CONPAR : CONcordance PARtitioning

CONPAR analyses a collection of dis/similarity (symmetric proximity) matrices of the kind otherwise considered by INDSCAL and seeks to identify groups of individual-subject matrices with concordant proximity structures.

 

Data:  3-way, 2-mode dissimilarities
Transform:  Nonmetric, invariant to monotone transformations            
Model:  Minimizing partition diameter for a specified number of groups

 

The program takes as input a set of N matrices each of which is a lower triangular matrix (of order p) of (dis)similarity judgments/measures between the p stimuli and looks for a set of best-fitting proximity matrices, by partitioning respondents into groups who share comparable proximity matrix structures. It is therefore best restricted to the exploration of data in which a number of distinct groups are supposed to exist.  

 

The INDSCAL model, by contrast, explains differences between subjects' cognitions by a variant of the distance model, where the stimuli are located in the same 'master' stimulus space which is perceived differentially by the subjects, requiring that the analyst has to provide interpretations for the different weights assigned to each of the dimensions of a single space. It has, however,  been argued that the INDSCAL common space may not be the best way to display group, as opposed to individual , differences. CONPAR offers an alternative approach which makes no assumptions about the dimensionality of the MDS solutions which can be derived from the pooled responses of each of the groups derived, or the appropriate metric (Euclidean, city block, etc.) It is offered as a preprocessing tool to be applied prior to fitting MDS models to potentially different groups of subjects.

The objective is to identify groups of subject dis/similarity matrices that have comparable structural properties. This is achieved by first collapsing three-way dissimilarity data into a subject x subject pairwise dissimilarity matrix, using a non-metric concordance measure based on gradient information to capture internal structural properties. This matrix is subsequently partitioned, identifying an upper bound by biased-sampling complete link cluster analysis, and then using a branch-and-bound algorithm to minimize the 'partition diameter', that is, the maximum of the cluster 'diameters' (pairwise dissimilarity values for the groups). The value of this process is that the individual subject matrices can then more safely be pooled for further analysis, and the groups to be considered are not constrained at all in their relative sizes. The same partitioning process can be directly applied to a single matrix using BBDIAM .   

When running CONPAR in NewMDSX, when partioning into the number of groups specified is complete, an option is automatically offered to apply MINISSA to the pooled group matrices. The fitting of an MDS model to each group is a separate and subsequent process, and it is possible that the appropriate dimensionality or the appropriate distance metric might vary among the identified groups.

As an alternative to pooling within the subgroups identified by CONPAR, it might instead be preferred to obtain separate INDSCAL  solutions for one or more of the groups.

USING CONPAR 

It is necessary to specify the number of SUBJECTS and the number of STIMULI represented in the individual subject matrices.

The number of CLUSTERS to be identified by the partitioning process must be given by the CLUSTERS command. Only one partitioning at a time is permitted. Occasionally CONPAR appears to be unable to reach an optimal partition into the number of clusters requested. In this case, you should halt the process and try again, requesting a different number of clusters.

The N dis/similarity matrices of order p must follow the READ MATRIX command sequentially, according to the DATA TYPE parameter specified. These will automatically be converted to dissimilarities for partitioning by CONPAR

DIMENSIONS
For input to CONPAR, the number of DIMENSIONS specified is employed for the subsequent application of MINISSA to the pooled matrices for the clusters identified. Usually between two and four dimensional solutions will be adequate for any reasonable data set.

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]                         A single integer or list, specifying
                       [number list]                      the dimensions for an optional MINISSA
                  [number] TO [number]             analysis of the groups identified

CLUSTERS         [number]                         A single integer, giving the number of
                                                               clusters to be defined in the partitioning
                                                               process (<N OF STIMULI)  

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

DATA TYPE                 1                            0: Lower triangle similarity matrix
                                                                  (without diagonals).
                                                               1: Lower triangle dissimilarity matrix
                                                                  (without diagonals).
                                                               2: Lower triangle euclidean distances
                                                                   (without diagonals).
                                                               3: Lower triangle correlation matrix
                               

Apart from PRINT DATA,  there are no PRINT or PLOT options for CONPAR.

PROGRAM LIMITS
Maximum no. of dimensions = 5
Maximum no. of stimuli = 100
Maximum no. of subjects = 300

See also The NewMDSX commands in full