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A Monte Carlo Evaluation Of Interactive Multidimensional Scaling

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A Monte Carlo Evaluation Of Interactive Multidimensional Scaling

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Content A MONTE CARLO EVALUATION OF
INTERACTIVE MULTIDIMENSIONAL SCALING
by
Roger Alan Girard
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In P artial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
( Psychology)
August 1973
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Xerox University Microfilms
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GIRARD, Roger A lan, 1935-
A M O N T E C A R L O EVALUATION O F INTERACTIVE
MULTIDIMENSIONAL SCALING.
U n iv e r s ity o f Southern C a lifo r n ia , Ph.D., 1973
P sych ology, g en era l
University Microfilms, A X E R O X Company, Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.
UNIVERSITY O F SO U TH ER N CALIFORNIA
THE GRADUATE SCHOOL
UNIVERSITY PARK
LOS A NGELES, CA LIFORN IA 9 0 0 0 7
This dissertation, written by
Roger Alan Girard
• is
under the direction of h Dissertation Com
mittee, and approved by all its members, has
been presented to and accepted by The Graduate
School, in partial fulfillment of requirements of
the degree of
D O C T O R OF P H IL O S O P H Y
Z
Dean
D ate..iM d
DISSERTATION TOMMI
v vht
Chairman
TABLE OF CONTENTS
Page
LIST OF TA B LES........................................................................... iv
LIST OF FIGURES........................................................................... v
Chapter
I. BACKGROUND ..................' ......................................... 1
Standard (Non-Interactive) Multidimensional
Scaling Procedures
Monte Carlo Studies in Non-Interactive MDS
Interactive Multidimensional Scaling
II. METHOD............................................................................ 23
True Configuration and True Distances
Relationship of True Distances to the
E rro r Component
Addition of E rror
Scaling Procedures
Statistical Indices
IH. RESULTS............................................................................ 46
Number of Judgments
IV. DISCUSSION AND CONCLUSIONS.............................. 59
Estimation of Dimensionality from
Indices of Apparent Fit
ii
Chapter Page
IV. DISCUSSION AND CONCLUSIONS CONTINUED
Comparability of Configurations
Number of Judgments
Conclusions
REFERENCES.................................................................................. 74
APPENDIX
A cronym s...................................................................................... 79
Basic 4-Dimensional Configuration......................................... 80
Correlation of Fit (S. C. M .) - Tables R1-R4......................... 81
Correlation of Fit (S. I. M .) - Tables R 5 -R 8 ......................... 85
Correlation of Fit (I.A.M .) - Tables R 9 -R 1 2 ...................... 89
Correlation of Fit (Pre-Set) - Tables R 1 3 -R 1 6 .................. 93
Standard E rror (S. C .M .) - Tables E 1 -E 4 ............................. 97
Standard E rror (S. I. M .) - Tables E 5 - E 8 .................................101
Standard E rror (I. A. M .) - Tables E9-E12 105
Standard E rror (Pre-Set) - Tables E13-E16............................. 109
Stress (S. C. M .) - Tables S1-S4...................................................113
Stress (S. I. M .) - Tables S 5 -S 8 ...................................................117
Stress (I. A. M .) - Tables S 9 -S 1 2 ...............................................121
Stress (Pre-Set) - Tables S13-S16...............................................125
iii
LIST OF TABLES
Table Page
1 Distributional Properties of
Distorted D istances............................................................. 33
2 True Correlation of Fit for Three
Scaling M ethods.................................................................... 49
3 True Correlation of Fit for I. A. M.......................................54
4 True and Apparent Correlation of
Fit for S. I. M...........................................................................56
5 Mean Number of Judgments Required
for I. A. M.................................................................................58
6 Apparent Fit for I. A. M. Stripe
Judgments............................................................................... 61
7 Apparent Fit for S. C. M......................................................... 62
8 Comparison of Adjusted Configurations . . . . . . . 65
9 Mean Square Deviation of Distorted
Distances from True D ista n c e s........................................ 66
iv
LIST OF FIGURES
Figure Page
1 Non-Metric Multidimensional Scaling
P ro ced u res....................................................................... 7
2 Procedures for Monte Carlo Studies in
Multidimensional S calin g ............................................... 24
3 Selection of Active and Passive Cells
for Incomplete M atrix...................................................... 36
4 Sample Matrix of Obtained Judgm ents............................... 40
5 Comparison of Scaling Methods in Four
True D im ensions................................................................. 48
6 Comparison of I. A. M ., Pre-Set Bases
and S. C. M.......................................................................... 53
v
CHAPTER I - BACKGROUND
Differences between unidimensional and multidimensional
scaling involve more than simply the question of the number of
dimensions. It is true that, in the most common scaling situations
of the unidimensional type (e .g ., those treated by Torgerson, 1958),
judgments are made with respect to a single known unidimensional
attribute, while in the multidimensional situation the stimuli being
judged differ with respect to several dimensions simultaneously.
However, in the multidimensional situation we know neither what the
dimensions are, nor how many there are. Judgments are made not
with explicit reference to some known continuum, but along a scale
of dissim ilarity. It is from the dissim ilarity judgments made on
pairs of pairs of the stimuli, rather than on direct judgments of each
stimulus taken singly, that the dimensional structure underlying the
judgments must be derived.
As an example of the task confronting a subject in a multi
dimensional scaling experiment, we define for the subject a scale of
dissim ilarity that ranges, say, from one to nine. One end of the scale
represents extreme sim ilarity and the other end extreme difference.
In order to scale N stimuli, all of the possible N(N-l)/2 pairs of these
stimuli are formed and presented to the subject, who is asked to rate
on the one to nine scale just described, the degree of difference
between the members of each of the pairs. The data provided by the
subject represents a partial ordering of the pairs of stimuli, from
which can be recovered orderings of the stimuli themselves with
respect to the dimensions used by the subject in ordering the pairs.
The mathematical model, by means of which we hope to
recover the dimensions and their associated stimulus orderings, is
a distance model. That is, the notion of sim ilarity-dissim ilarity
seems logically related to the idea of psychological closeness-
distance, which in turn seems to be logically related to the concept
of physical distance. Thus, we can represent the stimuli being
scaled by points in a "metric space " (a representation of psycholog
ical space), and the judgments of dissim ilarity can be represented by
the distances between the points in the space.
The formal properties of a m etric space seem to correspond
to what could be considered as logical expectations about psycholog
ical space. These properties are:
(1) That the distance between a point and itself is zero,
while the distance between a point and any other point is
positive.
(2) That the distances are symmetric (that is dj^ = d^j).
(3) That the triangular inequality holds (that is
It is also necessary to specify some function that relates a
distance in the space to the projections of the points on the coordi
nates of the space. The most common functions are the Minkowski
M etrics, in which the distance between two points is a function of the
absolute values of the differences between the projections of the
points on the coordinates:
The most common Minkowski Metric is the Euclidean distance
function (r=2), which, in 2-dimensional space when j lies on the first
coordinate and k lies on the second, reduces to the familiar
Pythagorean Theorem:
The only other of these m etrics to receive any great attention
is the "city-block" m etric (r=l), as described by Attneave (1950).
In all spaces where the distance function is a Minkowski
Metric, distances between points are not affected by shifts in the
origin of the space (translational invariance), and in an Euclidean
space, distances are also unaffected by rotations of the entire
coordinate system (rotational invariance).
Thus, we are ordinarily dealing in multidimensional scaling
with a mathematical model based on the representation of psycholog
ical dissim ilarity by the idea of distance in an Euclidean space,
where the distances are related to the coordinates of the space by the
function:
The location of the origin of the space, and the orientation of the
coordinates of the space are arbitrary and may be changed at will.
For amplification with respect to the scaling model, see
Coombs, Dawes & Tversky (1970, pages 51-76). In addition, excel
lent overviews of most aspects of multidimensional scaling are
provided by Green & Carmone (1970), and Shepard, Romney &
Nerlove (1972).
Standard (Non-Interactive) Multidimensional Scaling Procedures
The traditional multidimensional scaling model for recovering
the coordinates of an Euclidean space was that of Torgerson (1958,
pages 254-259). The dissim ilarities (distances) are first transformed
to "scalar products" as follows. IE j and k are any two points,
5
and i an arbitrary origin, then by the Law of Cosines:
where the term
d - - d - L c o s © - - l
ij ik j i k
is the scalar product of the vectors from the origin, i, to the two
points. Solving for the scalar product, bj^, we have
The matrix of scalar products is factored (e. g ., by a principal
components method yielding eigenvalues and eigenvectors), which in
turn yield a matrix of projections of the original points on a set of
orthogonal coordinates. These projections represent a set of scale
values for the stimuli with respect to these coordinates (dimensions),
and the dimensions are presumed to be those used in making the
judgments of dissimilarity.
The Torgerson model is termed a "fully metric" model in that
it assumes dissim ilarity judgments are related to interpoint distances
of the model on a ratio scale, or with the use of an appropriate con
stant, an interval scale. These assumptions are made for the cur
rent research even though less stringent assumptions have become
popular since 1962 — namely that the relationship between the dis
sim ilarities and the distances need only be monotonic.
This assumption has given rise to what are termed "non
m etric " techniques of multidimensional scaling, in which a metric
6
method may be used to generate an initial configuration (e.g ., Young
& Torgerson, 1967)-*-, followed by procedures to adjust distances
under the constraint of preserving the rank order correspondence
between distances and dissim ilarities.
This can be illustrated schematically as in Figure 1. There
is assumed to exist a set of true distances (A) among the points in a
true configuration. A set of judgments of dissim ilarity (C) are
obtained from a subject, which are related to the true distances by
some as yet unknown function (presumed to be monotonic), and which
are somewhat distorted by judgmental error. The multidimensional
scaling analysis recovers a configuration (D) which may be adjusted
(as described above) if non-metric assumptions apply, and there is
then some final configuration of points with an associated set of
"recovered distances" (E).
The model can be evaluated by statistical indices of goodness
of fit determined in one or more of the following ways:
(1) Indices calculated between all the recovered distances
and all of the dissim ilarity judgments. The most com
mon such index has been the Kruskal Stress (Kruskal,
There are other ways to choose an initial configuration, all
of them arbitrary (e. g ., Kruskal, 1964 a; Lingoes, 1966). The
m etric method is convenient to illustrate differences between metric
and non-metric MDS. (Acronyms appear on page 79 of Appendix.)
F I G U R E 1
NON-METRIC MULTIDIMENSIONAL SCALING PROCEDURES
TRUE CONFIGURATION "M" (Index of Metric Determinacy)
RECOVERED CONFIGURATION
TRUE DISTANCES
I ~ ’ C
RECOVERED DISTANCES
Unsystematic
Distortion
(Monotonic)
Stress
Non-metric
procedures
True Distances Distorted Beginning Configuration
only by Error B D
for Non-metric analysis
Systematic
Distortion
(Monotonic)
Metric MDS
Dissimilarity
Judgments
1964 a, b), which is defined as
\ I J E E M .
V I d j k
where the dj^ are distances recovered from the config
uration obtained as a result of the scaling procedures,
and the are a monotonic transformation of the dis
sim ilarity judgments (i. e ., between points B and E of
Figure 1).
Indices calculated between all the recovered distances
and all the true distances (i. e ., between points A and E
of Figure 1). These indices can be calculated only in an
artificial experiment, where all elements shown in
Figure 1 are known and under control of the experi
menter. The most common such index has been a cor
relative statistic (e. g ., the Index of Metric Determi-
nacy, Young, 1970), which is the squared correlation
calculated between recovered and true distances.
Indices calculated between a set of repeated dissim ilar
ity judgments not used in the scaling solution, and the
corresponding recovered distances. For example, a
reliability coefficient might be estimated for a given sub
ject by including some repeated judgments in the set.
These judgments would not be used as data for the MDS
9
procedures, but would be used as a check on the con
sistency of the judgments of the subject. Now, in addi
tion, these judgments can be used to predict recovered
corresponding distances, and the resulting correlation
is thus a type of validity index. Such an index is being
used in current studies in interactive scaling.
The issues of concern in the current research are assumed
not to hinge on whether m etric or non-metric constraints are
assumed. However, the indices of goodness of fit described above
were derived in a non-metric context, and in fact, virtually all of the
research in the last ten years has been with respect to non-metric
models and techniques (based on the reference studies of Shepard,
1962 a, b; Kruskal, 1964 a, b; and Torgerson, 1965). Indices
derived in these studies are inherently lower (Stress) or higher (Cor
relation of Fit) than they would have been under m etric constraints,
since some of the erro r variance in the m etric model becomes sys
tematic variance in the non-metric model. For this reason, it is
necessary to point out that, while the current research utilizes sta
tistical indices originally used in non-metric studies, the relevancy
of these studies is methodological rather than lying in a comparison
of results.
10 i
Monte Carlo Studies in Non-Interactive MDS
In the theoretical evaluation of a scaling procedure, or of
assumptions about a scaling model, a Monte Carlo investigation can
be initiated from a known configuration and its associated known
interpoint distances. These distances are those "true distances"
shown in Figure 1, and the evaluation is based on an index of good
ness of fit, measured from A to D. In such a study, if we were to
assume erro rless data and a m etric relationship between true dis
tances and dissim ilarities, and then perform a MDS analysis of the
Torgerson (metric) type, we should recover a set of distances that
would have a perfect correlation (R) with the true distances. The
Kruskal Stress (S) or other indices based on deviations of the recov
ered distances from the true or obtained data would be 0. 0.
The aim of the study is to simulate the judgment process by
distorting the true distances with known amounts of unsystematic
(random) erro r variance (and/or in the case of non-metric studies the
introduction of systematic distortion in the form of a known monotonic
transformation). The model is evaluated in term s of concomitant
changes in the indices of fit with the increases in the amount of dis
tortion added to the true distances.
Because of the emphasis on non-metric scaling methods, the
important Monte Carlo studies during the past decade have been per
formed in a non-metric context, some having been done with the
11
specific purpose of evaluating one or more non-metric models.
Shepard (1966) constructed 120 different configurations,
using data from Coombs & Kao (1960). These configurations varied
in the number of points (from three to forty-five), with ten replica
tions for each. Results indicated that the fit improved with the num
ber of points, becoming nearly perfect beyond about ten points. It
was concluded that a non-metric procedure could recover metric
information if there were a sufficient number of points. However, the
investigation was limited to 2-dimensional configurations, and more
importantly,involved completely errorless data.
Young (1970) varied the dimensionality of the true configura
tion and also introduced different amounts of erro r into the configura
tion. The number of points was also varied. Again, the object of the
study was to determine the goodness of fit that could be obtained using
non-metric procedures. Configurations of 6, 8, 10, 15 and 30 points
were used, with three dimensionalities (1, 2 and 3) and five levels of
erro r (. 00, . 10, .20, .35 and . 50). The erro r level was defined as
the ratio of the standard deviation of the normal deviate used to dis
tort the true distances to the standard deviation of the true distances
themselves. The experiment was replicated five different times using
different configurations constructed from the coordinates used by
Coombs & Kao (as had Shepard). As in the Shepard study, the
error-distorted distances were monotonically transformed and a
12
non-metric scaling procedure was used. Goodness of fit was defined
by the Index of Metric Determinacy (M) (the square of the Correlation
of Fit), and Stress was also reported. Results showed that fit
improved with number of points, decreased with increasing erro r
levels, and decreased with increasing dimensionality. It was noted
that, while fit as measured by M improved with the number of points,
the Stress statistic became poorer.
Sherman (1972) presents even more comprehensive data in a
replication and extension of Young. Using randomly generated con
figurations, data were generated over the same numbers of points,
true dimensionalities, and erro r levels as in the Young study. In
addition the effect of several of the Minkowski constants (1. 0, 2. 0
and 3. 0) was investigated. The study was non-metric and goodness
of fit was measured by both M and S.
An important feature of the study was a systematic investiga
tion of underextraction and overextraction (where the number of
dimensions extracted differed from the number of true dimensions).
This data is important since the number of true dimensions is
unknown in the practical situation, and in the absence of statistical
criteria, the best way to estimate the true dimensionality may be to
examine trends in the goodness of fit statistics for different numbers
of extracted dimensions.
In general, Sherman's results were poorer than Young's,
13
probably because of differences in the true configurations used,
and in the method of adding erro r. These differences in methodology
will be discussed in Chapter II. Sherman found that the non
correspondence between M and S noted by Young held true at higher
e rro r levels, but not at low e rro r levels. A comparison of true fit
as measured by ]/ 1-M, and apparent fit as measured by S, led him to
conclude that S overestimated fit in many instances, underestimated
'in some, but was an acceptable index of fit for low levels of error,
combined with a large number of points, and when extracted dimen
sionality equalled the true dimensionality.
Wagenaar and Padmos (1971) also varied both true and
extracted dimensionality, as well as erro r level and number of
points. They indicate that Stress values actually obtained in real
situations might be compared with tabled Stress values from Monte
Carlo studies in order to estimate true dimensionality and erro r
level. Their method of adding erro r was quite different from the
preceding studies, and is discussed in Chapter II.
Stenson & Knoll (1969), and KLahr (1969) both performed non
m etric analyses of random data. Klahr generated data for 6 to 16
points in dimensionalities from 1 to 5, while Stenson & Knoll's data
was for 10 to 60 points in dimensionalities from 1 to 10. Taken
together these studies provide base-line data for a "null hypothesis"
situation in which the data are pure error, with no underlying
14
dimensional structure. These data indicate that non-metric proce
dures will obtain a solution no m atter how bad the data are, and that
with small numbers of points the Stress statistic can be quite low and
therefore quite misleading. Wagenaar and Padmos (1971) also report
a sim ilar study.
Finally, Spence (1972) reports a Monte Carlo comparison of
several non-metric computing algorithms (computer programs):
M D SCAL (Kruskal, 1968); SSA-I (Lingoes & Roskam, 1971); and
TORSCA-9 (Young, 1968). These procedures differed both in the
method of selecting an initial configuration, and in the iterative pro
cedures used to optimize goodness of fit. Differences among the
methods were small with respect to the final solutions obtained, des
pite the fact that there were wide differences in the goodness of fit
statistics for the initial configurations. TORSCA-9, which selected
its initial configuration by the Torgerson method '(as previously des
cribed), produced the best statistics at this initial stage of the pro
gram. These initial statistics might be considered comparable to
those generated in the current research, since they represent the
only results of a purely m etric procedure. However, they are not
comparable on other bases, particularly the number of points, and
possibly also with respect to the magnitude of the erro r levels.
15
Interactive Multidimensional Scaling
In all of the traditional methods of multidimensional scaling,
both m etric and non-metric, the scaling of N stimuli utilizes judg
ments of dissim ilarity for N(N-l)/2 pairs of stimuli (i. e ., all of the
possible pairs). It is obvious that this number of judgments rapidly
becomes prohibitive, from the point of view of the practicality of the
task demanded of the subject, when N exceeds about twenty to twenty-
five stimuli. In addition, the larger the number of stimuli, the more
redundancy there is in the data (unless one wants to assume that more
stimuli necessarily result in more dimensions).
In order to develop methods of scaling numbers of stimuli
larger than 20 to 25, say on the order of 50 to 60, methods are
required that can obtain an acceptable solution based on some subset
of the N(N-l)/2 possible pairs. For example, the scaling of 25 stim
uli traditionally requires 300 judgments. If this is taken as the max
imum feasible number of judgments, then the scaling of 50 stimuli
would have to be accomplished with only a subset of less than 25 per
cent of the 1225 possible judgments.
Kruskal (1964 b, page 116) asserts that 25 per cent is a sufficient
number of judgments, provided that they are "properly distributed in
the m atrix of all possible judgments. " Thus, we might select a sub
set of judgments in some sort of "representative" manner, or perhaps
by random selection under the assumption that in the long run such a
selection would be representative. In either case, computer pro
grams currently in use can be used to scale an incomplete matrix.
Normal scaling procedures are performed after the mean of all the
available judgments has been substituted for all of the missing data
("passive cells") (e.g ., see Young, 1968).
One of the basic propositions of the current research, how
ever, is that a systematic method of selecting a subset of pairs is
superior to a representative or random selection procedure. The
specific systematic method proposed is "ISIS" (Interactive Scaling
with Individual Subjects), as described by Young & Cliff (1972). This
method is based on the assumption that, for erro rless data involving
N stimuli in R-dimensional space, as few as N(R+1)-(R+1) (R+2)/2
judgments are actually necessary to derive the N x R coordinate
matrix, assuming that the "right" pairs are used. This is an exten
sion of the geometrical fact that a set of N-2 points lying in a straight
line (an unidimensional space) may be fixed by knowing the distances
of-'each of these points from two other points lying in the line (plus the
distance of these two points from each other). In the general case, a
set of (N-R+l) points in R-dimensional space may be fixed by knowing
their distances from a set of (R+l) other points in the hyperplane
(plus the R(R+l)/2 distances of these (R+l) points from each other).
We should thus, for erro rless data, be able to obtain a perfect solu
tion by choosing some random set of (R+l) points and obtaining only
the judgments of these points with each other, and with all the other
points in the entire set. The m atrix of scalar products derived from
the judgments among the (R+l) points is the one that is factored
(rather than the full m atrix), and the R eigenvalues and eigenvectors
can be applied to the scalar products m atrix to obtain the N x R coor
dinates m atrix that constitutes the solution. Although we do not know
in advance the value of R (the dimensionality), we can always assume
a maximum value of R that can be expected to exceed the actual
dimensionality of the space, and still effect a tremendous savings in
the number of judgments required.
In the practical situation, however, the inevitable erro rs of
judgment make it impossible to obtain a good solution based on any
random set of (R+l) points. This is because the magnitude of judg
ment e rro r is not proportional to the scale value of the dissim ilar
ities scale, but probably remains relatively constant along the length
of the scale (that is the same magnitude of e rro r probably applies to a
judgment of eight or nine as to a judgment of one or two). This means
that sm aller distances are much more subject to distortion than
larger distances (i. e ., a greater proportion of sm aller distances is
likely to be error). Thus use of small distances as reference points
for all other stimuli is therefore undesirable (see Young & Cliff, page
393). To minimize the effect of error, then, in the selection of a set
of (R+l) points, these points must be maximally far apart. Young &
18 I
Cliff proposed an interactive procedure between subject and computer,
as a method of selecting an appropriate set of (R+l) stimuli on which
to base a scaling solution. They presented both a theoretical proce
dure and a computer algorithm, which are discussed as follows.
In the theoretical ISIS procedure, two stimuli are selected to
define a first dimension. These are the initial members of what is
called a "basis. " Next, each of the other stimuli is tested in turn for
inclusion in the basis set, the criterion for inclusion being whether
or not a new dimension is required to account for one or more of the
distances between the new stimulus and the stimuli in the basis set.
That is, if all these distances can be reconstructed from only their
projections on the dimensions defined by the old basis, then the stim
ulus does not define a new dimension, and is discarded. However, if
one or more distances between the new stimulus and the old basis set
show a residual length that is unaccountable in term s of projections of
the dimensions of the basis set, then that residual length is assumed
to represent a projection on a new dimension, and it is added to the
basis set.
The computing algorithm presented by Young & Cliff differs
somewhat from the theoretical procedure, in that a stimulus-by-
stimulus interactive process is foregone in place of one where sets of
approximately twenty-five judgments are obtained between each set of
computer computations (rather than the stimulus-by-stimulus
19 :
computations implied bythe theoretical procedure). The subject is pre
sented with a group of about eight stimuli called a "focus," and judg
ments are obtained on all possible pairs of these stimuli. The compu
te r then determines which of these stimuli to retain in the "basis "set,
adds additional stimuli to that basis to form a new focus, and obtains
judgments on this entire subset (if some of the judgments have already
been obtained in a previous cycle, they are not obtained again). The
process is repeated until all stimuli have been included in a focus.
This method results in a larger number of judgments than the
theoretical minimum number. This is because the judgments that are
of interest in the final scaling solution are those between the (R+l)
stimuli retained in the final basis and all of the other stimuli. Many
of these judgments may not have been obtained during the interactive
process, and therefore have to be obtained as a last step. Even if
there remain no additional judgments to be obtained, there will be a
small degree of redundancy in the form of non-essential judgments,
simply because the computing algorithm handles a subset of stimuli
(a "focus") on each cycle, rather than a stimulus at a time. This
does not substantially alter the large savings in the number of
required judgments, and is necessary because of judgment error,
which might distort the results of a stimulus-by-stimulus interactive
procedure.
The net result of the interactive procedure is a final basis set
20
of (R+l) or more reference stimuli, and a (R+l) x N m atrix of dis
sim ilarity judgments between the basis stimuli and all stimuli being
scaled (referred to as the STRIPE matrix). The STRIPE m atrix of
judgments is transformed into scalar products in reference to an
origin at the centroid of the final basis,
where m = the number of stimuli in the basis. (Note that one of the
subscripts of b varies from 1 to N, and the other from 1 to m .)
The m x m Basis Scalar Products Matrix is then factored by
principal components, and the R x N coordinate matrix obtained by:
- ^ i in
x mn s ^ m 1 0 mn v im
bm+1 = b V i m xmn
Where bm is the scalar product m atrix before extraction of the m^1
dimension, bm+- * - is the residual scalar product matrix after extrac
tion of the m^1 dimension, Am is the m ^ eigenvalue, and V is a
m atrix of eigenvectors. Distances are recovered, and indices of fit
calculated, after each dimension is extracted.
The foregoing description of ISIS and Standard MDS raise a
number of potential research issues. The current research is
21
prim arily concerned with the following:
(1) What are the base-line goodness of fit data for ISIS in
term s of the fam iliar MDS statistics (Correlation of Fit
and Stress)?
(2) Some sacrifice in goodness of fit would be expected for
ISIS in relation to Standard MDS procedures applied to a
complete m atrix of dissim ilarities. What is the degree
of difference, and how important is it,bearing in mind
that the Standard procedures are not feasible from the
point of view of number of required judgments ?
(3) What is the relationship between ISIS generated data and
Standard MDS procedures applied to a m atrix with a
large number of passive cells? The latter method would
appear to be the only present alternative to ISIS requir
ing a comparable number of judgments.
(4) Are there modifications to the Young & Cliff version of
ISIS that might substantially improve goodness of fit?
Such modifications might be suggested from the answers
to (1), (2) and (3) above.
The general approach to these research questions is one of
replicating, by Monte Carlo procedures, the three conditions des
cribed above: a simulated ISIS procedure, the Standard (Torgerson)
metric procedure on a complete m atrix of dissim ilarities, and the
22 ;
Standard procedure on an incomplete m atrix with a large number of
passive cells. Replications for each of these basic conditions are
made while varying other param eters of general relevance in multi
dimensional scaling studies, most importantly the e rro r level, tine
dimensionality of the basic configuration, and extracted dimension
ality. Issues that may be of interest, but which are not undertaken by
the research include the number of points in the true configuration,
qualitative differences among true configurations, and issues
involving m etric versus non-metric MDS.
CHAPTER II - METHOD
As a starting point for the Monte Carlo procedure, an arbi
trary configuration of fifty points in four dimensions was chosen,
based on a 4-dimensional configuration recovered from a subject in an
actual interactive situation. This configuration is shown on page 80
of the Appendix.
Four configurations were formed from the basic true config
uration described above as follows:
(1) Configuration 1TD - dimension 1 only.
(2) Configuration 2TD - dimensions 1 and 2.
(3) Configuration 3TD - dimensions 1, 2 and 3.
(4) Configuration 4TD - dimensions 1, 2, 3 and 4.
The Euclidean distance function
was used throughout the study in the generation of the true distances.
From this starting point, the generation of replications proceeds as
follows (also see the schematic diagram in Figure 2):
(1) Different amounts of random erro r are added to each set
of true distances (five e rro r levels for runs under the
Interactive Method, two e rro r levels for runs under
each of the Standard Methods). Note that systematic
(monotonic) distortion is not added.
23
FIGURE 2
PR O C ED U R ES FOR M O N T E CARLO ST U D IE S IN MULTIDIMENSIONAL S C A L IN G
TRUE CONFIGURATION RECOVERED DISTANCES
Correlation of Fit "" "*
TRUE DISTANCES RECOVERED CONFIGURATION
Addition of Error
(Distortion)
Standard Error [ Metric Multidimensional
and Stress V Scaling Procedure
DISTORTED DISTANCES
(Simulated Dissimilarity Judgements)
CO
(2) The distorted distances (simulated dissim ilarities) are
subjected to each of the three MDS methods under
investigation:
(a) Standard MDS of the Complete Matrix (S. C. M .).
(b) Standard MDS of the Incomplete Matrix (S. I. M .).
(c) Interactive MDS, a simulated procedure sim ilar
to ISIS (I. A .M .).
(3) Extracted dimensionality is varied by retaining the first
1, 2, 3 or 4 principal components of the factored m atrix
(the factored basis m atrix in the case of I. A. M .).
(4) The above procedures are replicated 3 times for S. C. M.
and S. I. M ., and 20 times for I. A. M. (Statistics gen
erated under Standard Methods are quite stable, and the
computer time required for these methods is extensive.)
(5) Dependent variables for each replication are 6 indices of
goodness of fit for S. C. M ., and 15 indices for S. I.M.
and I. A. M. The design may be summarized as follows:
(a) For S. C. M ., 4 extracted dimensions X 2 erro r
levels X 3 replications (plus 4 extracted dimen
sions X 1 replication for errorless data).
(b) For S. I. M ., 4 extracted dimensions X 2 erro r
levels X 3 replications (plus 4 extracted dimen
sions X 1 replication for errorless data).
26
(c) For I. A .M ., 4 extracted dimensions X 5 erro r
levels X 20 replications (plus 4 extracted dimen
sions X 1 replication for errorless data).
The entire design is repeated for each of the four configura
tions, which represents a total of 1840 scaling solutions. The
detailed procedures used in defining the variables of this design are
described as follows.
True Configuration and True Distances
A Monte Carlo procedure may be initiated from any configura
tion of points, and different methods have been used to decide on
appropriate configurations. Points may be located in the space by
randomly assigning each coordinate of each point (Sherman 1972,
Spence 1972). A closer simulation of reality might involve using a
"real" configuration selected for some specific purpose or character
istic, or one purposely constructed for the same purpose. Young
(1970) used the same configuration as had Shepard and Coombs &
Kao, for purposes of comparability.
The configurations used throughout the current study were
based on a single 4-dimensional configuration recovered from a sub
ject in an actual interactive situation. The subject's data were highly
reliable, but some variance was added to the third and fourth dimen
sions because goodness of fit statistics showed them to be weak. It is
27
recognized that results may be biased because of use of a specific
configuration, and that it may be desirable to replicate portions of
this study using configurations chosen by random means.
Another issue is whether there should be one configuration
(whether generated at random, or purposely chosen or constructed),
or whether each replication should begin with a new configuration.
The latter approach is that of both Sherman and Spence. The position
taken in this research is that we are attempting to simulate an actual
session with an individual subject, that it is logical to assume that
there is a single true configuration underlying his judgments, and that
all replications should therefore begin with this configuration. The
method of Sherman and Spence would seem to result in an additional
source of variability that ought to be considered in interpreting the
magnitude of the erro r component that is added subsequently.
Relationship of True Distances to the E rror Component
The use of the Euclidean distance function implies that the
average magnitude of the generated distances is a function of the
number of dimensions. This means that if an erro r component is
added to these distances, the relative amount of erro r will be an
28
inverse function of the number of dimensions, unless compensatory
adjustments are made.
Young (1970) added erro r to each coordinate and determined
the standard deviation of the erro r in relation to the standard devia
tion of the true distances. Young points out that this, in conjunction
with the method of generating the erro r component, confounded the
effective proportion of erro r variance with dimensionality (Young
1970, pages 462-463).
Spence (1972) and Sherman (1972) both added e rro r to each
coordinate, with the standard deviation of the erro r component deter
mined relative to the standard deviation of the coordinate rather than
to the standard deviation of the true distances. However, as Spence
pointed out, with coordinates uniformly generated on the interval
-1 to +1, the maximum possible distance was a function of the num
ber of dimensions (equal to 2 \pm, where m is the number of dimen
sions). Spence, therefore constrained all his points to lie within a
hypersphere of unit radius with the center at the origin, meaning that
the maximum distance was 2, regardless of dimensionality.
Wagenaar and Padmos (1971) handled the problem as follows.
A normally distributed e rro r component with mean 1. 0 and variance
e^ was generated (e varying from 0. 0 to 0. 40). This component, r,
was applied multiplicatively to the true distances, the quantity (r x d)
then serving as the distorted distance. While this method does ensure
29 !
that the magnitude of the erro r is proportional to the magnitude of the
true distances being distorted, it implies that sm aller judgments have
sm aller erro r components, an assumption rejected in this study.
It is assumed, however, that .the relative effect of erro r at
true dimensionalities of 1, 2, 3 or 4 should be comparable. Since, in
the current research, erro r is added to the true distances rather than
the coordinates, true distances were adjusted, prior to the addition of
erro r by the factors
( E d 2)* ( I d 2)n (E d 2)™
( E d 2)1 ' ( E d 2)1 ' ( E d 2)” respectively
where ( i d 2) 1, etc. are the sums of squared interpoint distances
when each distance is Euclidean in 1 dimension, 2 dimensions, 3
dimensions and 4 dimensions.
Addition of E rror
Young (1970), Sherman (1972) and Spence (1972) all added
erro r not to the true distances, but to the original coordinates.
Before each distance was computed, each coordinate was indepen
dently distorted by a random erro r component from a normal distrib
ution. Young points out that this process is equivalent to adding
directly to the true distances an erro r component having a non-central
chi-square distribution. His rationale for this method (as opposed to,
30
say, an addition of normally distributed e rro r directly to the dis
tances themselves) was that:
(1) The distorted distances cannot be negative numbers,
which would have violated the definition of a distance
between two points.
(2) Overestimates of distances are more likely than under
estimates, which is desirable for distances close to
zero.
(3) Most importantly, it represents the distribution of
distances when the coordinates of those distances are
independently normally distributed. This is a multi
dimensional analog of Thurstone's unidimensional
scaling model, each dimension representing an indepen
dent discriminal process.
In the current research, e rro r is added directly to the true
distances, which follows from the assumption that the behavior being
measured is a single judgment of dissim ilarity made without aware
ness of the multidimensional structure underlying the judgment. If
this is true, then a model based on the separate distortion of coor
dinates, known only to the experimenter on a post-hoc basis, would
be inappropriate.
It is further assumed that the subj ect is constrained not only
with respect to negative numbers, but is constrained at both ends of a
31
scale (ranging say from 1 to 9). We might logically expect distribu
tions of erro r to be more or less symmetrical at the middle of this
scale, positively skewed at the lower end, and negatively skewed at
the upper end. This was, in fact, found to be the case by Ekman and
Kuennapas (1969), in a study of a large number of sim ilarity esti
mates. They also found that the standard deviation was a maximum
when the mean was at the middle of the scale, and decreased as the
mean approached either end. Their data were continuous on a
0. 0 to 1.0 scale, but by dividing the scale into six discrete cate
gories, a binomial distribution on the scores 0 to 5 was found to
provide an excellent fit to the data.
It would appear that other distributions with sim ilar properties
(i. e ., skewness and mean/standard deviation relationship) could also
provide a good fit to the data, and a continuous distribution would be
preferable, since there is assumed to exist an underlying continuum .
The following method, therefore, was used to introduce the
erro r component:
(1) To each true distance (dj.), add a normally distributed
erro r component (e), where the dj. are on a scale from
1
It is true that a subject must make a discrete judgment even
if his perception should be on a continuous scale. It might be of
research interest to determine the erro r level associated with only
the rounding process in making discrete data from continuous data.
1.0 to 9.0 and the e are N (0, Five different
levels of erro r were used, with (T"k of 0. 20, 0. 40,
0. 60, 0. 80 and 1. 00.
(2) Transform (linearly) these distorted distances froin the
1 to 9 scale to a scale from -1. 0 to +1. 0. This dis
tribution is still normal and may have elements falling
beyond the lim its of the scale.
(3) Transform by the inverse of the Fisher Z transform
(i. e ., by the hyperbolic tangent) so that all values lie
between -1.0 and +1.0. This distribution resembles
that of a Pearson r, with the skewness depending on the
original value of the dj..
(4) Transform linearly again to obtain a 1.0 to 9. 0 scale.
The skewness of this distribution will be the same as in
(3), and will be completely within the lim its of the scale.
The net result of steps (1) to (4) is a transformation of the
form: d^ = 4. 0 tanh (0. 25(dj. + e - 5.0) ) + 5.0.
Table 1 shows the distributional properties of the d^ obtained
by this transformation. The statistics shown are the result of 12250
distortions for each of the 9 values of dj. from 1. 0 to 9.0. The orig
inal erro r component (N = 12250) has the following properties:
mean = -0. 005, standard deviation = 0.998, skewness (third
moment) = 0.005, kurtosis (fourth moment) = 2. 854.
TABLE 1
DISTRIBUTIONAL PROPERTIES OF DISTANCES DISTORTED BY
= 4.0 tanh (0. 25 (dj. + e - 5. 0) ) + 5.0
Magnitude of Original True Distance
1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00
Mean 2.03 2.55 3.23 4.07 5.00 5.92 6.76 7.45 7.97
Standard Deviation .45 .61 .78 .90 .95 .90 .78 .61 .45
Skewness .09 .18 .25 .21 .01 -.19 -.25 -.18 -.09
Kurtosis .17 .50 1.09 1.75 2.05 1.76 1.11 . 51 .18
CO
CO
3 4 ;
Scaling Procedures
For each of the configurations, each erro r level produces a
m atrix of distorted distances which simulate judgments of dissim ilar
ity. Each matrix was then analyzed by each of the three scaling
procedures being investigated: the Standard Method performed on the
complete m atrix (S. C. M .), the Standard Method performed on the
incomplete matrix (S. I.M .), and the Interactive Method (a fully
off-line simulation of ISIS, with a few modifications) (I.A .M .). Rep
lications were made of each procedure, the only difference among
replications being a new random number seed as a starting point for
the generation of the erro r component.
The standard method is that of Torgerson (1958, pages 254 -
259), as described in Chapter I of this study. The entire m atrix of
distorted distances (dissim ilarities) is analyzed (the S. C.M. condi
tion). Then the standard method is used again, but this time only
one-third of the cells of the 50 x 50 m atrix are "active" (i. e ., contain
usable dissim ilarities). The remaining two-thirds of the cells are
replaced by the mean of the active judgments and become "passive"
cells. The distances recovered from the analysis are analyzed again
by the standard method to obtain a better fit, and the distances
recovered from this analysis are analyzed a second time. Thus, the
recovered distances that constitute the final solution for S. I. M. are
the result of two scaling iterations of distances recovered from the
35 !
previous solution.
There are a large number of ways to decide which of the cells
in the matrix should be active and which should be passive. The con
cern in this research was to avoid choosing a subset that might pro
duce extreme results on a chance basis. For this reason, the choice
was not a random selection of 408 of the possible 1225 judgments, but
a selection of every third off-diagonal cell in the pattern shown in
Figure 3.
This method at least ensures that certain key stimuli are not
unintentionally unrepresented, and that all stimuli are equally repre
sented as both first and last members of a pair.
The decision to perform two iterative scaling procedures on
the recovered distances was based on some prelim inary work in which
two iterations appeared to offer a reasonable improvement from the
original solution. Further iterations appeared to provide diminishing
returns from the point of view of computer time involved, an impor
tant consideration with m atrices of this size.
The interactive method (I. A. M .) parallels that of Young &
Cliff, but with some modifications. The first eight stimuli (initial
"focus") are paired in all possible ways and dissim ilarities are
obtained for these 28 pairs. Any given pair is "obtained" by selection
from the full m atrix of 1225 dissim ilarities generated by the addition
of error, as previously described. (In the Monte Carlo situation,
FIGURE 3
36
SELECTION OF ACTIVE AND PASSIVE CELLS FOR INCOMPLETE MATRIX
OX— X— X— X— X— X— X— X— X— X— X— X— X— X— X— X— X
OX— X— X— X— X— X— X— X— X— X— X— X— X— X— X— X—
o x — X— X— X— X— X— X— X— X— X— X— X— X— X— X— X -
o x - -X - -X - -X - -X — X— X— X - -X — X - -X - -X - -X — X - -X — X
o x — X— X— X— X— X— X— X— X— X— X— X— X— X— X—
o x — X— X— X— X— X— X— X— X— X— X— X— X— X— X -
o x — X— X— X— X— X— X— X— X— X— X— X— X— X— X
o x - -X - -X - -X - -X — X— X - -X - -X — X - -X - -X — X— X—
OX— X— X— X— X— X— X— X— X— X— X— X— X— X -
OX— X— X— X— X— X— X— X— X— X— X— X— X— X
o x — X— X— X— X --X — X— X— X— X— X— X— X—
o x - -X - -X - -X - - X - -X - -X - -X — X— X— X - -X — X -
o x — X— X— X— X— X— X— X— X— X— X— X— X
o x — X— X— X— X— X— X— X— X— X— X— X—
o x — X— X— X— X— X— X— X— X— X— X— X -
o x - -X - -X — X - -X — X— X - -X — X— X - -X - -X
OX— X— X— X— X— X— X— X— X— X— X—
o x - -X - -X - -X - -X — X— X— X - -X - -X - -X -
OX— X— X— X— X— X— X— X— X— X— X
o x — X - -X - -X — X— X— X— X - - X - - X - -
o x — X - -X - -X - -X - -X — X— X - -X --X -
o x — X— X— X— X * — X— X— X— X— X
OX— X— X— X— X— X— X— X— X—
o x — X— X— X— X— X— X— X— X-
OX— X— X— X— X— X— X— X— X
OX— X— X— X— X— X— X— X—
ox— X — X — X — X — X — X -
ox— X — X — X — X — X — X
OX— X— X— X— X— X—
OX— X— X— X— X— X -
ox— X — X — X — X — X
ox— X— X— X— X—
OX— X— X— X— X -
ox— X — X — X — X
ox— X— X— X—
OX— X— X— X-
OX— X— X— X
OX— X— X—
OX— X --X -
X = Active Cells
- = P assive Cells
O = Diagonal
o x — X— X— X— X— X— X— X -
OX— X— X— X— X— X— X— X
o x - -X — X— X— X— X— X—
OX— X— X
o x — X—
OX— X -
O X --X
o x —
o x -
o x
37 i
we have what amounts to a pool of potential judgments, not all of
which are used in the interactive procedure, but which are available
if required.)
A computational routine is then called to determine which of
the eight stimuli should be retained as "basis" stimuli - that is,
which stimuli appear to define dimensions. This computational
procedure is described as follows:
(1) The first two stimuli (a and b) to be retained are
selected (the selection criteria are discussed later) and
used to define the first dimension (the first dimension
lies along da^ with the origin at da^/2, so that the coor
dinates of a and b with respect to the first dimension are
Xal = dab/2 ’ ^ Xbl = da t / 2'
(2) The loadings of each of the other (i) stimuli on this first
i2 i2
dimension are defined as: v . _ > a ~
x 11 -
- 2d
a b
A scalar products m atrix of residuals (after extraction
of the first dimension) is determined by:
I 2 2 2 2 2 2
b ij = — (d ia +d ib + d j a + d jb - d o b _ 2d jj) - xn xji
(3) If the sum of the diagonal elements of this m atrix is
positive (this criterion will be discussed later), it is
assumed that there is another dimension to extract, and
that this dimension is defined by that stimulus, m,
having the largest diagonal element. Loadings on this
new dimension are determined:
v - b im
* f
and the residual scalar product m atrix diminished
n z
b ij s b jj ~ Xi2 xjz
where b* is the residual scalar product matrix after
extraction of the first dimension, and b ^ is the scalar
product m atrix after extraction of the first two dimen
sions.
(4) Step (3) is repeated until either the sum of the diagonal
of b ^ becomes negative, or until the maximum number
of dimensions specified has been extracted. The ratio
nale for this criterion will be discussed later.
(5) The R+l stimuli retained are the "basis" stimuli for the
first cycle. These stimuli are those that best define the
R dimensions (within the context of the eight stimuli in
the first focus).
For the second cycle, only the basis stimuli from cycle one
are retained. To these are added several more stimuli to form a new
focus of nine stimuli. For example, if stimuli 1, 3, 4, 5 and 7 were
39
retained on cycle one, the focus for cycle two would be 1, 3, 4, 5, 7,
9, 10, 11, 12.
The computational procedure is called again and a new basis
is determined, which may or may not include stimuli from the orig
inal basis. The cyclic process is repeated until all of the fifty stim
uli have been included in a focus, which typically takes eleven or
twelve cycles. The basis retained on the final cycle is considered the
final basis; that is, these stimuli are presumed to be the best subset
of (R+l) stimuli that could be selected from the group of fifty to define
the R dimensions contained in the data.
Finally, before the final computational procedure is begun, it
is necessary to complete a m atrix of dissim ilarities between the
(R+l) stimuli from the final basis and all the other stimuli in the
configuration. This is called the STRIPE matrix. The final step in
the interactive procedure is to obtain any dissim ilarities in the
STRIPE that were not obtained as part of the cyclic process des
cribed above. When this final step is completed, there are available
a m atrix of (R+l) x N STRIPE judgments, plus a number of additional
judgments that are not part of the STRIPE, but that were obtained
during one of the cycles. The total of the STRIPE and additional
judgments are referred to as ALL OBTAINED judgments.
Figure 4 illustrates the cyclic process, with each cycle
delimited by vertical lines. The judgments indicated by X are those
40
FIGURE 4
SAMPLE MATRIX OF OBTAINED JUDGMENTS
oxxxxxxx
0 X X X X X X
0 X X X X X
0 X X X X
0 X X X
0 X X
0 X
0
Cycle 1
X X X X
E E
X X X X
X X X X
X X X X
E E
X X X X
E E
0 X X X
0 X X
0 X
Cycle 2 0
X X X X
X X X X
X X X X
X X X X
X X X X
b x x x
0 X X
0 X
Cycle 3 o
X X X X
X X X X
X X X X
X X X X
X X X X
•o X X X
0 X X
. 0 X
Cycle 4 0
X X X X
X X X X
X X X X
X X XX
X X X X
toxxx
0 X X
„ , r ox
Cycle 5 o
X X X X
X X X X
X X X X
X X X X
X X X X
<0 X X X
0 X X
Cycle 6 0 o
X = Judgments Obtained During
Cyclic Process
E = Extra Judgments
O = Diagonal
Cycle 7
E
E
X X X X X
E
X X X X X
E
E
E
E
XX X X X
E
X X X X X
X X X X X
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
0 X X X X
0 X X X
0 X X
0 X
0
X X X
X X X
X X X
X X X
X X X
X X X
0 X X
0 X
Cycle 8 0
X X X X
X X X X
X X X X
X X X X
X X X X
OX X X
0 X X
0 X
Cycle 9 0
X X X X
X X X X
X X X X
X X X X
X X X X
0 X X X
0 X X
0 X
Cycle 10 o
X X X X
X X X X
X X X X
X X X X
X X X X
0 X X X
0 X X
0 X
Cycle 11 0
0 X
0
Cycle 12
X X
X X
X X
X X
X X
41 ;
obtained, proceeding from left to right, during the cyclic process.
The final basis in this sample illustration consists of stimuli 3, 5,
10, 12 and 33. In order to complete the STRIPE matrix (the matrix
formed by pairing the five basis stimuli with all other stimuli) it is
necessary to go back and obtain certain extra judgments not obtained
during the cyclic process. These judgments are indicated by the
letter E. The X judgments plus the E judgments are ALL
OBTAINED judgments. The STRIPE matrix is formed from the five
completed columns/rows.
As will be discussed in Chapter T V , the number of judgments
required of a subject in I. A. M. is a function of which stimuli end up
as the final basis, as well as their number. If the stimuli tend to be
those retained on early cycles, then a small number of judgments tend
to be required. If the basis .consists of stimuli from later cycles, a
large number of judgments are required. This variation is because of
the necessity to go back and obtain extra judgments to fill the STRIPE
matrix (this is to ensure that all basis stimuli have been paired with
all stimuli in the configuration).
Therefore, in the interest of keeping the number of required
judgments small, it is desirable to make it more and more difficult
to replace basis stimuli as the cyclic process proceeds. On
succeeding cycles, the first two stimuli from the basis retained
on the previous cycle are used as a starting point, and all
42 ;
other pairs of stimuli having a dissim ilarity of nine are investigated
in relation to the first pair. However, the first pair is only replaced
if a new pair of stimuli result in a substantial reduction in the residual
sum of elements of the b - * - matrix. The meaning of "substantial" is
arbitrary. In the current research, a 30 per cent reduction in the
residual sum from that of the previous cycle was required before sub
stituting a new pair of stimuli to define the first dimension. In addi
tion, the basis was not changed on the very last cycle, since this
cycle often had few new stimuli remaining to be analyzed, and the
sm aller focus often led to unstable results.
The second problem involves the determination of when enough
stimuli have been retained in the basis (or determination of the
dimensionality of the focus under consideration). The theoretical
aspects of this question are discussed by Young & Cliff (1972, page
392). The criterion used in the current research derives from these
considerations, but has been slightly modified. During prelim inary
trial runs, it was often observed that fewer stimuli would be retained
in the basis on one cycle than on the proceding cycle. Since dimen
sions can be obtained that are entirely due to error, it would seem
desirable to have them disappear on subsequent cycles. On the other
hand, it would seem logical that most stimuli retained in a basis
define dimensions that have some foundation in reality, but which are
also error-distorted. The logic of the interactive process is that of
43
defining a basis and then seeing, on subsequent cycles, whether or
not there will be defined any additional dimensions. We are thus
looking for a maximum dimensionality, and should probably over
extract during the interactive process, on the grounds that goodness
of fit statistics can be used to decide, at the end of the process, on
the appropriate dimensionality (this philosophy would seem appropri
ate for non-interactive methods as well). For these reasons, the
procedure was modified so that each cycle was forced to retain at
least as many basis stimuli as had been retained on the previous
cycle, and the number of stimuli in the final basis are maximized.
The final STRIPE m atrix is analyzed as described in Chapter
I, the STRIPE matrix being transformed to scalar products, the
Basis Matrix factored, and resulting eigenvalues and eigenvectors
applied to the Stripe Scalar Product Matrix to yield a coordinate
matrix.
Statistical Indices
The dependent variables in this study consist of three statis
tical indices, each of which is calculated as an index of true fit
(between the recovered distances and the true distances), and as an
index of apparent fit (between the recovered distances and the dis
torted distances). A correlational index of true fit was of interest,
44
and the Pearson r calculated between the recovered distances and the
true distances (Rt) has been used satisfactorily in the past (Shepard
1966, Young 1970). This correlation can also be calculated between
recovered and distorted distances (Ra), and, in the actual scaling
situation, this index is available rather than the true Correlation of
Fit.
The Kruskal Stress has been used for a number of years as an
index of apparent fit, having been originally derived (Kruskal 1964 a,
b) as the criterion to be minimized in non-metric MDS procedures.
In this study it has been calculated as both a true and apparent index
where the d^ are the distorted distances.
Since Stress can often be a misleading indicator (see Sherman
1972), we have introduced, for comparative purposes, a statistic
which is analogous to a standard erro r of estimate for predicting
either the d^ or the d^ from the dr . This statistic is called simply
the Standard E rror, and has been calculated as both a true and an
apparent index:
In the interactive situation, these six indices may be calcu
lated in more than one way, because the interactive method does not
of fit:
45
use all of the judgments possible. As previously discussed, the
interactive solution is based on a STRIPE m atrix consisting of the
pairs formed by the (R+l) retained basis stimuli, and all the N stim
uli in the original configuration. Thus, it is of interest to examine
indices of fit derived from only these (R+l) x N distances. In addi
tion to the STRIPE, other judgments were obtained during the cyclic
process that were not used in the final solution. There are always
available, therefore, a larger number of judgments than just the
STRIPE, and indices have been calculated over ALL OBTAINED
judgments.
In I. A. M ., all the N(N-l)/2 possible distances can be
recovered from the obtained configuration, as in S. C. M. and S. I. M.
Purposes of comparability require that I. A. M. indices of fit be cal
culated over ALL POSSIBLE judgments, since there are no analogs
in S.C.M. and S. I.M. to the STRIPE and to ALL OBTAMED judg
ments.
CHAPTER HI - RESULTS
The design summarized at the beginning of Chapter H is based
on 1840 different scaling solutions. For each solution generated
using the Standard-Complete Method (S. C. M .), six indices of fit were
determined as dependent variables, while eighteen indices were
obtained for each solution using the Standard-Incomplete Method
(S. I.M .) and the Interactive Method (I. A .M .). Taking means over
replications for these indices results in 2880 tabulated mean values,
a complete presentation of which can be found in the Appendix.
The structure of the Appendix, as shown in its Table of Con
tents, is as follows. Tables R1 to R16 contain data on Correlation of
Fit, Tables El to E l6 contain data on the Standard E rror statistic,
and Tables SI to SI6 contain data on Stress. Each of these sets
(Tables R, E and S) is divided according to method (Tables 1-4 for
S. C. M ., Tables 5-8 for S. I. M ., Tables 9-12 for I. A. M ., and Tables
13-16 for a supplementary study of I.A.M . to be discussed). Within
each of these sets of four tables, there is a separate table for each of
the four configurations, and each individual table is partitioned into
two sections, one dealing with indices of true fit, and one with appar
ent fit. Within each section of each table, the appropriate index of fit
is tabulated with respect to erro r level and extracted dimensionality.
Comparisons of the three multidimensional scaling methods
46
47
investigated by this research could be made in as many as seventy-two
ways for each configuration (6 statistical indices X 4 extracted dimen
sions X 3 erro r levels). Some bases of comparison, however, are not
as good as others. For reasons to be discussed, the data are not
directly comparable across configurations, and in addition, Standard
E rror and Stress do not appear to measure fit as well as the Correla
tion coefficient. The most meaningful comparison of methods, there
fore, is that based on True Correlation of Fit for distances generated
from configuration 4TD, as shown in Figure 5. This figure shows True
Correlation of Fit (ordinate) as a function of E rror Level (abscissa),
for each of the four extracted dimensionalities. For each mean cor
relation, a range of +1 standard erro r is shown, based on twenty
replications for I. A. M ., and three replications for S. C. M. and
S. I. M. Correlations shown for I. A. M. are calculated over ALL
POSSIBLE judgments. Most of the data on which Figure 5 is based
are presented in Table 2.
Since the data in Figure 5 are entirely based on a configuration
having four true dimensions, it would be expected that the Correlation
of Fit should improve with the extraction of each dimension (as one
moves along the ordinate at a given erro r level). It is also to be
expected that for any given extracted dimensionality, there should be
decrements in the Correlation of Fit with increasing erro r levels (as
one follows one of the contours from left to right). It can be seen that
CORRELATION OF F IT (TRUE)
F IG U R E 5
C O M P A R I S O N O F S C A L I N G M E T H O D S IN 4 TRUE D I M E N S I O N S
( N u m b e r s 1, 2 , 3 , 4 , r e f e r to E x t r a c t e d D i m e n s i o n s )
S T A N D A R D M ETH O D
COMPLETE
LOO
0.90
0.80
0.70
0.60
0.50
0.2 0.6
ERROR LEVEL
S T A N D A R D M E T H O D INTERACTIVE M E T H O D
INCOM PLETE
4
*
_i
1. 0 0.2 0.6 1.0 0.2 0.6
ERROR LEVEL ERROR LEVEL
00
49
TABLE 2
TRUE CORRELATION OF FIT
FOR THREE SCALING METHODS - CONFIGURATION 4TD
Extracted Dimensions
1 2 3 4
E rror Level 0. 00
Standard Complete . 692 .870 .963 1.000
Standard Incomplete .632 .771 .844 .859
Interactive .718 .833 .946 1.000
E rror Level 0. 60
Standard Complete . 582 .856 .942 .977
Standard Incomplete . 579 .700 .751 .807
Interactive .651 .767 .830 .855
E rro r Level 1. 00
Standard Complete . 573 . 829 .916 .948
Standard Incomplete .565 .701 .751 .788
Interactive . 575 . 685 .726 .672
50
the data conform to these expectations, with the exception of the fourth
extracted dimension in the Interactive Method at high erro r levels.
After extraction of four dimensions, S. C. M. achieves a
uniformly excellent fit, falling off to . 948 at the highest erro r level.
This result is according to expectations.
Perfect fit is achieved by I. A.M. for errorless data, which is
empirical confirmation that the ISIS model as used in I. A. M. works
as predicted. However, the rate of decrease in Correlation of Fit
with increasing erro r is greater than desirable, falling to . 855 at
erro r level 0. 60, to . 766 at e rro r level 0. 80, and to . 672 at erro r
level 1. 00. Furtherm ore, at these highest two erro r levels, I. A. M.
fit is poorer for four extracted dimensions than for three dimensions,
indicating that the fourth dimension is not recoverable at these erro r
levels. These results led to a further investigation into possible
modifications to the interactive procedures that might result in better
indices of fit.
It was possible that the poor performance by I. A. M. at high
erro r levels could be because some of the bases were poor. It was
clear that I. A. M. statistics were less stable than either S. C. M. or
S. I. M ., and the instability indicated that it was at least possible
to obtain bases that resulted in acceptable indices of fit, even
though inconsistently. One avenue of approach might have been
to change the interactive procedure to try and obtain more
uniformly good bases. However, a more direct approach was to
determine the performance of I. A.M. when the basis was constant,
and was known to be good.
The "best possible basis" would appear to be that obtained
when there was no error; this basis had led to a perfect fit. There
fore, using this as a "pre-set" basis, twenty replications were made
for each of the four extracted dimensionalities times five erro r levels
(using only configuration 4TD). The complete results of these runs
are shown in the Appendix, Tables R13, E13 and S13, from which it is
clear that, while the optimum basis offers a small degree of improve
ment over the variable bases selected by I. A. M ., it remains true
that the 4-dimensional solution is worse than the 3-dimensional solu
tion at the 1. 00 erro r level. It is noteworthy that the comparison of
the optimum basis to the variable bases is not too unfavorable to the
latter - - that is, even though the selection of variable bases results
in some instability in the indices of fit, the general level of fit is not
a great deal worse than one might expect to achieve under the best
possible circumstances.
Another possible way to improve I. A.M. was to increase the
number of stimuli in the basis. Interactive scaling is based on the
prem ise that there is a large amount of redundancy in a complete
m atrix of data, when the number of stimuli is much greater than the
number of dimensions. On the other hand, there may not be sufficient
52
redundancy in a basis that exceeds the dimensionality by a small
amount. Therefore, using the p re-set basis of five stimuli just des
cribed as a starting point, additional stimuli were added to form
bases of seven, nine and ten stimuli. These additional stimuli had
been retained in some basis on an early cycle of the interactive pro
cedure, but later dropped. They had, then, for at least one focus
during the cyclic process, been good enough to define a dimension.
Twenty replications were run using each of these augmented
bases, with complete results shown in the Appendix, Tables R14 -
R16, E14 - El 6 and S14 - SI 6. A summary of the results is pre
sented in Figure 6 and shows that augmentation of the basis leads to a
definite improvement in the True Correlation of Fit. This improve
ment is manifested both by a general, across-the-board improvement
in the level and stability of the index (see Table 3), and by a clear def
inition of the fourth dimension at erro r levels 0. 8 and 1. 0, where no
fourth dimension could be identified with a 5-stimulus basis.
Results for the Standard-Incomplete Method in Table 2 and
Figure 5 show that this method does not attain an excellent fit at any
erro r level (even with errorless data), and that it is inferior to
I. A. M. at the lower e rro r levels. This would also be true at higher
erro r levels, if I. A. M. statistics were defined as those associated
with the augmented bases of Figure 6.
Additional problems with S. I. M. are shown in Table 4. For
CORRELATION O F F IT (TRUE)
Figure 6
C O M PA R ISO N O F PRE-SET BASES A N D S.C.M.
C O N F I G U R A T I O N 2 4 T D . EXTRACTED D I M E N S I O N S
( P 5 - P 1 0 REFER TO P R E -S E T B A S E S
O F 5, 7 , 9, A N D 10 STIM U LI)
0
S C M
0 . 9
P10
0.8
0 . 7
I A M
0.6
0 . 0 0 .2 0 . 4 0 . 6 0 .8 1.0
ERROR LEVEL
TABLE 3
TRUE CORRELATION OF FIT FOR I.A.M . (ALL POSSIBLE JUDGMENTS)
CONFIGURATION = 4TD ERROR LEVEL = 1. 00 EXTRACTED DIMENSIONS = 4
Statistics Calculated Over 20 Replications
Mean S. D. S. E. Max. Min.
Interactive Method (Variable Basis) .672 .072 .017 .804 .459
Pre-Set Basis (5 Stimuli) .709 .099 .022 .814 .337
Pre-Set Basis (7 Stimuli) .800 .041 .009 .855 .677
Pre-Set Basis (9 Stimuli) .846 .022 .005 .884 .806
Pre-Set Basis (10 Stimuli) .860 .020 .004 .901 .813
55 ;
all configurations, fit is not improved by a large amount by iterating
on the distances recovered from the initial solution. This tendency is
most noticeable for configuration 4TD where there is no improvement
at all. Apparent fit, however, would lead to the opposite conclusion,
since this index is a gross underestimate of true fit on the initial
solution, and is an overestimate thereafter.
Table 4 also shows that the correlation of apparent fit rather
uniformly overestimates the true dimensionality, with few exceptions.
This does not occur with the other methods, and from further exam
ination of the data, it appears that the true fit continues to improve
somewhat with over-extraction, a surprising result.
In any case, it is clear from these data that S. I. M ., as con
ceived in this research, produces indices of fit which are quite m is
leading. It may be that different statistics could be derived that might
alleviate some of these problems, perhaps, for example, statistics
calculated over only active cells.
Number of Judgments
The Standard Method of MDS used in this study requires all the
N(N-l)/2 possible dissim ilarities. The comparison of S. I. M. and
I. A. M. has been made on the basis that they are alternative methods
of obtaining a solution based on subsets of judgments of comparable
56
TABLE 4
TRUE AND APPARENT CORRELATION OF FIT FOR S.I.M .
ERROR LEVEL = 1. 00
PASSIVE CELLS REPLACED BY MEAN OF ACTIVE CELLS
Extracted Conflq. 1TD Confiq. 2TD Confiq. 3TD Confiq. 4TD
Dimensions True Apparent True Apparent True Apparent True Apparent
1 .897 . 532 . 763 . 462 . 676 . 405 . 617 . 367
2 .840 .510 .767 .494 .713 .470 .740 .499
3 . 838 . 513 . 825 . 540 . 762 . 519 . 771 .525
4 . 832 . 535 . 813 . 562 . 809 . 546 . 797 . 547
FIRST ITERATION ON RECOVERED DISTANCES
Extracted Confiq. 1TD Confiq. 2TD Confiq. 3TD Confiq. 4TD
Dimensions True Apparent True Apparent True Apparent True Apparent
1 .955 . 894 . 761 .766 . 659 . 635 . 517 . 563
2 .930 933 .830 .848 .706 .755 .682 . . 756
3 .913 .922 .869 .902 .769 .857 .724 .837
4 . 900 . 938 . 878 . 937 . 820 . 912 . 781 .897
SECOND ITERATION ON RECOVERED DISTANCES
Extracted Confiq. 1TD Confiq. 2TD Confiq. 3TD Confiq. 4TD
Dimensions True Apparent True Apparent True Apparent True Apparent
1 .955 . 920 . 798 . 789 . 696 . 682 . 565 . 638
2 .934 .951 .887 .891 .762 .818 .701 .793
3 .932 .964 .892 .941 .801 .899 .751 . .870
4 .920 .978 .901 .967 .830 .941 .788 .931
size. For S. I. M ., every third dissim ilarity was chosen as an
"available" judgment (active cell), and the rem ainder of the data
discarded. The number of judgments required of the hypothetical
subject under this condition was therefore 408, even though 1225
(including passive cells) were actually used in the scaling.
For I. A. M ., the number of judgments varies from replication
to replication. The minimum theoretical number of judgments is
(R+l) X N - (R+l) (R+2)/2, which is equal to 235 when R=4 and
N=50. In practice, for reasons which will be discussed, a good many
more judgments are actually required. Mean numbers of judgments
for this study (averaged over replications) are shown in Table 5.
Each mean was obtained over twenty replications having ranges of
roughly + 60 judgments from the mean values shown.
TABLE 5
MEAN NUMBER OF JUDGMENTS REQUIRED FOR I.A.M .
E rror Level
0.20 0.40 0.60 0.80 1.00 Average
Configuration 1TD 394 381 388 382 378 385
Configuration 2TD 366 369 363 351 354 361
Configuration 3TD 370 373 369 356 358 365
Configuration 4TD 363 365 367 361 355 362
CHAPTER IV - DISCUSSION AND CONCLUSIONS
Estimation of Dimensionality from Indices of Apparent Fit
In the practical situation, one has available only indices of
apparent fit to use as a means of estimating true dimensionality
underlying a set of sim ilarity judgments. Previous studies, all per
formed in the non-metric context, have tended to use only Stress as
an index of apparent fit, and only the Correlation or its square as an
index of true fit. Attempts have been made to relate these two statis
tics, even though they are quite different.
In this study, Correlation of Fit, Stress, and a related statis
tic called Standard E rror have been used in computing indices of both
true fit and of apparent fit. The question is then to judge how well true
fit is reflected in apparent fit. If there is a satisfactory correspondence,
then it should be possible to determine true dimensionality by exam
ining the apparent indices of fit obtained in several dimensionalities,
looking for a leveling off or a deterioration in one or more of the
indices. It has already been pointed out in Chapter III that S. I. M.
cannot be adequately evaluated in practice because of the complete
lack of correspondence between apparent and true fit, and because of
the failure of indices of fit to accurately reflect the true dimensionality.
Table 6 shows the indices of apparent fit for I. A. M. for each
configuration, which suggest that the Correlation of Fit is the most
59
60
reliable index. That is, it is true more often for this index than for
Standard E rror or Stress, that a discernible maximum is reached
when the extracted dimensionality corresponds to the true dimension
ality. Both Standard E rro r and Stress tend to overestimate the true
dimensionality. Table 7 shows the same data for S. C. M. where this
tendency is much more obvious.
Perhaps these results are not surprising when one considers
how the Standard E rror and Stress are determined. For a given set
of distorted (or true) distances, the quantity Z (dr - d^)^ is a para
bolic function of dr , which in turn is a function of the number of
dimensions extracted. As more and more dimensions are extracted,
Z (dr - d^)^ decreases, regardless of the covariance between dr
and d^, up to the point where the dr become larger than the d^. An
Z
O
(dr - d^) only occurs beyond this point, and the data
would indicate that by this time the true dimensionality has been over
estimated. This tendency should be accentuated for the Stress statis
tic, since the quantity Z d | appearing in the denominator continues
to increase with extracted dimensionality, even after the dr and d^
are equal.
If these observations are valid, it would seem inadvisable to
recommend either Standard E rror or Stress as a useful statistic for
either true fit or apparent fit, at least in the context of the current
research.
TABLE 6
Correlation of Fit
Standard Error
Stress
APPARENT FLT FOR I.A.M . STRIPE JUDGMENTS
ERROR LEVEL = 1. 00
Extracted
Dimensions 1TD
1 .969
2 .962
3 .951
4 .865
1 1.35
2 0.83
3 0.93
4 1.78
Configuration
2TD 3TD 4TD
. 733 . 555 . 545
.933 . 756 . 676
. 941 . 899 . 824
. 889 . 888 . 849
2. 61 2. 86 2.99
1.00 1. 52 1. 79
0. 80 0. 82 1. 05
1.32 1.02 1.14
1
2
3
4
.290
.161
.167
.273
671
189
136
200
.816
.325
.146
.163
867
390
195
174
O i
t- 1
TABLE 7
Correlation of Fit
Standard E rror
Stress
APPARENT FIT FOR S.C.M.
ERROR LEVEL = 1.00
Extracted
Dimensions 1TD
1 .972
2 .963
3 .957
4 .957
1 1.46
2 1.07
3 0.89
4 0.82
Configuration
2TD 3TD 4TD
. 782 . 722 . 534
.941 . 836 . 767
. 937 .906 . 860
.939 .910 .894
2.31 2.27 2. 63
1.06 1.44 1.66
0. 89 0.92 1.13
0.79 0. 82 0. 86
1
2
3
4
.426
.288
.228
.202
801
275
218
184
.867
.407
.229
.192
1.120
. 503
.299
.207
CO
D O
63 ;
Unsatisfactory results using the Stress statistic have been
noted in previous studies (e. g ., Young 1970, Sherman 1972). Young
found that Stress increased with the number of points, while Sherman
confirmed this relationship only at high erro r levels, showing some
decrease in Stress at lower erro r levels. Both studies found the
apparent contradiction that both Stress and Index of Metric Deter-
minacy decreased with increasing dimensionality.
These problems may also be due in part to the same problems
of comparability encountered in this research. It would seem that,
pending a comprehensive investigation of the behavior of Stress
and/or Standard E rror, correlative indices offer a better criterion
for determining fit and dimensionality.
Comparability of Configurations
One aspect of this research was the attempt to make compari
sons of configurations in one, two, three and four dimensions. A
basic configuration in four dimensions was chosen and configurations in
one, two and three dimensions were formed using the appropriate sub
sets of dimensions. The configurations were therefore dependent, as
opposed to the completely random configurations of, for example,
Sherman (1972).
The average magnitude of the true distances generated from
these four configurations differs according to the number of dimen
sions. This can lead to confounding of erro r level with dimensionality
(see Young, 1970, pages 462-463), unless compensations can be made
in the erro r component, the true distances or both. In this research,
the true distances were adjusted by ratios of the £ d^ as described in
Chapter II. These adjustments, however, were not satisfactory.
Table 8 shows, for each configuration and each erro r level, the pro
portion of true variance resulting in each set of true distances after
random distortion (but before the hyperbolic tangent transform
described in Chapter II). The larger standard deviations shown for
distances reflecting a low dimensionality, mean that the relative
distortion for these distances is much less than in the case of the
higher dimensionalities.
Also shown in Table 8 are correlations which are analogous to
reliability coefficients. These were obtained by correlating pairs of
sets of distorted distances, thus simulating repeated measurements of
a subjects dissim ilarity judgments. Ten replications for each coef
ficient, over 1225 judgments each, showed great stability. These
reliability figures correspond quite well to the proportions of true
variance shown in Table 8, and this is as expected. The differences
that do occur would be presumed to be due to the hyperbolic tangent
transfo rmation.
Additional data of interest is shown in Table 9. These data
65
TABLE 8
COMPARISON OF ADJUSTED CONFIGURATIONS
Configuration
1TD 2TD 3TD 4TD
True Distances
Mean 2. 56 3.27 3.63 3.73
Standard Deviation 3.34 2.65 2.13 1.96
Variance 11.16 7.02 4. 54 3.84
Proportion of True
Variance after
Distortion3 ,
E rror Level 0. 20 .996 .994 .991 .990
E rror Level 0.40 .986 .978 .966 .960
E rro r Level 0. 60 .969 .951 .926 .914
E rror Level 0. 80 .945 .916 . 876 . 857
E rro r Level 1. 00 .918 . 875 .819 .793
Reliability*3
E rror Level 0. 20 .998 .996 .993 .991
E rror Level 0. 40 .993 .984 .971 .964
E rro r Level 0. 60 .985 .965 .936 .922
E rror Level 0. 80 .973 .938 . 891 . 868
E rror Level 1. 00 .956 .905 . 837 . 807
a Distortion refers to addition of normally distributed erro r
component prior to the hyperbolic tangent transformation.
b Reliability defined as a correlation between two sets of Distorted
Distances, with distortion now including the hyperbolic tangent
transformation.
66
TABLE 9
MEAN SQUARE DEVIATION
OF DISTORTED DISTANCES FROM TRUE DISTANCES
Configuration
1TD 2TD 3TD 4TD
RMS Deviation of
Time Distances from
Distorted Distances
E rror Level 0. 00 1.19 0.75 0. 51 0.44
E rror Level 0. 20 1.20 0. 76 0. 54 0.47
E rror Level 0. 40 1.22 0. 80 0. 60 0. 55
E rror Level 0. 60 1.25 0. 86 0. 70 0. 66
E rror Level 0. 80 1.29 0.95 0. 81 0. 77
E rror Level 1. 00 1.35 1.05 0.93 0.92
represent mean square deviations of the distorted distances from the
true distances. One might expect a rough correspondence between
these deviations and the standard deviations of the normally distrib
uted erro r component (error levels), with allowances for the com
ponent introduced by the hyperbolic tangent transformation. This is
most nearly true for configuration 4TD, and becomes less true with
decreasing dimensionality. The differences are probably attributable
again to the higher variance of the true distances at low dimension
alities and to the fact that extreme true distances are forced, by the
transformation, into the same scale. This results in large differ
ences between true and distorted distances, but these differences are
systematic and should not contribute materially to unreliability or
poor fit. It should be noted that the transformation itself results in a
certain amount of distortion, even with almost errorless data, but
that, again, this systematic component has little effect on reliability
or on true and apparent fit.
In spite of the lack of comparability of the data from the four
configurations, data generated within the context of each separate
configuration is perfectly valid. The data obtained with configuration
1TD, however, is not of much practical use, since there is very little
variability in the real level of error, with uniformly high reliabilities
at each level. On the other hand, data obtained from configuration
4TD, which was the basis for all the comparisons of scaling methods,
68 :
and the studies dealing with augmentation of the basis, were quite
satisfactory with respect to these problems.
It is clear that the "error level" reported in this research,
and in previous research as well, is an arbitrary indicator of the
amount of introduced distortion. A better index would be some
analogue of the reliability correlation of Table 8. Such a statistic
could not, however, be obtained from situations where each replica
tion is begun with the generation of a new configuration (e. g ., Sherman
and Spence). A satisfactory method also needs to be found to compen
sate adequately for the differences in the distributions of true dis
tances, when different numbers of dimensions underlie these dis
tances. Adjustments in the variance can be made, but problems
remain or are accentuated with respect to the means and to extreme
values. All Monte Carlo research seems lacking with respect to one
or another of these problems.
Number of Judgments
As previously pointed out, the number of judgments required
for I. A. M. varies from replication to replication. The minimum
theoretical number of judgments is (R+l) X N - (R+l)(R+2)/2, and
is equal to 235 when R=4 and N=50. In theory, it should be possible
to drastically reduce the number of judgments if the space is of a
69 :
sm aller dimensionality (190, 144 and 97 judgments respectively for
3, 2 and 1 dimension). Several factors make it unlikely that these
low numbers can be achieved. F irst, when there is e rro r in the data,
more stimuli are often retained in the basis than are necessary to
define the true dimensions. In addition, one current modification of
the ISIS procedure is that the number of stimuli in the basis can never
decrease on successive cycles. Very few of the replications in the
entire study, therefore, had a final basis of less than five stimuli.
As previously discussed, the interactive procedure differs
from the theoretical algorithm in that it presents sets of new stimuli
(the focus) rather than a single stimulus at a time. This procedure
results in a predictable degree of redundancy. In the procedure out
lined in Chapter II, the first 8 stimuli require 28 judgments on the
first cycle, and if 5 basis stimuli are retained, and 4 new stimuli
added on each subsequent cycle, there would be 10 further cycles of
26 judgments each. A final short cycle of 10 judgments would result
in 298 judgments, and this can be considered about the minimum
number for this research.
A minimum number of judgments, however, would only be
attained if the basis obtained on the first cycle were carried through
and became the final basis. Only in this case would it not be neces
sary to obtain extra judgments to ensure that each basis stimuli had
been paired with each of the other forty-five stimuli. At the other
70 !
extreme, if the final basis is composed entirely of stimuli from the
last cycle, over 200 extra judgments would have to be obtained by
pairing these stimuli with stimuli from all preceding cycles. We
could probably expect, on the average replication, that there would
be about 100 extra judgments required.
As discussed in Chapter n, however, the current method
includes features to favor retention of stimuli from early cycles,
making it more difficult to replace basis stimuli as the procedure
progresses. We would thus have reason to expect somewhat less than
100 extra judgments and therefore less than 400 judgments overall.
The mean values shown in Table 5 appear to bear out this prediction,
even though individual replications varied considerably (ranging from
296 to 454). There do not appear to be noteworthy trends in the data,
except the higher number of judgments required for configuration
1TD. This may be attributable to the fact that bases of five stimuli,
when there was only one true dimension, were probably fairly
unstable, with a greater likelihood of dropping early basis stimuli in
favor of later ones.
Conclusions
The major conclusions from this research may be summarized
as follows:
71 '
(1) Using a procedure based on ISIS (Young & Cliff, 1972), it
is possible to recover configurations whose indices of fit
are an improvement over those obtained from a scaling
method based on an unsystematically selected subset of
dissim ilarities. The Interactive Method, as described
in Chapter II, appears capable of achieving such an
improvement when erro r levels are comparable to a
subject reliability of about . 90 or more. For reliabil
ities of less than . 90, equivalent performance can be
achieved by augmenting the basis with from two to five
additional stimuli, as described in Chapter III.
(2) I. A. M ., as expected, is not as good as the Standard
(Torgerson) MDS procedure applied to the full m atrix of
dissim ilarities. However, the efficacy of I. A. M. is
based on the prem ise that S. C. M. is im practical for a
large number of stimuli. Statistics approaching those of
S. C. M. can be attained, if desired, by adding stimuli to
the basis.
(3) S. I. M. yields indices of apparent fit that do not ade
quately reflect true fit, and are misleading with respect
to the degree of improvement possible with iterating
procedures. Neither apparent fit nor true fit adequately
reflects true dimensionality.
A correlative index more accurately reflects true fit
than an index based on deviations of recovered distances
from dissim ilarities (i. e ., Stress or Standard Error).
It is not a straightforward task to achieve comparability
of erro r levels when working with configurations with
different true dimensionalities. Further study is
necessary in this area.
A review should be made of several param eters that
were not investigated in this study.
(a) The number of stimuli was held constant at fifty
throughout the study. Data from Stenson & Knoll
(1969) indicate that fit changed very little from
forty to sixty stimuli, and even though the context
of their study was quite different (pure e rro r and a
non-metric program), there is little reason to
predict different results here.
(b) Differences among the characteristics of config
urations could be an important variable in Monte
Carlo studies, and in MDS in general. A relevant
study may be found in Tschudi (1972).
(c) Metric versus non-metric considerations have not
been an issue in this research. It has been
assumed that m etric constraints applied during the
interactive procedure and the off-line recovery of
a configuration. However, one might well use the
m etric configuration resulting from the inter
active procedure as a starting configuration for
further non-metric analysis, if he is so inclined.
Finally, the emphasis in this research has been on
how well the Interactive Method perform s and can
potentially perform relative to Standard Methods.
The issue of how poorly the Interactive Method
might perform under extreme circumstances has
not been pursued.
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APPENDIX
78
79
ACRONYMS
MDS Multidimensional Scaling
ISIS Interactive Scaling with Individual Subjects
I. A. M. Interactive Method
S. C. M. Standard-Complete Method
S. I.M. Standard-Incomplete Method
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B A S IC D IM E N SIO N A L CONFIGUBATION
TABLE R1
CORRELATION OF FIT. STANDARD METHOD (COMPLETE) CONFIGURATION=lTD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S T A N C E S )
1 1 . 0 0 0 0 . 9 9 0 0 . 9 8 7
2 1 . 0 0 0 0 . 9 8 8 0 . 9 7 8
3 1 . 0 0 0 0 . 9 8 2 0 . 9 6 8
4 1 . 0 0 0 0 . 9 8 1 0 . 9 6 7
APPARENT F I T (BETWEEN RECOVEREC AND D IST O R T ED D I S T A N C E S )
1 1 . 0 0 0 0 . 9 8 7 0 . 9 7 2
2 1 . 0 0 0 0 . 9 8 2 0 . 9 6 3
3 1 . 0 0 0 0 . 9 7 9 0 . 9 5 7
4 1 . 0 0 0 0 . 9 7 8 0 . 9 5 7
0 0
TABLE R2
CORRELATION OF FIT. STANDARD METHOD (COMPLETE) CONFIGURATI0N=2TD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S T A N C E S )
1 0 . 8 1 8 0 . 8 0 3 0 . 8 0 2
2 1 . 0 0 0 0 . 9 8 8 0 . 9 7 9
3 1 . 0 0 0 0 . 9 8 9 0 . 9 7 0
4 1 . 0 0 0 0 . 9 8 5 0 . 9 6 4
APPARENT F I T (BETWEEN RECOVERED AND D I S T O RTED D I S T A N C E S )
1 0 . 8 1 8 0 . 8 0 4 0 . 7 8 2
2 1 . 0 0 0 0 . 9 7 3 0 . 9 4 1
3 1 . 0 0 0 0 . 9 7 5 0 . 9 3 7
4 1 . 0 0 0 0 . 9 7 4 0 . 9 3 9
0 0
CORRELATION OF FIT.
TABLE R3
STANDARD METHOD (COMPLETE) CONFIGURATION=3TD
D IM E N S IO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D IS T A N C E S )
1 0 . 7 9 7 0 . 7 9 9 0 . 7 6 8
2 0 . 9 3 2 0 . 9 0 9 0 . 8 9 3
3 1 . 0 0 0 0 . 9 8 5 0 . 9 6 4
4 1 . 0 0 0 0 . 9 8 2 0 . 9 5 8
APPARENT F I T (BETWEEN RECOVERED AND D I S T ORTED D I S T A N C E S )
1 0 . 7 9 7 0 . 7 8 3 0 . 7 2 2
2 0 . 9 3 2 0 . 8 6 7 0 . 8 3 6
3 1 . 0 0 0 0 . 9 5 9 0 . 9 0 6
4 1 . 0 0 0 0 . 9 6 1 0 . 9 1 0
TABLE R4
CORRELATION OF FIT. STANDARD METHOD (COMPLETE) CGNFIGURATI0N=4TD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D ISTAN CES!
1 0 . 6 9 2 0 . 5 8 2 0 . 5 7 3
2 0 . 8 7 0 0 . 8 5 6 0 . 8 2 9
3 0 . 9 6 3 0 . 9 4 2 0 . 9 1 6
4 1 . 0 0 0 0 . 9 7 7 0 . 9 4 3
APPARENT F I T (BETWEEN RECOVEREC AND D IST OR T ED D I S T A N C E S ?
0 . 5 3 4
0 . 7 6 7
0 . 8 6 0
0 . 8 9 4
1 0 . 6 9 2 0 . 5 6 2
2 0 . 8 7 0 0 . 8 2 8
3 0 . 9 6 3 0 . 9 1 8
4 1 . 0 0 0 0 . 9 5 3
CORRELATION OF FIT.
TABLE R5
STANDARD METHOD (INCOMPLETE) CONFIGURATION=lT D
D IM E N SIO N S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S TANCES)
P A S S I V E C E L L S REPLACED BY MEAN
F I R S T IT E R A T IO N
SECOND I T E R A T IO N
1 0 . 8 6 9 0 . 8 9 9 0 . 8 9 7
2 0 . 8 0 0 0 . 8 4 2 0 . 8 4 0
3 0 . 8 1 3 0 . 8 3 8 0 . 8 3 8
4 0 . 8 0 9 0 . 8 3 0 0 . 8 3 2
1 0 . 9 2 0 0 . 9 5 7 0 . 9 5 5
2 0 . 9 3 2 0 . 9 3 3 0 . 9 3 0
3 0 . 9 2 5 0 . 9 1 5 0 . 9 1 3
4 0 . 9 0 1 0 . 8 9 1 0 . 9 0 0
1 0 . 8 7 9 0 . 9 5 8 0 . 9 5 5
2 0 . 9 2 9 0 . 9 3 9 0 . 9 3 4
3 0 . 9 4 1 0 . 9 3 8 0 . 9 3 2
4 0 . 9 1 4 0 . 9 1 6 0 . 9 2 0
APPARENT F I T (BETWEEN RECOVERED AND D IS T O R T E D D I S T A N C E S )
P A S S I V E C E L L S REPLACED BY MEAN
1
2
3
4
F I R S T IT E R A T IO N
1
2
3
4
SECOND IT E R A T I O N
1
2
3
4
0 . 5 1 8 0 . 5 4 0 0 . 5 3 2
0 . 4 8 1 0 . 5 1 4 0 . 5 1 0
0 . 4 9 8 0 . 5 1 7 0 . 5 1 3
0 . 5 1 6 0 . 5 3 8 0 . 5 3 5
0 . 8 6 1 0 . 8 9 8 0 . 8 9 4
C . 9 1 1 0 . 9 3 9 0 . 9 3 3
0 . 9 0 9 0 . 9 2 7 0 . 9 2 2
0 . 9 2 6 0 . 9 4 5 0 . 9 3 8
0 . 8 6 3 0 . 9 1 9 0 . 9 2 0
0 . 9 2 7 0 . 9 5 0 0 . 9 5 1
0 . 9 4 2 0 . 9 6 4 0 . 9 6 4
0 . 9 7 1 0 . 9 8 3 0 . 9 7 3
CORRELATION OF FIT.
TABLE R6
STANDARD METHOD (INCOMPLETE) CONF I GURATI0N=2TD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D I S TANCES)
P A S S I V E C EL L S REPLACED BY MEAN
F I R S T IT E R A T IO N
SECOND I T E R A T IO N
1 0 . 7 5 2 0 . 7 6 6 0 . 7 6 3
2 0 . 7 7 7 0 . 7 7 3 0 . 7 6 7
3 0 . 8 3 7 0 . 8 3 4 0 . 8 2 5
4 0 . 8 3 0 0 . 8 1 8 0 . 8 1 3
1 0 . 8 0 0 0 . 7 6 0 0 . 7 6 1
2 0 . 8 4 8 0 . 8 3 7 0 . 8 3 0
3 0 . 9 1 2 0 . 8 8 8 0 . 8 6 9
4 0 . 9 1 3 0 . 8 9 0 0 . 8 7 8
1 0 . 7 7 0 0 . 8 2 8 0 . 7 9 8
2 0 . 9 3 2 0 . 9 0 0 0 . 8 8 7
3 0 . 9 4 6 0 . 9 1 5 0 . 8 9 2
4 0 . 9 4 1 0 . 9 1 6 0 . 9 0 1
A P P ARENT F I T (BETWEEN RECOVERED AND D IST O R T ED D I S T AN C E S )
P A S S I V E CEL L S REPLACED BY MEAN ~ “
F I R S T IT E R A T IO N
SECOND I T E R A T I O N
1 0 . 4 6 2 0 . 4 7 4 0 . 4 6 2
2 0 . 4 9 9 0 . 5 0 6 0 . 4 9 4
3 0 . 5 4 9 0 . 5 5 4 0 . 5 4 0
4 0 . 5 7 2 0 . 5 7 5 0 . 5 6 2
1 0 . 8 0 5 0 . 7 7 7 0 . 7 6 6
2 0 . 8 6 3 0 . 8 6 0 0 . 8 4 8
3 0 . 9 1 6 0 . 9 1 7 0 . 9 0 2
4 0 . 9 4 3 0 . 9 4 8 0 . 9 3 7
1 0 . 7 6 6 0 . 8 2 5 0 . 7 8 9
2 0 . 9 2 1 0 . 9 0 3 0 . 8 9 1
3 0 . 9 6 6 0 . 9 5 4 0 . 9 4 1
4 0 . 9 8 1 0 . 9 7 6 0 . 9 6 7
00
C 51
CORRELATION OF FIT.
TABLE R7
STANDARD METHOD (INCOMPLETE) CONFIGURATI0N=3TD
D IM E NSION S
EXTRACTED
STANOARD D E V IA T IO N OF ERROR
0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWE EN R E COVERED AND TRUE D I S TANCES)
P A S S I V E CELLS REPLACED BY MEAN ~
F I R S T I T E R A T IO N
SECOND IT E R A T I O N
1 0 . 6 9 2 0 . 6 8 8 0 . 6 7 6
2 0 . 7 2 1 0 . 7 2 4 0 . 7 1 3
3 0 . 7 8 5 0 . 7 8 0 0 . 7 6 2
4 0 . 8 5 0 0 . 8 3 8 0 . 8 0 9
1 0 . 7 4 8 0 . 6 7 4 0 . 6 5 9
2 0 . 7 9 5 0 . 7 2 6 0 . 7 0 6
3 0 . 8 4 8 0 . 7 8 0 0 . 7 6 9
4 0 . 8 9 3 0 . 8 4 4 0 . 8 2 0
1 0 . e 2 7 0 . 7 5 2 0 . 6 9 6
2 0 . 8 7 2 0 . 7 8 4 0 . 7 6 2
3 0 . 9 0 9 0 . 8 2 9 0 . 8 0 1
4 0 . 9 2 7 0 . 8 5 8 0 . 8 3 0
APPARENT F I T ( BETWEEN RECOVERED AND D I S T ORTED D I S T A NCES )
P A S S I V E CELLS REPLACED BY MEAN
F I R S T IT E R A T IO N
SECOND I T E R A T IO N
1 0 . 4 2 3 0 . 4 2 9 0 . 4 0 5
2 0 . 4 7 4 0 . 4 8 8 0 . 4 7 0
3 0 . 5 3 8 0 . 5 4 1 0 . 5 1 9
4 0 . 5 7 3 0 . 5 7 2 0 . 5 4 6
1 0 . 7 2 3 0 . 6 6 5 0 . 6 3 5
2 0 . 8 4 8 0 . 7 8 7 0 . 7 5 5
3 0 . 9 0 9 0 . 8 8 2 0 . 8 5 7
4 0 . 9 5 5 0 . 9 3 8 0 . 9 1 2
1 0 . 8 2 1 0 . 7 3 1 0 . 6 8 2
2 0 . 8 8 9 0 . 8 3 6 0 . 8 1 8
3 0 . 9 4 0 0 . 9 1 4 0 . 8 9 9
4 0 . 9 7 4 0 . 9 6 0 0 . 9 4 1
0 0
-a
TABLE R8
CORRELATION OF FIT. STANDARD METHOD (INCOMPLETE) CONFIGURATI0N=4TD
D IM E N S IO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D IS T A N C E S )
P A S S I V E C EL L S REPLACED BY MEAN
1 0 . 6 1 4 0 . 6 2 9 0 . 6 1 7
2 0 . 7 3 7 0 . 7 5 2 0 . 7 4 0
3 0 . 7 7 4 0 . 7 8 3 0 . 7 7 1
4 0 . 8 2 5 0 . 8 2 3 0 . 7 9 7
F I R S T IT E R A T IO N
1 0 . 6 3 7 0 . 5 9 6 0 . 5 1 7
2 0 . 7 5 8 0 . 6 9 8 0 . 6 8 2
3 0 . 8 0 1 0 . 7 3 0 0 . 7 2 4
4 0 . 8 4 3 0 . 7 9 6 0 . 7 8 1
SECOND I T E R A T IO N
1 0 . 6 3 2 0 . 5 7 9 0 . 5 6 5
2 0 . 7 7 1 0 . 7 0 0 0 . 7 0 1
3 0 . 8 4 4 0 . 7 5 1 0 . 7 5 1
4 0 . 8 5 9 0 . 8 0 7 0 . 7 8 8
APPARENT F I T (BETWEEN RECOVERED AND D IST O R T ED D IS T A N C E S )
P A S S I V E CEL L S REPLACED BY MEAN
1 0 . 3 7 5 0 . 3 8 6 0 . 3 6 7
2 0 . 5 0 6 0 . 5 2 2 0 . 4 9 9
3 0 . 5 3 3 0 . 5 4 5 0 . 5 2 5
4 0 . 5 6 7 0 . 5 7 1 0 . 5 4 7
F I R S T IT E R A T IO N
1 0 . 6 9 2 0 . 6 0 6 0 . 5 6 3
2 0 . 8 5 2 0 . 7 9 6 0 . 7 5 6
3 0 . 8 8 6 0 . 8 6 1 0 . 8 3 7
4 0 . 9 3 1 0 . 9 1 3 0 . 8 9 7
SECOND I T E R A T IO N
1 0 . 6 9 1 0 . 6 3 2 0 . 6 3 6
2 0 . 8 5 2 0 . 8 0 4 0 . 7 9 3
3 0 . 9 3 1 0 . 8 8 6 0 . 8 7 0
4 0 . 9 5 3 0 . 9 4 6 0 . 9 3 1
TABLE R9
CORRELATION OF FIT. INTERACTIVE METHOD CONFIGURATION=l TD
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T (B E TW E E N RECOVERED AND TRUE P I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 1 . 0 0 0 0 . 9 9 3 0 . 9 8 8 0 . 9 8 6 0 . 9 8 3 0 . 9 7 8
2 1 . 0 0 0 0 . 9 9 1 0 . 9 8 5 0 . 9 7 9 0 . 9 7 2 0 . 9 6 0
3 1 . 0 0 0 0 . 9 9 2 0 . 9 8 2 0 . 9 6 9 0 . 9 6 6 0 . 9 4 1
4 1 . 0 0 0 0 . 9 9 4 0 . 9 7 4 0 . 9 3 8 0 . 9 0 9 0 . 8 5 1
1 1 . 0 0 0 0 . 9 8 8 0 . 9 8 6 0 . 9 8 4 0 . 9 8 4 0 . 9 7 8
2 1 . 0 0 0 0 . 9 8 7 0 . 9 8 1 0 . 9 7 5 0 . 9 6 7 0 . 9 5 3
3 1 . 0 0 0 0 . 9 8 7 0 . 9 7 5 0 . 9 6 2 0 . 9 5 4 0 . 9 2 2
4
1 . 0 0 0 0 . 9 8 6 0 . 9 5 3 0 . 9 3 1 0 . 8 6 9 0 . 8 2 2
1 1 . 0 0 0 0 . 9 8 8 0 . 9 8 7 0 . 9 8 4 0 . 9 8 2 0 . 9 7 7
2 1 . 0 0 0 0 . 9 8 6 0 . 9 7 6 0 . 9 6 7 0 . 9 5 3 0 . 9 3 8
3
1 . 0 0 0 0 . 9 8 5 0 . 9 6 7 0 . 9 4 2 0 . 9 2 7 0 . 8 7 7
4 1 . 0 0 0 0 . 9 8 3 0 . 9 4 6 0 . 9 0 6 0 . 8 1 2 0 . 7 7 3
A PPARENT F I T (BETWEEN RECOVERED AND D I S T O R T ED D I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 1 . 0 0 0 0 . 9 9 5 0 . 9 9 0 0 . 9 8 5 0 . 9 8 0 0 . 9 6 9
2 1 . 0 0 0 0 . 9 9 2 0 . 9 8 7 0 . 9 8 1 C . 9 7 4 0 . 9 6 2
3 1 . 0 0 0 0 . 9 9 4 0 . 9 8 6 0 . 9 7 5 0 . 9 7 2 0 . 9 5 1
4 1 . 0 0 0 0 . 9 9 6 0 . 9 8 0 0 . 9 4 6 0 . 9 1 8 0 . 8 6 5
1 1 . 0 0 0 0 . 9 9 4 0 . 9 9 0 0 . 9 8 5 0 . 9 8 0 0 . 9 6 9
2 1 . 0 0 0 0 . 9 8 5 0 . 9 8 1 0 . 9 7 5 0 . 9 6 6 0 . 9 5 0
3 1 . 0 0 0 0 . 9 8 7 0 . 9 7 7 0 . 9 6 3 0 . 9 5 5 0 . 9 2 3
4 1 . 0 0 0 0 . 9 8 6 0 . 9 6 5 0 . 9 3 4 0 . 8 7 2 0 . 8 2 4
1 1 . 0 0 0 0 . 9 9 4 0 . 9 9 0 0 . 9 8 3 0 . 9 7 6 0 . 9 6 3
2 1 . 0 0 0 0 . 9 8 3 0 . 9 7 3 0 . 9 6 3 0 . 9 4 6 0 . 9 2 5
3 1 . 0 0 0 0 . 9 8 2 0 . 9 6 4 0 . 9 3 8 0 . 9 1 9 0 . 8 6 5
4 1 . 0 0 0 0 . 9 8 0 0 . 9 4 3 0 . 9 0 2 0 . 8 0 4 0 . 7 6 3
03
CD
CORRELATION OF FIT.
TABLE RIO
INTERACTIVE METHOD
CONFIGURATION=2TD
D IM E NSION S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 • CO 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D IS T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 5 3 5 0 . 6 8 4 0 . 6 8 4 0 . 6 8 3 0 . 7 4 2 0 . 7 3 9
2 1 . 0 0 0 0 . 9 7 7 0 . 9 7 1 0 . 9 6 4 0 . 9 5 5 0 . 9 3 5
3 1 . 0 0 0 0 . 9 8 6 0 . 9 7 9 0 . 9 6 5 0 . 9 5 0 0 . 9 2 3
4 1 . 0 0 0 0 . 9 8 8 0 . 9 7 5 0 . 9 5 2 0 . 9 2 3 0 . 8 5 6
1 0 . 6 1 5 0 . 7 2 3 0 . 7 1 6 0 . 7 2 1 0 . 7 6 3 C . 7 5 0
2 1 . 0 0 0 0 . 9 7 7 0 . 9 7 2 0 . 9 6 4 0 . 9 5 4 0 . 9 3 7
3 1 . 0 0 0 0 . 9 8 8 0 . 9 7 9 0 . 9 6 1 0 . 9 4 3 0 . 9 1 6
4 1 . 0 0 0 0 . 9 8 8 0 . 9 7 1 0 . 9 4 1 0 . 9 0 4 0 . 8 2 5
1 0 . 6 4 7 0 . 8 0 0 0 . 7 7 3 0 . 7 5 9 0 . 7 8 1 0 . 7 5 2
2 1 . 0 0 0 0 . 9 8 3 0 . 9 7 6 0 . 9 6 2 0 . 9 4 5 0 . 9 2 4
3 1 . 0 0 0 0 . 9 8 9 0 . 9 7 7 0 . 9 4 5 0 . 9 1 7 0 . 8 8 4
4 1 . 0 0 0 0 . 9 8 6 0 . 9 5 9 0 . 9 0 2 0 . 8 4 1 0 . 7 5 1
AP P A RENT F I T (BETWEEN RECOVERED AND D I S T O R TED D I S T A N C E S )
S T R I P E JUDGMENTS *
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 5 3 5 0 . 6 9 2 0 . 6 8 9 0 . 6 8 6 0 . 7 4 0 0 . 7 3 3
2 1 . 0 0 0 0 . 9 8 2 0 . 9 7 5 0 . 9 6 8 0 . 9 5 7 0 . 9 3 3
3 1 . 0 0 0 0 . 9 9 1 0 . 9 8 5 0 . 9 7 6 0 . 9 6 5 0 . 9 4 1
4 1 . 0 0 0 0 . 9 9 4 0 . 9 6 7 0 . 9 6 9 0 . 9 4 7 0 . 8 8 9
1 0 . 6 1 5 0 . 7 2 8 0 . 7 1 7 0 . 7 1 8 0 . 7 5 7 0 . 7 3 6
2 1 . 0 0 0 0 . 9 8 1 0 . 9 7 3 0 . 9 6 2 0 . 9 4 7 0 . 9 2 3
3 1 . 0 0 0 0 . 9 8 8 0 . 9 7 9 0 . 9 6 3 0 . 9 4 3 0 . 9 1 4
4 1 . 0 0 0 0 . 9 8 9 0 . 9 7 4 0 . 9 4 7 0 . 9 0 9 0 . 8 3 2
1 0 . 6 4 7 0 . 8 0 3 0 . 7 7 1 0 . 7 5 2 0 . 7 6 4 0 . 7 2 7
2 1 . 0 0 0 0 . 9 7 7 0 . 9 6 8 0 . 9 5 0 0 . 9 2 4 0 . 8 9 1
3 1 . 0 0 0 0 . 9 8 2 0 . 9 6 9 0 . 9 3 5 0 . 8 9 9 0 . 8 5 6
4 1 . 0 0 0 0 . 9 8 0 0 . 9 5 2 0 . 8 9 3 0 . 8 2 5 0 . 7 3 0
CD
O
TABLE R 11
CORRELATION OF F I T . IN T E R A C T IV E METHOD C O N F IG U R A T I0 N = 3 T D
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
TRUE F I T (B ETW EEN RECOVERED AND _TRUE D I S T A N C E S )
S T R I P E JUDGMENTS ~
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 7 0 0 0 . 7 1 6 0 . 7 0 0 0 . 6 4 4 0 . 5 5 1 0 . 5 4 8
2 0 . 9 2 9 0 . 8 5 6 0 . 8 5 2 0 . 8 0 8 0 . 7 3 2 0 . 7 2 6
3 1 . 0 0 0 0 . 9 7 6 0 . 9 5 8 0 . 9 2 2 0 . 8 8 6 0 . 8 5 0
4 1 . 0 0 0 0 . 9 8 4 0 . 9 6 5 0 . 9 3 1 0 . 8 7 6 0 . 8 0 8
1 0 . 6 8 3 0 . 6 8 6 0 . 6 8 0 0 . 6 3 5 0 . 5 8 2 0 . 5 7 0
2 0 . 8 3 1 0 . 3 5 9 0 . 8 4 9 0 . 8 1 6 0 . 7 6 1 0 . 7 4 7
3 1 . 0 0 0 0 . 9 7 8 0 . 9 5 9 0 . 9 1 8 0 . 8 8 1 0 . 8 3 6
4 1 . 0 0 0 0 . 9 8 3 0 . 9 5 8 0 . 9 1 8 0 . 8 6 4 0 . 7 6 8
1 0 . 7 1 3 0 . 7 0 3 0 . 6 9 0 0 . 6 5 1 0 . 6 1 5 0 . 5 8 6
2 0 . 8 4 3 0 . 8 8 6 0 . 8 6 4 0 . 8 3 6 0 . 7 8 2 0 . 7 4 3
3 1 . 0 0 0 0 . 9 8 2 0 . 9 5 9 0 . 9 1 0 0 . 8 5 8 0 . 7 9 2
4 1 . 0 0 0 0 . 9 8 2 0 . 9 4 4 0 . 8 9 8 0 . 8 1 7 0 . 6 8 4
A P P A R E N T _ F I T . i B E T W E E N RECOVERED AND D IS T O R T ED D IS T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 7 0 0 0 . 7 1 3 0 . 6 9 4 0 . 6 4 0 0 . 5 5 7 0 . 5 5 5
2 0 . 9 2 9 0 . 8 5 5 0 . 8 5 3 0 . 8 0 9 0 . 7 5 2 0 . 7 5 6
3 1 . 0 0 0 0 . 9 8 1 0 . 9 6 7 0 . 9 3 9 0 . 9 1 7 0 . 8 9 9
4 1 . 0 0 0 0 . 9 9 1 0 . 9 8 1 0 . 9 6 5 0 . 9 2 9 0 . 8 8 8
1 0 . 6 8 3 0 . 6 8 3 0 . 6 7 3 0 . 6 2 4 0 . 5 7 6 0 . 5 5 6
2 0 . 8 3 1 0 . 8 5 6 0 . 8 4 4 0 . 8 0 5 0 . 7 5 2 0 . 7 3 6
3 1 . 0 0 0 0 . 9 7 9 0 . 9 5 7 0 . 9 1 5 0 . 3 8 2 0 . 8 4 0
4 1 . 0 0 0 0 . 9 8 4 0 . 9 6 0 0 . 9 2 4 0 . 8 7 6 0 . 7 9 2
1 0 . 7 1 3 0 . 7 0 2 0 . 6 8 6 0 . 6 3 7 0 . 5 9 4 0 . 5 5 4
2 0 . 8 4 3 0 . 8 8 1 0 . 8 5 7 0 . 8 1 7 0 . 7 5 1 0 . 7 0 1
3 1 . 0 0 0 0 . 9 7 5 0 . 9 4 7 0 . 8 9 1 0 . 8 2 9 0 . 7 5 4
4 1 . 0 0 0 0 . 9 7 6 0 . 9 3 6 0 . 8 8 2 0 . 7 9 5 0 . 6 5 9
C O
l — 1
CORRELATION OF FIT
TABLE R12
INTERACTIVE METHOD CONFIGURATION=4TD
D IM E NSION S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0
1 . 0 0
TRUE F I T ( BETWEEN RECOVERED AND TRUE D I S T A N C E S)
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 7 2 9 0 . 7 3 4 0 . 6 9 8 0 . 6 5 6 0 . 5 9 8 0 . 5 5 3
2 0 . 8 8 3 0 . 3 3 0 0 . 8 0 7 0 . 7 3 9 0 . 6 9 9 0 . 6 4 1
3 0 . 9 6 0 0 . 9 3 2 0 . 9 0 3 0 . 8 7 0 0 . 8 2 0 0 . 7 6 1
4 1 . 0 0 0 0 . 9 7 0 0 . 9 4 6 0 . 9 1 4 0 . 8 5 5 0 . 7 6 8
1 0 . 7 1 6 0 . 7 3 4 0 . 6 9 4 0 . 6 4 6 0 . 6 0 8 0 . 5 6 7
2 0 . 8 3 8 0 . 8 1 3 0 . 7 9 5 0 . 7 3 5 0 . 7 0 7 0 . 6 6 9
3 0 . 9 2 5 0 . 9 0 5 0 . 8 8 3 0 . 8 3 9 0 . 8 0 3 0 . 7 5 5
4 1 . 0 0 0 0 . 9 5 5 0 . 9 3 1 0 . 8 8 5 0 . 8 2 5 0 . 7 4 0
1 0 . 7 1 8 0 . 7 2 9 0 . 6 8 9 0 . 6 5 1 0 . 6 2 4 0 . 5 7 5
2 0 . 8 3 3 0 . 8 4 3 0 . 8 1 9 0 . 7 6 7 0 . 7 2 6 0 . 6 8 5
3 0 . 9 4 6 0 . 9 1 8 0 . 8 8 0 0 . 8 3 0 0 . 7 8 1 0 . 7 2 6
4 1 . 0 0 0 0 . 9 5 8 0 . 9 1 5 0 . 8 5 5 0 . 7 6 6 0 . 6 7 2
APPARENT F I T ( BETWEEN R ECOVERED AND D I S TORTED P I S TANC E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 7 2 9 0 . 7 2 3 0 . 6 8 4 0 . 6 4 8 0 . 5 8 6 0 . 5 4 5
2 0 . 8 8 3 0 . 8 2 2 0 . 8 0 0 0 . 7 4 2 0 . 7 1 3 0 . 6 7 6
3 0 . 9 6 0 0 . 9 3 4 0 . 9 1 3 0 . 8 9 2 0 . 8 5 6 0 . 8 2 4
4 1 . 0 0 0 0 . 9 7 8 0 . 9 6 5 0 . 9 5 2 0 . 9 2 2 0 . 8 4 9
1 0 . 7 1 6 0 . 7 2 6 0 . 6 8 2 0 . 6 3 3 0 . 5 8 6 0 . 5 4 4
2 0 . 8 3 8 0 . 8 04 0 . 7 8 4 0 . 7 2 5 0 . 6 9 6 0 . 6 6 4
3 0 . 9 2 5 0 . 9 0 3 0 . 8 8 1 0 . 8 3 9 0 . 8 0 4 0 . 7 6 5
4 1 . 0 0 0 0 . 9 5 6 0 . 9 3 4 0 . 8 9 2 0 . 8 4 4 0 . 7 5 - 9
1 0 . 7 1 8 0 . 7 3 6 0 . 6 8 5 0 . 6 3 7 0 . 5 9 7 0 . 5 3 4
2 0 . 8 3 3 0 . 8 3 9 0 . 8 0 8 0 . 7 4 7 0 . 6 9 7 0 . 6 4 2
3 0 . 9 4 6 0 . 9 1 4 0 . 8 7 1 0 . 8 1 2 0 . 7 5 2 0 . 6 8 6
4 1 . 0 0 0 0 . 9 5 5 0 . 9 0 8 0 . 8 3 9 0 . 7 4 4 0 . 6 4 0
CD
DO
CORRELATION OF FIT.
TABLE R13
PRE-SET BASIS C 5 STIMULI) CONFIGURATI0N=4TD
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S T ANCES)
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 7 2 9 0 . 7 0 0 0 . 7 0 2 0 . 6 9 8 0 . 6 6 4 0 . 6 3 8
2 0 . 8 8 3 0 . 8 6 8 0 . 8 5 7 0 . 8 3 8 0 . 8 04 0 . 7 4 3
3 0 . 9 6 0 0 . 9 3 4 0 . 9 1 3 0 . 8 8 4 0 . 8 4 6 0 . 8 0 1
4 1 . 0 0 0 0 . 9 7 9 0 . 9 6 3 0 . 9 3 3 0 . 8 8 3 0 . 7 9 9
1 0 . 7 2 9 0 . 7 0 0 0 . 7 0 2 0 . 6 9 8 0 . 6 6 4 0 . 6 3 8
2 0 . 8 8 3 0 . 8 6 8 0 . 8 5 7 0 . 8 3 9 0 . 8 0 4 0 . 7 4 3
3 0 . 9 6 0 0 . 9 3 4 0 . 9 1 3 0 . 8 8 4 0 . 8 4 6 0 . 8 0 1
4 1 . 0 0 0 0 . 9 7 9 0 . 9 6 3 0 . 9 3 3 0 . 8 8 3 0 . 7 9 9
1 0 . 7 1 8 0 . 6 8 9 0 . 6 7 7 0 . 6 5 7 0 . 6 1 7 0 . 5 8 3
2 0 . 8 3 3 0 . 8 2 6 0 . 8 0 7 0 . 7 7 8 0 . 7 4 2 0 . 6 9 9
3 0 . 9 4 6 0 . 9 4 1 0 . 9 0 7 0 . 8 6 5 0 . 8 1 2 0 . 7 5 1
4 1 . 0 0 0 0 . 9 7 4 0 . 9 4 2 0 . 8 8 9 0 . 8 1 1 0 . 7 0 9
APPARENT F I T (BETWEEN RECOVERED AND D I S T ORTED D I S T A N C E S )
S T R I P E JUDGMENTS ~
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 7 2 9 0 . 6 86 0 . 6 8 5 0 . 6 7 8 0 . 6 4 8 0 . 6 2 5
2 0 . 8 6 3 0 . 8 7 0 0 . 8 5 8 0 . 8 4 0 0 . 8 0 9 0 . 7 5 6
3 0 . 9 6 0 0 . 9 3 3 0 . 9 1 9 0 . 9 0 1 0 . 8 7 6 0 . 8 5 0
4 1 . 0 0 0 0 . 9 8 5 0 . 9 7 8 0 . 9 6 4 0 . 9 3 4 0 . 8 7 2
1 0 . 7 2 9 0 . 6 8 6 0 . 6 8 5 0 . 6 7 8 0 . 6 4 8 0 . 6 2 5
2 0 . 8 6 3 0 . 8 7 0 0 . 8 5 8 0 . 8 4 0 0 . 8 0 9 0 . 7 5 6
3 0 . 9 6 0 0 . 9 3 3 0 . 9 1 9 0 . 9 0 1 0 . 8 7 6 0 . 8 5 0
4 1 . 0 0 0 0 . 9 8 5 0 . 9 7 8 0 . 9 6 4 0 . 9 3 4 0 . 8 7 2
1 0 . 7 1 8 0 . 6 9 6 0 . 6 7 7 0 . 6 4 6 0 . 5 9 3 0 . 5 4 5
2 0 . 8 3 3 0 . 8 3 1 0 . 8 0 4 0 . 7 6 4 0 . 7 1 4 0 . 6 5 5
3 0 . 9 4 6 0 . 9 3 4 0 . 8 9 7 0 . 8 4 5 0 . 7 8 0 0 . 7 0 8
4 1 . 0 0 0 0 . 9 7 0 0 . 9 3 4 0 . 8 7 1 0 . 7 3 3 0 . 6 7 1
CD
00
TA8L E R 1 4
CORRELATION OF F I T . P R E - S E T B A S I S ( 7 S T I M U L I ) C O N F IG U R A T I0 N = 4 T D
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D I S TANCES)
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 6 8 7 0 . 6 4 5 0 . 6 4 8 0 . 6 4 4 0 . 6 3 2 0 . 6 0 9
2 0 . 7 8 6 0 . 7 9 4 0 . 7 7 2 0 . 7 5 7 0 . 7 3 2 0 . 7 0 1
3 0 . 9 2 5 0 . 9 2 4 0 . 8 9 9 0 . 8 6 8 0 . 8 2 6 0 . 7 8 4
4 1 . 0 0 0 0 . 9 8 1 0 . 9 6 4 0 . 9 3 5 0 . 8 9 4 0 . 8 3 9
1 0 . 6 8 7 0 . 6 4 5 0 . 6 4 8 0 . 6 4 4 0 . 6 3 2 0 . 6 0 9
2 0 . 7 8 6 0 . 7 9 4 0 . 7 7 2 0 . 7 5 7 0 . 7 3 2 0 . 7 0 1
3 0 . 9 2 5 0 . 9 2 4 0 . 8 9 9 0 . 8 6 8 0 . 8 2 6 0 . 7 8 4
4 1 . 0 0 0 0 . 9 8 1 0 . 9 6 4 0 . 9 3 5 0 . 8 9 4 0 . 8 3 9
1 0 . 7 1 7 0 . 7 0 7 0 . 6 9 8 0 . 6 8 1 0 . 6 5 5 0 . 6 2 1
2 0 . 8 4 5 0 . 8 4 5 0 . 8 3 0 0 . 8 0 2 0 . 7 6 8 0 . 7 3 0
3 0 . 9 4 7 0 . 9 4 5 0 . 9 1 6 0 . 8 8 0 0 . 8 3 4 0 . 7 8 5
4 1 . 0 0 0 0 . 9 8 4 0 . 9 6 1 0 . 9 2 3 0 . 8 7 0 0 . 8 0 0
AP P A R E NT F I T (BETWEEN RECOVERED AND D I S T O RTED P I ST A NC E S )
S T R I P E JUDGMENTS "
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 6 8 7 0 . 6 3 3 0 . 6 3 0 0 . 6 2 0 0 . 6 0 3 0 . 5 7 7
2 0 . 7 8 6 0 . 7 8 6 0 . 7 6 4 0 . 7 5 3 0 . 7 3 1 0 . 7 0 4
3 0 . 9 2 5 0 . 9 2 0 0 . 9 0 0 0 . 8 7 5 0 . 8 4 1 0 . 8 1 1
4 1 . 0 0 0 0 . 9 8 5 0 . 9 7 3 0 . 9 5 2 0 . 9 2 2 0 . 8 8 5
1 0 . 6 8 7 0 . 6 3 3 0 . 6 3 0 0 . 6 2 0 0 . 6 0 3 0 . 5 7 7
2 0 . 7 8 6 0 . 7 8 6 0 . 7 6 4 0 . 7 5 3 0 . 7 3 1 0 . 7 0 4
3 0 . 9 2 5 0 . 9 2 0 0 . 9 0 0 0 . 8 7 5 0 . 8 4 1 0 . 8 1 1
4 1 . 0 0 0 0 . 9 8 5 0 . 9 7 3 0 . 9 5 2 0 . 9 2 2 0 . 8 8 5
1 0 . 7 1 7 0 . 7 1 3 0 . 6 97 0 . 6 6 8 0 . 6 2 9 0 . 5 8 2
2 0 . 8 4 5 0 . 8 4 2 0 . 8 2 0 0 . 7 8 5 0 . 7 3 9 0 . 6 8 7
3 0 . 9 4 7 0 . 9 3 7 0 . 9 0 4 0 . 8 5 8 0 . 8 0 0 0 . 7 3 9
4 1 . 0 0 0 0 . 9 7 9 0 . 9 5 0 0 . 9 0 2 0 . 8 3 8 0 . 7 5 8
C O
TABLE R 1 5
CORRELATION OF F I T . P R E - S E T B A S I S ( 9 S T I M U L I ) C O N F IG U R A T 10N =4T D
D IM E NSION S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
TRUE F I T ( BETWEEN RECOVERED AND TRUE P I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 7 1 5 0 . 6 6 5 0 . 6 3 8 0 . 6 0 3 0 . 5 6 6 0 . 5 2 8
2 0 . 7 8 4 0 . 7 8 0 0 . 7 6 2 0 . 7 4 0 0 . 7 1 4 0 . 6 7 7
3 0 . 9 24 0 . 9 1 9 0 . 9 0 3 0 . 8 7 8 0 . 8 4 1 0 . 8 0 5
4 1 . 0 0 0 0 . 9 83 0 . 9 6 7 0 . 9 4 2 0 . 9 0 7 0 . 8 6 3
1 0 . 7 1 5 0 . 6 6 5 0 . 6 3 8 0 . 6 0 3 0 . 5 6 6 0 . 5 2 8
2 0 . 7 8 4 0 . 7 8 0 0 . 7 6 2 0 . 7 4 0 0 . 7 1 4 0 . 6 7 6
3 0 . 9 2 4 0 . 9 1 9 0 . 9 0 3 0 . 8 7 8 0 . 8 4 1 0 . 8 0 5
4 1 . 0 0 0 0 . 9 8 3 0 . 9 6 7 0 . 9 4 2 0 . 9 0 7 0 . 8 6 3
1 0 . 7 4 9 0 . 7 5 2 0 . 7 2 6 0 . 6 9 2 0 . 6 5 6 0 . 6 1 8
2 0 . 8 7 5 0 . 8 7 6 0 . 8 5 7 0 . 8 3 1 0 . 7 9 8 0 . 7 5 9
3 0 . 9 4 0 0 . 9 4 1 0 . 9 2 3 0 . 8 9 6 0 . 8 6 1 0 . 8 2 2
4 1 . 0 0 0 0 . 9 8 7 0 . 9 6 7 0 . 9 3 6 0 . 8 9 5 0 . 8 4 6
APPARENT F I T ( BETWEEN RECOVERED AND P I S T ORTED D I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 7 1 5 0 . 6 5 3 0 . 6 2 3 0 . 5 8 4 0 . 5 4 5 0 . 5 0 7
2 0 . 7 8 4 0 . 7 6 6 0 . 7 4 5 0 . 7 2 1 0 . 6 9 4 0 . 6 5 8
3 0 . 9 2 4 0 . 9 1 2 0 . 8 9 5 0 . 8 7 1 0 . 8 3 5 0 . 8 0 4
4 1 . 0 0 0 0 . 9 8 5 0 . 9 6 9 0 . 9 4 6 0 . 9 1 5 0 . 8 8 0
1 0 . 7 1 5 0 . 6 5 3 0 . 6 2 3 0 . 5 8 4 0 . 5 4 4 0 . 5 0 7
2 0 . 7 8 4 0 . 7 6 5 0 . 7 4 5 0 . 7 2 0 0 . 6 9 4 0 . 6 5 8
3 0 . 9 2 4 0 . 9 1 2 0 . 8 9 5 0 . 8 7 1 0 . 8 3 5 0 . 8 0 4
4 1 . 0 0 0 0 . 9 8 5 0 . 9 6 9 0 . 9 4 6 0 . 9 1 5 0 . 8 8 0
1 0 . 7 4 9 0 . 7 5 1 0 . 7 1 8 0 . 6 7 4 0 . 6 2 7 0 . 5 7 8
2 0 . 8 7 5 0 . 8 6 7 0 . 8 4 1 0 . 8 0 4 0 . 7 5 8 0 . 7 0 6
3 0 . 9 4 0 0 . 9 3 1 0 . 9 0 9 0 . 8 7 3 0 . 8 2 6 0 . 7 7 3
4 1 . 0 0 0 0 . 9 8 1 0 . 9 5 6 0 . 9 1 6 0 . 8 6 3 0 . 8 0 1
CD
C J ]
CORRELATION OF FIT
TABLE R16
PRE-SET BASIS (10 STIMULI) CONFIGURATI0N=4TD
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D I S T ANCES)
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 7 2 4 0 . 6 8 2 0 . 6 6 4 0 . 6 3 4 0 . 6 0 0 0 . 5 6 5
2 0 . 8 2 0 0 . 8 4 6 0 . 8 2 1 0 . 7 8 6 0 . 7 5 3 0 . 7 1 3
3 0 . 9 3 6 0 . 9 2 8 0 . 9 1 5 0 . 8 9 6 0 . 8 6 9 0 . 8 3 7
4 1 . 0 0 0 0 . 9 8 6 0 . 9 7 3 0 . 9 5 1 0 . 9 2 2 0 . 8 8 5
1 0 . 7 2 4 0 . 6 8 2 0 . 6 6 4 0 . 6 3 4 0 . 6 0 0 0 . 5 6 5
2 0 . 8 2 0 0 . 8 4 6 C . 8 2 1 0 . 7 8 6 0 . 7 5 3 0 . 7 1 3
3 0 . 9 3 6 0 . 9 2 8 0 . 9 1 5 0 . 8 9 6 0 . 8 6 9 0 . 8 3 7
4 1 . 0 0 0 0 . 9 8 6 0 . 9 7 3 0 . 9 5 1 0 . 9 2 2 0 . 8 8 5
1 0 . 7 4 4 0 . 7 4 4 0 . 7 2 2 0 . 6 9 0 0 . 6 5 5 0 . 6 2 1
2 0 . 8 8 1 0 . 8 9 6 0 . 8 7 3 0 . 8 4 0 0 . 8 0 3 0 . 7 6 4
3 0 . 9 4 5 0 . 9 4 4 0 . 9 2 7 0 . 9 0 2 0 . 8 7 0 0 . 8 3 3
4 1 . 0 0 0 0 . 9 8 8 0 . 9 7 0 0 . 9 4 1 0 . 9 0 4 0 . 8 6 0
APPARENT F I T (BETWEEN RECOVERED AND P I S T ORTED D IS T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 7 2 4 0 . 6 7 2 0 . 6 5 0 0 . 6 1 6 0 . 5 7 8 0 . 5 4 1
2 0 . 8 2 0 0 . 8 3 7 0 . 8 0 7 0 . 7 6 8 0 . 7 3 2 0 . 6 9 5
3 0 . 9 3 6 0 . 9 2 1 0 . 9 0 7 0 . 8 8 6 0 . 8 5 9 0 . 8 2 8
4 1 . 0 0 0 0 . 9 8 7 0 . 9 7 3 0 . 9 5 1 0 . 9 2 3 0 . 8 9 1
1 0 . 7 2 4 0 • 6 7 2 0 . 6 5 0 0 . 6 1 6 0 . 5 7 8 0 . 5 4 1
2 0 . 8 2 0 0 . 8 3 7 0 . 8 0 7 0 . 7 6 8 0 . 7 3 2 0 . 6 9 5
3 0 . 9 3 6 0 . 9 2 1 0 . 9 0 7 0 . 8 8 6 0 . 8 5 9 0 . 8 2 8
4 1 . 0 0 0 0 . 9 8 7 0 . 9 7 3 0 . 9 5 1 0 . 9 2 3 0 . 8 9 1
1 0 . 7 4 4 0 . 7 4 3 0 . 7 1 4 0 . 6 7 3 0 . 6 2 6 0 . 5 7 9
2 0 . 8 8 1 0 . 8 8 9 0 . 8 5 9 0 . 8 1 4 0 . 7 6 5 0 . 7 1 1
3 0 . 9 4 5 0 . 9 3 4 0 . 9 1 2 0 . 8 7 8 0 . 8 3 3 0 . 7 8 3
4 1 . 0 0 0 0 . 9 8 2 0 . 9 5 9 0 . 9 2 0 0 . 8 7 0 0 . 8 1 3
CD
G i
TABLE El
STANDARD E R R O R . STANDARD METHOD (C O M PL ET E ) C O N F IG U R A T I O N = l T D
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S TANCES)
1
o
•
o
0 . 9 1 3 0 . 9 4 0
2 o
•
o
0 . 7 4 3 0 . 8 7 4
3
o
•
o
0 . 8 4 5 1 . 0 3 7
4
o
•
o
0 . 9 0 4 1 . 1 3 8
APPARENT F I T (BETWEEN RECOVERED AND D I S T O R TED D I S T A N C E S )
1
o
•
o
1 . 3 9 0 1 . 4 5 5
2
o
•
o
1 . 0 0 5 1 . 0 7 0
3
o
•
o
0 . 8 2 7 0 . 8 9 3
4 o
•
o
0 . 7 3 0 0 . 8 1 6
CD
-O
STANDARD ERROR
TABLE E2
STANDARD METHOD (COMPLETE) CONFIGURATION=2TD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE P I S T A N C E S )
1 1 . 9 6 1 2 . 1 2 6 2 . 0 9 4
2 0 . 0 0 . 5 4 0 0 . 6 3 7
3 0 . 0 0 . 4 6 4 0 . 7 0 5
4 0 . 0 0 . 5 6 9 0 . 8 4 5
APPARENT F I T (BETWEEN RECOVERED AND D IS T O R T ED D I S T A N C E S )
1 1 . 9 6 1 2 . 2 9 0 2 . 3 0 8
2
o
•
o
0 . 9 2 4 1 . 0 5 6
3
o
•
o
0 . 7 4 1 0 . 8 8 7
4
o
•
o
0 . 6 0 5 0 . 7 8 7
CD
CO
TABLE E3
STANDARD ERROR. STANDARD METHOD (COMPLETE) CONE IGURATI0N=3TD
DIMENSIONS STANDARD DEVIATION OF ERROR
EXTRACTED 0.00 0.60 1.00
TRUE FIT (BETWEEN RECOVERED AND TRUE DISTANCES)
1 1 .977 2.027 2.063
p
0.974 1.131 1.160
3
o
•
o
0.432 0.585
4
o
•
o
0.427 0.643
APPARENT FIT (BETWEEN RECOVERF0 AND DISTORTED DISTANCES)
1 1.977 2.165 2.266
2 0.974 1 .332 1.441
3
o
•
o
0.718 0.919
4
o
•
o
0.606 0.820
CD
CD
TABLE E4
STANDARD ERROR. STANDARD METHOD (COMPLETE) CONFIGURATI0N=4TD
D IM E N S IO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
T RUE F I T (BETWEEN RECOVEREC AND TRUE D I S T ANCES)
1 2 . 6 6 3 2 . 4 8 6 2 . 4 7 6
2 1 . 3 1 9 1 . 3 9 8 1 . 4 2 0
3 0 . 6 4 7 0 . 8 1 9 0 . 8 7 7
4
o
•
o
0 . 4 4 8 0 . 6 3 0
AP P A RENT F I T (BETWEEN RECOVEREC AND D I S T O R T E D D I S T A N C E S )
t
1 2 . 6 6 3 2 . 5 8 2 2 . 6 3 7
2 1 . 3 1 9 1 . 5 5 5 1 . 6 5 6
3 0 . 6 4 7 1 . 0 0 1 1 . 1 2 7
4
o
•
o
0 . 6 5 2 0 . 8 5 7
100
TABLE E5
STANDARD ERROR. STANDARD METHOD (INCOMPLETE) CONFIGURATION=lTD
D IM E N S IO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
T R UE F IT (BETWEEN RECOVER E D AND TRUE P I S T A NCES)
P A S S I V E CEL L S REPLACED BY~MEAN_
F I R S T IT E R A T I O N
SECOND I T E R A T IO N
1 1 . 9 2 6 2 . 1 1 1 2 . 1 2 2
2 2 . 0 3 9 1 . 9 0 5 1 . 9 1 6
3 2 . 2 7 7 1 . 9 3 6 1 . 9 4 0
4 2 . 6 2 4 2 . 1 13 2 . 1 1 4
1 1 . 6 2 4 1 . 8 1 2 1 . 8 1 5
2 1 . 2 3 2 1 . 3 9 5 1 . 4 1 3
3 1 . 3 7 0 1 . 4 1 1 1 . 4 3 9
4 1 . 8 4 7 1 . 6 7 5 1 . 6 3 0
1 1 . 7 5 7 1 . 7 3 8 1 . 7 5 0
2 1 . 2 5 7 1 . 3 1 7 1 . 3 4 8
3 1 . 3 0 8 1 . 2 3 0 1 . 2 8 1
4 1 . 8 6 2 1 . 4 9 2 1 . 4 7 0
A PPA RENT F I T ( BETWEEN RECOVERED AND D IST OR T ED D I S T A N C E S )
P A S S I V E C E L L S REPLACED BY MEAN
F I R S T IT E R A T IO N
SECOND ITERATION
1 2 . 3 3 9 2 . 4 6 3 2 . 4 7 9
2 2 . 5 1 8 2 . 0 9 8 2 . 1 0 2
3 2 . 8 1 4 2 . 0 2 3 2 . 0 1 7
4 3 . 1 1 5 2 . 0 6 2 2 . 0 4 7
1 2 . 3 3 2 2 . 0 7 3 2 . 0 6 0
2 1 . 6 9 0 1 . 4 4 1 1 . 4 4 7
3 1 . 4 1 3 1 . 1 5 8 1 . 1 6 3
4 1 . 2 2 9 0 . 8 2 5 0 . 8 7 3
1 2 . 2 0 1 1 . 8 5 9 1 . 8 2 6
2 1 . 4 3 8 1 . 2 3 2 1 . 1 8 8
3 1 . 1 8 5 0 . 9 1 9 0 • 3 7 3
4 0 . 9 3 5 0 . 4 8 7 0 . 5 5 1
STANDARD E R R O R .
TABLE E6
STANDARD METHOD ( I N C O M P L E T E ) C 0 N F I G U R A T I 0 N = 2 T D
D IM E N SIO N S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 6 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D I S TA NCES)
P A S S I V E C EL L S REPLACED BY MEAN
F I R S T IT E R A T IO N
SECOND IT E R A T IO N
1 2 . 4 8 4 2 . 5 6 8 2 . 5 6 8
2 1 . 8 5 6 1 . 9 7 6 1 . 9 7 8
3 1 . 4 5 6 1 . 5 3 1 1 . 5 6 0
4 1 . 5 2 0 1 . 5 3 8 1 . 5 5 5
1 2 . 2 9 9 2 . 4 1 6 2 . 4 1 0
2 1 . 6 2 3 1 . 7 3 1 1 . 7 3 5
3 1 . 1 2 1 1 . 3 1 0 1 . 3 8 2
4 1 . 1 0 5 1 . 2 5 8 1 . 3 1 7
1 2 . 4 2 3 2 . 2 6 3 2 . 3 0 0
2 1 . 1 9 1 1 . 4 4 3 1 . 4 7 4
3 0 . 8 9 9 1 . 1 7 3 1 . 2 6 9
4 0 . 9 2 0 1 . 1 2 0 1 . 2 1 8
A P PARENT F I T (BETWEEN RECOVERED AND D I S T O RTED D IS T A N C E S )
P A S S I V E C E L L S REPLACED BY MEAN
F I R S T IT E R A T IO N
SECOND ITERATION
1 2 . 4 3 4 2 . 7 2 2 2 . 7 3 5
2 2 . 0 4 0 2 . 1 9 6 2 . 1 9 1
3 1 . 8 8 5 1 . 8 5 5 1 . 8 6 5
4 1 . 3 6 4 1 . 6 7 2 1 . 6 8 6
1 2 . 1 3 7 2 . 0 7 7 2 . 1 0 2
2 1 . 4 3 5 1 . 3 7 5 1 . 3 8 2
3 0 . 9 5 2 0 . 8 5 2 0 . 9 0 1
4 0 . 7 8 6 0 . 6 5 1 0 . 7 1 3
1 2 . 3 4 2 2 . 0 4 4 2 . 1 1 4
2 1 . 1 3 7 1 . 1 9 6 1 . 2 2 1
3 0 . 6 5 8 0 . 7 0 0 0 . 7 4 9
4 0 . 4 8 7 0 . 4 5 9 0 . 5 3 1
102
STANDARD ERROR.
TABLE E7
STANDARD METHOD (INCOMPLETE) CONFIGURATI0N=3TD
D IM E N SIO N S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 6 0 1 00
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S T A N C E S )
P A S S I V E CEL L S REPLACED BY MEAN- ~
F I R S T IT E R A T I O N
SECOND IT E R A T IO N
1 2 . 8 3 3 2 . 8 5 6 2 . 8 4 9
2 2 . 0 8 7 2 . 1 3 8 2 . 1 2 6
3 1 . 5 8 9 1 . 6 7 0 1 . 6 7 7
4 1 . 2 1 2 1 . 3 0 8 1 . 3 6 1
1 2 . 5 0 4 2 . 6 7 6 2 . 6 8 2
2 1 . 7 8 7 1 . 9 4 7 1 . 9 4 8
3 1 . 2 8 1 1 . 4 9 0 1 . 4 9 9
4 0 . 9 7 9 1 . 1 8 5 1 . 2 4 6
1 2 . 2 3 9 2 . 4 3 4 2 . 4 7 7
2 1 . 4 7 1 1 . 7 0 8 1 . 7 1 0
3 1 . 0 2 9 1 . 3 1 0 1 . 3 5 7
4 0 . 8 1 4 1 . 1 1 9 1 . 1 9 6
APP A RENT F I T (BETWEEN RECOVERED AND D I S TORTED D I S T A N C E S )
P A S S I V E C E L L S REPLACED BY MEAN
F I R S T IT E R A T IO N
SECOND ITERATION
1 2 . 7 6 9 2 . 9 6 3 2 . 9 6 9
2 2 . 1 7 9 2 . 3 2 8 2 . 3 1 7
3 1 . 7 9 8 1 . 9 2 1 1 . 9 1 8
4 1 . 5 7 0 1 . 6 3 3 1 . 6 4 4
1 2 . 0 6 6 2 . 1 7 3 2 . 2 4 0
2 1 . 2 7 4 1 . 3 9 0 1 . 4 5 4
3 0 . 7 9 9 0 . 8 5 4 0 . 9 3 7
4 0 . 6 1 1 0 . 6 1 6 0 . 7 2 1
1 1 . 9 1 7 2 . 0 5 8 2 . 1 3 7
2 1 . 1 4 3 1 . 2 4 6 1 . 2 7 2
3 0 . 6 8 5 0 . 7 3 4 0 . 7 8 1
4 0 . 4 6 0 0 . 4 7 8 0 . 5 8 4
103
STANDARD ERROR.
TABLE E8
STANDARD METHOD (INCOMPLETE) CONFIGURATI0N=4TD
D IM E N SIO N S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T ( BETWEEN RECOVERED AND TR UE P I S T A N C E S )
P A S S I V E C E L L S REPLACED BY MEAN ~
F I R S T I T E R A T IO N
SECOND IT E R A T I O N
P A S S I V E CELLS REPLACED BY MEAN
F I R S T IT E R A T I O N
SECOND ITERATION
1 2 . 8 3 0 2 . 9 2 3 2 . 9 1 8
2 2 . 0 5 5 2 . 0 9 3 2 . 0 7 6
3 1 . 6 6 6 1 . 6 9 6 1 . 6 7 2
4 1 . 3 0 7 1 . 3 6 6 1 . 3 8 4
1 2 . 8 2 6 2 . 9 4 8 2 . 8 7 0
2 1 . 9 0 6 2 . 0 1 9 2 . 0 0 3
3 1 . 4 4 8 1 . 6 2 3 1 . 5 9 6
4 1 . 1 7 2 1 . 2 7 4 1 . 2 8 5
1 2 . 7 0 7 2 . 7 6 1 2 . 6 2 8
2 1 . 8 0 2 1 . 9 1 9 1 . 8 4 0
3 1 . 2 3 9 1 . 4 9 8 1 . 4 4 2
4 1 . 0 7 0 1 . 1 9 6 1 . 2 3 0
D AND D IST O R T E D D I S T A N C E S )
1
1 2 . 7 8 3 3 . 0 0 7 3 . 0 1 8
2 2 . 141 2 . 2 8 6 2 . 2 8 2
3 1 . 8 8 4 1 . 9 6 6 1 . 9 4 3
4 1 . 6 3 2 1 . 6 9 2 1 . 6 8 2
1 2 . 2 3 4 2 . 3 5 7 2 . 3 1 3
2 1 . 2 6 2 1 . 3 5 5 1 . 4 3 7
3 0 . 8 8 6 0 . 9 2 6 0 . 9 9 5
4 0 . 6 7 2 0 . 6 9 2 0 . 7 5 4
1 2 . 1 5 4 2 . 2 1 5 2 . 1 3 0
2 1 . 2 1 9 1 . 3 1 2 1 . 3 1 1
3 0 . 6 5 4 0 . 8 1 2 0 . 8 5 2
4 0 . 5 4 8 0 . 5 2 7 0 . 6 0 3
104
TABLE E9
STANDARD ERROR IN T E R A C T IV E METHOD CONFIGURATION=lT D
DIM ENSION S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
TRUE F I T ( BET WEEN RECOVERED AND TRUE P I S TANCES )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 0 1 . 1 8 1 1 . 2 3 9 1 . 2 8 2 1 . 2 9 8 1 . 3 2 3
2 0 . 0 1 . 0 1 9 1 . 0 7 0 1 . 1 5 3 1 . 2 4 1 1 . 3 6 9
3 0 . 0 1 . 0 8 7 1 . 1 9 6 1 . 3 5 2 1 . 4 0 6 1 . 6 9 1
4 0 . 0 1 . 1 5 3 1 . 3 5 8 1 . 7 0 7 2 . 0 0 1 2 . 4 6 7
1 0 . 0 1 . 1 85 1 . 1 9 6 1 . 2 4 6 1 . 2 2 4 1 . 2 5 9
,2------
0 . 0 0 . 9 7 2 1 . 0 4 4 1 . 1 3 8 1 . 2 2 7 1 . 3 5 9
3 0 . 0 1 . 0 1 9 1 . 1 7 9 1 . 3 6 4 1 . 4 4 0 1 . 7 8 6
4 0 . 0 1 . 0 7 6 1 . 3 8 3 1 . 7 6 7 2 . 2 4 8 2 . 7 8 8
1 0 . 0 0 . 9 84 0 . 9 8 1 1 . 0 3 5 1 . 0 3 3 1 . 0 5 9
2 0 . 0 0 . 7 9 3 0 . 9 1 8 1 . 0 3 2 1 . 1 6 6 1 . 3 1 2
3 ■ 0 . 0 0 . 8 1 7 1 . 0 6 4 1 . 3 5 4 1 . 4 6 8 1 . 9 3 4
4 0 . 0 0 . 8 6 2 1 . 3 3 3 1 . 8 5 2 2 . 4 9 9 3 . 1 7 2
APPARENT F I T (BETWEEN RECOVER ED AND D IS T O R T ED D I S T A N C E S )
S T R I P E JUDGMENTS ~
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 0 1 . 2 3 7 1 . 2 3 8 1 . 2 9 4 1 . 2 9 9 1 . 3 5 1
2 0 . 0 0 . 7 7 0 0 . 7 1 2 0 . 7 4 3 0 . 7 6 8 0 . 8 2 8
3 0 . 0 0 . 5 3 4 0 . 5 1 9 0 . 6 5 6 0 . 6 6 5 0 . 9 3 4
4 0 . 0 0 . 3 6 5 0 . 5 4 9 0 . 9 5 3 1 . 2 9 7 1 . 7 8 3
1 0 . 0 1 . 3 6 1 1 . 3 4 6 1 . 3 9 8 1 . 3 9 3 1 . 4 4 8
2 0 . 0 0 . 8 0 1 0 . 7 9 5 0 . 8 5 1 0 . 8 8 8 0 . 9 9 6
3 0 . 0 0 . 6 6 3 0 . 6 7 9 0 . 8 1 9 0 . 8 6 0 1 . 1 6 1
4 0 . 0 0 . 5 8 2 0 . 7 5 4 1 . 1 4 8 1 . 6 1 9 2 . 2 0 6
1 0 . 0 1 . 4 4 1 1 . 4 2 3 1 . 4 6 8 1 . 4 5 8 1 . 4 8 8
2 0 . 0 0 . 9 1 9 0 . 9 2 5 0 . 9 7 9 1 . 0 0 9 1 . 1 3 0
3 0 . 0 0 . 8 5 4 0 . 8 4 1 0 . 9 8 3 1 . 0 5 1 1 . 4 1 1
4 0 . 0 0 . 7 9 8 0 . 9 1 8 1 . 3 5 6 1 . 9 1 1 2 . 6 2 2
105
TABLE E10
STANDARD E R R O R .
IN T E R A C T IV E METHOD C Q N F IG U R A T I0 N = 2 T D
DIM ENSION S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 80 1 . 0 0
-------------------------------------------------------------------------------------------------
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S T A N C E S )
S T R I P E JUDGMENTS
1 3 . 1 6 2 3 . 3 0 3 3 . 1 8 3 3 . 1 0 8 2 . 8 3 8 2 . 7 6 8
2 0 . 0 1 . 0 4 3 1 . 0 1 5 0 . 9 7 9 1 . 0 1 8 1 . 1 1 2
3 0 . 0 0 . 6 9 2 0 . 7 1 7 0 . 8 4 0 0 . 9 7 7 1 . 1 3 7
4 0 . 0 0 . 6 0 1 0 . 7 4 5 0 . 9 9 2 1 . 2 4 3 1 . 6 7 2
ALL OBTAINED JUDGMENTS
1 2 . 8 9 6 3 . 0 4 8 2 . 9 7 9 2 . 8 9 5 2 . 6 7 8 2 . 6 7 0
2 0 . 0 0 . 9 0 0 0 . 9 1 1 0 . 9 4 3 1 . 0 1 2 1 . 1 0 7
3 0 . 0 0 . 6 1 1 0 . 6 8 8 0 . 8 5 8 1 . 0 1 3 1 . 2 2 0
4 0 . 0 0 . 5 6 4 0 . 7 8 8 1 . 1 0 0 1 . 3 9 9 1 . 9 5 5
ALL P O S S I B L E JUDGMENTS
1 2 . 4 0 2 2 . 1 70 2 . 2 1 5 2 . 2 3 6 2 . 1 3 3 2 . 1 8 6
2 0 . 0 0 . 6 0 9 0 . 6 8 2 0 . 8 1 2 0 . 9 2 9 1 . 0 5 6
3 0 . 0 0 . 4 5 5 0 . 6 1 2 0 . 8 9 7 1 . 0 9 7 1 . 3 5 0
4 0 . 0 0 . 5 1 7 0 . 8 7 9 1 . 3 5 7 1 . 7 0 9 2 . 3 9 8
A PP A R ENT F I T (BETWEEN RECOVERED AND D I S T O R TED D I S T ANCE S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 3 . 1 6 2 3 . 0 8 0 2 . 9 9 2 2 . 9 1 1 2 . 6 6 2 2 . 6 0 9
2 0 . 0 0 . 9 0 4 0 . 9 2 8 0 . 8 7 1 0 . 9 1 1 0 . 9 9 9
3 0 . 0 0 . 4 4 6 0 . 4 7 0 0 . 5 1 7 0 . 6 1 5 0 . 7 9 7
4 0 . 0 0 . 2 9 6 0 . 3 7 7 0 . 6 1 8 0 . 8 4 5 1 . 3 2 4
1 2 . 8 9 6 2 . 9 2 6 2 . 8 7 6 2 . 7 9 8 2 . 5 9 8 2 . 5 8 9
2 0 . 0 0 . 8 9 1 0 . 9 2 6 0 . 9 6 4 1 . 0 3 0 1 . 0 9 9
3 0 . 0 0 . 5 5 9 0 . 5 9 6 0 . 6 7 9 0 . 8 1 4 0 . 9 9 8
4 0 . 0 0 . 4 2 9 0 . 5 3 6 0 . 7 9 5 1 . 0 8 8 1 . 6 6 4
1 2 . 4 0 2 2 . 2 9 3 2 . 3 2 6 2 . 3 4 9 2 . 2 5 2 2 . 3 0 1
2 0 . 0 0 . 9 1 8 0 . 9 4 7 1 . 0 4 6 1 • 1 0 5 1 . 1 9 0
3 0 . 0 0 . 7 4 9 0 . 7 5 1 0 . 8 5 8 1 . 0 0 7 1 . 1 9 7
4 0 . 0 0 . 6 1 2 0 . 7 0 7 1 . 0 4 1 1 . 4 0 8 2 . 0 5 4
O
OS
STANDARD ERROR..
TABLE Ell
INTERACTIVE METHOD
CONFIGURAT10N=3T D
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T ( BETWEEN RECOVERED AND TRUE D I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 2 . 6 4 4 2 . 9 2 9 2 . 9 4 1 3 . 0 0 1 3 . 0 4 2 2 . 9 3 9
2 0 . 8 3 8 1 . 6 2 1 1 . 6 2 9 1 . 6 4 4 1 . 7 1 4 1 . 6 5 0
3 0 . 0 0 . 5 8 9 0 . 6 7 5 0 . 8 2 7 0 . 9 3 6 1 . 0 5 3
4 0 . 0 0 . 4 2 0 0 . 5 2 5 0 . 7 0 1 0 . 9 8 2 1 . 2 9 8
1 2 . 7 4 9 2 . 9 4 6 2 . 9 4 6 2 . 9 8 5 2 . 9 6 9 2 . 9 0 7
2 1 . 3 7 3 1 . 6 5 6 1 . 6 7 9 1 . 7 1 3 1 . 7 5 6 1 . 7 2 8
3 0 . 0 0 . 5 9 1 0 . 7 0 6 0 . 8 9 6 1 . 0 2 3 1 . 1 7 5
4 0 . 0 0 . 4 5 7 0 . 6 1 5 0 . 8 3 8 1 . 1 2 6 1 . 5 3 2
1 2 . 3 7 6 2 . 4 8 5 2 . 4 9 2 2 . 4 8 3 2 . 4 8 3 2 . 4 9 1
2 1 . 3 3 8 1 . 3 8 7 1 . 4 4 7 1 . 4 6 9 1 . 5 4 4 1 . 6 1 7
3 0 . 0 0 . 5 0 6 0 . 6 6 7 0 . 8 9 5 1 . 0 9 1 1 . 3 0 9
4 0 . 0 0 . 4 5 3 0 . 7 3 2 0 . 9 9 9 1 . 3 9 5 1 . 9 4 6
APPARENT F I T (BETWEEN RECOVERED AND D IST O R T ED D IS T A N C E S )
S T R I P E JUDGMENTS - - _
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 2 . 6 4 4 2 . 8 4 8 2 . 8 6 2 2 . 9 1 1 2 . 9 3 6 2 . 8 5 5
2 0 . 8 3 8 1 . 5 4 8 1 . 5 6 0 1 . 5 5 5 1 . 5 7 8 1 . 5 1 6
3 0 . 0 0 . 5 0 7 0 . 5 6 1 0 . 6 8 4 0 . 7 3 2 0 . 8 1 6
4 0 . 0 0 . 2 7 7 0 . 3 3 0 0 . 4 5 3 0 . 7 2 2 1 . 0 1 7
1 2 . 7 4 9 2 . 9 0 3 2 . 9 1 6 2 . 9 5 4 2 . 9 3 1 2 . 9 0 1
2 I . 3 7 3 1 . 6 2 7 1 . 6 6 7 1 . 6 9 6 1 . 7 1 1 1 . 7 0 6
3 0 . 0 0 . 5 7 5 0 . 6 8 5 0 . 8 5 3 0 . 9 4 1 1 . 0 8 0
4 0 . 0 0 . 4 0 9 0 . 5 2 7 0 . 7 1 4 0 . 9 7 3 1 . 3 5 6
1 2 . 3 7 6 2 . 5 7 8 2 . 5 9 3 2 . 5 8 7 2 . 5 9 9 2 . 6 3 6
2 1 . 3 3 8 1 . 5 0 2 1 . 5 6 2 1 . 5 7 9 1 . 6 3 9 1 . 7 3 2
3 0 . 0 0 . 6 8 3 0 . 7 8 3 0 . 9 3 4 1 . 0 9 5 1 . 2 9 1
4 0 . 0 0 . 5 6 0 0 . 6 9 0 0 . 8 8 6 1 . 2 2 5 1 . 7 4 6
107
TABLE E l 2
STANDARD E R R O R . IN T E R A C T IV E METHOD C O N F IG U R A T I0 N = 4 T D
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 2 . 9 4 3 3 . 1 2 8 3 . 1 0 8 3 . 1 0 3 3 . 0 9 5 3 . 0 6 0
2 1 . 6 3 0 1 • 7 9 2 1 . 7 8 3 1 . 8 5 3 1 . 8 7 1 1 . 9 2 2
3 0 . 6 7 4 0 . 9 5 7 1 . 0 4 9 1 . 0 4 5 1 . 1 4 7 1 . 2 7 2
4 0 . 0 0 . 5 5 1 0 . 6 4 4 0 . 7 4 2 0 . 9 4 9 1 . 3 6 8
1 2 . 9 9 2 3 . 0 8 8 3 . 0 8 7 3 . 1 0 3 3 . 0 8 8 3 . 0 7 8
2 1 . 8 8 2 1 . 9 5 9 1 . 9 5 7 2 . 0 3 5 2 . 0 2 9 2 . 0 5 2
3 0 . 9 3 6 1 . 1 7 4 1 . 2 4 2 1 . 2 7 3 1 . 3 1 3 1 . 4 1 2
4 0 . 0 0 . 7 2 6 0 . 7 7 9 0 . 9 0 6 1 . 1 2 0 1 . 5 1 7
1 2 . 5 7 8 2 . 6 4 1 2 . 6 4 5 2 . 6 5 1 2 . 6 1 3 2 . 6 3 8
2 1 . 7 5 6 1 . 7 0 2 1 . 7 2 5 1 . 8 0 3 1 . 8 1 3 1 . 8 5 2
3 0 . 7 9 7 1 . 0 7 0 1 . 1 7 6 1 . 2 4 4 1 . 3 0 8 1 . 4 1 7
4 0 . 0 0 . 6 8 8 0 . 8 2 9 1 . 0 2 5 1 . 3 2 9 1 . 7 5 8
A PPARENT F I T (BETWEEN RECOVERED AND D IST OR T ED D IS T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 2 . 9 4 3 3 . 0 4 9 3 . 0 2 7 3 . 0 2 7 3 . 0 2 2 2 . 9 8 7
2 1 . 6 3 0 1 . 7 0 5 1 . 6 9 6 1 . 7 5 9 1 . 7 6 8 1 . 7 8 1
3 0 . 6 7 4 0 . 8 4 9 0 . 9 2 8 0 . 8 9 8 0 . 9 7 9 1 . 0 5 0
4 0 . 0 0 . 4 1 9 0 . 4 6 1 0 . 5 1 5 0 . 6 8 9 1 . 1 4 2
1 2 . 9 9 2 3 . 0 5 1 3 . 0 5 1 3 . 0 7 7 3 . 0 7 8 3 . 0 7 2
2 1 . 8 8 2 1 . 9 2 1 1 . 9 2 3 2 . 0 0 4 2 . 0 0 6 2 . 0 1 5
3 0 . 9 3 6 1 . 1 34 1 . 1 9 8 1 . 2 2 5 1 . 2 6 1 1 . 3 3 2
4 0 . 0 0 . 6 8 9 0 . 7 1 2 0 . 8 1 8 0 . 9 9 6 1 . 4 0 9
1 2 . 5 7 8 2 . 7 4 2 2 . 7 4 9 2 . 7 6 7 2 . 7 4 8 2 . 8 0 2
2 1 . 7 5 6 1 . 8 0 7 1 . 8 3 2 1 . 9 1 2 1 . 9 3 6 2 . 0 0 0
3 0 . 7 9 7 1 . 1 8 3 1 . 2 6 6 1 . 3 1 7 1 . 3 7 1 1 . 4 8 6
4 0 . 0 0 . 8 1 2 0 . 8 6 7 0 . 9 9 9 1 . 2 4 7 1 . 6 5 8
108
TABLE E1 3
STANDARD ERROR. PRE-SET BASIS ( 5 STIMULI) CONFIGURATI0N=4TD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERE D AND TRUE D I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 2 . 9 4 3 3 . 3 5 0 3 . 3 1 6 3 . 2 8 3 3 . 2 7 2 3 . 2 4 1
2 1 . 6 30 1 . 9 6 1 1 . 9 3 9 1 . 9 1 4 1 . 9 3 0 1 . 9 9 8
3 0 . 6 7 4 1 . 0 2 4 1 . 0 9 9 1 . 1 5 9 1 . 2 3 7 1 . 3 2 4
4 0 . 0 0 . 5 3 5 0 . 6 0 0 0 . 7 2 1 0 . 9 1 5 1 . 3 3 4
1 2 . 9 4 3 3 . 3 5 0 3 . 3 1 6 3 . 2 8 3 3 . 2 7 2 3 . 2 4 1
2 1 . 6 3 0 1 . 9 6 1 1 . 9 3 9 1 . 9 1 4 1 . 9 3 0 1 . 9 9 8
3 0 . 6 7 4 1 . 0 2 4 1 . 0 9 9 1 . 1 5 9 1 . 2 3 7 1 . 3 2 4
4 0 . 0 0 . 5 3 5 0 . 6 0 0 0 . 7 2 1 0 . 9 1 5 1 . 3 3 4
1 2 . 5 7 8 2 . 7 1 3 2 . 6 8 7 2 . 6 6 0 2 . 6 5 4 2 . 6 3 5
2 1 . 7 5 6 1 . 8 8 2 1 . 8 5 1 1 . 8 2 4 1 . 8 1 1 1 . 8 2 0
3 0 . 7 9 7 0 . 9 9 4 1 . 0 9 4 1 . 1 6 8 1 . 2 5 3 1 . 3 5 6
4 0 . 0 0 . 5 9 4 0 . 7 0 0 0 . 9 0 5 1 . 2 2 9 1 . 8 4 2
APPARENT F I T (BETWEEN RECOVERED AND D I S TORTED D I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 2 . 9 4 3 3 . 2 9 6 3 . 2 6 1 3 . 2 2 6 3 . 2 0 2 3 . 1 6 4
2 1 . 6 3 0 1 . 9 2 3 1 . 8 9 1 1 . 8 4 7 1 . 8 4 3 1 . 8 8 9
3 0 . 6 7 4 0 . 9 4 1 0 . 9 9 3 1 . 0 1 7 1 . 0 5 6 1 . 1 0 4
4 0 . 0 0 . 4 3 2 0 . 4 2 8 0 . 4 7 7 0 . 6 3 0 1 . 0 4 4
1 2 . 9 4 3 3 . 2 9 6 3 . 2 6 1 3 . 2 2 6 3 . 2 0 2 3 . 1 6 4
2 1 . 6 3 0 1 . 9 2 3 1 . 8 9 1 1 . 8 4 7 1 . 8 4 3 1 . 8 8 9
3 0 . 6 7 4 0 . 9 4 1 0 . 9 9 3 1 . 0 1 7 1 . 0 5 6 1 . 1 0 4
4 0 . 0 0 . 4 3 2 0 . 4 2 8 0 . 4 7 7 0 . 6 3 0 1 . 0 4 4
1 2 . 5 7 8 2 . 8 1 4 2 . 7 9 5 2 . 7 8 3 2 . 7 9 5 2 . 8 0 3
2 1 . 7 5 6 1 . 9 9 5 1 . 9 6 4 1 . 9 4 4 1 . 9 4 6 1 . 9 7 5
3 0 . 7 9 7 1 . 1 1 8 1 . 2 0 5 1 . 2 6 8 1 . 3 4 7 1 . 4 4 7
4 0 . 0 0 . 7 3 8 0 . 7 7 5 0 . 9 1 1 1 . 1 8 7 1 . 7 6 9
TABLE E 1 4
STANDARD E R R O R . P R E - S E T B A S I S ( 7 S T I M U L I ) C O N F IG U R A T I0 N = 4 T D
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
I R U E _ E I I _ ( BETWEEN.RECOVERED AND TRUE D I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 3 . 0 3 0 3 . 3 1 1 3 . 2 7 5 3 . 2 3 9 3 . 2 1 2 3 . 2 0 3
2 1 . 8 2 1 2 . 0 3 1 2 . 0 2 7 2 . 0 2 1 2 . 0 1 8 2 . 0 2 5
3 0 . 8 7 8 1 . 0 9 9 1 . 1 6 1 1 . 2 1 2 1 . 2 9 0 1 . 3 5 3
4 0 . 0 0 . 5 1 7 0 . 5 8 9 0 . 7 0 2 0 . 8 4 8 1 . 0 1 6
1 3 . 0 30 3 . 3 1 1 3 . 2 7 5 3 . 2 3 9 3 . 2 1 2 3 . 2 0 3
2 1 . 8 2 1 2 . 0 3 1 2 . 0 2 7 2 . 0 2 1 2 . 0 1 8 2 . 0 2 5
3 0 . 8 7 8 1 . 0 9 9 1 . 1 6 1 1 . 2 1 2 1 . 2 9 0 1 . 3 5 3
4
0 . 0 0 . 5 1 7 0 . 5 8 9 0 . 7 0 2 0 . 8 4 8 1 . 0 1 6
1 2 . 5 9 5 2 . 6 4 9 2 . 6 2 9 2 . 6 1 2 2 . 6 0 1 2 . 5 9 9
2 1 . 5 2 4 1 . 6 7 4 1 . 6 7 9 1 . 7 1 3 1 . 7 3 6 1 . 7 5 9
3 0 . 7 7 2 0 . 9 1 5 1 . 0 1 1 1 . 0 9 1 1 . 1 8 8 1 . 2 8 2
4 0 . 0 0 . 4 1 9 0 . 5 6 7 0 . 7 6 7 0 . 9 9 9 1 . 2 4 2
A £ P A R £ N T _ F I I (.BETWEEN RECOVERED AND D IS T O R T ED D IS T A N C E S )
S T R I P E JUDGMENTS “ ~
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 3 . 0 3 0 3 . 2 6 1 3 . 2 2 8 3 . 1 9 7 3 . 1 7 4 3 . 1 6 8
2 1 . 8 2 1 1 . 9 6 9 1 . 9 6 0 1 . 9 4 8 1 . 9 4 2 1 . 9 4 8
3 0 . 8 7 8 1 . 0 3 2 1 . 0 7 7 1 . 1 1 1 1 • 1 7 3 1 . 2 2 2
4 0 . 0 0 . 4 1 3 0 . 4 5 8 0 . 5 4 6 0 . 6 7 2 0 . 8 2 1
1 3 . 0 30 3 . 2 6 1 3 . 2 2 8 3 . 1 9 7 3 . 1 7 4 3 . 1 6 8
2 1 . 8 2 1 1 . 9 6 9 1 . 9 6 0 1 . 9 4 8 1 . 9 4 2 1 . 9 4 8
3 0 . 8 7 8 1 . 0 3 2 1 . 0 7 7 1 . 1 1 1 1 . 1 7 3 1 . 2 2 2
4 0 . 0 0 . 4 1 3 0 . 4 5 8 0 . 5 4 6 0 . 6 7 2 0 . 8 2 1
1 2 . 5 9 5 2 . 7 4 8 2 . 7 3 7 2 . 7 3 6 2 . 7 4 8 2 . 7 7 3
2 1 . 5 2 4 1 . 7 7 3 1 . 7 8 5 1 . 8 3 0 1 . 8 7 2 1 . 9 2 0
3 0 . 7 7 2 1 . 0 3 7 1 . 1 1 9 1 . 1 9 4 1 . 2 9 2 1 . 3 9 4
4 0 . 0 0 . 5 6 9 0 . 6 4 9 0 . 7 9 8 0 . 9 9 8 1 . 2 2 4
TABLE El 5
STANDARD ERROR* PRE-SET BASIS { 9 STIMULI) CONFIGURATI0N=4TD
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 * 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE P I S T A N C E S )
S T R I P E JUDGMENTS “ “
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 2 . 9 3 9 3 . 0 1 4 3 . 0 3 7 3 . 0 7 0 3 . 1 0 0 3 . 1 2 6
2 1 . 7 0 8 1 . 8 7 3 1 . 8 9 9 1 . 9 2 3 1 . 9 4 9 1 . 9 9 4
3 0 . 8 3 7 1 . 0 3 0 1 . 0 6 4 1 . 1 1 3 1 . 1 9 0 1 . 2 5 3
4 0 . 0 0 . 4 7 6 0 . 5 4 9 0 . 6 5 8 0 . 7 8 8 0 . 9 2 8
1 2 . 9 3 9 3 . 0 1 4 3 . 0 3 7 3 . 0 7 0 3 . 1 0 0 3 . 1 2 6
2 1 . 7 0 8 1 . 8 7 3 1 . 8 9 9 1 . 9 2 3 1 . 9 4 9 1 . 9 9 4
3 0 . 8 3 7 1 . 0 3 0 1 . 0 6 4 1 . 1 1 3 1 . 1 9 0 1 . 2 5 3
4 0 . 0 0 . 4 7 6 0 . 5 4 9 0 . 6 5 8 0 . 7 8 8 0 . 9 2 8
1 2 . 5 0 8 2 . 4 6 7 2 . 4 8 6 2 . 5 0 9 2 . 5 2 9 2 . 5 4 6
2 1 . 3 5 8 1 . 4 6 1 1 . 4 9 1 1 . 5 3 2 1 . 5 8 0 1 . 6 4 0
3 0 . 8 0 3 0 . 9 2 1 0 . 9 5 7 1 . 0 1 7 1 . 0 9 4 1 . 1 8 1
4 0 . 0 0 . 4 0 1 0 . 5 3 1 0 . 6 9 9 0 . 8 8 2 1 . 0 6 7
APPARENT F I T (BETWEEN R EC O VERED AND D I STORTED D I S T A N C E S )
S T R I P E JUDGMENTS '
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 2 . 9 3 9 2 . 9 6 0 2 . 9 8 8 3 . 0 2 8 3 . 0 6 7 3 . 1 0 5
2 1 . 7 0 8 1 . 8 3 5 1 . 8 6 5 1 . 8 9 4 1 . 9 2 7 1 . 9 7 9
3 0 . 8 3 7 0 . 9 9 0 1 . 0 2 1 1 . 0 6 8 1 . 1 4 7 1 . 2 1 0
4 0 . 0 0 . 4 1 5 0 . 4 7 6 0 . 5 7 4 0 . 6 9 7 0 . 8 3 1
1 2 . 9 3 9 2 . 9 6 0 2 . 9 8 8 3 . 0 2 8 3 . 0 6 7 3 . 1 0 5
2 1 . 7 0 8 1 . 8 3 5 1 . 8 6 5 1 . 8 9 4 1 . 9 2 7 1 . 9 7 9
3 0 . 8 3 7 0 . 9 9 0 1 . 0 2 1 1 . 0 6 8 1 . 1 4 7 1 . 2 1 0
4 0 . 0 0 . 4 1 5 0 . 4 7 6 0 . 5 7 4 0 . 6 9 7 0 . 8 3 1
1 2 . 5 0 8 2 . 5 61 2 . 5 8 8 2 . 6 2 6 2 . 6 6 8 2 . 7 1 4
2 1 . 3 5 8 1 . 5 7 2 1 . 6 0 8 1 . 6 6 2 1 . 7 3 1 1 . 8 1 8
3 0 . 8 0 3 1 . 0 4 0 1 . 0 7 0 1 . 1 2 9 1 . 2 1 4 , 1 . 3 1 2
4 0 . 0 0 . 5 6 4 0 . 6 3 3 0 . 7 5 7 0 . 9 1 8 1 . 0 9 4
111
STANDARD ERROR.
TABLE El 6
PRE-SET BASIS (10 STIMULI) CONFIGURATICN=4TD
D IM E NSION S
EXTRACTED
STANDARD D E V IA T IO N OF ERROR
0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F IT (BETWEEN R E COVERED AND TRUE P I S T A N C E S )
S T R I P E ~ J U D G M E N T S ~
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 2 . 9 0 7 2 . 9 8 1 2 . 9 9 0 3 . 0 1 1 3 . 0 3 2 3 . 0 4 7
2 1 . 6 6 3 1 . 7 3 1 1 . 7 7 9 1 . 8 4 1 1 . 8 8 5 1 . 9 3 1
3 0 . 7 9 5 0 . 9 9 9 1 . 0 2 5 1 . 0 6 5 1 . 1 1 9 1 . 1 8 3
4 0 . 0 0 . 4 5 1 0 . 5 2 6 0 . 6 3 5 0 . 7 6 2 0 . 8 9 8
1 2 . 9 0 7 2 . 9 8 1 2 . 9 9 0 3 . 0 1 1 3 . 0 3 2 3 . 0 4 7
2 1 . 6 6 3 1 . 7 3 1 1 . 7 7 9 1 . 84 1 1 . 8 8 5 1 . 9 3 1
3 0 . 7 9 5 0 . 9 9 9 1 . 0 2 5 1 . 0 6 5 1 . 1 1 9 1 . 1 8 3
4 0 . 0 0 . 4 5 1 0 . 5 2 6 0 . 6 3 5 0 . 7 6 2 0 . 8 9 8
1 2 . 5 1 0 2 . 4 7 6 2 . 4 8 9 2 . 5 0 9 2 . 5 2 8 2 . 5 4 3
2 1 . 3 4 2 1 . 4 1 6 1 . 4 5 9 1 . 5 2 3 1 . 5 8 4 1 . 6 5 5
3 0 . 7 6 3 0 . 8 9 9 0 . 9 3 9 0 . 9 9 9 1 . 0 7 3 1 . 1 5 7
4 0 . 0 0 . 3 9 3 0 . 5 1 7 0 . 6 7 5 0 . 8 4 5 1 . 0 1 5
:d an d D IST OR T ED D IS T A N C E S )
i 2 . 9 0 7 2 . 9 5 1 2 . 9 6 5 2 . 9 9 4 3 . 0 2 5 3 . 0 5 3
2 1 . 6 6 3 1 . 7 1 9 1 . 7 7 1 1 . 6 3 9 1 . 8 8 6 1 . 9 3 3
3 0 . 7 9 5 0 . 9 7 8 1 . 0 0 2 1 . 0 4 1 1 . 0 9 8 1 . 1 6 6
4 0 . 0 0 . 4 2 9 0 . 4 8 4 0 . 5 7 5 0 . 6 9 1 0 . 8 2 1
1 2 . 9 0 7 2 . 9 5 1 2 . 9 6 5 2 . 9 9 4 3 . 0 2 5 3 . 0 5 3
2 1 . 6 6 3 1 . 7 1 9 1 . 7 7 1 1 . 8 3 9 1 . 8 8 6 1 . 9 3 3
3 0 . 7 9 5 0 . 9 7 8 1 . 0 0 2 1 . 0 4 1 1 . 0 9 8 1 . 1 6 6
4 0 . 0 0 . 4 2 9 0 . 4 8 4 0 . 5 7 5 0 . 6 9 1 0 . 3 2 1
1 2 . 5 1 0 2 . 5 7 0 2 . 5 9 2 2 . 6 2 8 2 . 6 6 9 2 . 7 1 3
2 1 . 3 4 2 1 . 5 3 6 1 . 5 8 5 1 . 6 6 0 1 . 7 4 0 1 . 8 3 6
3 0 . 7 6 3 1 . 0 2 1 1 . 0 5 8 1 . 1 2 0 1 . 2 0 4 1 . 3 0 2
4 0 . 0 0 . 5 6 1 0 . 6 3 0 0 . 7 5 0 0 . 9 0 1 1 . 0 6 6
112
STRESS
TABLE SI
STANDARD METHOD (COMPLETE) CONFIGURATIQN=1TD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERE D AND TRUE D I S T ANCES)
1
o
•
o
0 . 2 6 6 0 . 2 7 5
2
o
•
o
0 . 2 0 2 0 . 2 3 6
3
o
•
o
0 . 2 2 2 0 . 2 6 6
4 o
.
o
0 . 2 3 2 0 . 2 8 1
APPARENT F I T (BETWEEN RECOVEREC AND D I S T ORTED D I S T A N C E S )
1 0 . 0 0 . 4 0 4 0 . 4 2 6
2 0 . 0 0 . 2 7 3 0 . 2 8 8
3 0 . 0 0 . 2 1 8 0 . 2 2 8
4 0 . 0 0 . 1 8 8 0 . 2 0 2
t - *
00
TABLE S2
STRESS. STANDARD METHOD (COMPLETE) CONFIGURATI0N=2TD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE P I S T A N C E S )
1 0 . 6 2 5 0 . 7 3 8 0 . 7 2 7
2 0 . 0 0 . 1 4 1 0 . 1 6 7
3 0 . 0 0 . 1 1 6 0 . 1 7 3
4 0 . 0 0 . 1 3 7 0 . 1 9 7
A P PA R E NT F IT (BETWEEN RECOVERED AND D IST O R T ED D I S T A N C E S )
1 0 . 6 2 5 0 . 7 9 5 0 . 8 0 1
2 0 . 0 0 . 2 4 1 0 . 2 7 5
3 0 . 0 0 . 1 8 5 0 . 2 1 8
4 0 . 0 0 . 1 4 6 0 . 1 8 4
114
TABLE S3
STRESS. STANDARD METHOD (COMPLETE) CONFIGURATI0N=3TD
D IM E N S IO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVEREC AND TRUE P I S T A N C E S )
1 0 . 7 3 1 0 . 7 8 2 0 . 7 9 0
2 0 . 2 6 4 0 . 3 2 2 0 . 3 2 8
3 0 . 0 0 . 1 0 9 0 . 1 4 6
4 0 . 0 0 . 1 0 3 0 . 1 5 1
APPARENT F I T (BETWEEN RECOVERED AND D IST OR T ED D I S T A N C E S )
1 0 . 7 3 1 0 . 8 3 5 0 . 8 6 7
2 0 . 2 6 4 0 . 3 7 9 0 . 4 0 7
3 0 . 0 0 . 1 8 0 0 . 2 2 9
4 0 . 0 0 . 1 4 6 0 . 1 9 2
115
T A BLES D I
STRESS ( STANDARD METHOD (COMPLETE) CONFIGURATI0N=4TD
D IM E N SIO N S
EXTRACTED
STANDARD D E V IA T IO N OF ERROR
0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D IS T A N C E S )
1 1 . 1 0 1 1 . 0 6 9 1 . 0 4 7
2 0 . 3 9 0 0 . 4 3 0 0 . 4 3 2
3 0 . 1 6 8 0 . 2 2 2 0 . 2 3 3
4 0 . 0 0 . 1 1 1 0 . 1 5 3
APPARENT F I T (BETWEEN RECOVERED AND D ISTORTED D IS T A N C E S )
1 1 . 1 0 1 1 . 1 1 1 1 . 1 1 5
2 0 . 3 9 0 0 . 4 7 9 0 . 5 0 3
3 0 . 1 6 8 0 . 2 7 1 0 . 2 9 9
4 0 . 0 0 . 1 6 2 0 . 2 0 7
0 5
TABLE S5
STRESS. STANDARD METHOD (INCOMPLETE) CONFIGURATION=lTD
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D IS T A N C E S )
P A S S I V E CELLS REPLACED BY MEAN
F I R S T IT E R A T IO N
SECOND I T E R A T IO N
1 0 . 6 5 3 0 . 8 6 1 0 . 8 6 8
2 0 . 5 0 9 0 . 5 8 5 0 . 5 9 1
3 0 . 4 8 2 0 . 5 0 5 0 . 5 0 8
A 0 . 5 0 3 0 . 4 9 8 0 . 4 9 8
1 0 . 5 1 6 0 . 6 9 4 0 . 6 9 7
2 0 . 3 0 6 0 . 4 1 8 0 . 4 2 5
3 0 . 3 0 7 0 . 3 8 1 0 . 3 3 8
4 0 . 3 8 3 0 . 4 2 3 0 . 4 1 1
1 0 . 5 3 3 0 . 6 4 9 0 . 6 5 6
2 0 . 2 9 7 0 . 3 8 6 0 . 3 9 5
3 0 . 2 7 9 0 . 3 2 9 0 . 3 4 1
4 0 . 3 7 0 0 . 3 7 6 0 . 3 6 7
APPARENT F I T (BETWEEN RECOVERED AND D I S T ORTED P I S T A N C E S )
P A S S I V E CELLS REPLACED~BY MEAN
F I R S T IT E R A T IO N
SECOND ITERATION
1 0 . 7 9 3 1 . 0 0 4 1 . 0 1 4
2 0 . 6 2 8 0 . 6 4 5 0 . 6 4 8
3 0 . 5 9 6 0 . 5 2 8 0 . 5 2 9
4 0 . 5 9 7 0 . 4 8 4
0 . 4 8 3
1 0 . 7 4 0 0 . 7 9 4
0 . 7 9 1
2 0 . 4 1 9 0 . 4 3 2 0 . 4 3 5
3 0 . 3 1 7 0 . 3 1 3 0 . 3 1 4
4 0 . 2 5 5 0 . 2 0 8 0 . 2 2 1
1 0 . 6 6 7 0 . 6 9 4 0 . 6 8 5
2 0 . 3 4 0 0 . 3 6 1 0 . 3 4 8
3 0 . 2 5 3 0 . 2 4 6 0 . 2 3 2
4 0 . 1 8 6 0 . 1 2 3 0 . 1 3 6
117
TABLE S6
STRESS. STANDARD METHOD (INCOMPLETE) CCNFIGURATI0N=2TD
D IM E N S IO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 6 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D I S T A NCES )
P A S S I V E C E L L S REPLACED BY MEAN
F I R S T I T E R A T IO N
SECOND IT E R A T I O N
1 1 . 0 8 3 1 . 2 1 3 1 . 2 1 5
2 0 . 5 7 7 0 . 6 7 3 0 . 6 7 2
3 0 . 3 7 6 0 . 4 3 7 0 . 4 4 4
4 0 . 3 5 9 0 . 3 9 9 0 . 4 0 2
1 0 . 9 6 1 1 . 0 6 3 1 . 0 6 2
2 0 . 5 0 0 0 . 5 6 8 0 . 5 6 7
3 0 . 2 9 4 0 . 3 6 6 0 . 3 8 3
4 0 . 2 6 6 0 . 3 2 0 0 . 3 3 2
1 0 . 9 7 6 0 . 9 7 0 0 . 9 7 7
2 0 . 3 5 0 0 . 4 5 3 0 . 4 5 9
3 0 . 2 3 3 0 . 3 2 4 C . 3 4 5
4 0 . 2 1 9 0 . 2 8 3 0 . 3 0 2
AP PARENT F I T (BETWEEN RECOVEREC AND D IS T O R T E D D I STAN C ES)
P A S S I V E C EL L S REPLACED BY MEAN ~
F I R S T IT E R A T I G N
SECOND ITERATION
1 1 . 0 6 1 1 . 2 8 6 1 . 2 9 4
2 0 . 6 3 4 0 . 7 4 7 0 . 7 4 4
3 0 . 4 8 7 0 . 5 2 9 0 . 5 3 1
4 0 . 4 4 1 0 . 4 3 3 0 . 4 3 5
1 0 . 8 9 3 0 . 9 1 3 0 . 9 2 6
2 0 . 4 4 2 0 . 4 5 1 0 . 4 5 2
3 0 . 2 5 0 0 . 2 3 7 0 . 2 5 0
4 0 . 1 8 9 0 . 1 6 6 0 . 1 8 0
1 0 . 9 4 4 0 . 8 7 6 0 . 8 9 8
2 0 . 3 3 4 0 . 3 7 5 0 . 3 8 0
3 0 . 1 7 0 0 . 1 9 3 0 . 2 0 3
4 0 . 1 1 6 0 . 1 1 6 0 . 1 3 2
118
STRESS.
TABLE S7
STANDARD METHOD (INCOMPLETE)
CONFIGURATI0N=3TD
D IM E N SIO N S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 6 0 1 . 0 0
I SyE_E.IT.- (.BETWEEN RECOVERED AND TRUE D I S T A N C E S )
P A S S I V E C E L L S REPLACED BY MEAN
F I R S T I T E R A T IO N
SECOND I T E R A T IO N
1 1 . S 3 4 1 . 5 8 8 1 . 5 7 3
2 0 . 7 4 6 0 . 8 1 0 0 . 7 9 7
3 0 . 4 8 0 0 . 5 3 4 0 . 5 2 8
4 0 . 3 3 2 0 . 3 7 7 0 . 3 8 6
1 1 . 1 8 6 1 . 3 5 1
1 . 3 5 8
2 0 . 6 0 4 0 . 6 8 2 0 . 6 7 8
3 0 . 3 6 6 0 . 4 4 1
0 . 4 3 9
4 0 . 2 5 2 0 . 3 1 7 0 . 3 3 0
1 0 . 9 7 8 1 . 1 2 7 1 . 1 2 4
2 0 . 4 7 1 0 . 5 6 5 0 . 5 5 5
3 0 . 2 8 5 0 . 3 7 3 0 . 3 8 0
4 0 . 2 0 4 0 . 2 9 0 0 . 3 0 5
A PPAR EN I _ E 1 T - ( B E T W E E N ,R ECOVERED AND D IST O R T ED D I S T A N C E S )
P A S S I V E C EL L S REPLACED BY MEAN
F I R S T I T E R A T IO N
SECOND ITERATION
1 1 . 5 0 0 1 . 6 4 7 1 . 6 3 9
2 0 . 7 7 8 0 . 8 8 2 0 . 8 6 9
3 0 . 5 4 3 0 . 6 1 4 0 . 6 0 4
4 0 . 4 2 9 0 . 4 7 0 0 . 4 6 6
1 0 . 9 7 8 1 . 0 9 7 1 . 1 3 4
2 0 . 4 3 0 0 . 4 8 7 0 . 5 0 6
3 0 . 2 2 8 0 . 2 5 3 0 . 2 7 5
4 0 . 1 5 7 0 . 1 6 5 0 . 1 9 2
1 0 . 8 3 7 0 . 9 5 4 0 . 9 7 0
2 0 . 3 6 6 0 . 4 1 2 0 . 4 1 3
3 0 . 1 9 0 0 . 2 0 9 0 . 2 1 9
4 0 . 1 1 5 0 . 1 2 4 0 . 1 4 9 CD
STRESS.
TABLE S8
STANDARD METHOD (INCOMPLETE) CONFIGURATI0N=4TD
D IM E N SIO N S
EXTRACTED
STANDARD D E V IA T IO N OF ERROR
0 . 0 0 0 . 6 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S TANCE S )
P A S S I V E CEL L S REPLACED BY MEAN
F I R S T IT E R A T I O N
SECOND IT E R A T IO N
1 1 . 4 2 1 1 . 6 2 6 1 . 6 2 0
2 0 . 7 5 2 0 . 8 0 9 0 . 7 9 1
3 0 . 5 1 7 0 . 5 5 7 0 . 5 4 0
4 0 . 3 6 9 0 . 4 0 5 0 . 4 0 2
1 1 . 4 3 7 1 . 7 0 2 1 . 5 1 7
2 0 . 6 7 4 0 . 7 4 3 0 . 7 2 4
3 0 . 4 3 8 0 . 5 0 9 0 . 4 9 0
4 0 . 3 2 4 0 . 3 5 4 0 . 3 5 0
1 1 . 2 7 8 1 . 3 6 6 1 . 2 0 9
2 0 . 6 1 0 0 . 6 6 6 0 . 6 1 6
3 0 . 3 5 6 0 . 4 4 6 0 . 4 1 4
4 0 . 2 8 4 0 . 3 1 8 0 . 3 2 0
APPARENT F I T ( BETWEEN RECOVERED
P A S S I V E CEL L S REPLACED- BY MEAN
F I R S T IT E R A T IO N
AND P I S TORTED D IS T A N C E S )
SECOND ITERATION
1 1 . 3 9 7 1 . 6 7 3 1 . 6 7 6
2 0 . 7 8 3 0 . 9 8 4 0 . 8 7 0
3 0 . 5 8 5 0 . 6 4 6 0 . 6 2 8
4 0 . 4 6 1 0 . 5 0 2 0 . 4 8 9
1 1 . 1 3 6 1 . 3 6 1 1 . 2 2 4
2 0 . 4 4 6 0 . 4 9 9 0 . 5 1 9
3 0 . 2 6 8 0 . 2 9 0 0 . 3 0 5
4 0 . 1 8 6 0 . 1 9 2 0 . 2 0 5
1 1 . 0 1 7 1 . 0 9 7 0 . 9 8 0
2 0 . 4 1 3 0 . 4 5 5 0 . 4 3 9
3 0 . 1 8 8 0 . 2 4 2 0 . 2 4 5
4 0 . 1 4 5 0 . 1 4 0 0 . 1 5 7
120
TABLE S9
STRESS. I N T E R A C T IV E METHOD CONFIGURATION=lTD
DIM ENSION S
EXTRACTED 0 . 0 0
STANDARD D E V IA T I O N OF ERROR
0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T ( BETWEEN RECOVERED AND TRUE D I S T A N C E S )
S T R I P E JUDGMENTS ~
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 0 0 . 2 5 9 0 . 2 6 6 0 . 2 7 9 0 . 2 8 0 0 . 2 8 2
2 0 . 0 0 . 2 1 1 0 . 2 1 4 0 . 2 3 2 0 . 2 4 6 0 . 2 6 5
3 0 . 0 0 . 2 2 3 0 . 2 3 3 0 . 2 5 9 0 . 2 6 8 0 . 3 0 5
4 0 . 0 0 . 2 3 4 0 . 2 5 8 0 . 3 0 4 0 . 3 4 2 0 . 3 8 8
1 0 . 0 0 . 2 8 2 0 . 2 8 0 0 . 2 9 4 0 . 2 9 0 0 . 2 9 8
2 0 . 0 0 . 2 1 4 0 . 2 2 6 0 . 2 4 6 0 . 2 6 5 0 . 2 8 7
3 0 . 0 0 . 2 2 3 0 . 2 4 7 0 . 2 7 8 0 . 2 9 5 0 . 3 4 5
4 0 . 0 0 . 2 3 3 0 . 2 7 9 0 . 3 2 7 0 . 3 9 7 0 . 4 4 4
1 0 . 0 0 . 2 8 9 0 . 2 8 9 0 . 3 0 8 0 . 3 0 7 0 . 3 1 4
2 0 . 0 0 . 2 1 5 0 . 2 4 5 0 . 2 7 6 0 . 3 0 8 0 . 3 3 6
3 0 . 0 0 . 2 1 9 0 . 2 7 1 0 . 3 3 2 0 . 3 5 5 0 . 4 2 9
4 0 . 0 0 . 2 2 8 0 . 3 2 0 0 . 3 9 2 0 . 4 8 5 0 . 5 3 8
D AND D IS T O R T ED D IS T A N C E S )
1 0 . 0 0 . 2 7 4 0 . 2 6 8 0 . 2 8 3 0 . 2 8 2 0 . 2 9 0
2 0 . 0 0 . 1 6 1 0 . 1 4 4 0 . 1 5 1 0 . 1 5 3 0 . 1 6 1
3 0 . 0 0 . 1 1 1 0 . 1 0 1 0 . 1 2 6 0 . 1 26 0 . 1 6 7
4 0 . 0 0 . 0 7 5 0 . 1 0 3 0 . 1 6 3 0 . 2 1 8 0 . 2 7 3
1 0 . 0 0 . 3 2 6 0 . 3 1 7 0 . 3 3 1 0 . 3 3 1 0 . 3 4 3
2 0 . 0 0 . 1 7 8 0 . 1 7 3 0 . 1 8 5 0 . 1 9 3 0 . 2 1 1
3 0 . 0 0 . 1 4 6 0 . 1 4 3 0 . 1 6 7 0 . 1 7 6 0 . 2 2 4
4 0 . 0 0 . 1 2 7 0 . 1 5 1 0 . 2 0 7 0 . 2 8 3 0 . 3 4 4
1 0 . 0 0 . 4 2 4 0 . 4 1 9 0 . 4 3 6 0 . 4 3 3 0 . 4 4 1
2 0 . 0 0 . 2 4 9 0 . 2 4 8 0 . 2 6 3 0 . 2 6 7 0 . 2 9 1
3 0 . 0 0 . 2 2 9 0 . 2 1 5 0 . 2 4 2 0 . 2 5 5 0 . 3 1 2
4 0 . 0 0 . 2 1 1 0 . 2 2 0 0 . 2 8 1 0 . 3 6 7 0 . 4 3 7
DO
TABLE S10
STRESS. INTERACTIVE METHOD CONFIGURATI0N=2TD
D IM E N SIO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T R U E _ F I T _ ( B E T W E E N RECOVEREC AND TRUE D IS T A N C E S )
S T R I P E JUDGMENTS
1 0 . 6 8 3 0 . 8 0 1 0 . 7 9 5 0 . 7 5 0 0 . 7 1 1 0 . 7 1 1
2 0 . 0 0 . 1 90 0 . 1 9 0 0 . 1 7 9 0 . 1 9 0 0 . 2 1 0
- 3 - • 0 . 0 0 . 1 2 0 0 . 1 2 6 0 . 1 4 4 0 . 1 7 0 0 . 1 9 5
4 0 . 0 0 . 1 0 2 0 . 1 2 7 0 . 1 6 3 0 . 2 0 1 0 . 2 5 6
ALL OBTAINED JUDGMENTS
1 0 . 7 4 0 0 . 8 3 0 0 . 8 2 8 0 . 7 8 4 0 . 7 4 9 0 . 7 4 8
2 0 . 0 0 . 1 7 9 0 . 1 8 4 0 . 1 9 0 0 . 2 0 7 0 . 2 2 5
3 0 . 0 0 . 1 1 6 0 . 1 3 2 0 . 1 6 2 0 . 1 9 3 0 . 2 2 3
4 0 . 0 0 . 1 0 5 0 . 1 4 5 0 . 1 9 6 0 . 2 4 2 0 . 3 0 7
ALL P O S S I B L E JUDGMENTS
1 0 . 8 5 0 0 . 7 8 3 0 . 8 0 7 0 . 8 0 8 0 . 7 8 0 0 . 8 1 0
2 0 . 0 0 . 1 5 9 0 . 1 7 9 0 . 2 1 4 0 . 2 4 5 0 . 2 7 6
3 0 . 0 0 . 1 1 4 0 . 1 5 2 0 . 2 1 8 0 . 2 6 1 0 . 3 0 7
4 0 . 0 0 . 1 2 6 0 . 2 0 5 0 . 2 9 8 0 . 3 5 0 0 . 4 3 2
APPARENT F I T (BETWEEN RECOVERED AND D IST OR T ED D I S T A N C E S )
S T R I P E JUDGMENTS
1 0 . 6 8 3 0 . 7 4 5 0 . 7 4 8 0 . 7 0 2 0 . 6 6 7 0 . 6 7 1
2 0 . 0 0 . 1 6 5 0 . 1 7 4 0 . 1 6 0 0 . 171 0 . 1 8 9
3 0 . 0 0 . 0 7 8 0 . 0 8 3 0 . 0 8 9 0 . 1 0 7 0 . 1 3 6
4 0 . 0 0 . 0 5 0 0 . 0 6 4 0 . 1 0 1 0 . 1 3 5 0 . 2 0 0
ALL OBTAINED JUDGMENTS
1 0 . 7 4 0 0 . 7 9 7 0 . 8 0 0 0 . 7 5 8 0 . 7 2 6 0 . 7 2 6
2 0 . 0 0 . 1 7 7 0 . 1 8 8 0 . 1 9 4 0 . 2 1 1 0 . 2 2 3
3 0 . 0 0 . 1 06 0 . 1 1 4 0 . 1 2 8 0 . 15 4 0 . 1 8 3
4 0 . 0 0 . 0 7 9 0 . 0 9 8 0 . 1 4 1 0 . 1 8 8 0 . 2 6 0
ALL P O S S I B L E JUDGMENTS
1 0 . 8 5 0 0 . 8 2 7 0 . 8 4 7 0 . 8 4 9 0 . 8 2 3 0 . 8 5 2
2 0 . 0 0 . 2 4 0 0 . 2 4 8 0 . 2 7 6 0 . 291 0 . 3 1 1
3 0 . 0 0 . 1 8 8 0 . 1 3 6 0 . 2 0 9 0 . 2 4 0 0 . 2 7 2
4 0 . 0 0 . 1 4 8 0 . 1 6 4 0 . 2 2 8 0 . 2 8 8 0 . 3 6 8
122
TABLE Sll
STRESS• INTERACTIVE METHOD CONFIGURATI0N=3TD
D IM E N S IO N S STANDARD D E V IA T I O N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 C C . 8 0 1 . 0 0
TRkJE_ F H „ t BETWEEN RECOVER ED AND TRUE D IS T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 6 7 3 0 . 8 5 7 0 . 8 6 1 0 . 8 7 8 0 . 8 8 5 0 . 8 4 0
2 0 . 1 5 7 0 . 3 5 4 0 . 3 5 4 0 . 3 5 6 0 . 3 7 1 0 . 3 5 3
3 0 . 0 0 . 1 1 0 0 . 1 2 6 0 . 1 5 3 0 . 1 6 9 0 . 1 8 9
4 0 . 0 0 . 0 7 6 0 . 0 9 3 0 . 1 2 1 0 . 1 6 3 0 . 2 0 9
1 0 . 8 1 8 0 . 9 5 8 0 . 9 7 6 0 . 9 8 6 0 . 9 6 6 0 . 9 3 1
2 0 . 2 9 4 0 . 3 9 3 0 . 4 0 1 0 . 4 0 9 0 . 4 2 0 0 . 4 1 0
3 0 . 0 0 . 1 1 8 0 . 1 4 3 0 . 1 8 1 0 . 2 0 1 0 . 2 2 9
4 0 . 0 0 . 0 8 9 0 . 1 1 8 0 . 1 5 6 0 . 2 0 1 0 . 2 6 1
1 1 . 0 1 5 1 . 1 1 8 1 . 1 1 9 1 . 0 9 0 1 . 0 6 2 1 . 0 4 0
2 0 . 3 8 5 0 . 4 3 0 0 . 4 4 9 0 . 4 5 2 0 . 4 7 2 0 . 4 9 1
3 0 . 0 0 . 1 30 0 . 1 7 2 0 . 2 2 6 0 . 2 6 7 0 . 3 1 6
4 0 . 0 0 . 1 1 3 0 . 1 7 4 0 . 2 2 9 0 . 2 9 8 0 . 3 7 7
APPARENT F I T ( BETWEEN RECOVERED AND D IS T O R T ED D IS T A N C E S )
S T R I P E JUDGMENTS ~
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 6 7 3 0 . 8 3 3 0 . 8 3 9 0 . 8 5 2 0 . 8 5 5 0 . 8 1 6
2 0 . 1 5 7 0 . 3 3 9 0 . 3 3 9 0 . 3 3 7 0 . 3 4 2 0 . 3 2 5
3 0 . 0 0 . 0 9 5 0 . 1 0 5 0 . 1 2 7 0 . 1 3 2 0 . 1 4 6
4 0 . 0 0 . 0 50 0 . 0 5 8 0 . 0 7 8 0 . 1 1 9 0 . 1 6 3
1 0 . 8 1 8 0 . 9 4 4 0 . 9 6 6 0 . 9 7 5 0 . 9 5 4 0 . 9 2 9
2 0 . 2 9 4 0 . 3 8 6 0 . 3 9 8 0 . 4 0 5 0 . 4 0 9 0 . 4 0 4
3 0 . 0 0 . 1 1 5 0 . 1 3 9 0 . 1 7 2 C . 1 8 4 0 . 2 1 1
4 0 . 0 0 . 0 7 9 0 . 1 0 1 0 . 1 3 3 0 . 1 7 4 0 . 2 3 0
1 1 . 0 1 5 1 . 1 6 0 1 . 1 6 4 1 . 1 35 1 . 1 1 2 1 . 1 0 1
2 0 . 3 8 5 0 . 4 6 5 0 . 4 3 5 0 . 4 8 6 0 . 5 0 1 0 . 5 2 6
3 0 . 0 0 . 1 76 0 . 2 0 2 0 . 2 3 6 0 . 2 6 8 0 . 3 1 2
4 0 . 0 0 . 1 3 9 0 . 1 6 5 0 . 2 0 3 0 . 2 6 2 0 . 3 3 8
123
TABLE SI 2
STRESS. IN T E R A C T IV E METHOD CONFIGURATIGN=4TD
D IM ENSION S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T (BETWEEN RECOVE RED AND TRUE D I S T A N C E S ;
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 7 4 4 0 . 8 9 5 0 . 9 0 2 0 . 9 2 7 0 . 9 2 5 0 . 8 8 8
2 0 . 3 2 9 0 . 3 9 5 0 . 3 9 2 0 . 4 1 5 0 . 4 1 6 0 . 4 2 1
3 0 . 1 2 1 0 . 1 8 3 0 . 2 0 3 0 . 2 0 1 0 . 2 1 7 0 . 2 3 6
4 0 . 0 0 . 0 9 8 0 . 1 1 4 0 . 1 3 0 0 . 1 6 2 0 . 2 1 1
1 0 . 8 8 8 1 . 0 1 5 1 . 0 2 9 1 . 0 6 0 1 . 0 5 1 1 . 0 2 2
2 0 . 4 4 0 0 . 4 9 0 0 . 4 8 9 0 . 5 1 8 0 . 5 1 0 0 . 5 0 8
3 0 . 1 8 8 0 . 2 5 1 0 . 2 6 8 0 . 2 7 2 0 . 2 7 4 0 . 2 8 8
4 0 . 0 0 . 1 4 4 0 . 1 5 2 0 . 1 7 2 0 . 2 0 7 0 . 2 5 5
1 1 . 1 3 6 1 . 2 7 9 1 . 2 9 2 1 . 2 7 9 1 . 2 3 4 1 . 2 2 6
2 0 . 5 8 9 0 . 5 9 2 0 . 5 9 6 0 . 6 2 7 0 . 6 1 4 0 . 6 1 7
3 0 . 2 1 3 0 . 3 1 0 0 . 3 4 1 0 . 3 5 1 0 . 3 5 6 0 . 3 7 1
4 0 . 0 0 . 1 8 3 0 . 2 1 3 0 . 2 4 7 0 . 3 0 2 0 . 3 5 8
APPARENT F I T (BETWEEN RECOVERED AND D I S T ORTED D IS T A N C E S )
S T R I P E JUDGMENTS “
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 7 4 4 0 . 8 7 3 0 . 8 7 9 0 . 9 0 4 0 . 9 0 3 0 . 8 6 7
2 0 . 3 2 9 0 . 3 7 6 0 . 3 7 3 0 . 3 9 4 0 . 3 9 3 0 . 3 9 0
3 0 . 1 2 1 0 . 1 6 2 0 . 1 7 9 0 . 1 7 3 0 . 1 8 5 0 . 1 9 5
4 0 . 0 0 . 0 7 5 0 . 0 8 2 0 . 0 9 0 0 . 1 1 7 0 . 1 7 4
1 0 . 8 8 8 1 . 0 0 2 1 . 0 1 7 1 . 0 5 1 1 . 0 4 7 1 . 0 2 0
2 0 . 4 4 0 0 . 4 8 0 0 . 4 8 1 0 . 5 1 0 0 . 5 0 4 0 . 4 9 9
3 0 . 1 8 8 0 . 2 4 2 0 . 2 5 8 0 . 2 6 2 0 . 2 6 4 0 . 2 7 2
4 0 . 0 0 . 1 3 7 0 . 1 3 9 0 . 1 5 6 0 • 1 8 4 0 . 2 3 6
1 1 . 1 3 6 1 . 3 2 8 1 . 3 4 2 1 . 3 3 4 1 . 2 9 8 1 . 3 0 2
2 0 . 5 8 9 0 . 6 2 8 0 . 6 3 3 0 . 6 6 4 0 . 6 5 5 0 . 6 6 6
3 0 . 2 1 3 0 . 3 4 3 0 . 3 6 7 0 . 3 7 1 0 . 3 7 3 0 . 3 9 0
4 0 . 0 0 . 2 1 5 0 . 2 2 3 0 . 2 4 1 0 . 2 8 4 0 . 3 3 8
124
TABLE SI 3
STRESS PRE-SET BASIS ( 5 STIMULI) CONFIGURATI0N=4TD
D IM E N SIO N S STANDARD D E V IA T IO N OF ERROR
EXTRACTED 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
TRUE F I T (BETWEEN RECOVERED AND TRUE D I S T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 7 4 4 0 . 9 6 1 0 . 9 5 7 0 . 9 5 1 0 . 9 5 9 0 . 9 5 0
2 0 . 3 2 9 0 . 4 3 4 0 . 4 2 9 0 . 4 2 3 0 . 4 2 7 0 . 4 4 1
3 0 . 121 0 . 1 9 9 0 . 2 1 4 0 . 2 2 4 0 . 2 3 7 0 . 2 5 1
4 0 . 0 0 . 0 9 7 0 . 1 0 8 0 . 1 2 7 0 . 1 5 6 0 . 2 0 3
1 0 . 7 4 4 0 . 9 6 1 0 . 9 5 7 0 . 9 5 1 0 . 9 5 9 0 . 9 5 0
2 0 . 3 2 9 0 . 4 3 4 0 . 4 2 9 0 . 4-23 0 . 4 2 7 0 . 4 4 1
3 0 . 121 0 . 1 9 9 0 . 2 1 4 0 . 2 2 4 0 . 2 3 7 0 . 2 5 1
4 0 . 0 0 . 0 9 7 0 . 1 0 8 0 . 1 2 7 0 . 156 0 . 2 0 3
1 1 . 1 3 6 1 . 2 7 9 1 . 2 5 9 1 . 2 3 4 1 . 2 2 7 1 . 1 9 8
2 0 . 5 8 9 0 . 6 7 5 0 . 6 5 3 0 . 6 2 9 0 . 6 1 0 0 . 5 9 8
3 0 . 2 1 3 0 . 2 8 9 0 . 3 1 9 0 . 3 3 4 0 . 3 4 7 0 . 3 6 1
4 0 . 0 0 . 1 5 6 0 . 1 7 9 0 . 2 1 9 0 . 2 7 5 0 . 3 4 7
APPARENT F I T (BETWEEN RECOVERED AND D I S T O RTED D IS T A N C E S )
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 7 4 4 0 . 9 4 5 0 . 9 4 1 0 . 9 3 5 0 . 9 3 8 0 . 9 2 7
2 0 . 3 2 9 0 . 4 2 6 0 . 4 1 8 0 . 4 0 8 0 . 4 0 7 0 . 4 1 7
3 0 . 1 2 1 0 . 1 83 0 . 1 9 4 0 . 1 9 7 0 . 2 0 3 0 . 2 1 0
4 0 . 0 C . 0 7 8 0 . 0 7 6 0 . 0 8 4 0 . 1 0 7 0 . 1 5 5
1 0 . 7 4 4 0 . 9 4 5 0 . 9 4 1 0 . 9 3 5 0 . 9 3 8 0 . 9 2 7
2 0 . 3 2 9 0 . 4 2 6 0 . 4 1 8 0 . 4 0 8 0 . 4 0 7 0 . 4 1 7
3 0 . 1 2 1 0 . 1 8 3 0 . 1 9 4 0 . 1 9 7 0 . 2 0 3 0 . 2 1 0
4 0 . 0 0 . 0 7 8 0 . 0 7 6 0 . 0 8 4 0 . 1 0 7 0 . 1 5 5
1 1 . 1 3 6 1 . 3 2 7 1 . 3 1 0 1 . 2 9 0 1 . 2 9 2 1 . 2 7 4
2 0 . 5 8 9 0 . 7 1 6 0 . 6 9 3 0 . 6 7 1 0 . 6 5 5 0 . 6 4 8
3 0 . 2 1 3 0 . 3 2 5 0 . 3 5 1 0 . 3 6 2 0 . 3 7 3 0 . 3 8 6
4 0 . 0 0 . 1 9 4 0 . 1 9 8 0 . 2 2 1 0 . 2 6 6 0 . 3 3 2
125
STRESS,
TABLE S14
PRE-SET BASIS ( 7 STIMULI)
CONFIGURATI0N=4TD
D IM E N SIO N S
EXTRACTED
STANDARD D E V IA T I O N OF ERROR
0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T (BETWEEN RECOVERED AND TRUE D I S TANCE S ]
S T R I P E JUDGMENTS ~
ALL OBTAINED JUDGMENTS
ALL P O S S I B L E JUDGMENTS
1 0 . 8 3 2 1 . 0 1 0 1 . 0 0 1 0 . 9 9 3 0 . 9 8 7 0 . 9 8 5
2 0 . 4 0 5 0 . 4 9 3 0 . 4 8 8 0 . 4 8 3 0 . 4 7 8 0 . 4 7 4
3 0 . 1 6 7 0 . 2 2 6 0 . 2 3 9 0 . 2 4 8 0 . 2 6 2 0 . 2 7 2
4 0 . 0 0 . 0 9 7 0 . 1 C 9 0 . 1 2 9 0 . 1 54 0 . 1 8 3
1 0 . 8 3 2 1 . 0 1 0 1 . 0 0 1 0 . 9 9 3 0 . 9 8 7 0 . 9 8 5
2 0 . 4 0 5 0 . 4 9 3 0 . 4 3 8 0 . 4 8 3 0 . 4 7 8 0 . 4 7 4
3 0 . 1 6 7 0 . 2 2 6 0 . 2 3 9 0 . 2 4 8 0 . 2 6 2 0 . 2 7 2
4 0 . 0 0 . 0 9 7 0 . 1 0 9 0 . 1 2 9 0 . 1 5 4 0 . 1 8 3
1 1 . 1 5 0 1 . 2 5 6 1 . 2 4 0 1 . 2 2 2 1 . 2 0 5 1 . 1 8 8
2 0 . 4 8 7 0 . 5 8 0 0 . 5 7 8 0 . 5 8 6 0 . 5 8 5 0 . 5 8 1
3 0 . 2 0 5 0 . 2 6 0 0 . 2 8 8 0 . 3 0 6 0 . 3 2 8 0 . 3 4 5
4 0 . 0 0 . 1 0 6 0 . 1 4 0 0 . 1 8 5 0 . 2 3 2 0 . 2 8 0
APPARENT F I T (BETWEEN RECOVERED AND D I ST O RTED D IS T A N C E S >
S T R I P E JUDGMENTS
ALL OBTAINED JUDGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 3 3 2 0 . 9 9 5 0 . 9 8 7 0 . 9 8 0 0 . 9 7 5 0 . 9 7 5
2 0 . 4 0 5 0 . 4 7 7 0 . 4 7 2 0 . 4 6 6 0 . 4 6 0 0 . 4 5 6
3 0 . 1 6 7 0 . 2 1 3 0 . 2 2 2 0 . 2 2 7 0 . 2 3 6 0 . 2 4 6
4 0 . 0 0 . 0 7 7 0 . 0 8 5 0 . 1 0 0 0 . 1 2 2 0 . 1 4 7
1 0 . 8 3 2 0 . 9 9 5 0 . 9 8 7 0 . 9 8 0 0 . 9 7 5 0 . 9 7 5
2 0 . 4 0 5 0 . 4 7 7 0 . 4 7 2 0 . 4 6 6 0 . 4 6 0 0 . 4 5 6
3 0 . 1 6 7 0 . 2 1 3 0 . 2 2 2 0 . 2 2 7 0 . 2 3 8 0 . 2 4 6
4 0 . 0 0 . 0 7 7 0 . 0 8 5 0 . 1 0 0 0 . 1 22 0 . 1 4 7
1 1 . 1 5 0 1 . 3 0 3 1 . 2 9 1 1 . 2 8 1 1 . 2 7 3 1 . 2 6 8
2 0 . 4 8 7 0 . 6 1 4 0 . 6 1 4 0 . 6 2 6 0 . 6 3 0 0 . 6 3 4
0 . 2 0 5 0 . 2 9 4 0 . 3 1 8 0 . 3 3 5 0 . 3 5 6 0 . 3 7 5
4 0 . 0 0 . 1 4 4 0 . 1 6 1 0 . 1 9 2 0 . 2 3 2 0 . 2 7 7
126
TABLE SI 5
STRESS. PRE-SET BASIS ( 9 STIMULI) CONFIGURATI0N=4TD
D I M E N S I O N S STANDARD D E V I A T I O N OF ERR OR
E X T R A C T E D 0 . 0 0 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 C 1 . 0 0
T RUE F I T (B E T W E E N R E C O V E R ED
S T R I P E JU DGM EN TS
ALL O B T A I N E D JU DGM ENTS
ALL P O S S I B L E JU DGM ENTS
AND TRUE D I S T A N C E S )
1 0 . 8 8 6 1 . 0 0 9 1
2 0 . 3 8 6 0 . 4 5 7 0
3 0 . 1 6 5 0 . 2 1 7 0
4 0 . 0 0 . 0 9 2 0
1 0 . 8 8 6 1 . 0 0 9 1
2 0 . 3 6 6 0 . 4 5 7 0
3 0 . 1 6 5 0 . 2 1 7 0
4 0 . 0 0 . 0 9 2 0
1 1 . 1 1 6 1 . 1 6 5 1
2
0 . 4 1 7 0 . 4 7 5 0
3 0 . 2 1 4 0 . 2 6 1 0
4 0 . 0 0 . 1 0 1 0
. 0 1 5 1 . 0 2 3 1 . 0 2 8 1 . 0 3 0
. 4 6 3 0 . 4 6 9 0 . 4 7 4 0 . 4 8 4
. 2 2 3 0 . 2 3 3 0 . 2 4 7 0 . 2 5 8
. 1 0 6 0 . 1 2 5 0 . 1 4 9 0 . 1 7 3
. 0 1 5 1 . 0 2 3 1 . 0 2 8 1 . 0 3 0
. 4 6 3 0 . 4 6 9 0 . 4 7 4 0 . 4 8 4
. 2 23 0 . 2 3 3 0 . 2 4 7 0 . 2 5 8
. 1 0 6 0 . 1 2 5 0 . 1 4 9 0 . 1 7 3
. 1 6 8 1 . 1 6 9 1 . 1 6 6 1 . 1 5 9
. 4 8 4 0 . 4 9 6 0 . 5 0 9 0 . 5 2 5
. 2 6 8 0 . 2 8 1 0 . 2 9 8 0 . 3 1 6
. 1 3 3 0 . 1 7 1 0 . 2 1 1 0 . 2 4 9
A P P A R E N T F I T ( B E T W E E N R E C O V E R E D AND D I S T O R T E D D I S T A N C E S )
S T R I P E JU D G M EN TS
ALL O B T A I N E D JU DGM ENTS
ALL POSSIBLE JUDGMENTS
1 0 . 8 8 6 0 . 9 9 0 0 . 9 9 9 1 . 0 0 9 1 . 0 1 7 1 . 0 2 3
2 0 . 3 8 6 0 . 4 4 7 0 . 4 5 5 0 . 4 6 2 0 . 4 6 8 0 . 4 8 0
3 0 . 1 6 5 0 . 2 0 8 0 . 2 1 4 0 . 2 2 3 0 . 2 3 8 0 . 2 4 9
4 0 . 0 0 . 0 8 0 0 . 0 9 2 0 . 1 1 0 0 . 1 3 2 0 . 1 5 5
1 0 . 8 8 6 0 . 9 9 0 0 . 9 9 9 1 . 0 0 9 1 . 0 1 7 1 . 0 2 3
2 0 . 3 8 6 0 . 4 4 7 0 . 4 5 5 0 . 4 6 2 0 . 4 6 8 0 . 4 8 0
3 0 . 1 6 5 0 . 2 0 8 0 . 2 1 4 0 . 2 2 3 0 . 2 3 8 0 . 2 4 9
4 0 . 0 0 . 0 8 0 0 . 0 9 2 0 . 1 1 0 0 . 1 3 2 0 . 1 5 5
1 1 . 1 1 6 1 . 2 0 9 1 . 2 1 6 1 . 2 2 4 1 . 2 3 0 1 . 2 3 5
2 0 . 4 1 7 0 . 5 1 1 0 . 5 2 2 0 . 5 3 8 0 . 5 5 7 0 . 5 8 2
3 0 . 2 1 4 0 . 2 9 4 0 . 3 0 0 0 . 3 1 3 0 . 3 3 1 0 . 3 5 1
4 0 . 0 0 . 1 4 3 0 . 1 5 8 0 . 1 8 6 0 . 2 2 0 0 . 2 5 6
127
TABLE SI 6
STRESS, PRE-SET BASIS <10 STIMULI)
C0NFIGURATI0N=4TD
D I M E N S I O N S
E X T R A C T E D
STANDARD D E V I A T I O N OF ER ROR
O . O C 0 . 2 0 0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0
T RUE F I T ( B E T W E E N R E C O V E R E D AND TRUE D I S T A N C E S )
S T R I P E JU D G M E N T S ~
ALL O B T A I N E D JU DGM EN TS
ALL P O S S I B L E JU DGM EN TS
S T R I P E JU DGM ENTS
ALL O B T A I N E D JU DGMENTS
ALL POSSIBLE JUDGMENTS
1 0 . 9 1 0 1 . 0 2 7 1 . 0 2 9 1 . 0 3 3 1 . 0 3 5 1 . 0 3 3
2 0 . 3 9 0 0 . 4 3 3 0 . 4 4 5 0 . 4 6 1 0 . 4 7 2 0 . 4 8 4
3 0 . 1 6 1 0 . 2 1 7 0 . 2 2 2 0 . 2 2 9 0 . 2 3 9 0 . 2 5 0
4 0 . 0 0 . 0 9 0 0 . 1 0 5 0 . 1 2 5 0 . 1 4 8 0 . 1 7 3
1 0 . 9 1 0 1 . 0 2 7 1 . 0 2 9 1 . 0 3 3 1 . 0 3 5 1 . 0 3 3
2 0 . 3 9 0 0 . 4 3 3 0 . 4 4 5 0 . 4 6 1 0 . 4 7 2 0 . 4 8 4
3 0 . 1 6 1 0 . 2 1 7 0 . 2 2 2 0 . 2 2 9 0 . 2 3 9 0 . 2 5 0
4 0 . 0 0 . 0 9 0 0 . 1 0 5 0 . 1 2 5 0 . 1 4 8 0 . 1 7 3
1 1 . 1 0 9 1 . 1 5 6 1 . 1 5 7 1 . 1 5 8 1 . 1 5 7 1 . 1 5 0
2 0 . 4 1 1 0 . 4 5 9 0 . 4 7 3 0 . 4 9 3 0 . 5 1 3 0 . 5 3 7
3 0 . 2 0 2 0 . 2 5 3 0 . 2 6 3 0 . 2 7 7 0 . 2 9 3 0 . 3 1 1
4 0 . 0 0 . 0 9 9 0 . 1 2 9 0 . 1 6 6 0 . 2 0 4 0 . 2 4 0
:d AND D I S T O R T E D D I S T A N C E S )
1 0 . 9 1 0 1 . 0 1 7 1 . 0 2 0 1 . 0 2 7 1 . 0 3 3 1 . 0 3 5
2 0 . 3 9 0 0 . 4 3 0 0 . 4 4 4 0 . 4 6 1 0 . 4 7 2 0 . 4 8 4
3 0 . 1 6 1 0 . 2 1 2 0 . 2 1 7 0 . 2 2 4 0 . 2 34 0 . 2 4 6
4 0 . 0 0 . 0 8 6 0 . 0 9 6 0 . 1 1 3 0 . 1 3 4 0 . 1 5 8
1 0 . 9 1 0 1 . 0 1 7 1 . 0 2 0 1 . 0 2 7 1 . 0 3 3 1 . 0 3 5
2 0 . 3 9 0 0 . 4 3 0 0 . 4 4 4 0 . 4 6 1 0 . 4 7 2 0 . 4 8 4
3 0 . 1 6 1 0 . 2 1 2 0 . 2 1 7 0 . 2 2 4 0 . 2 3 4 0 . 2 4 6
4 0 . 0 0 . 0 8 6 0 . 0 9 6 0 . 1 1 3 0 . 1 3 4 0 . 1 5 8
1 1 . 1 0 9 1 . 2 0 0 1 . 2 0 5 1 . 2 1 4 i . 2 2 1 1 . 2 2 7
2 0 . 4 1 1 0 . 4 9 8 0 . 5 1 3 0 . 5 3 8 0 . 5 6 3 0 . 5 9 6
3 0 . 2 0 2 0 . 2 8 8 0 . 2 9 6 0 . 3 1 0 0 . 3 2 9 0 . 3 5 0
4 0 . 0 0 . 1 4 2 0 . 1 5 7 0 . 1 8 5 0 . 2 1 8 0 . 2 5 2
D O
03

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Asset Metadata

Creator Girard, Roger Alan (author)

Core Title A Monte Carlo Evaluation Of Interactive Multidimensional Scaling

Degree Doctor of Philosophy

Degree Program Psychology

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,psychology, general

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Cliff, Norman (committee chair), Abrahamson, Stephen (committee member), Lovell, Constance (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-714516

Unique identifier UC11364293

Identifier 7401666.pdf (filename),usctheses-c18-714516 (legacy record id)

Legacy Identifier 7401666

Dmrecord 714516

Document Type Dissertation

Rights Girard, Roger Alan

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA