INDSCAL
provides internal metric analysis of a "stack" of dis/similarity (or
correlation) matrices in terms of a weighted distance model, such that each "individual" (or
data-source) has a set of dimensional weights which systematically "distort"
the Group Stimulus space to produce a "Private" space.
Data:
3-way, 2-mode dis/similarities (or correlations)
Transform: Linear
Model: Weighted Euclidean Distance (or
Canonical Decomposition)
INDSCAL was originally developed to explain the relationship between subjects' differential cognition of a set of stimuli. Suppose that there are N subjects and p stimuli. The program takes as input a set of N matrices each of which is a square symmetric matrix (of order p) of (dis)similarity judgments/measures between the p stimuli. The model explains differences between subjects' cognitions by a variant of the distance model. The stimuli are thought of as points positioned in a 'Group' or 'master' Stimulus space. This space is perceived differentially by the subjects in that each of them affords a different salience or weight to each of the dimensions of the space. In the graphic displays of the 2 dimensional subject space , an additional menu item Vectors optionally enables you to plot the subjects as vectors and the line representing equal weighting. The transformation which is assumed to take the data into the solution is a linear one. Note that on closing the graphic displays of subject spaces, it is also possible to submit arc-distances in the space to further analysis using SUBJSTAT. Note that a subject space, by convention, is always represented in the positive quadrant of the plotted space, i.e. the coordinate values are all positive.
The INDSCAL model is a special case of CANDECOMP (where the second and third 'ways' of the data matrix are identical), and is also akin to the P1 model in the PINDIS hierarchy of models.INDSCAL is an
expressly dimensional model and produces a unique orientation of the axes of
the Group Space, in the sense that any rotation will destroy the optimality of
the solution and will change the values of the subject weights. Moreover, the
distances in the Group Space are weighted Euclidean, whereas those in the
private spaces are simple Euclidean. Because of this, it is not legitimate to
rotate the axes of a Group Space to a more 'meaningful' orientation, as is
commonly done both in factor analysis and in the basic multidimensional scaling
model. It has generally been found that the recovered dimensions yield readily
to interpretation. In many uses of INDSCAL, the third way consists not of individual "subjects" but of aggregated
subgroups ("pseudo-subjects"), and indeed different replications, time-series, methods,
places, etc.
The
N (dis)similarity matrices of order p must follow the READ MATRIX
command sequentially.
At
the beginning of an INDSCAL analysis each input matrix of dis/similarities,
or distances is converted into a matrix of scalar products. To equalize each
subject's influence on the analysis these data are normalized by scaling each
scalar products matrix so that its sum of squares equals one. Data input as product-moment
covariances or correlations are already scalar products and do not need
converting in this way. Thus
it is essential to signal this type of input by means of the DATA TYPE
parameter (see below).
DIMENSIONS
Some experimentation is generally needed to determine how many dimensions are
appropriate for a given set of data. This involves analyzing the data in spaces
of different dimensionality. For each space of r dimensions the program uses as
a starting configuration the solution in (r + 1) dimensions less the dimension
accounting for the least variance. Usually between two and four dimensional
solutions will be adequate for any reasonable data set.
The
starting configuration
The analysis begins with an initial configuration of stimulus points. This may
be supplied by the user following the READ CONFIG command. The configuration
should contain stimuli coordinates in the maximum dimensionality required
Alternatively
the program can generate a pseudo-random starting configuration if the value of
the parameter RANDOM is 0. If RANDOM is assigned a non-zero value
this is used as a seed to generate the random numbers. Since sub- optimal
solutions are not uncommon with this method users are strongly recommended to
make several runs with different starting configurations. A series of similar
(or identical) solutions may be taken to indicate that a true 'global' solution
has been found.
Alternatively,
the user may wish to minimize this particular difficulty by submitting as an
initial configuration one obtained from, say, a MINISSA
run in which the averaged judgements have been analysed. This method will also
reduce the amount of machine time taken to reach a solution.
READ
CONFIG / FIX POINTS
It is sometimes useful to determine only subject weights for some previously
determined stimulus configuration, such as a previous INDSCAL solution,
or, some known configuration. This makes it possible to use INDSCAL in an
external mode. This configuration may be supplied following the READ
CONFIG command.
The
full set of data should follow READ MATRIX but FIX POINTS should
be set to 1 in the PARAMETERS command and the program will then solve
only for the subject weights.
This
option is particularly useful when the user has more data than the program is
capable of handling. The user can input the configuration obtained either from
a MINISSA analysis of averaged judgments or from
an INDSCAL analysis of some judiciously (or randomly) selected subset of
subjects and fit to it any number of subjects' weights.
If
the user wishes to constrain the solution as closely as possible to
orthogonality (i.e. in the sense that the correlation between the coordinates
is zero) then the parameter SOLUTIONS should be set to 1 in the PARAMETERS
statement. Users are warned that this will necessarily produce a suboptimal
solution.
INPUT
COMMANDS
Keyword Function
N OF STIMULI [number] Number
of stimuli for analysis
N
OF SUBJECTS [number] Number
of subjects for which
data
are to be input
DIMENSIONS [number]
[number
list] Dimensions
for analysis
[number]
TO [number]
LABELS [followed by a
series
Optionally identify the
stimuli,
of labels (<= 65
chars)
followed by the subjects, as
each on a
separate
required. All labels should
be
line]
entered, without omissions.
PARAMETERS
Keyword Default
Value Function
SOLUTIONS 0 0:
Compute all dimensions simultaneously
1:
Compute separate one-dimensional
solutions.
FIX
POINTS 0 0:
Iterate and solve for all matrices.
1:
Solve for subject weights only.
RANDOM 0
Random
number seed for generating the
initial
configuration. (When the user does
not
provide an initial configuration
using
the READ CONFIG command)
DATA
TYPE 1 0:
IDIOSCAL starting configuration.
1:
Lower triangle similarity matrix
(without
diagonals).
2:
Lower triangle dissimilarity matrix
(without
diagonals).
3:
Lower triangle euclidean distances
(without
diagonals).
4:
Lower triangle correlation matrix
(without
diagonals).
5:
Lowerhalf covariance matrix
(with
diagonals).
6:
Full symmetric similarity matrix
(diagonals
ignored).
7:
Full symmetric dissimilarity matrix
(diagonals
ignored).
CRITERION 0.005 Sets
criterion value for termination of
iterations.
MATFORM 0 0:
Input configuration is stimuli
(rows)
by dimensions (columns).
1:
Input configuration is dimensions
(rows)
by stimuli (columns).
Valid
only with READ CONFIG.
PRINT
options
(to the main output file)
Option Form Description
INITIAL N
x r Three
matrices are output:
p
x r 1.
the initial estimates of the subject
weights.
2.
& 3. separate estimates of the
stimulus
configuration.
FINAL N
x r Two
matrices are output:
p
x r the
matrix of subject weights and the
coordinates
of the group space.
These
are followed by the correlation
N between
each subject's data and
solution
and the matrix of cross-
r
x r products
between the dimensions.
HISTORY An
iteration by iteration history
of
the overall correlation. (The final
(3)
matrices at convergence are also
output).
SUMMARY Summary
of results produced at the end
of
each analysis.
By
default only the solution matrices and the final overall
correlation are output.
PLOT
options
(to the main output file)
Option Description
INITIAL The
initial configuration may be
plotted
only if one is input by the user.
CORRELATIONS The
correlations at each iteration are plotted.
GROUP Up
to r(r-1)/2 plots of the p stimulus points.
SUBJECTS Up
to r(r-1)/2 plots of the Subject Space.
Note, however, that it is mistaken to
regard
these spaces
as Euclidean.
SUBJSTAT offers
an arc-distance measure for the analysis of
distances between items in subject spaces.
By
default the Subject and Group Spaces will be plotted.
PUNCH
options
(to a secondary output file)
Option Description
FINAL Outputs
the final configuration
and
the subject correlations in
the
following order:
-
each subject is followed by the
coordinates
of its weight on
each
dimension;
-each
stimulus point is followed
by
its coordinates on each dimension.
CORRELATIONS The
overall correlation at each
iteration
is output in a fixed format.
SCALAR PRODUCTS the scalar product matrix
is saved.
By default, no secondary output file is generated.
PROGRAM LIMITS
Maximum no. of dimensions = 5
Maximum no. of stimuli = 200
Maximum no. of subjects = 200
N OF SUBJECTS x N OF STIMULI x N
OF STIMULI = 8,000,000
See also