takes as its input data a number of configurations. These will normally be the
result of some previous scaling analysis, although any technique giving
dimensional output (such as factor or components analysis) is also suitable. The points
in each of the configurations must be the same although the dimensionality of
the spaces is allowed to differ between configurations. Most importantly,
PINDIS is not a single model, but
a hierarchy.
The
simplest model (P0) is often used as it stands to perform a simple Procrustes analysis,
which puts a set of configurations into maximum conformity with each other by
performing "permissible" operations (rigid rotation, reflection, uniform
re-scaling) which leave the relative distances unchanged (see below). The
remaining models allow operations such as dimensional weighting and vector
displacement, which do not keep distances unchanged and are in this sense "impermissible",
but which give a simple systematic way of putting the configurations into
maximal conformity.
The hierarchy is actually a partial
order, with two branches :
P5:
Hybrid Dimensional & Vector model
(Dimensional) (Vector)
P2 (idiosyncratic rotation
& P4
(idiosyncratic origin &
dimensional weighting )
vector
weighting)
P1 (dimensional weighting)
P3
(vector weighting)
P0: Simple Procrustes
(P1
is the PINDIS equivalent of INDSCAL, and P2 corresponds to IDIOSCAL in the Bell
Labs. Hierarchy)
Input
data often consist of final configurations from a set of scalings done under
different conditions (e.g. obtained by different methods of data-collection; using different transformations; using different models;
studies replicated in different countries, and even an equivalent of INDSCAL where
each configuration results from the scaling of a given individual's judgements
of a set of stimuli).
The
maximum number of dimensions in any one configuration is given in the DIMENSIONS
statement, the number of configurations, by N OF SUBJECTS. The number of
points in the configuration is given by N OF STIMULI and the data are
read by the READ CONFIGS command. They may be input either stimuli
(rows) by dimensions (columns) or vice versa (in which case MATFORM(1)
should be specified in the PARAMETERS command). The INPUT FORMAT
specification, if used, should read the longest row of the configurations. By
default, free format input is assumed.
The basic
Procrustean model (P0) : Similarity transformation (Unit
weighting), relies on the assumption that that MDS configurations are unique
under translation, rotation, and reflection and rescaling of axes by uniform
stretching or shrinking. The program's first step is to take each pair of
configurations in turn and, by applying the permissible similarity
transformations, move them into maximum conformity with each other. This
effectively eliminates any differences in the configurations due to the
conventions of the program producing them and leaves the substantive
differences - due to random error and differential cognition. A centroid
configuration is formed simply by taking the average position of each point
over all the configurations. The model at this stage implies that individual "subjects"
(or data-sources) make no
systematic distortions of the group space (the centroid).
Instead
of estimating the Centroid configuration, the user may input an external reference (or hypothetical) configuration
of the same stimuli in the same number of dimensions (such as an original study
of which the data are replications), or repeat the configuration for one of the
subjects to which the others should be compared, introduced by READ
HYPOTHESIS.
The
higher order models in PINDIS allow that subjects may
systematically distort this basic configuration. It is the mode of
distortion which differs in these models. The basic configuration is
usually allowed to be rotated to an optimal position for each subject before further
transformations are applied as this may be expected to result in
substantively more interpretable solutions. The ROTATE(0) parameter
setting allows for this. If, however, the user wishes to fix the centroid at
P0, or has input a hypothesis configuration with 'meaningful' axes, then
ROTATE(1) should be specified in the PARAMETERS command, so that
this becomes fixed as the starting point for further transformations.
In
contrast to earlier versions, the default settings in PINDIS now
report only the basic Procrustean solution P0.
To see the results of further transformations, make
sure PRINT SUBJECTS is present in the
input RunScript.
The first
mode of distortion reported (P1) is analogous to that of the INDSCAL model in that subjects, in arriving at their
perceptual spaces, are thought of as applying differential weights to the
dimensions of the group space (the centroid).
In
the next model, allowing different dimensional salience with idiosyncratic
orientation (P2 ), each subject is thought of as distorting the centroid
by first rotating the axes of the configuration to his/her own preferred
orientation and then applying differential weights to these new axes. (If ROTATE
(1) has been specified this solution will be identical to P1 .)
The
perspective model with fixed origin (vector weighting) ( P3 ) allows
differential stretching or shrinking for each subject configuration of each
stimulus vector drawn from the origin of the space. It is sometimes called the
"unscrambling" model since a weight applied to a stimulus vector
moves the position of that stimulus, for a given subject, in the direction
indicated by that vector in the space.
The
perspective model with idiosyncratic origin ( P4 ) additionally allows
the subjects to move the origin of the centroid space to an idiosyncratic
position before the vector weighting operations are performed. If the centroid
configuration has a rational origin and it does not make sense to shift it
about in this manner, then the user should specify TRANSLATE (1) in the PARAMETERS
command.
The double weighted (dimension and vector weighting) model ( P5 ) allows both dimensional and vector weighting simultaneously. Although the number of free parameters in this model is large, it has been found that the goodness-of-fit of this particular model is often surprisingly low. This may indicate that the geometrical processes which define it have little psychological rationale, though other substantive applications may find one.
Note that on using the graphics facility to display "subject spaces", it is possible to submit arc-distances in the space to further analysis using SUBJSTAT. A subject space, by convention, is always represented in the positive quadrant of the plotted space, i.e. the coordinate values are all positive.
INPUT
COMMANDS
Keyword
Function
N OF STIMULI [number]
Number of stimuli in the analysis
N OF SUBJECTS [number]
Number of subject configurations to be
compared
in the analysis
DIMENSIONS [number]
Maximum number of dimensions in
any one
configuration
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.
READ CONFIGS Input
the subject configurations
immediately
following this command.
READ HYPOTHESIS (Optional)
Input a configuration to
use
as a reference for comparisons;
if
not present, the centroid
is
used by default.
PARAMETERS
Keyword Default
Function
ROTATE 1
0: Idiosyncratic rotations of the
centroid (P2
) are performed.
1: Idiosyncratic rotations are not
performed, i.e. fit to P0 only
is reported .
TRANSLATE 0
0: Idiosyncratic translations of the origin
(P4 ) are performed.
1: Idiosyncratic translations of the origin
are not performed .
SUPPRESS 0
0: Double-weighted solution ( P5)
is
performed.
1: Double-weighted solution ( P5)
is not performed .
ORIGIN 0
0: The origin is situated at the centroid
of the space.
n n:
Gives the number of the point to be
regarded as the origin.
MATFORM 0
0: The input configurations are input
stimuli(rows)
by dimensions (columns).
1: The input configurations are input
dimensions(rows)
by stimuli (columns).
NOTES
1. The READ CONFIGS command is obligatory in PINDIS.
2. READ MATRIX is not valid with PINDIS.
3. For ROTATE, TRANSLATE and SUPPRESS,
parameter value 0 means
the option is performed - value 1, that it is not performed.
PRINT
options
(to main output file)
Option
Form Description
CENTROID p x r
The centroid configuration is printed
at each phase, with the
results of applying
the basic similarity
transformations
P0 for each subject.
SUBJECTS N
The subject
matrices are printed at
each phase showing the results
of
all PINDIS transformations
applied.
By
default, CENTROID only is applied.
PLOT
options
(to main output file) By
default, CENTROID only is applied. PUNCH
options
(to secondary output file) PROGRAM
LIMITS
See also
Option
Description
CENTROID
The CENTROID (or HYPOTHESIS) configuration,
whichever
is being used for reference,
is
plotted at each phase.
SUBJECT The subject space is also plotted. Note,
however,
that it is mistaken to
regard these spaces as
Euclidean.
SUBJSTAT provides an appropriate
arc-distance measure for the analysis of distances
between items in subject spaces.
Option
Description
CENTROID The
coordinates of the centroid
configuration
are output.
Maximum no. of subjects = 100
Maximum no. of stimuli = 100
Maximum no. of dimensions = 6