The model implemented in MDSORT is designed specifically for the direct analysis of sorting data, and was developed to generate a joint representation of objects and subjects' categories, which simultaneously scales and represents the sorting data. Takane's (1980) model takes the data as a matrix F consisting of a set of N row vectors, one for each respondent i, arrayed so that each column refers to a given object j and where the entry f(i,j) consists of the category/group number in which the object is located. The categories are in a sequential (but arbitrary) numbering, that is:
The data matrix F is then expanded into a set of individual matrices G, each of which is of size p rows and q categories, where q may differ from subject to subject in free-sorting, consisting of the values
The intention is to obtain a configuration of stimulus/object points in such a way that the sum of squared inter-category distances (averaged over subjects) is maximized under suitable normalization restrictions. MDSORT determines a matrix X of coordinates of the n objects in a minimal, user-chosen dimensionality, r. The squared distances between category centroids are related by definition to the trace of the product-moment of X, which is determined so that tr(X'BX) is maximized, where B is the mean of the sum of the subject-specific similarity matrices. It is important to note that these similarity values are scaled according to the size of the categories on which they are based, so that the similarity between two objects sorted into the same group is inversely related to the size of the category. The raw co-occurrence counts may also be output, and may be submitted for comparison to other scaling routines within NewMDSX.
With the added restriction for the multidimensional case:
INPUT for MDSORT
The number of principal components to be listed must be restricted by the number given in
the DIMENSIONS statement. The number of columns in the input data is given
by N OF STIMULI and the number of rows by N OF SUBJECTS.
The input matrix is read by the READ DATA command. By default, input is
assumed to be in free format, but if an INPUT FORMAT specification is used, it
should be specified to read a line of integer values corresponding to the
N OF STIMULI given. LABELS, followed by a series of labels (<=
65 char) each on a separate line, optionally identifies the
stimuli, in order, with no omissions.
OUTPUT
PRINT options (to main output file)
Option
Description
SIMILARITIES
Outputs the matrix of similarities B between
the stimuli, derived from the input data.
CO-OCCURRENCES Outputs the matrix of raw cooccurrences
in categories of the stimuli.
CLUSTERS
Outputs the set of individual cluster centroids
corresponding to the overall similarities.
PLOT options (to main output file)
Option
Description
STIMULI
Plots the stimulus configuration, representing
the number of normalized principal components
specified by the DIMENSIONS statement.
CLUSTERS Plots the set of individual cluster centroid configurations.
If the N OF SUBJECTS is more than a small,
number, this option may produce a
rather large amount of output.
NOTES
1. The READ DATA command is obligatory in MDSORT.
2. No secondary output file is produced by MDSORT.
3. No PARAMETERS are used by MDSORT.
PROGRAM LIMITS
Maximum no. of subjects = 200
Maximum no. of stimuli = 200
Maximum dimensions = 8
See also