BO EKEHAMMAR. Psychological Laboratories, University of Stockholm, Sweden. ABSTRACT. The purpose of this study was to compare the outcomes of some ...
36 (1972) 79-84;
Acta Psychologica
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VECTOR
SUBJECTIVE
Publishing
TO
MODELS
SIMlLARITY
*
BO EKEHAMMAR Psychological
Laboratories,
University
of Stockholm,
Sweden
ABSTRACT The purpose of this study was to compare the outcomes of some multidimensional scaling methods, which could be derived from four different vector models for subjective similarity. The analysis was made on both theoretical and empirical material. The differences in the results between the various methods could be regarded as negligible in practical contexts.
Multidimensional scaling may be viewed as a summarizing term for different techniques aiming at revealing the underlying dimensionality of a perceptual, emotional or other subjective nature (EKMAN, 1970). Dimension analyses are often based on measures of dissimilarity, proximity or distance, which directly or indirectly, are obtained from an individual’s responses to stimuli. Perhaps the most usual technique is to allow the individual to perform some kind of similarity judgment of stimuli. The different types of such scaling methods may be regarded as being based on different kinds of models for subjective similarity or proximity. One group of such models has been called ‘content models’ by EKMAN and SJ~BERC (1965), since similarity in these cases is regarded as the degree of common content (‘communality’) in relation to total content (‘totality’) for the percepts compared. By describing such a model in vector terms (see e.g. EKMAN, 1963), at least four different geometrical models can be obtained by defining communality and totality differently (EKEHAMMAR, 1972). Of these four theoretical cases, in the following called cases I-IV, case I was reported first by EKMAN and LINDMAN (1961), and case II by EKMAN et al. (1964). If only qualitative variation is assumed to be present between stimuli, i.e. the stimuli are experienced * The study was supported by a research grant to Professor D. Magnusson from the Swedish Council for Social Science Research. 79
80
B. EKEHAMMAR
as being equally intensive, the similarity equations for the four theoretical models mentioned may be expressed mathematically as: Sij=COS
cos
sijx $!lij
$qj
(case 11)
($Qj/2) ’
COS
Sij=COS
(case I)
y?ij,
COS
Sfj=COS
($Oij/2),
(case ru) (case Iv)
fjlij,
where sij is the similarity estimate and pii is the angle between stimulus vectors i and j. It will be observed that case I and case IV give mathematically identical expressions, which, in order of size, are between the values obtained according to case II and case III respectively. To obtain methods as a basis for multidimensional scaling, the above equations may be solved for cos cpij. By these formulae, the estimates of similarity can then be transformed into cosines, which may be treated with conventional factor analytical methods. The purpose of this study was to compare the outcomes of the different scaling methods that can be derived from the theoretical similarity equations mentioned. Since everyone of these equations had some empirical support for at least some stimulus material (EKMAN and LINDMAN, 1961; EKMAN et al., 1964; EKEHAMMAR, 1972) it may be of practical interest to study differences in results obtained by the different methods. This analysis was made theoretically by comparing the different similarity functions, and empirically from the different
methods
by comparing the factor structures
for the same data.
THEORETICAL ANALYSIS Of the four theoretical
similarity equations for cases I-IV, there are
two (cases I and III) in which cos vij # Sij. Solving the similarity equation in case II for cos qii gives, according COS fJJfj=
By solving equation
the similarity
is obtained
to EKMAN (1970):
‘2 (Sij +
equation
j/S$-
si.j’).
(1)
in case III for cos qai a third-degree
: COS3
$?ij+
COS’
paj=2Sij2,
(2)
81
MULTIDIMENSIONAL SCALING
which was solved iteratively for different values of sii by the NewtonRaphson method (see e.g. HILDEBRAND, 1956, p. 447). To demonstrate the differences between the methods, cos qq =.f(s~) is reported graphically for each case in fig. 1.
0.6 -
0.6 -
0.0 Fig. 1.
0.2
0.4
0.6
0.6
1.0
sij
Cosine values (cos q 1.0 were rotated to simple structure according to the Varimax principle. The resulting factor matrices according to case II and case III are reported in table 2. An inspection of the factor matrices shows that the same structure is present, but with higher factor loadings for case III, since the analysis there is based on systematically higher cosines. The maximum difference between corresponding factor loadings for this material was only 0.08. To obtain a quantitative expression for similarity in factor structure between the various methods, coefficients of congruence were calculated according to Tucker (HARMAN, 1960, p. 257) for corresponding factors. The results are given in table 3. The values in table 3 indicate almost complete factorial agreement.
MULTIDIMENSIONAL
SCALING
83
TABLE2 Rotated factor matrix based on cos q,tj according to formula (l), case II, and formula (2), case III (within parentheses) for the same empirical material.
Stimulus
Factor B
A
I II III IV V VI VII VIII IX X
0.65 0.24 0.05 0.82 0.77 0.70 0.08 0.24 0.14 0.05
(0.73) (0.30) (0.10) (0.84) (0.82) (0.73) (0.14) (0.30) (0.21) (0.10)
0.34 0.68 0.85 0.15 0.06 0.00 0.72 0.17 0.14 0.16
(0.39) (0.74) (0.87) (0.21) (0.10) (0.05) (0.74) (0.24) (0.20) (0.21)
C - 0.06 0.08 0.16 0.24 0.10 0.37 0.29 0.72 0.75 0.64
(0.00) (0.14) (0.23) (0.30) (0.17) (0.46) (0.36) (0.73) (0.78) (0.72)
TABLE3 Similarity in factor structure, calculated as coefficients of congruence for corresponding factors, obtained according to cases I-IV for the same empirical material.
Factor
Case I (IV) vs. case II
Case I (IV) vs. case III
Case II vs. case III
I II III
0.9993 0.9989 0.9986
0.9991 0.9990 0.9989
0.9969 0.9960 0.9950
COMMENTS
Both the theoretical and empirical analyses show that the multidimensional scaling methods, based on different theoretical vector models for subjective similarity, give only negligible differences in outcomes for the same data. Provided that one or another of the similarity models described (cases I-IV) gives a satisfactory approximation of the similarity mechanism for a certain stimulus material, this implies that it is not very important to know which of the models that best describes the similarity mechanism. This is of interest for those who wish to apply the method reported. Without any great inconvenience, the mathematically simplest principle, cos 976,= sij, can be used generally, i.e. the dimension analysis can be made directly on similarity data (possibly after linear trans-
B. EKEHAMMAR
84 formation
to values between 0 and 1). The last-named mode of procedure
has also been
used in some
later investigations
EKMAN, 1969 ; EKEHAMMAR, 1971;
(e.g.
BRATFISCH and
MAGNUSSON, 197 I ; MAGNUSSON
and EKEHAMMAR, 1972; MAGNUSSON and EKMAN, 1970), in which also intuitively
meaningful
results were obtained. (Accepted
December
14, I 97 1.)
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