Multidimensional Scaling in Economic Research

A C T A U N I V E R S I T A T I S L O D Z I E N S I S

F O L IA O E C O N O M IC A 175, 2004

A d a m B i e l a *

M U L T ID IM E N S IO N A L S C A L IN G IN E C O N O M IC R E S E A R C H

Abstract. A relationship between the theoretical terms and the observational ones, called also a perceptual or observational, is essential for scientific research o f empirical type, including social sciences and econ om ic sciences. This relationship cannot be clarified in terms o f a com plete definition but only by a partial definition. This m ethodological truth is well known since R. C arnap’s works. Later on it was developed in m eth od ology o f sciences by the Polish logicians: Przełęcki, Poznański and Kamiński.

M ultivariable techniques are necessary when one wants to define the relationships between variables in econ om ic and social sciences. However, the results obtained in such analysis are often unsatisfactory because the residual variance is too large. M ultidim ensional scaling proposes quite a different m ethodological approach for seeking the relationship between the theoretical terms and the observational ones.

T his paper aims: (1) to sh ow w hat kind o f m ethodological proposition is m ultidim ensional scaling; (2) to show what are the possible directions o f applying m ultidim ensional scaling to social and econ om ic analysis; (3) to define the m ultidim ensional character o f decision analysis.

Key words: m ultidim ensional scaling, theory o f data decision analysis.

1. M E T H O D O L O G IC A L E SSE N C E O F M U L T ID IM E N SIO N A L SC A L IN G

M ultidim ension al scaling is a m ethod o f scientific en quiry w hich is based on inductive inference schem a. It is based on assu m p tio n th a t reality which is a n object o f the enquiry, is o f a different level o f com plexity, i.e. m ultidim ensional. H u m a n being (e.g. researcher, price analyst, expert), w hen expressing his relation to the reality (cognitively, preferentially, behaviorally), op erates using th e d im en sio n s (in terp reted usually as v ariab les) w hich enable him a cognitive “ possession” o f th a t reality. A ccording to this assum ption, one can say th a t h u m an being is m u ltidim ensio nally scaling the reality, i.e. he is schem atizing and categorizing th e reality in acco rd an ce with som e learned style w hich is corresponding to the defined m ethodological, cultural o r professional p a tte rn .

* Prof., Chair for D ecisio n A n alysis, C ath olic U niversity o f Lublin, e-m ail: biela- ada@ kul.Iublin.pi.

In social life situ atio n s, in econom ic dom ains, in professional practice and in research activities we are facing a kind o f a n a tu ra l scaling, i.e. intuitively o b v io u s c o m p a rin g and system atizin g th e pcrccivcd ob jccts, events, situations, concepts or ideas. T h e object o f this scaling is a reality which is existing really, hypothetically, intentionally, o r ideally.

M u ltidim ensional scaling techniques propo se a re co n stru ctio n o f a m u l­ tidim ensional space o f the investigated reality w here the co m p ared objccts will be located. It is a system atizing space which co n tain s the analyzed objects. In decision situ atio n it is a preference space; in ev alu atin g situ atio n s it is an ev alu atio n space; in econom ic situations it is a b ehav ioral space; etc. A m o d e o f existence o f the system atization space and strictly speaking the defined co n fig u ratio n , i.e. location o f the considered objects in this space, depends on tw o factors: (1) behavior o f the subjects, and (2) the logical and m ath em atica l assum ptions o f the co m p u ter p ro g ram which reconstructs the system atizing space and the co n fig u ratio n s o f the co m p ared objects in this space.

In o rd e r to clarify the assum ption and aim o f m ultid im en sio n al scaling, let us give an illustrative exam ple. Let us im agine th a t a g eo g rap h er lost a m ap o f a region bu t he know s the direct distances betw een the p artic u la r cities in this region. In such a situ atio n m ultidim ensio nal scaling enables us to reco n stru ct a m a p o f this region.

In a sim ilar way, having the stated “closeness” o r sim ilarities betw een the pairs o f different co n su m p tio n goods, it is possible to recon stru ct their lo catio n in c o n su m e rs’ space. T h ere are m a n y p ossible w ays o f using m ultidim en sional scaling in social and econom ic sciences, to recon s­ tru ct various system atizing spaces for econom ic behav iors o f individuals or com panies. Scaling m ight be also used to system atize v arious g rou ps o f technologies, resources, p ro d u cts, services, ind ustrial w aste, behaviors o f p ersons em ployed in th e p articu lar cycles o f technological process, tax p ay ers, b eh a v io r o f tra d e union m em bers, o f m a n a g e rs, etc. O th e r exam ples o f app lying m ultidim ensional scaling could be research on se­ m an tic space o f social concepts, preference space o f consum ers, space o f the perceived smells, tastes, tra d e m arks. T his m eth o d o lo g y can be also used to in v estig ate p u b lic o p in io n , preferences, a ttitu d e s an d po litical opinions.

It is w o rth y to underline also a possibility o f using the techn iq ues o f m u ltid im e n sio n a l sca lin g in th e m e th o d o lo g y o f social an d e c o n o m ic sciences, and p articu larly in system atizing o f theories, m odels, p aradig m s, research m eth o d s, in d icato rs, coefficients and term s. T h e subjects are here the scientists, experts, analy sts and scholars.

M u ltid im ensional scaling m igh t be used to investigate: (1) individual persons, events, situ atio n s, processes (e.g. typical o r n o t typical, rare goods), (2) differences betw een the individuals; (3) differences b etw eengroups o f persons, classes o f events, situ atio n s o r processes; (4) differences betw een the groups. A n exam ple m ight be scaling o f efficiency o f the defined econom ic activities by the individual experts; perceiving o f sim ilarities o f com panies, w ork positions, professions by the individual advisors, em ployees, unem ployed p ersons; classificatio n o f th e co n cep ts o r th e o rie s by th e individual academ icians; environm ental risk perception in th e p artic u la r technology o r investm ent by the individual experts o r by the in h a b ita n ts o f local co m m unity. In tu rn , an exam ple o f investigation w hich aim s to state the m ain tendencies in a g ro u p m ight lead to perceiving the m u lti­ dim ensional characteristics o f o n e’s own com p any , professional g rou p, regional g roup, tra d e group; expressing o pinion in econom ic d o m ain , on technological ch aracteristics, on m oral issue - by the defined social g ro up , professional g ro u p , tra d e union people, etc.

W hat kind o f d a ta could be the object o f m ultid im en sio n al scaling? Directly, fo r m ultid im en sio n al scaling m ay be used th e d a ta w hich define sim ilarity re la tio n sh ip (closeness or distance) betw een the elem ents o f any n-elemcnt set, w here for n is defined a co ndition 4 < n < 100. T h e co n d itio n for n depends on a concrete co m p u ter program fo r scaling.

Sim ilarity is defined as d a ta for scaling and can be m easu red o n an interval scale o r o n a q u o tie n t scale. F o r exam ple, th e d a ta fo r scaling can be obtained by co m p arin g consum er relation betw een the co n su m p tio n goods (e.g. distan ce betw een various p roducts o f the sam e catego ry o r between the p a rtic u la r kinds o f services) o r the perceived re latio n betw een the p otential investors, the fu tu re shareholders, the officers from central institutions (e.g. C om m ission fo r V aluable Papers and Stocks), closeness between the stock com panies. T h e d a ta should be prep ared as a b atch file for the co m p u ter p ro g ram for m ultidim ensional scaling in a sh ape o f a d a ta triangle o f n — 1 row s and n — 1 colum ns w hich com es from m easu rin g on an interval closeness scale betw een the com bined in p airs the co m p ared elem ents o f a set.

H ow ever, indirectly fo r m ultidim ensional scaling m ay be also used the d a ta w hich com e from m easu rin g on independent interv al scales o r even on ran k in g scales o f the individual elem ents o f the analyzed set. Between these m easu res m a y be sta te d th e a p p ro p ria te c o rre la tio n co efficien ts dependent on a type o f a m easu rin g scale. T h u s, m ultidim ensio nal scaling m ay be used fo r m easures o f independent objects o n internal scale (five-point scale), b u t also the outcom es o f the o rd e r scale (e.g. ra n k in g the elem ents o f a set).

T he obtained correlations will be interpreted as the m easures o f connection (e.g. sim ilarity, closeness) and as such m ay be com bined in a shap e o f a d a ta trian g le o f n - 1 row s and n - 1 colum ns, they m ay also be used as a b atch file in m ultidim en sional scaling.

2. T H E O R Y OK » А Г А A S A LO G IC A L BASE FOR M U L T ID IM E N S IO N A L SC A L IN G

A theory o f d a ta by C. H . C o o m b s (1964) can be rccognizcd as a logical b ack g ro u n d fo r m ultidim ensional scaling. T h is th eo ry seeks an unifying system which w ould allow to system atize the d a ta o b tain ed by using various techniques and research m ethods. F ro m a d ev elop m ent o f behav io ral sciences p o in t o f view (w here belong, am o n g o th e rs, such disciplines like sociology, econom ics, and psychology) elaboration o f a unified and logically co h e ren t system o f d a ta classification is o f a great th eoretical and practical im portance, because it shows how to systematize the background o f behavior m easu rem en t itself.

A startin g p o in t fo r a theory o f d a ta is various k in ds o f recordings which are th e o u tc o m e o f co n c rete techn iq ues an d research m eth o d s. A nalysis o f the form al stru ctu re o f d a ta enables us to state th a t a deeper analo gical co n n e c tio n co u ld be recognized betw een som e o f th e d a ta . A base for this co n n ectio n is a relational intrinsic stru c tu re o f the d ata. A ccording to C. H . C o o m b s, each behavioral d a ta w hich is a result o f em pirical research, is n o t a directly observed b ehav io r bu t a relatio nal character, as its essence is a relationship betw een the stim uli and the individuals, o r betw een the stim uli them selves, if it is assum ed th a t the same individuals are reactin g to the sam e stim uli.

C. H . C o o m b s distinguishes the three phases o f scientific en q u iry w hen his th eory o f d a ta is considered (see Fig. 1).

P hase 1: th e sch o lar is sep aratin g the recorded o b serv atio n s from an universum o f the p o ten tial set o f in fo rm atio n by applying the designed research and m easu rem en t procedures.

Phase 2: the p rim ary observatio ns are system atized into the d a ta by finding relational b o n d s betw een a p p ro p ria te stim uli and the individuals.

Phase 3: re co n stru ctio n o f the m -dim ensional system atizing space in which are located the analyzed elem ents o f the n-elem ent set (the inferred classification o f th e in d iv id u a l subjects and th e stim uli recogn ized by a co m p u ter p rogram ).

Phase 1 Phase 2 Phase 3

Fig. 1. The phases o f construction o f data in behavioral sciences (according to C o o m b s 1964)

E ach elem ent o f a set (stim ulus, object, behavior, event) is presented in m ultidim ensional scaling as a p o in t in m -dim ensional space. A n u m b er o f dim ensions o f a space d epends on the traits and p ro p erties o f the co m p ared elem ents o f a seta as perceived by the subject. T h e subject (i.e. observer, analyst, expert, sch olar) w ho is perceiving the stim uli, m ay also be presented in this space in such a way th a t the p oint representing each individual being investigated, m ean s a m axim um preference o f the stim uli presented. A ccording to such in te rp re ta tio n , the relation c o n stitu tin g individual d a ta is indicated by the distance between the points or a sector o f a m -dim ensional space.

T h e m o st ad vanced m ethodologically are the techniques o f collecting the preference d a ta (o f type: choose r elem ents form n-elem ent set; o r o f type: ra n k n elem ents acco rd in g to the defined attrib u te).

T h e m o st fu n d a m e n ta l m a th e m a tic a l p ro b lem fo r m u ltid im e n sio n a l scaling is to find a w ay o f tran sferrin g the prim ary m easu rem en t d a ta o n to the distances in a space. In o rd e r to reach this, th e co n fig u ratio n s for the points (th a t is a set o f co o rd in ates) should be fixed in m ultidim ension al space, w hich co rresp o n d to the analyzed objects, decisional o p tio n s, etc. H ow ever, this tra n sfo rm a tio n should be d o n e in such a way th a t th e ra n k order o f the in p u t d a ta (batch file) co rresponds to the ra n k o rd e r o f the distances in m ultid im en sio n al space which com es from the defined c o n ­ fig u ra tio n o f th e p o in ts o f th e space. T h is fittin g sh o u ld m im im alize a ra n d o m fu n ctio n called a stress. T h ere are tw o ways o f defining this function: (a) a p ro c ed u re o f m o n o to n ic regression o f K ru sk a l, an d (b) a pro ced u re o f the im agined ra n k ord er o f G u ttm a n . B oth proced u res are used in a sta n d a rd version o f a co m p u ter o u tp u t, e.g. in the p ro g ram M IN IS S A which is an analytical stage o f scaling.

D ifferent types o f space arc usued in m ultidim ensional scaling, dep en din g on the assum ed needs. T h ere are in a usage: the city block m atrix , the Euclidean m atrix , the M in k o w sk i’s m atrixes (am ong which arc distinguished so called d o m in an ce m atrix es or m axim ize m atrixes).

T h e first stage o f scaling is o f an analytical ch a racter. T h is stage is based on re co n stru ctio n o f system atizing space for the analyzed (perceived o r valuatcd) objects, i.e. the elem ents o f a set. T h ere m ay be system atizing spaces for the individual persons (analysts, experts, decision m ak ers) or for the g rou p o f subjects (e.g. consum ers o f a certain age, social status). An exam ple o f a c o m p u ter p ro g ram which enables to reco n stru ct this kind o f system atizing space is th e M 1N ISSA .

T h e second stage o f m ultidim ensional scaling is a synthesis o f the scaling outcom es o b tain ed in the analytical stage. T h e p ro g ram P IN D IS can be used for this kind o f scaling which aim s to co m p are th e individual config u ratio n s in system atizing space. F o r exam ple, a co m p an y produces n asso rtm en ts o f a com m o d ity for r various m arkets. Is is im p o rta n t for the strategy o f m a rk e tin g m an agem ent to state w h at arc the preferences for n asso rtm en ts in each o f the m arkets, w hat can be o b tain ed using the M IN IS S A pro g ram (an analytical one), and then on e can ask a qu estio n how are these spaces m u tu ally related (how they fit th e targ e t m a rk e t which is the m o st significant fo r a com pany). In o rd e r to reach this p u rp o se the P IN D IS p ro g ram can be used for which the in p u t d a ta arc the individual configuratio ns (i.e. the co o rd in ates o f the points) o b tain ed e.g. in the program M IN IS S A (see: B i e l a 1992, 1995).

3. M U L T ID IM E N SIO N A L IT Y IN D E C ISIO N A N A L Y SIS

D ecision situ atio n m ay be one o f the d om ain s w here m ultidim ension al analysis can be applied. H ere an explorative possibilities are very extensive. If we define the decision situ atio n as an ord ered five:

DS = ( A , H, {p(hj)}, uy, 1}

where:

A = ( a {, a 2, ..., a i; ..., a„) - a finite set o f possible altern ativ e actions; H = ( h lt h 2, ..., hj, hn) - a finite set o f possible states o f the w orld (hypothesis);

{p(hj)} - p ro b a b ility d istrib u tio n on hj th a t d epends on alternative actions a (;

utJ - th e C artesia n p ro d u c t A x H, so th a t u,j = a, x hy,

I = O i, i2> it, ik) - the set o f actions which en ab le the subject to o b ta in new in fo rm a tio n a b o u t th e utility o f a c tio n s o r a b o u t the probability o f the states o f the w orld.

D ecision analyst o r decision m ak er perceives m an y aspccts, planes, dim ensions in a decision situ atio n , how ever, no t at the sam e tim e because o f his b o u n d ed cognitive capacity. In decision analysis one ca n find the “ tra n sitio n ” m om en t from one dim ension into a n o th e r one, w h at explains so called in tran sitiv ity o f the preferences in m any contexts. In acco rdance to the ra tio n a lity p o stu la te one can expect th at: if a p erson w ants to choose A, th an В in a decision situ atio n , and if this person w ants to cho ose В th an C. T h u s w hen these o p tio n s are presented in pairs, the sam e p erso n w ants to choose A ra th e r th a n C, when A and С are co m p ared in a new pair. F orm ally, this situ atio n m ay be form ulated as

[(A > B ) n ( B > С)] —* (A > C).

T his p o stu late is one o f the axiom s o f the classical utility theory. U n fo rtu n ately , b ehavioral research says th a t this axiom w hich seems to be a fu ndam en tal for ra tio n a lity o f hu m an behavior, is n o t fulfilled in p eo p le’s decision m ak in g situ atio n s.

W h a t is the reason o f intran sitiv ity in h u m an preferences? Isn ’t it an evident lack o f ra tio n a lity in hu m an behavior? It m ay be th a t people are no t ratio n al beings. H ow ever, it seems th a t the reason on in tran sitiv ity in preferences is n o t a lack o f logic in h u m an th in k in g o r em o tio n al instability in people. A lack o f ra tio n a lity seems to be here only a very surfice phenom en on. A principle o f transitivity in preferences w ould be fulfilled when people w ould o p e ra te a sim ple, one-dim ensional utility scale w hich is assum ed in th e axiom s o f the classical utility theory. In such a case lack o f transitivity in preferences w ould m ean a lack o f ra tio n a lity in h u m an behavior in decision situ atio n s. H ow ever, people are o p eratin g w ith a m u l­ tidim ensional utility scale w hen considering various altern ativ e o p tio n s in decision m ak in g situ a tio n s ( H u b e r 1983). In tran sitiv ity o f preferences in m any situ atio n s m ay be explained by decision m a k e r’s “ tra n s it” in his analysis in to a n o th e r d im en sio n ra th e r th a n th a t w ith w hich he was operating whem co m p arin g the previous alternatives. T h u s, w hen considering A w ith В an d В w ith C , th e p erson considered som e o th e r d im en sion o f utility scale th a n w hen this p erson com p ared A w ith C.

O peratin g w ith m ultid im en sio n al scale requires n o t only scaling on the particular dim ensions, bu t also evaluating the im po rtance o f these dim ensions, i.e. their weighting. A fundam ental ontological assum ption o f m ultidim ensional

scaling is th a t people in cognitive processes, evaluative processes an d in decision m ak in g are: w eighting the dim ensions an d scaling th e objects on the p artic u la r dim ensions. In teg ratin g o f these elem entary fun ction s aim s to state a co n fig u ratio n o f the com p arin g objects, i.e. th eir lo catio n in a system atization space. B oth the tw o elem entary fu nctio ns (w eighting the dim ensions and evaluating th e objects o n the dim ensions) and th e in teg ratio n o f these functions, aim to cognitive system atization o f the considered elem ents, i.e. co m p an y su rro u n d in g s, segm ents o f a m a rk e t, co n su m ers’ preferences.

Such a sy stem atizatio n is a kind o f cognitive “ p o ssessio n ” o f the analyzed reality. A need for cognitive “ possession” o f the situ atio n is a m o tiv atio n al base for system atization. Satisfying this need reduces a fear o f chaos, random ness in activity and prevents from n o t ch o osin g the best alternative in a given decision situation. O p eratin g w ith m ultid im ension al scale in decision m ak in g situations was no t an object o f m any m ethodological analysis. T h e first were the a u th o rs w orking in decision m ak in g analysis (e.g. H u b e r 1983; Ł ukasik -G o szczy ń sk a 1974). A ssu m p tio n a b o u t o p eratin g with m ultidim ensional scale requires in consequence to accept a hy po th etical con stru ct, i.e. a th eo retical con cep t w hich deno tes a system atizing space o f cognition o f the defined econom ic environm ent. D ependently on w hat is the object o f analysis, the system atizing space m ay deal w ith a m a rk e t o f p ro ducts and services, a capital m ark e t, or a lab o r m ark e t. T a b le 1 show s som e possibilities o f m ultidim ensional scaling which aim to reco n stru ct various system atizing spaces in m a rk e t econom y env iro n m en t. T h e exam ples indicated in T ab . 1 show various d o m ain s o f econom ic reality. T h ese m ay be the object o f m ultid im en sio n al scaling in decision m ak in g situ atio n s o f a m an ag e r w ho is considering strategic decisions for a co m p an y w hich is functioning in d o m ain s o f m ark e t p ro d ucts and services, capital m a rk e t or labo r m arket.

T a b l e 1

Examples o f spaces in m ultidim ensional scaling for market econ om y pillars

Pillars o f market econom y Content o f system atizing space 1. Products and services

market

A . Consumer spaces 1. Consumer needs space

2. D eclared consumer preferences space 3. Consumer behaviors space

4. M arket segments space

B. Producers and tenderers space 1. General standing o f producers space 2. Participation in market space

T able 1 (condt.)

Pillars o f market econ om y Content o f system atizing space 2. Capital market A . investors space

1. Investors segments space 2. Short-term allocations space 3. Long-term allocations space 4. Investors preference space 5. Portfolio allocations space

B. Capital market offers space 1. Investm ent risk o f stock com panies 2. Stock com panies space

C. Capital market institutions space

D . Spaces system atizing the investors in capital market 3. Labor market A . Work offers spaces

1. Part time work offers spaces 2. Full time work offers spaces

B. Unem ploym ent spaces 1. A ctual unem ploym ent spaces 2. Unem ploym ent segm ents spaces

4. FINA L REM ARKS

M ultid im ensional scaling is u n d o u b ted ly a m eth o d which m ay enrich economical analysis, and particularly decision m aking analysis by co n trib u tin g new m eth o d s dealing w ith m easuring m ultivariab ility. G o o d exam ple are the m anag erial dim ensions in decision m aking. T his exam ple illu strates how rich can be th e extension o f system atizing spaces w ithin one d o m ain o f analysis.

O f course, m ultid im en sio n al scaling should not be trea ted as a p an acea which can solve all econ o m etric o r psychom etric problem s. F o r exam ple, this m ethod can n o t be used to substitute for statistics w hich are a p p ro p ria te to test causal co n n ectio n s an d to verify research hypothesis.

M u ltid im en sio n al scaling can u n d o u b ted ly be useful in first stages o f reasoning, ev a lu a tin g , an a ly tica l p ro c ed u re s or ap p ly in g - w hen it is necessary to system atize the collected d a ta and then to form ulate a hypothesis, diagnosis, ju d g em en ts o r evaluations. If m anagerial decision situ atio n is the case, m ultid im en sio n al scaling can essentially help in shap ing an d designing decision analysis.

T h ere is also on e m o re attra ctiv e way w hen m u ltid im en sio n al scaling can be used, th a t is an in teg ratin g various o pinion s and ev a lu atio n s w hich deals w ith one issue. In th a t case m ultidim ensional scaling , w hen firstly the p ro g ram M IN IS S A (or som e o th er o f this kind) an d then the P IN D IS

are used, can be a to o l fo r building m ethod olo gical consensus in a w orld o f experts, specialists and au th o rities w ho represent v ario u s ap p ro ach es, m eth o d s, techniques, concepts and schools and behave like people in the T ow er o f Babel.

R E FER EN C ES

B i e l a A. (1992), Skalow anie wielowym iarowe ja k o m etoda badań naukow ych, T ow arzystw o N au kow e K U L , Lublin.

B i e l a A. (1995), Skalow anie w ielow ym iarow e w analizach behawioralnych i ekonom icznych, Norbertinum , Lublin.

B i e l a A . (2001), W ym ia ry d ecyzji m enedżerskich, Tow arzystw o N au kow e K U L , Lublin. C o o m b s C. H. (1964), A T heory o f D ata, John W iley, N ew York.

H u b e r O. (1983), D om inance am ong som e cognitive strategies f o r m ultidim ensional decisions, [in:] W. L. S j o b e r g , T. T y s z k a , J. A. W i s e (eds.), H um an D ecision M a king, D oxa, Karishamn, 228-242.

L u k a s i k - G o s z c z y ń s k a M. (1974), Badania strategii podejm ow ania d ecyzji w ielow ym ia­ rowych, UW , Warszawa.

A d a m B ie la

W IE L O W Y M IA R O W E SK A L O W A N IE W B A D A N IA C H E K O N O M IC Z N Y C H

W badaniach naukow ych typu em pirycznego (do których należą również nauki społeczne i ekonom iczne) istotne znaczenie m a określenie związku pom iędzy terminami teoretycznym i a terminami em pirycznymi. Związku tego nie da się ustalić w postaci definicji zupełnych, lecz tylko i w yłącznie przez definicje cząstkow e. T a prawda znana jest ju ż od czasu prac R. Carnapa, a została utrw alona i rozwinięta w m etodologii nauk przez polskich logików: Przełęckiego, Poznańskiego, K am ińskiego. W określaniu zw iązków pom iędzy analizow anym i zm iennym i w nau kach sp ołeczn ych i ekonom icznych kon ieczn e jest stosow an ie technik wielozm iennow ych. Wyniki uzyskanych analiz nie są jednak zadawalające z uwagi na ich zbyt wielką wariancję resztową. N ie co inne podejście m etodologiczne w poszukiw aniu związku m iędzy term inam i teoretycznym i i empirycznym i proponuje sk alow anie w ielow ym iarow e. Artykuł om aw ia założenia m etodologiczne skalow ania w ielow ym iarow ego, teorię danych C. H. C o o m b s a (1964) jako pod staw ę logiczną tego skalow ania oraz przydatność tej m etody w analizie decyzyjnej. W skazano, iż skalow anie w ielow ym iarowe m oże okazać się przydatne w pierwszych etapach pracy badawczej, eksperckiej, analitycznej czy aplikacyjnej, gdy należy usystem atyzow ać zebrane dane i na tej podstaw ie przystąpić dopiero d o form ułow ania hipotez, sądów , d iagn oz, ocen . Istnieje jeszcze jedna m ożliw ość w yk orzystan ia sk alow ania w ielo ­ wymiarowego, a jest nią m ianowicie integrowanie różnych opinii oraz ekspertyz w przedmiotowej kwestii.