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The MDS Procedure |
Multidimensional scaling (MDS) is a class of methods that estimates the coordinates of a set of objects in a space of specified dimensionality that come from data measuring the distances between pairs of objects. A variety of models can be used that include different ways of computing distances and various functions relating the distances to the actual data. The MDS procedure fits two- and three-way, metric and nonmetric multidimensional scaling models. PROC MDS shares many of the features of the ALSCAL procedure (Young, Lewyckyj, and Takane 1986; Young 1982), as well as some features of the MLSCALE procedure (Ramsay 1986). Both PROC ALSCAL and PROC MLSCALE are described in the SUGI Supplemental Library User's Guide, Version 5 Edition.
The data for the MDS procedure consist of one or more square symmetric or asymmetric matrices of similarities or dissimilarities between objects or stimuli (Kruskal and Wish 1978, pp. 7-11). Such data are also called proximity data. In psychometric applications, each matrix typically corresponds to a subject, and models that fit different parameters for each subject are called individual difference models. Missing values are allowed. In particular, if the data are all missing except within some off-diagonal rectangle, the analysis is called unfolding. There are, however, many difficulties intrinsic to unfolding models (Heiser, 1981). PROC MDS does not perform external unfolding; for analyses requiring external unfolding, use the TRANSREG procedure instead.
The MDS procedure estimates the following parameters by nonlinear least squares:
Depending on the LEVEL= option, PROC MDS fits either a regression model of the form
or a measurement model of the form
where
For an introduction to multidimensional scaling, refer to Kruskal and Wish (1978) and Arabie, Carroll, and DeSarbo (1987). A more advanced treatment is given by Young (1987). Many practical issues of data collection and analysis are discussed in Schiffman, Reynolds, and Young (1981). The fundamentals of psychological measurement, including both unidimensional and multidimensional scaling, are expounded by Torgerson (1958). Nonlinear least-squares estimation of PROC MDS models is discussed in Null and Sarle (1982).
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