In data
collection using the method of triads, subjects are presented with sets of 3
objects drawn from a larger set and asked to judge the relative (dis)similarity
of the objects involved. In its most comprehensive form, the subject
is presented with all possible triads, but BIBDs (balanced incomplete block
designs) exist to reduce this number and still satisfy desirable statistical
properties.
Any triad of objects [1 2 3] contains three constituent
pairs: namely [1 2], [2 3] and [1 3] and the triadic judgment is often described
as "contextual" since the subject is asked to make a judgment of dis/similarity
in the presence of differing third objects. When presented with a triad, there
are two distinct methods of data collection:
PARTIAL: "which is the most
similar [MS] pair of these three ?" (This is the most common
form)
COMPLETE
:
"which is the most similar [MS] pair and which the least similar pair [LS]
of these three ?". The reason why this is termed "complete" is that it
follows from the MS and LS judgment that the third pair is intermediate between
the MS and LS pair.
TRISOSCAL
interprets these (dis)similarities as distances between the
objects and then represents the points in a space of minimum dimensionality.
Because triadic data are more fine-grain than (say) paired comparisons, there is
a
greater possibility of inconsistency (e.g. [12] may be judged as most similar
in the presence of 3, but not as most similar in the context of 4). TRISOSCAL is
unique among triadic scaling programs in that it makes it possible either
to average over all instances when estimating "the" dissimilarity of a
pair -- the "local" (0) option in
the STRESS parameter, which in
effect washes out contextual effects. In other triadic scaling programs, this
generates a "vote-count" matrix, which forms the input data for analysis But there is an alternative and more
rigorous approach, which keeps the
integrity of the triad, and penalising by increasing badness-of-fit if there are inconsistencies - this is
the more demanding "global" option (1) in STRESS. This option is unique to the
NewMDSX programs, and was developed by Mike Prentice at Edinburgh University. Obviously, global stress
will be higher than local stress, and if a set of data contains different
subjects' data, it may be considerably higher. If there are data from different
subjects, it is often better to scale each subject's data using STRESS(1) and
then compare the overall data by inputting the subjects' configurations in
PINDIS.
If
the first of the above methods has been used in obtaining the data, the
user should specify ORDER (0) in the PARAMETERS command. If the
method producing a strict ordering has been used then ORDER (1) should
be specified.
By default,
the data will be interpreted as similarities. By specifying DATA
TYPE (1) in the PARAMETERS command the data will be interpreted
as dissimilarities.
The
number of objects to be positioned as points in the space is specified by N
OF STIMULI (or N OF POINTS), and the number of actual
triads presented to the program by N OF TRIADS.
Each
object should be identified by an integer, and thus each triad
consists of three integers, say [5 2 4], which are
interpreted in the following way:
ORDER
(0)
The pair which is chosen as the most similar is designated by the first pair of
numbers of the three. Thus in our example the pair [5 2] is that chosen.
If the subject has been asked which pair is the most dissimilar then the pair
chosen should again be the pair defined by the first two numbers, but in this
case the parameter DATA TYPE should be given the value 1 in the PARAMETERS
command.
ORDER
(1)
When the subject has been asked to
choose both the most (dis)similar and the least (dis)similar pair, then the
triad must be capable of interpretation in the following way: The
first pair of numbers defines the pair chosen as the most (dis)similar. The pair
consisting of the first and last number is that chosen as the least (dis)similar
and the pair consisting of the second and third numbers is thus the "middle"
pair. For example, for the triad [5 2 4], the pair [5 2] is the most
similar, the pair [2 4] the next most similar and the pair [5 4] the least
similar. In contrast to the method with ORDER(0)
, the ordering of the first two items in the triad is also
significant.
Distances
in the configuration
The user may choose the way in which the distance between the points in the
configuration is measured by means of the MINKOWSKI parameter. The
default value 2 provides for the ordinary Euclidean metric where the distances
between two points will be the length of the line joining them. The user may
specify any value for the parameter. Commonly used values, however, include 1,
the so-called 'city-block' or 'taxi-cab' metric where the distance between the
two points is the sum of the differences between their co-ordinates on the axes
of the space, and infinity (in TRISOSCAL approximated by a large
number(>25)), the so-called 'dominance' metric when the largest difference
on any one axis will eventually come to dominate all others. (Users are warned
that high values of the MINKOWSKI parameter are liable to produce
program failure due to overflow).
The
initial configuration
A vote count matrix is formed which is used to generate an initial
configuration in the same way as the Guttman-Lingoes-Roskam MINI programs. This
configuration uses only the ordinal properties of the vote count matrix and has
certain desirable properties such as avoiding local minima.
If
the user wishes to supply an initial configuration then this is input following
the READ CONFIG command, (and an associated INPUT FORMAT if the
data cannot be read in free format). The configuration must be in the maximum
dimensionality to be used in the solution.
Setting STRESS(1)
gives an option offered in no other program, to keep GLOBAL
monotonicity over the traidic judgments in obtaining a solution.
PARAMETERS
Keyword Default Value Function
DATA TYPE 0 0:
Input data are similarities
1:
Input data are dissimilarities.
MINKOWSKI 2.0 (Any
positive number) sets the
Minkowski parameter
for
determination of distances
in
the configuration.
ORDER
0 0: Partial order is input
1:
Full order is input
STRESS 0
0: STRESS calculated using "local"
approach.
1:
STRESS calculated using "global"
approach.
NOTES
1. N OF POINTS , or N OF STIMULI, is mandatory in TRISOSCAL.
2. N OF TRIADS is needed if input is less than the complete
set of possible triads of these points.
3. The optional command LABELS improves the identification of stimuli
in the output. The expected number of variable labels (<= 65
characters) should follow on successive lines of input, and should identify all
variables, without omissions.
PRINT
options
Option Form Description
INITIAL p x r The
co-ordinates of the points in the
initial
configuration are output.
FINAL p
x r The solution matrix,
the co-ordinates
of
the stimulus points in the final
configuration,
is output.
DISTANCES p x p The
matrix of interpoint distances in
(lower
triangle the final configuration is output.
only)
FITTING p x p The
matrix of fitting values is output.
(lower
triangle
only)
RESIDUALS p x p The
matrix of residuals (distances-
(lower triangle fitting
values) is output.
only)
HISTORY A
detailed history of the iterative
process
is output.
COUNT p x p The
vote-count matrix as derived from
(lower
triangle the triadic comparisons is
output.
only)
GRADIENT p x r The
matrix at gradients as applied
to
the final configuration is output.
By
default only the final configuration is output.
PLOT
options
Option Description
INITIAL The
initial configuration is plotted as
r(r-1)/2
two-way plots.
FINAL The
solution is plotted as r(r-1)/2
two-way
plots.
SHEPARD The
Shepard diagram of data against
distances
is plotted.
POINT A
histogram of the contribution to
STRESS
of each point is plotted.
RESIDUALS A
histogram of residual values is
produced.
STRESS A
histogram of the STRESS values at each
iteration
is produced.
By
default only the Shepard diagram and the FINAL configuration are plotted.
PUNCH
options
(to secondary output file)
Option Description
FINAL The
solution configuration is output,
indexed
in a fixed format.
SPSS The
following are output in a fixed format:
I
= row index
J
= column index
VOTE
= entry in votecount matrix
to
I,J
DIST
= the correspondiong distance
FITTING
= the corresponding fitted value
RESID
= the corresponding residual value
STRESS An
iteration by iteration history of
STRESS
values is output in a fixed
format.
By
default, no secondary output file is produced.
PROGRAM
LIMITS
Maximum no. of points = 50
Maximum no. of triads = 3333
Maximum no. of dimensions = 8
See also