Subject Space and Private spaces (the former a way of representing patterns of dimensional weights as a vector; the latter being the result of applying a subject’s weight to the Group Space).
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More significant is the extension of the basic model to the Third Way, allowing sets, collections of data, to be analysed together and related to each other and the model thus allows the introduction of time, space, different conditions, contextual variables to be brought into the analysis.
Originally developed with the idea of individual differences in mind, the basic INDSCAL model (3W2M) posits a Reference Space for the variables/objects whose dimensions include all those which the individuals use; INDSCAL is pre-eminently a dimensional model. Individuals are thought of as sharing this common configuration, but attaching different weight to each dimension, from ignoring it entirely (0) to giving it a supreme salience (1) ... and everything in between.
An earlier model, called “Points of View” (PoV) developed by Tucker and Messick, was the first viable method for representing “individual differences” in scaling. In the original Tucker-Messick model, the 3W2M data were first factor analysed to give an empirical clustering of the subjects, and the averaged dissimilarity matrix for each cluster (“viewpoint”) was then separately scaled. Note that no allowance is made in this model for an overall Group Stimulus Space and the axes of the scalings are not uniquely oriented, as in the INDSCAL model which succeeded PoV. In this program there are two stages:
(i) Creating an efficient clustering of subjects: The BBDIAM procedure divides the subjects into a set of clusters of individuals and produces a dissimilarity matrix for each cluster.
(ii) Scaling each group (“point of view) separately. Each of the pooled group matrices are scaled using MINISSA, thus offering a series of representations of the stimulus space, one for each of the groups identified. There is thus no assumption that the subjects’ data refer to a single, common Group Space, and the group scalings may be of different dimensionality and have no dimensional uniqueness properties (unlike INDSCAL). PoV is therefore primarily an exploratory process used to detect homogeneous groups whose members share a common viewpoint.We shall also spend time examining the hierarchy of increasingly complex models introduced by PINDIS, where actual configurations from earlier scalings form the input rather than sets of 2W1M data, and each model provides different ways of finding out where the individual differences lie – in weighted common dimensions, to be sure, but in other rather different ways, too. The same is true also of PREFMAP, which takes a given (external) configuration, and proceeds to map individuals’ (or, as in INDSCAL, “pseudo-subjects’” ) rankings or ratings into that configuration, according to a set of increasingly complex models.
The key concepts are:
| Three-way Decomposition (which is the triple scalar-products version of INDSCAL) |
| Weighted distance models (referring to a common set of dimensions .... or not!) |
| Subject Space and Private spaces (the former a way of representing patterns of dimensional weights as a vector; the latter being the result of applying a subject’s weight to the Group Space). |
The material in this module is covered primarily in TUG, chapter 7 and the definitive article by Carroll and Chang is contained in the downloads.
The programs are:
| NewMDSX’s INDSCAL-S (and CANDECOMP for the general –way case); CONPAR (for Points of View), PINDIS and PREFMAP |
| SPSS’s PROXSCAL also provides metric (and non-metric) variants of INDSCAL. |
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