Procrustean Individual Differences Scaling : PINDIS

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)
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.

By default, CENTROID only is applied.

PUNCH options (to secondary output file)
Option                     Description
CENTROID            The coordinates of the centroid
                            configuration are output.

PROGRAM LIMITS
Maximum no. of subjects = 100
Maximum no. of stimuli = 100
Maximum no. of dimensions = 6

See also

  • SUBJSTAT - statistics of the subject space
  • The NewMDSX commands in full