimensions(Cox,1982),
a metric,
andsotherefore,
any
offormdistance
interpreis used in order to obtain a Bayesiansolutionof(1977)
the
distance
interpretation
offormsuch
diagrams
drawn
unwarranted.
be moreCASK
dokindnotaTometric,andprecise,
therefore,
any
As
Gower
pointsout,
the elementsof S isskt
XΓKX, unwarranted. To be
S sk =
tationofsuchdiagrams
so
drawn
is
stimulus
configuration.
According
to
them,
their
BMDS
,
γ
,
γ
,
γ
,
·
·
·
,
(0)),
and
X
is
an
where
Γ
is
a
diagonal
matrix
of
order
n
such
that
Γ
=
diag(γikelihood surface will tend to have many more local distanceminimawhen
there areofmore
dimensions,
1 1 that
2 To2ofbe
thestandard
symplectic
basis
therefore
it is diﬀerent
froma moreEuclidianspace,
asor
interpretationsuch anddiagrams
so drawn
is unwarranted.
precise,CASK
more precise, CASK employs the standard symplectic basis
provided a much better fit to the data than did the classical, γ1 ,that
γ2 , γ2of, · a· · Euclidian, (0)), and space,
X is anasor
whereΓ
a diagonal
matrixoforder
n such that=matrix
ofis
order
n.
Moreover,K
is atherefore
specialmatrixof diag(γorder 1n.
ng a good initial estimate will be correspondingly morediﬃcult.
Shiraiwa(1993)
point
out.
the
standardbasistherefore
and
it is Γdiﬀerent
and
it is different
from that fromof a Euclidian
space,MDS and a moderately better fit than did
ALSCALinsymplectic
allmatrixof(1993)ordern.
Moreover,isaShiraiwa
specialmatrixofado
order
notn.forma metric,
therefore,As
Gower(1977)
pointsout,
theelements
of Spoint
eir BMDS a Euclidian
distanceanda Gaussian
measurement
error
isK1978,
assumedin proposedsk
By
contrast,
Chino(1977,
1990)
similarmethodwhichandis diﬀerent
in anyphShiraiwa
point
out.
as Chino&(1993)
out.
the examples
theymodel
tested.isAused
simpleBayesiancriterion
calleddo
not formaTometric,andtherefore,As
Gower(1977)
theelements
of S
Byout,
contrast,(1977,
1978,
1990)
proposedabe
similar
MDSIC
which
is basedon Carlo
the Bayes
factoror
ratio
of intedistanceinterpretation
suchdiagrams
drawn
is
more
CASK
skaunwarranted.
rved dissimilarity.
Markov
Chain
Monte(MCMC)
algorithm
isversion
usedpoints
inof
order
to
obtainasoproposed
TheBy
generalized(Chino,1990)
isChinocalledtheGenenalized
Inner
Product
multidimensional
contrast,
Chino(1977,
1978,
1990)
similarmethodwhich
is precise,diﬀerent
in anyphmethodwhich
isdifferentinphilosophy.
The
generalized
grated likelihood is used in order to chooseanappropriate
distance
interpretationofsuchdiagrams
so
drawn
is
unwarranted.
To
be
more
precise,CASK
thestandard
symplectic
basis
and
therefore
itis
diﬀerent
from
that
of
a
Euclidianspace,
assolution of the stimuli conﬁguration. According to them,
their
BMDS
providedamuchbetterabbreviatedas GIPSCAL.
The GIPSCAL
model
expressed asfollows:
The
generalizedversion (Chino,1990) is called
theisGenenalized
Inner
Product multidimensional S
version (Chino, 1990) is called the Genenalized Inner
dimension.
thestandard
symplectic
basis
and
therefore
itis
diﬀerent
fromthatof a Euclidian space, as
Shiraiwa(1993)
pointout.
e data than did theclassical
MDS
anda
moderately
betterﬁt
than
did
ALSCALinalltheabbreviatedas GIPSCAL.
The GIPSCAL
model isSCALing
expressed
as follows:
Product
multidimensional
abbreviatedas GIPAll the symmetric MDS models discussed
above assumetttShiraiwa(1993)
point
By
contrast,
Chino(1977,
1978,
1990)
proposedasimilarmethodwhich
is diﬀerent in phis symmetric.
For
exSCAL.
Thefactor
GIPSCALmodel
isexpressedasfollows:
the Bayesiandissimilarity
data matrixs they tested. A that
simplecriterion
called MDSIC
which
is basedon out.
the
Bayes
or
∆ = aZZ + bZLq Z + c1n 1n ,ample,
nonsymmetry
ofdimension.
dissimicontrast,
Chino (Chino,(1977, 1978,
proposedmethodwhich(9)is diﬀerent in phS
tthe Genenalizedtt Product
TheBygeneralizedversion1990)1990)
is=called
multidimensional
ntegrated likelihood
is Kruskal
used in (1964a)order todiscusses
choose anappropriate+LbZLaisq similarZ+ c1InneraZZ
n 1skew-symmetricn ,,
a specialmatrix of ord
an n × q conﬁguration ∆matrix,andqlarities in his seminal paper on nonmetricwhere
MDS. Z
HeisrecomThethe
generalizedversion1990)
is called
theisGenenalized
Inner
Product multidimensional
abbreviatedas GIPSCAL.
The GIPSCAL
model
expressed asfollows:
he symmetric MDS models discussed above assume thatdissimilarity
data(Chino,matrix∆ is symmeasurementsa spewherematrix,mends to average ä i j and ä j i if they areiswheren ×represents
q configuration
Z is anmatrix,
L q isandapparent
that GIPSCAL
part
the asymmetricpa
is asymmetricspecialandskew-symmetricmatrix of ord
Z isfromanonneq.
× q(9)
conﬁgurationand Lqthe
abbreviatedas GIPSCAL.
TheGIPSCAL
expressed
asisfollows:
For example, Kruskal(1964a)discusses
nonsymmetry
ofonlydissimilarities
seminalpaper on model
the same
underlying
quantity,and differ
becauseof in his cialskew-symmetricmatrixis oforder q. It
apparent fromprecise,the fromskew-symmetricsimultaneously
a conﬁguration
space.
isapparent
eq. (9) thatpart)
GIPSCAL
representsinthe
symmetric partand the asymmetric patstatistical tofluctuation.
we shall frequently eneq. (9) that GIPSCAL+ bZLqthe
Z t symmetric+ c1n 1tn , part and the∆ = aZZrepresents
c MDS. He recommends
average δHowever,ij and δji if they are measurements on the same underlyKiers&beTakake
simpliﬁedand (togeneralized
in sucha way
that it ﬁts the dat
theskew-symmetricpart) simultaneously
in aGIPSCAL
conﬁguration
space.
counter asymmetric relationships whichprecise,
might
not
con- (1994)
asymmetricpart
be precise,the skew-symmetricpart)∆ = aZZ t + bZLq Z t + c1n 1tn ,tity, and diﬀer only because of statistical ﬂuctuation.where
However,we
shallfrequently
encounterisaspecialskew-symmetricmatrixoforde
ord
Z
is
an
n
×
q
conﬁguration
matrix,andLsidered as statistical fluctuation in daily life.
simultaneously
in generalized
a configuration
space.
Trendaﬁlov(2002) (1994)
reformulated
GIPSCALas an qinitialvalueinproblem
for thatcertainﬁrst
Kiers & Takake
simpliﬁed
and
GIPSCALsuch a way
it ﬁtsthedat
ric relationships which
might
not
be
consideredasstatisticalﬂuctuation
in
a
daily
life.
Asymmetric MDS goes back to the workof Young
Kiers& Takake(1994)and generalized
GIP- foris asymmetricspecialskew-symmetricmatrixwhere
Zdiﬀerentialis(1975).
an neq.
×reformulatedq(9)conﬁguration
matrix,andL
isapparent
from
that
GIPSCAL
represents
the
part
and
asymmetricpaqordinaryequations,which
results
in aansimplified
globally
convergent
algorithm
solvingit.ordTrendaﬁlov(2002)GIPSCALasinitialvalue problem
forthecertainﬁrst oforde
Heback
proposeda weighted
modelisHe
SCAL
in
such
a
way
that
it
fits
the
data
better.
Trendafilov
mmetric MDS goesto thework of EuclidianYoung (1975).
proposedaweighted
Euclidianmodel
apparent
from
eq. equations,(9) that
GIPSCAL
represents
symmetric
and the andasymmetricpa
precise,theskew-symmetricpart)
simultaneously
in thea conﬁguration
space.
Harshman(Harshman,
1978;
Harshmanet aal.,1982)
proposedpartaalgorithm
diﬀerent
simple asy
ordinary
diﬀerential
results
inglobally�(2002) whichreformulated
GIPSCALas anconvergent
initial value
problem for solving it.
� r
precise,theskew-symmetricpart) Harshman
simultaneously
a conﬁguration
space.
Kiers& Takake(1994)
andgeneralized
GIPSCALin sucha way thatitthe dat�∑
model
calledthe DEcomposition
into
DIrectional
COMponents
asﬁts
DEDICOM
Harshman(Harshman,
1978;
et al.,inordinary
1982)
proposedaabbreviateddiﬀerent
andsimpleasy
forsimpliﬁed
certain
first
ordermatrixdifferential
equations,
2 MDS
wia (xia − xja ) , wia ≥ 0, 
(7)
dij = �(7)
Kiers&
Takake(1994)
simpliﬁedandgeneralized
GIPSCAL
in
such
a
way
that
it
ﬁts
the
dat
which results
in intoa globally
convergent
algorithm
forsolving
Trendaﬁlov
reformulated
GIPSCALas an
initialvalue
problemfor certain
ﬁrst ordeis expressedasMDS
model (2002)
called theDEcomposition
DIrectional
COMponents
abbreviatedas DEDICOM
a=1
it.
t
Trendaﬁlov(2002)
reformulated
GIPSCALvalue problem
for certain
ﬁrst orde
ordinary
diﬀerential
equations, which
results asinalgorithm
for solving
it.
∆ aan
=globally
Yinitial
AY ,convergent
isexpressed
as it theto handle the problem
of dissimilarities,
andASYMSCAL
in order of
to nonsymmetry
handle the problem
of nonsymmetry
ofcalleddissimiHarshmanmodel.
(Harshman, 1978; Harshman et al., 1982) proordinary
diﬀerential
equations,which
results etinaal.,
t convergent
Harshman(Harshman,
1978;
Harshmanaalgorithm
diﬀerent forandsolving
simple it.
asy, proposed∆simpler=globally
Y 1982)
AYasymmetricand called it the ASYMSCAL model.
WeYshall
refer
posed
a different
MDS ofmodel
call it simply aslarities,
ASYMSCAL.
where
is a conﬁguration
matrix,and A isandthe
DEDICOM”core” matrix
order p giving the diHarshman(Harshman,
1978;
Harshmanet al.,into
1982)
proposedCOMponents
aabbreviateddiﬀerent andasy
MDS
model called
the DEcomposition
into DIrectional
COMponents
as simple
DEDICOM
it simply
as ASYMSCAL.
calledtheHowever,DEcomposition
DIrectional
us asymmetric MDS
methods
have been proposed by relationshipawherebodyYofisresearchers
since
then.
the
basic
p typesaamong
conﬁguration
matrix,andorAdimensions.
is the DEDICOM ”core” matrix of order p giving the di
Various asymmetric MDS methods havebeen
proposedabbreviated as DEDICOM,
which isexpressed as abbreviated as DEDICOM
MDS
model
called the DEcomposition
into DIrectional
COMponents
isexpressed
ascase of ASYMSCAL by Young, most of them have merely
concerned
howptorepresent
Itbeenwas&with
Grorud(1980)
that
proposed a similar method to GIPSCAL, but with s
relationship
basic
types
or
dimensions.
by a body of researchers since then. However,asin Escouﬁer
theamongcase thetis expressed as(10)∆ = Y AY ,,
ric relationshipsof
betweenstimulibyinYoung,a multidimensional
continuum
until
In otherwords,ASYMSCAL
most of themcurious
until
have recently.
formulation
which
they
calledthe complex
Theydecomposethe observed
data som
It recently
wasEscouﬁer&
Grorud(1980)
that
proposedcoding.
a similar
methodto GIPSCAL,
but witht, A is the DEDICOM∆ =matrix,
Y AY andwhereisaconfiguration
merelybeen
concerned
with
how
to
represent
asymmetricY
whereis a conﬁguration
matrix,andtheA iscomplex
the DEDICOM”core”
matrixp givingdata
the dimthem have merely discussed representation models. curious
into
theYformulation
symmetric
partandthecalledskew
symmetric
part,andThey
compilethem ofasorderwhich
they
coding.
decomposethe observed“core” matrix of order p giving the directional relationshiprelationships between stimuli in a multidimensional continwhere
Ysymmetricis aamong
conﬁguration
matrix,andis the DEDICOM
relationship
the
basic
p typesorAdimensions.
xample, Gower and Constantine (Constantine & Gower,
1978;
Gower,
1977)
discuss
several
the
the
skew
part,
and ”core”
compilematrix
them ofasorder p giving the di
among
the
basicsymmetricor S
dimensions.
uum. In other words, most of them haveinto
merelydiscussed part andp typesH=
s + iS sk ,relationship
among
the
basic
p(1980)types
ordimensions.
It wasEscouﬁer& Grorudthat
proposeda similar
GIPSCAL, but with s
methods for analyzing
asymmetriccanonical
analysisofIt asymmetry,a&was
EscoufierGrorud (1980)
thatmethod
proposedtoa similar
representation
models. data including SVD,
+
iS
,
H
=
S
s
sk
methodto
GIPSCAL,withsomewhatcurious
For example, Gower and Constantine(Constantine&thewhich
It wasEscouﬁer&imaginary
Grorudthat
proposeda asimilar
methodto formuGIPSCAL,
with
curiousthey(1980)
calledthe Itcomplex
coding.
They
decomposethe observed
data sm
whereiformulation
denotesnumber.
isbut
apparentthat
thiscomplexmatrixis abut
Hermitia
5
lationwhich
they
calledthecomplexcoding.
They
decomGower, 1978; Gower, 1977) discuss
severalpossible
methcurious
which
they
calledcoding.
They
decomposethe observed
data minto
theiformulation
symmetric
part
and
the
skew thesymmetric
part,compilethemaasofordern.
They thesolvedtheeigenvalue
problemofthe andmatrix,traditional
methodt
wheredenotes
imaginary
number.
Itcomplex
is apparent
that
thisutilizing
complexmatrixis a Hermitia
into the symmetric partpose the observed data matrix
ods for analyzing asymmetric data including SVD, canoniinto
thesymmetric
part
and
the
skew
symmetricpart,andcompilethem
asof ordern. on.
They
the theeigenvalue
problempart,of the matrix,
utilizing
skew symmetriccompilethem
as a traditional method t
cal analysis of asymmetry, a cyclone model,
and soWe solved and+ iS
H = S sandsk ,6shall hereafter abbreviate the canonical analysis of asym(11)H = S s + iS sk ,,metry as CASK (e.g., Chino & Shiraiwa, where
1993), ialthough
6denotesitthe imaginary number. It is apparent
that this complex matrix is a Hermitia
where i denotes the imaginary number. It is apparent thatis sometimes called the Gower diagram.
denotes
imaginary
number.problem
It is apparent
that thisutilizing
complexa matrix
is a method
Hermitiat
ofwhereorderi n.
They thesolvedthe eigenvalueof the matrix,
traditional
this complex matrix is a Hermitian matrix of order n. TheyIn CASK, the observed dissimilarity data matrix is deof order n. They solved the eigenvalue problem of the matrix, utilizing a traditional method t6
─ 104 ─6

Asymmetric Multidimensional Scaling

solved the eigenvalue problem of the matrix, utilizing a tradistance function is asymmetric, that is, the Minkowski metditional method to obtain it’s eigenvalues and eigenvectors
ric function. It should be noticed that the well-known
by computing a special kind of real symmetric matrix of orMinkowski’ r-metric which is, for example, used init’s eigenvalues
andeigenvectors
bycomputingaspecial
kind
of
real
symmetricof orderder 2n.
Kruskal’s matrix
nonmetric
MDS, 2n.
is a symmetric distance functhe
ﬁrst
eigenvalue
andthecorresponding
eigenvector,
theytion.
approximate
symmetric
partthe definition of asymUsing
first kindeigenvalueandthe corresponding
Ten Bergethe(1997)
considerednvectors byUsing
computinga thespecialof realsymmetric
matrixof eigenorder 2n.
vector,they
approximate
thesymmetric
part
and
the
skewmetry
impliedinGower’s
approach,which might be viewed
andthethecorrespondingskew-symmetricpart ofathedata
as follows:valueeigenvector,
theyapproximate
symmetric
partof order 2n.
valuesandandeigenvectors
by computing
specialkindof real thesymmetric
matrixsymmetric part of the data as follows:
as a method of reducing asymmetry by rank-one matrices,theeigenvalue
data as follows:gpart
the ofﬁrst
and the corresponding eigenvector, they approximate thesymmetric
suggested parta different method.
Tobler (1976–1977) pro(12)sjk (s) ∼ λ1 (uj1 uk1 + vj1 vk1 ), sjk (sk) ∼ λ1 (vj1 uandk1 − uj1 vk1 ),skew-symmetric part of the data as follows:
posed a unique method for analyzing the interaction bes) ∼ λ1 (uj1 uk1 + vj1 vk1 ), sjk (sk) ∼ λ1 (vj1 uk1 − uj1 vk1 ),
(12)
(12)
tweenj geographicalareas,
which is often represented bywhere (uj1 , vj1 ) and (uk1 , vk1 ) are, respectively, the coordinates of stimuli
and k in a real
space.
∼ λ(u+ vj1∼ λ1 (vj1 u− uj1 vk1 ), “from-to” tables.
(12)
sjk (s)where) and(uvkk1
respectively,thev k1) sare,
1 (ujk (sk)k1 coordij1,j1vuj1k1
1, ),contrast, Chino& Shiraiwa(Chino,1991a, b; Chino & Shiraiwa, 1993) proposed the same model
respectively,
the coordinates
stimuli
1 , vk1 ) are,Bynates of stimuli
space.j and k in a real space.
It is interesting to note that the asymmetric MDS methj and k in aofreal
valuesandeigenvectors
bycomputingaspecialkindofproposedreal
symmetric
matrixoforder
2n. abovethat
Escouﬁer
andGrorudindependently,
andcalledittheHermitianCanonical
Model
abbreviated&uj1
Shiraiwa(Chino,1991a,b;
Chino&
Shiraiwa,
1993)
the
same
model
, vj1 )asand
(uof
,
v
)
are,
respectively,
thecoordinatesofstimuli
j
and
k
in
a
real
space.
By
contrast,
Chino&
Shiraiwa(Chino,1991a,b;Chino
ods
discussed
haveremained to be descriptive until
k1 k1&
Shiraiwa,
1993)
proposed
the
same
model
as
that
of
Escquiterecently.
AlthoughChino(1992) proposed an ML
gontrast,
theﬁrst
eigenvalue
andthecorresponding
eigenvector,
theyapproximate
thesymmetricpart
HCM.
appliedthe iteigenvalue
problem
to1993)the abbreviatedHermitianin Eq.
(11), and obtained
Grorud as
independently,
and(Chino,
calledthe Hermitian
Canonical
Model
Chino
& TheyShiraiwa
1991a,
b; Chino& directly
Shiraiwa,
proposedmatrix
the samemodel
oufierandindependently,
and called it the Hermiasymmetric MDS which is a natural extension of MAXSskew-symmetricpart equation:oftheGroruddata
asfollows:
the following
heeigenvalue
problemdirectly
to theHermitianmatrixEq. (11), Canonicaland obtainedf Escouﬁer
andGrorudindependently,
and called
it theinHermitian
Model abbreviatedtian Canonical Model abbreviated as HCM. They applied
CAL by Takane (1981), it has yet remained to be completed.
ttXΩ(13) it further, and called it∆= XΩ
s X + matrix
sk X
They applied theeigenvalue
problemdirectly
totheHermitianmatrixin, Eq. (11),and obtained(2008) developedtheeigenvalue
problemdirectly
totheHermitianin(12)
sjk (s) ∼ λ1 (uj1 uk1 + vj1 vk1 ), sjk (sk) ∼ λ1 (vj1 uk1 − uj1 vk1 ), Saburi and Chino
Eq. XΩ(11),sandobtainedthet ,following equation:
ASYMMAXSCAL. In ASYMMAXSCAL, three submodels,
X tare+XΩ(13)wing equation:
sk XsymmetricΩskspecialand skew-symmetric matrices,
respectively.
where ∆
Ωs=andi.e.,thespace.
representation model, the error model, and the ret coordinatest of stimuli j and k in a
uj1 , vj1 ) and (uk1 , vk1 ) are, respectively,
the
real
,
(13)
X
+
XΩ
X
,
∆
=
XΩ
s
sk
At this andpoint
it is natural to matrices,ask whether
the conﬁguration of stimuli
obtainedby(13)
theassumed
complexascoding
pecial symmetricskew-symmetricrespectively.
sponse modelarein MAXSCAL. Any asymontrast, Chino &where
Shiraiwaand(Chino,are1991a,
b;
Chino
&
Shiraiwa,
1993)
proposed
the
same
model
specialsymmetric
andskew-symmetmetric
MDS
modelsdevelopedupss k and
or
HCM
has
a
metricproperty
or
not.
Chino
&
Shiraiwa(1993)
attacked
thisproblem,andfoundtheto now can be basicallytoaskwhetherthe
conﬁgurationofstimuliobtainedbythecomplexcodingandΩ
arespecialsymmetricskew-symmetricmatrices,
respectively.
Ωural
s
sk
f Escouﬁer and Grorud
independently,
ric matrices,
respectively.and called it the Hermitian Canonical Modelchosenabbreviatedas representation models in ASMMAXSCAL, they
andtosuﬃcientconditionthat the dissimilarity
data
expressible
in terms
of a certain metricoperty
or
Chino
&ask
Shiraiwa(1993)
attackedthisof problem,
and isfoundispointnecessary
it not.
is natural
whetherthe conﬁguration
stimuli obtainedby the complex
codingAt this pointproblem
it is natural
to asktowhether
the configuration
pickedupobtained
the OI model in their paper. In order to compare
They applied the eigenvalue
directly
the Hermitianmatrix in Eq. (11),andmutualspace.
To
precise,necessaryand suﬃcientcondition
that
the
set
condition
thatproperty
thedissimilarity
data
in terms
of a certain
metricjk gives
has a metricor obtained
not.theChino&expressible
Shiraiwa(1993)
attacked
thisfoundtheof bestimuliby isthecomplexcodingorHCM
hasa problem,the dandgoodnessofthe
fitof thedistances
several models, ASYMMAXSwing equation:
metricor not.
&(1993)
attackedutilizes
AIC.H is p.s.d.. The
of a andrealcondition(true)property
setthatof points
inChinoathe
ﬁnite-dimensional
Hilbertis thatthe(complex)
mutualdistancessuﬃcient
condition
that
setShiraiwa
djk gives
ye necessary
and suﬃcientthedissimilarity
data
is expressible
in terms
ofCAL
aspacecertainmetrict+ XΩskandX tsufficient,(13)∆ found= XΩthes Xnecessary
thisproblem,andcondiAnotheroutstanding
featurespace.
of ASYMMAXSCAL is thattheorem
issuﬃcient
clearlyHilbertanextensionof thatthe Young-Householder
theoremto the complex
oints
inChino-Shiraiwa
a ﬁnite-dimensional(complex)
spacethatis
H disjkp.s.d..
Themutual
gives thedistanceso be precise,the necessaryandcondition
set
tion that the dissimilarity data is expressible in terms of ait enables us to check whether the data is sufficiently asymand Ωsetare
specialinof
symmetric
and skew-symmetricmatrices,
respectively.
Ωiss(true)In
contrastwiththemodelsdiscussed
above,
some
researchershaveaugmented distancesk
clearlyan
Young-Householder
theoremtothecomplexspace.
ofextension
pointsathe
ﬁnite-dimensional(complex)
Hilbertspace
H isproposed
p.s.d..
The
certain metricspace. To be precise,thenecessaryandsuffi-is thatmetric.
As in symmetric
MDS’s, asymmetry had been preismodels
pointmodels
itdiscussed
is naturaltoaskwhetherofstimuliobtainedbythecomplexcodinglikeYoung’s
ASMSCAL(e.g.,
Krumhansl,
1978;
Okada
&
Imaizumi,
1987,
1997;
Saito,
1991;
gives
the
mutualdistancesof
cient
condition
thatthetheset
above,
some
researchers
have
proposedaugmented
distance
supposed
without
examinationin applying any asymmetricdthehiraiwa
theoremis clearlyanextensionofconﬁguration
Young-Householder
theoremtothecomplexspace.
jkareal(true)
setofpointsinafinite-dimensional(complex)
MDS
to
data.
By
contrast,
ASYMMAXSCALcompareshas a metricnot.
&& Shiraiwa(1993)
attacked
thisproblem,theSaito&property
Takenda,
1990;
Weeks
1982;1987,
ZielmanHeiser,1996).andIn foundthese
models, the Okada
MSCAL(e.g.,theKrumhansl,
1978;Chino
Okada&Bentler,
Imaizumi,
1997;&
Saito,
1991;
ntrastwithmodelsordiscussed
above,
some
researchers
have
proposedaugmented
distance
Hilbert space is that H is p.s.d.. The Chino-Shiraiwa theoany asymmetric MDS model with a saturated model under
yWeeks
suﬃcient
condition
thatabbreviatedthe&dissimilarity
expressible
inthe
terms
of
a certain
metricandImaizumi
modelsas1978;the data
OI
modelsunique
in
the
sense
that
these
handle the ordinal& Bentler,
1982;
ZielmanHeiser,1996).
Inisthesemodels,Okada
keand
Young’s
ASMSCAL(e.g.,
Krumhansl,
Okada
& are
Imaizumi,
1987,
1997;
Saito,
1991;
rem is clearly an extension of the Young-Householder theothe symmetry hypothesis and so on using AIC. However,gives
themodels,mutualdistanceso be precise,
theremnecessary
anduniquesuﬃcient
condition
that
the1996).
set
djkForasymmetricdata
onKruskal’s
monotone
regression.
example,theone-mode
two-wayversionbreviatedastheOI
modelsarein Zielmanthesense&that
thesehandleordinalTakenda,
1990;
Weeks
&basedBentler,
1982;
Heiser,
Inthe
these
the
Okadamerely
to thecomplexspace.
such
comparisonsserve
as indirect examinations ofIn
contrastwiththemodelsdiscussed
above,
somereasymmetry.
Moreover,information
criteria such as AIC do(true)
settheofmonotone
pointsinregression.
aisas
ﬁnite-dimensional
spacethat handle
H is p.s.d..
The
of
OI
modelsexpressed
asexample,
on
Kruskal’s
the one-mode
two-wayversionzumi
modelsabbreviatedthe OIFormodelsare(complex)
uniqueinHilbert
the sense
thatisthese
the ordinalsearchers have proposed augmented ∗distance models likenot generally consider the nature as well as features of thehiraiwa
is clearlyan extension
the Young-Householder
theoremto thetwo-way
complexversion
space.
= example,
djk − rj +therk ,one-mode
(14)
djk
essed
astheorem
ric data
based onKruskal’s
monotone ofregression.
For
Young’s ASMSCAL (e.g., Krumhansl, 1978; Okada & Imdata. To overcome this difficulty, ASYMMAXSCAL exam∗withthe ∗aizumi,
modelssome Saito
researchers
have 1990;
proposed (14)augmented
distance=
ddiscussed
rj + rabove,,
Intrastmodelsis expressed
as 1987,
jk − 1997;
kSaito,jk the1991;
Takenda,ines the featuresof the data by utilizing the traditional test
augmented
distance
between &stimulij and k.
where
djkdis∗ke Young’s ASMSCAL
Krumhansl,
1978;
Okada
&
Imaizumi,
1987,
1997;
Saito,
1991;
Weeks &(e.g.,
Bentler,
1982;
Zielman&Heiser,1996).
Inthese
forsymmetryandrelated tests such as the test for quasisym=
d
−
r
+
r
,
(14)
d
jk
j
k models Sato (1988), Ten Berge (1997), andWebetween
shall brieﬂy
refer
to jk
some
Tobler (1976nted distancestimulij andk. miscellaneousmodels,
the
Okada
and
Imaizumi
modelsabbreviated
as
the
metry
(Caussinus,1965), and so on (Chino, 2008; Chino &Takenda,
1990;
Weeks
&
Bentler,
1982;
Zielman&Heiser,1996).
In
these
models,the
Okada∗to some1977).
Sato(1988)anMDS
modelof which
distancefunctionmodelsSato (1988),
Ten
Berge
(1997),
andtheTobler(1976augmented
distance
betweenj and
OI models
areproposed
uniqueinstimuli
the asymmetricsensethat k.
thesehandleorSaburi,
2006,2008). is asymmetric,
jk is the miscellaneous
zumi models abbreviated as the OI models are unique in the sense that these handle the ordinal
dinal
asymmetricdataofbased
on Kruskal’s
monotone
regresForTobler
example,Chino and r-metric
Saburi (2006) proposed to adthat
is, theMinkowski
metricfunction.
It (1988),
shouldbe noticed
that aandwell-known
Minkowski’
posedanasymmetric
MDS
model
which
distancefunctionis asymmetric,hall
brieﬂyrefer
to somemiscellaneousmodels
Sato
Ten
Berge
(1997),(1976ric data based on
Kruskal’s
monotone
regression.
For example,
one-modeminister
two-wayversion test for the quasi-symmetry hypothesion.
For example,
the one-mode
two-wayversion of thethe OIasequential
whichproposed
is, forexample,in Kruskal’s
nonmetric
MDS,
is a symmetricfunction. Ten Berge
metric(1988)
function.
It should
be usednoticedthat amodelwell-known
Minkowski’
r-metricSato
an asymmetric
MDSof which
distancefunction isdistanceasymmetric,modelsasis expressed assis, the marginal homogeneity hypothesis, the quasi-indeI models is expressed(1997) considered
theMDS,deﬁnitionof asymmetry
impliedGower’smight thebe viewed
as a hypothesis, and somesed
Kruskal’s
nonmetrica symmetricdistance
Ten approach,Bergethe inMinkowski
metricfunction.
Itis should
be noticed
thatfunction.
a inwell-known
Minkowski’
r-metric
pendentwhich
hypothesis,
independent
(14)
(14)
d∗jk = djk − rj + rk ,,
methodof reducing
asymmetryrank-onematrices,andbesuggested
Tobler (1976the log-linear
hypothesis,
prior to analyze asymmetricﬁnitionof asymmetry
in nonmetric
Gower’sbyapproach,mightviewed function.
asaof
adiﬀerent
for example,
used
in implied
Kruskal’s
MDS, is which
a symmetricdistance
Tenmethod.
Berge
*
∗
is
the
augmented
distance
betweenstimuli
and
where
j
d
relational
data. However,some questions arise in testing
kuniqueaugmented
distance
betweenstimuliandk. themethod.
1977)
proposedajof
methodfor analyzing
interaction
betweengeographical
mmetrybyrank-onematrices,andsuggestedajindiﬀerent
Tobler(1976onsideredthedeﬁnition
asymmetry
impliedGower’s
approach,
which
might
be viewed areas,
as a which is often
jk is thek.
these hypotheses concerning (1) how to arrange the order ofhall
brieﬂyto somemiscellaneousmodels geographicalSato(1988), Ten
Berge(1997),and Tobler (1976representedby ”from-to”
tables.
methodforreferanalyzing
interaction
betweenareas,
which
is
often
of
reducing
asymmetrybyrank-onematrices,andmiscellaneoussuggesteda diﬀerent
method.
We theshall
briefly refer
to somemodels,testing these symmetry related hypotheses, (2) whether we(1988)
anforasymmetric
MDS
model
ofbetween
which
distance
asymmetric,
It proposed
is Satointeresting
to
note
that
the
asymmetric
MDS
methodsfunction
discussed
aboveisthehave
to kindbe (and possibly, that of”Satotables.
oposeda uniquemethodanalyzingtheinteraction
geographicalareas,
which
often(1988),Ten
Berge(1997),and Tobler(1976–1977).
caniscontrol
errorremainedof the firstSato(1988)
proposedanasymmetric
MDS
model
of
which
the
secondkind)(Chino,2008).
the
Minkowski
metricIt should
be noticed
thata well-known
r-metric MDS which is
descriptive
untilfunction.
quite
AlthoughChino(1992)
proposedtoMinkowski’
anote bythatthe asymmetric
MDSrecently.
methods
discussed
above
have
remained
beML asymmetricted”from-to”
tables.

for example,
inKruskal’s
nonmetric
MDS,
is asymmetrica symmetricdistancefunction.
Ten Berge
ecently.
Although(1992)
proposed
an ML
MDS
which
is remainedinteresting
to usednote Chino
thattheasymmetric
MDS
methods
discussed
above
have
to be
7
onsideredthe deﬁnition
asymmetry
impliedin Gower’s
approach,might
be viewed
as is
a
─ 105
─ MDS
ve until quite
recently. ofAlthough
Chino(1992)
proposedan ML whichasymmetricwhich
7of reducing asymmetry by rank-one matrices, and suggested a diﬀerent method. Tobler (1976-

7
oposed a unique method for analyzing the interaction
between geographical areas, which is often

Naohito CHINO

Recently, Chino, Tomizawa, & Saburi (Chino & Saburi,
2009, 2010; Tomizawa, 2010, personal communication)
have almost proven that the three likelihood ratio statistics
pertaining to symmetry, i.e., those for testing the quasisymmetry hypothesis, a symmetry hypothesis, and a marginal homogeneity hypothesis are mutually independent
stochastically. This result enables us to control at least theerror of the first kind in some sequential test for the symmetry and related hypothesis. In proving the above problem,
theorems due to Hogg & Craig (1956) and Lehmann (1983)
are utilized.
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─ 107 ─Final version received, December 31, 2010