the case of primary comparisons, the distance difference plotted against the confidence is the

average _{of ld!i - �kl} and ld;k - ��-

c 0 .. L Cl Q E 0 u ·- ·· . . ..•. ... .. .. • . . 0 11 u c 11 "tl c 0 u ... . ... , . . . . • ·'·'· · . ,_h

Figure 3.2 Distance difference against confidence

Chapter 2 discussed some theoretical problems with the vote-count procedure. In practice, it may

be that the issue is academic, with any distortions it adds to the rank order disappearing again

during the scaling process. We find that inserting

vcu

into (2.9) produces a value of0.0026 for

Stress (these low Stresses reflect the small, marginal numbers of stimuli and triads). Embedding

vciJ

in a two-dimensional space produces

d*iJ•

the distances of figure 3 . l (b), which yield 0.0075

when inserted in (2.9). Finally, the Stress for the

diJ

of the MTRIAD solution is 0.0004.

It seems that the more elements, and the more evenly they are distributed in the configuration, the better the vote-count approximation (here I am generalising from observations, rather than a systematic exploration).

Systematic distortions might not show up in correlation statistics. I conducted a Monte Carlo experiment, generating synthetic triadic data for 1 00 random configurations and reconstructing them with the vote-count procedure. N was 1 5 . Radial coordinates of elements are accumulated and plotted, for the actual and reconstructed configurations, in the left and right panels

respectively of figure 3.3. There are signs of a "centrifugal effect": points tend to migrate toward the periphery. This agrees with Gladstones' obervation of " ... the tendency of the approximate methods [for analysing triadic data] to overestimate distances between stimuli near the centroid." [ 1 962b, page 200], and also with observations of the distortions produced by low-resolution dissimilarity estimates [Green

&

Rao, 1 97 1 ; also Borg

&

Lingoes, 1 987]. Viewed with a sufficiently unskeptical eye, figure 3 . 1 (b) reveals the same distortion.

Figure 3.3

= 6

= 1 3

Tendencies for elements to appear in particular radial coordinate bands, for 1 00 randomised configurations (left) and their vote-counted reconstructions (right)

Radial d istribut ions ( points per equa l-area annulus>

Another approach to analysing triadic data, suitable when the triads have been judged repeatedly (not necessarily by the same judge), comes to us from Torgerson [Gladstones, 1 962a, 1 962b; Torgerson, 1 952, 1 956]. Torgerson' s name is also linked with a variant triad procedure, but I prefer to describe his method of processing the data separately from its collection. The method combines variability in the replications with arguments borrowed from one-dimensional scaling theory, to achieve interval-level data, grist for metric MDS. Briefly, if two dyads (i.j) and (j, k) have been repeatedly compared, Torgerson invokes the Law of Comparative Judgment and applies a pro bit transform (the inverse of the cumulative density function). The result is an interval relationship between the two distances:

dij

=

0k

+ a(p ), where p is the proportion of the comparisons in which (i,j) « (j, k) and the function a(p) is known. 1

For

i,j

and k, points defining a triangle in a mental map which we wish to reconstruct, responses for a single triad are a statement about the triangle's orientation: which corner is acutest and which most obtuse. With this approach we learn more about its geometry.

These interval relationships form a set of simultaneous equations. The second phase of Torgerson's analysis performs a least-squares calculation to produce an approximate, low­ dimensional solution for the coordinates X;p·

As a result, triadic data could be scaled before the advent of non-metric tools rendered proximity data equally tractable. This, as much as the advantages described earlier, accounts for the

popularity of the triadic method during the 1 950s and 1 960s. Torgerson' s method is of more than historical interest, and is summarised here to foreshadow the use of similar arguments in Chapter 5.

After that digression, one commonly-used alternative analysis remains to be described; MINITRI [Roskam, 1 970], also implemented as TRISOSCAL, part of the Cambridge package for MDS [Coxon & Jones, 1 978; MacRae, Howgate & Geelhoed, 1 990].

MINITRI is a version of the KYST algorithm, tailored to handle triadic data: the averaging of distances which violate the rank-order of the data, yielding disparities, is performed only within

1 Compare with Klingberg's use of the legit function, to transform vote-count estimates (summed over many respondents) into a form suitable for metric analysis (principal coordinates).

triads. Reconstructed distances are simultaneously adjusted to improve their fit to a separate list of disparities for each triad in the data set. It is instructive to inspect the stress contributions from each triad in isolation. Let the /-th triad be <i,j, k>, with its elements ordered so that

(i,

k)

)) (i.j)

)) (j, k).

The corresponding l-th list of disparities is

o;kof• oiJ.I• oJkJ

When the rank order of reconstructed distances is a complete reversal of the data,

d;k < diJ < �k·

then

and the contribution to raw stress from triad l is then

= {

d;k - (d;k

+

diJ

+

0k) I 3 } 2

+ {

diJ - (d;k

+

diJ

+

0k) I 3 } 2

+ {

0k - (d;k

+

diJ

+

0k) I 3 } 2

= {

(d;k - dij)l3

+

(d;k - 0k)l3 } 2

+ {

(dij - d;k)l3

+

(dij - 0k)l3 } 2

+

{ (�k - d;k)

13 +

C0k - du-)13 } 2

= 219

(d;k - dui

₊ 219

(d;k - 0k)2

+ 219

(diJ - �-ki

+ 219

(d;k - dij) (d;k - �k)

+ 219

(d;k - dij) (dij - �-k)

+ 219

(d;k - 0k) (dij - 0k)

(3.3)

(3.3)

is equal (apart from the factor 113

)

to the contribution of such a triad to the MTRlAD stress. If the reconstructed configuration clashes with just one of the inequalities - the first, for example - then:

and the contribution to raw stress becomes

(dk - o-k.1\2

I I , / + (d. -I)

o--.1)2

1),

(3 .4)

Again, (3.4) boils down to the same contribution to Stress (and therefore to restorative forces), albeit with a different scale factor.

For confirmation of the closeness of the two programs, I applied MTRIAD to data from a triadic study of odours (kindly provided by MacRae ) , yielding a solution which agreed with that

published by MacRae et al. [ 1 992] . A second source of confirmation is the POOC data, which includes triads from 4 7 subjects for 1 3 names of occupations. MTRIAD duplicates the Coxon and Jones solution [ 1 978, p. 77]. In this case, the responses varied from subject to subj ect, and often were fraught with internal contradictions within one subj ect's data, leading to higher Stresses. The Stress from inserting vciJ into (2.9) is 0. I 1 1 6. This becomes 0. 1 498 for d* ii• the result of scaling the vc!J, i.e. embedding them in two dimensions (figure 3 .4(b)). Finally, the Stress in the MTRIAD solution, figure 3 .4(a), is 0. 1 33 3 .

figure 3 .4 MTRIAD (a, left) and vote-count (b, right) solutions for occupational-title data (Coxon et a[)

""

Key to the occupations:

In document Mapping the mind with broken theodolites : contributions to multidimensional scaling methodology, with special application to triadic data, and the sorting and hierarchical sorting methods : a dissertation submitted in partial fulfillment of the requireme (Page 36-40)